Stratified Lie groups and potential theory for their sub-Laplacians

Springer Monographs in Mathematics A. Bonfiglioli • E. Lanconelli • F. Uguzzoni Stratified Lie Groups and Potential ...

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Springer Monographs in Mathematics

A. Bonfiglioli • E. Lanconelli • F. Uguzzoni

Stratified Lie Groups and Potential Theory for their Sub-Laplacians

A. Bonfiglioli Università Bologna, Dip.to Matematica Piazza di Porta San Donato 5 40126 Bologna, Italy e-mail: [email protected]

F. Uguzzoni Università Bologna, Dip.to Matematica Piazza di Porta San Donato 5 40126 Bologna, Italy e-mail: [email protected]

E. Lanconelli Università Bologna, Dip.to Matematica Piazza di Porta San Donato 5 40126 Bologna, Italy e-mail: [email protected]

Library of Congress Control Number: 2007929114

Mathematics Subject Classification (2000): 43A80, 35J70, 35H20, 35A08, 31C05, 31C15, 35B50, 22E60

ISSN 1439-7382 ISBN-10 3-540-71896-6 Springer Berlin Heidelberg New York ISBN-13 978-3-540-71896-3 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the author and VTEX using a Springer LATEX macro package Cover design: WMXDesign, Heidelberg, Germany Printed on acid-free paper

SPIN: 12029525

VA 41/3100/VTEX - 5 4 3 2 1 0

To Professor Bruno Pini and to our Families

Preface

With this book we aim to present an introduction to the stratified Lie groups and to their Lie algebras of the left-invariant vector fields, starting from basic and elementary facts from linear algebra and differential calculus for functions of several real variables. The second aim of this book is to perform a potential theory analysis of the sub-Laplacian operators m Xj2 , L= j =1

where the Xj ’s are vector fields, i.e. linear first order partial differential operators, generating the Lie algebra of a stratified Lie group. In recent years, these operators have received considerable attention in literature, mainly due to their basic rôle in the theory of subelliptic second order partial differential equations with semidefinite characteristic form.

1. Some Historical Overviews General second order partial differential equations with non-negative and degenerate characteristic form have appeared in literature since the early 1900s. They were first studied by M. Picone, who called them elliptic-parabolic equations and proved the celebrated weak maximum principle for their solutions [Pic13,Pic27]. The interest in this type of equations in application fields was originally found by A.D. Fokker, M. Planck and A.N. Kolmogorov. They discovered that partial differential equations with non-negative characteristic form arise in the mathematical modeling of theoretical physics and of diffusion processes [Fok14,Pla17,Kol34]. Since then, over the past half-century, this type of equations appeared in many other different research fields, both theoretical and applied, including geometric theory of several complex variables, Cauchy–Riemann geometry, partial differential equations, calculus of variations, quasiconformal mappings, minimal surfaces and convexity in sub-Riemannian settings, Brownian motion, kinetic theory of gases,

VIII

Preface

mathematical models in finance and in human vision. We report a short list of references for these topics at the end of this preface. A first systematic study of boundary value problems for wide classes of ellipticparabolic operators was performed by G. Fichera. In 1956 [Fic56a,Fic56b], he proved existence theorems of weak solutions of the “Dirichlet problem” and found the right subset of the boundary on which the data have to be prescribed. Some years later, several existence and regularity results for elliptic-parabolic operators were proved by O.A. Ole˘ınik and E.V. Radkeviˇc and by J.J. Kohn and L. Nirenberg (see the monograph [OR73] for a presentation and a wide survey on this subject). The methods used by these authors required particular assumptions on the Fichera boundary set and led to regularity results strongly depending on the regularity of the boundary data. 1.1. L. Hörmander’s Theorem The investigations of the local regularity properties of the solutions to ellipticparabolic equations, that is, regularity properties only depending on the given operator, have produced more interesting results. The most beautiful ones have been obtained for elliptic-parabolic equations with underlying algebraic-geometric structures of sub-Riemannian type. The milestone of these research field is a celebrated theorem of L. Hörmander proved in 1967. Theorem 1 (L. Hörmander, [Hor67]). Let X1 , . . . , Xm and Y be smooth vectors fields, i.e. linear first order partial differential operators with smooth coefficients in the open set Ω ⊆ RN . Suppose rank Lie{X1 , . . . , Xm , Y }(x) = N ∀ x ∈ Ω. (P.1) Then the operator L=

m

Xj2 + Y

(P.2)

j =1

is hypoelliptic in Ω, i.e. every distributional solution to Lu = f is of class C ∞ whenever f is of class C ∞ . Condition (P.1) simply means that at any point of Ω one can find N linearly independent differential operators among X1 , . . . , Xm , Y and all their commutators (the Lie algebra generated by {X1 , . . . , Xm , Y }). Hörmander’s work opened up a research field, the most remarkable contributions to which have been given by G.B. Folland, L.P. Rothschild and E.M. Stein. They developed and applied to (P.2) the singular integral theory in nilpotent Lie groups.1 1 The application of this theory also occurs in the developments started from the works by

¯ J.J. Kohn on the ∂-Neumann problem and the ∂¯b complex.

1. Some Historical Overviews

IX

By using these techniques, in 1975, G.B. Folland accomplished a functional analytic study of sub-Laplacians on stratified Lie groups [Fol75]. One year later L.P. Rothschild and E.M. Stein proved their celebrated lifting theorem (see [RS76]), enlightening the basic rôle played by the sub-Laplacians in the theory of second order partial differential equations which are sum of squares of vector fields. In force of this theorem, indeed, we can roughly say that: m 2 Every operator L = j =1 Xj satisfying the Hörmander rank condition “as close as we want” to a sub(P.1) can be lifted to an operator L Laplacian. 1.2. The Rank Condition The geometrical meaning of the rank condition (P.1) is clarified by the C. Carathéodory, W.L. Chow and P.K. Rashevsky theorem: If (P.1) is satisfied, then given two points x, y ∈ Ω, sufficiently close, there exists a piecewise smooth curve, contained in Ω and connecting x and y, which is the sum of integral trajectories of the vector fields ±X1 , . . . , ±Xm , ±Y . The appearance of (P.1) in Hörmander’s theorem seems to be suggested by some deep properties of the Kolomogorov operators (see also the Introduction in [Hor67]), which we now aim to discuss. In studying diffusion phenomena from a probabilistic point of view, A.N. Kolmogorov showed that the probability density of a system with 2n degrees of freedom satisfies an equation with non-negative characteristic form Ku = 0 in R2n × R, where R2n is the phase-space of the system. A prototype for K is the following operator K=

n j =1

∂x2j +

n

xj ∂yj − ∂t ,

(P.3)

j =1

where x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) denote the velocity and the position vectors of the system, respectively. The operator K is “very degenerate”: its second order part only contains derivatives with respect to the variables x1 , . . . , xn . Nevertheless, as Kolmogorov showed, it has a fundamental solution Γ which is smooth out of its pole. This implies that K is hypoelliptic, that is, every distributional solution to Ku = f is of class C ∞ whenever f is of class C ∞ . The explicit expression of Γ is given by Γ (z, t; ζ, τ ) = γ ζ − E(t − τ )z, t − τ , z = (x, y), ζ = (ξ, η), (P.4) where γ (z, t) = 0 if t ≤ 0, and

(4π)n 1 −1 γ (z, t) = √ exp − C z, z if t > 0. 4 det C(t)

(P.5)

X

Preface

Here, ·, · stands for the usual inner product in R2n ; E(t) and C(t), respectively, denote the 2n × 2n matrices

t 0 0 , C(t) = E(s)A E(s)T ds. E(t) = exp −t In 0 0 Moreover, In denotes the identity matrix of order n and A = I0n 00 . We explicitly remark that C(t) > 0 for every t > 0. (P.6) This condition makes expression (P.5) meaningful and can be restated in geometrical– differential terms. Indeed, denoting Xj = ∂xj and Y = nk=1 xk ∂yk − ∂t , it can be proved that (P.6) is equivalent to the following rank condition: rank Lie{X1 , . . . , Xn , Y }(z, t) = 2n + 1 ∀ (z, t) ∈ R2n+1 .

(P.7)

It is also worthwhile to note that the Kolmogorov operator K can be written as K=

n

Xj2 + Y.

(P.8)

j =1

1.3. The Left Translation and Dilation Invariance The structure (P.4) of Kolmogorov’s fundamental solution suggests the relevance that a Lie group theoretical approach has in the analysis of Hörmander operators. Indeed, from the explicit expression of Γ one realizes that Γ (z, t; ζ, τ ) = γ (ζ, τ )−1 ◦ (z, t) , where ◦ is the following composition law making K := (R2n × R, ◦) a noncommutative Lie group (z, t) ◦ (z , t ) := z + E(t ) z, t + t , i.e. more explicitly, (x, y, t) ◦ (x , y , t ) = x + x , y + y + t x, t + t . In K one has (ζ, τ )−1 = (−E(−t) ξ, −τ ). It is easy to check that K is invariant w.r.t. the left translations on K and commutes with the following dilations: dλ (z, t) := (λx, λ3 y, λ2 t),

λ > 0.

For every λ > 0, dλ is an automorphism of K, so that (R2n × R, ◦, dλ ) is a homogeneous Lie group. It can be seen that its Lie algebra is the one generated by the vector fields Xj = ∂xj and Y = nk=1 xk ∂yk − ∂t appearing in (P.8).

1. Some Historical Overviews

XI

1.4. The Elliptic Counterpart: Stratified Groups and Sub-Laplacians For a proper comprehension and appreciation of this type of “parabolic”-type operators such as the above Kolmogorov operator K, it is crucial to possess a deep knowledge of their “elliptic” counterpart. This seems unavoidable, also bearing in mind that the underlying algebraic–geometric structures of these two different classes of operators are almost identical. Let us go back again, for a moment, to the Kolmogorov operator (P.3). If in that operator we square the term Y = nj=1 xj ∂xj +n − ∂t , we obtain the following “sum of square”-operator (which we may refer to as the “elliptic counterpart” of K): L :=

n j =1

∂x2j +

n

2 xj ∂xj +n − ∂t

(P.9)

.

j =1

The characteristic form of L is a non-negative quadratic form with non-trivial kernel. Then L has to be considered as a degenerate elliptic operator. However, it is hypoelliptic: the Hörmander rank condition (P.7) does not distinguish between L and K! Moreover, L is left-invariant on (R2n × R, ◦) (as we already know, so are the ∂xj ’s and Y ) but, this time, it commutes with the dilations δλ (z, t) = (λx, λ2 y, λ2 t),

λ > 0.

Also these dilations are automorphisms of (R2n × R, ◦), and G := (R2n × R, ◦, δλ ) becomes a stratified Lie group whose generators are the vector fields ∂xj ’s and Y . Then, according to our general agreement, L is a sub-Laplacian2 on G. 1.5. The Heisenberg Group In the lower-dimensional case n = 1, the operator (P.9) is ∂x2 + (x ∂y − ∂t )2 ,

(x, y, t) ∈ R3 .

(P.10)

Up to a change and a relabeling of the variables, this can be written as follows: (∂x2 + 2 y ∂t )2 + (∂y − 2 x ∂t )2 ,

(x, y, t) ∈ R3 ,

which, in turn, is the lower-dimensional version of the celebrated sub-Laplacian on the Heisenberg group. The Heisenberg group Hn is the stratified Lie group (R2n+1 , ◦) whose composition law is given by (z, t) ◦ (z , t ) = z + z , t + t + 2 Im z, z . (P.11) Here we identify R2n with Cn , and we use the notation 2 All these notions will be properly introduced in Chapter 1.

XII

Preface

(z, t) = (z1 , . . . , zn , t) = (x1 , y1 , . . . , xn , yn , t) for the points of Hn . In (P.11), ·, · stands for the usual Hermitian inner product in Cn . The dilation δλ (z, t) = (λz, λ2 t) is an automorphism of

Hn

(P.12)

and the vector fields

Xj = ∂xj + 2 yj ∂t ,

Yj = ∂yj − 2 xj ∂t

are left-invariant on (Hn , ◦). One readily recognizes that the following commutation relations hold: [Xj , Yj ] = −4 ∂t

(P.13a)

[Xj , Xk ] = [Yj , Yk ] = [Xj , Yk ] = 0 ∀ j = k.

(P.13b)

and

Identity (P.13a) is the canonical commutation relation between momentum and position in quantum mechanics. From (P.13a) it follows that rank Lie X1 , . . . , Xn , Y1 , . . . , Yn , ∂t = 2n + 1 at any point of R2n+1 . Then, by Hörmander’s Theorem 1, the sub-Laplacian on Hn Δ

Hn

:=

n

(Xj2 + Yj2 )

j =1

is hypoelliptic. The Heisenberg group and its Lie algebra originally arose in the mathematical formalizations of quantum mechanics (see H. Weyl [Weyl31]). Today, they appear in many research fields such as several complex variables, CR geometry, Fourier analysis and partial differential equations of subelliptic type. The Heisenberg sub-Laplacian is undoubtedly the most important prototype of the sub-Laplacians on non-commutative stratified Lie groups. 1.4. The Lifting Theorem Obviously, a generic Hörmander operator sum of squares of vector fields is not, in general, a sub-Laplacian on some stratified Lie group. Just consider, as an example, M = ∂x2 + (x ∂y )2

in R2 .

This operator satisfies the Hörmander rank condition, hence it is hypoelliptic. However, there is no Lie group structure in R2 making M left-invariant on it. Nevertheless, adding the new variable t, M can be lifted to the operator in (P.10) which is the sub-Laplacian on (a group isomorphic to) the Heisenberg group H1 . This is the source idea of the lifting theorem by L.P. Rothschild and E.M. Stein, which states, roughly speaking, that any Hörmander operator sum of squares of vector fields can be approximated by a sub-Laplacian on a stratified group. This result emphasizes the major rôle played by the sub-Laplacians in the theory of second order PDE’s with non-negative and degenerate characteristic form.

2. The Contents of the Book

XIII

1.5. Stratified Groups in Sub-Riemannian Geometry Stratified groups also appear naturally in sub-Riemannian geometry (frequently referred to as “Carnot” geometry). Roughly speaking, stratified groups play a rôle, for sub-Riemannian manifolds, analogous to that played by Euclidean vector spaces for Riemannian manifolds. More precisely, once it has been provided a suitable notion of tangent space at a point of a sub-Riemannian manifold, it turns out that (at a regular point) this tangent space is naturally endowed with a structure of nilpotent Lie group with dilations, a stratified Lie group (see J. Mitchell [Mit85] and A. Bellaïche in [BR96]). Furthermore, the analysis of a left invariant sub-Laplacian on a connected nilpotent Lie group (or more generally on a Lie group of polynomial growth) and the geometry at infinity of this group is described by a canonically associated dilationinvariant sub-Laplacian on a stratified Lie group. See G. Alexopoulos [Ale92,Ale02], S. Ishiwata [Ish03] and N.Th. Varopoulos [Varo00].

2. The Contents of the Book: An Overview A glance at the contents of the book and at our approach to the subjects is in order. The book is divided into three parts, and every part is, in its turn, subdivided in several chapters plus some appendices, if necessary. 2.1. Part I The first four chapters of Part I are devoted to an elementary and self-contained introduction to the stratified Lie groups in RN . Our presentation does not require a specialized knowledge neither in algebra nor in differential geometry. The approach is completely elementary, “constructive” whenever possible, abundant in examples and intended to be understandable by readers with basic backgrounds only in linear algebra and differential calculus in RN . Subsequently, we present the formal and abstract approach to the stratified Lie groups commonly used in literature, and we prove the equivalence of the abstract notion of stratified group to the “constructive” notion of homogeneous Carnot group. This equivalence is also provided in Part I. A very special emphasis is given to the examples. We introduce and discuss a wide range of explicit stratified Lie groups of arbitrarily large dimension and step. Some of them have been known in specialized literature for several years, such as the Heisenberg–Kaplan groups, the filiform groups and the Métivier groups. Many others have only appeared very recently, in particular what we shall call the Kolmogorov-type groups and the Bony-type groups. Other examples are completely new, some extracted from geometric control theory. Our long list of examples is also intended to be appreciated by readers working in geometry and analysis on Carnot groups. It provides a valuable benchmark set to

XIV

Preface

test new special properties of the groups, to exhibit explicit examples and counterexamples of the “pathologies” and the special features of Carnot groups. It is also payed a special attention to the Lie algebras of the groups by stressing their links with second order partial differential operators of Hörmander type (sum of squares of vector fields). In particular, given such an operator, we show necessary and sufficient conditions for it to be a sub-Laplacian on a suitable homogeneous stratified Lie group, and we explicitly show how to construct the related composition law. As a byproduct, this enables the reader to build up another plenty of examples of stratified groups and sub-Laplacians. Chapter 5 of Part I is dedicated to the analysis of the fundamental solution for the sub-Laplacians, a central topic of Part I. Here, the mainly used analytic tools are integration by parts and coarea-formulas. We start from the hypoellipticity of sub-Laplacians, easy consequence3 of the Hörmander Theorem 1. From this “assumption” on hypoellipticity, and with the aid of the strong maximum principle, whose proof is postponed to the Appendix of Chapter 5, we deduce the existence of a gauge function d for any given sub-Laplacian L, i.e. the existence of a positive non-constant homogeneous function d such that d 2−Q is L-harmonic away from the origin. Here Q stands for the homogeneous dimension of the group on which the sub-Laplacian lives (we always assume that Q ≥ 3). This property is one of the most striking analogies between L and the classical Laplace operator. We show that this leads to suitable mean value formulas on the d-balls, extending to this new setting the well-known Gauss theorem for classical harmonic functions. We then use these formulas (which will play a crucial rôle throughout the book) to prove Liouville-type theorems, Harnack-type inequalities, and a Sobolev–Stein embedding theorem. Furthermore, three sections are devoted to the following topics: some remarks on the analytic-hypoellipticity of sub-Laplacians, L-harmonic approximations and an integral representation formula for the fundamental solution. 2.2. Part II Part II of the book contains an exhaustive potential theory for the sub-Laplacians. Basically, our only starting point is the theorem by G.B. Folland asserting the existence for these operators of a homogeneous and smooth fundamental solution with pole at the origin. This key result allows us to perform a complete potential theory that parallel the one of the classical Laplace operator. The lack of explicit Poisson integral formulas forces us to follow an abstract approach to this theory. For this reason, in Chapter 6 we present some topics from abstract harmonic space theory, mainly inspired by the ones developed by H. Bauer [Bau66] and C. Costantinescu and A. Cornea [CC72]. This chapter mainly involves Perron–Wiener–Brelot method for the Dirichlet problem, harmonic minorants and majorants and balayage theory. 3 We do not go into the proof of this theorem, for it would require techniques very far from

the ones developed in the book.

2. The Contents of the Book

XV

Next, in Chapter 7 we show that every sub-Laplacian equips RN with a structure of harmonic space satisfying the axioms of the theory presented in Chapter 6. This is accomplished by using the Harnack-type theorem proved in Chapter 5, and then by showing the existence of a basis of the topology of RN formed by L-regular sets, i.e. by sets for which the Dirichlet problem for L is solvable in the usual classical sense (here we follow an idea by J.-M. Bony [Bon69]). In the subsequent chapters of Part II, we use the full strength of the abstract theory, together with the remarkable properties of the fundamental solution for L to deal with the arguments listed below: a) sub-mean characterizations of the L-subharmonic functions, and applications to the notion of convexity in Carnot groups; b) Green functions and Riesz representation theorems for L-subharmonic functions, with applications (among which Bôcher-type theorems); c) maximum principles on unbounded domains; d) L-capacity and L-polar sets, with applications: the Poisson–Jensen formula and the so-called fundamental convergence theorem; e) L-thinness and L-fine topology, with applications to the Dirichlet problem (and the derivation of Wiener’s criterion); f) the links between the Hausdorff measure naturally related to the gauge d and the capacity for L. In writing this part of the book we were also partially inspired by some monographs on potential theory for the classical Laplace operator—in particular the beautiful books by L.L Helms [Helm69], by W.K. Hayman and P.B. Kennedy [HK76] and by D.H. Armitage and S.J. Gardiner [AG01]. 2.3. Part III In Part III, we take up further topics on the algebraic and analytic theory of Carnot groups. In particular, this part of the book provides: a) the study of free Lie algebras; b) clear and complete proofs, in several contexts, of the fundamental and remarkable Campbell–Hausdorff formula4 ; c) the equivalence of sub-Laplacians under diffeomorphisms; d) the Rothschild–Stein lifting theorem and Folland’s lifting theorem (for stratified or homogeneous vector fields); e) the study of the algebraic structure of Heisenberg–Kaplan-type groups (also providing an explicit characterization of them) with a special emphasis to the remarkable form of their fundamental solutions, discovered by G.B. Folland and A. Kaplan (we also present the inversion and the Kelvin transform in the H-type groups of Iwasawa type); 4 In Chapter 15, we collect four theorems for the Campbell–Hausdorff formula: one for ho-

mogeneous vector fields, two for formal power series and one for general smooth vector fields.

XVI

Preface

f) the Carathéodory–Chow–Rashevsky connectivity theorem (for stratified vector fields) with applications; g) Taylor’s formula (with Lagrange and with integral remainder) on Carnot groups. The difficulty of finding “easy” and complete proofs of some of the above mentioned results in the existing literature is well known. By working with stratified vector fields we are able to overcome some of the lengthy steps of the proofs, while maintaining a good amount of generality in the final results.

3. How to Read this Book Besides Ph.D. students, the book is addressed to young and senior researchers. Indeed, one of the main efforts in presenting the material is to use an elementary approach and to reach, step by step, the level of current researches. Many parts of the book may be used for graduate courses and advanced lectures. The first four chapters of Part I are addressed to non-specialists in Lie group and Lie algebra theory. The first two chapters can be skipped by the readers having familiarity with the basics of differential geometry and Lie group theory. The reader already acquainted with Carnot group theory can pass directly to Chapter 5. In any case, beginners and specialists in the theory of stratified groups can exploit the first four chapters as a source for examples. Part II is the core of the monograph. The reader with some background in potential theory (and interested in the main case of sub-Laplacians) can pass directly to Chapter 7 and proceed throughout Part II, leaving Chapters 10 and 13 as a further reading. Part III is thought of as a more specialized lecture. Nonetheless, a deep understanding of, e.g. the Campbell–Hausdorff formula or of Heisenberg-type groups are amongst the main goals of this monograph. The book provides 21 illustrative figures, 250 exercises (each chapter has its own section of exercises) and an index of the basic notation. For the reading convenience, we furnish a synoptic diagram of the structure of the book on page XVII.

3. How to Read this Book

The synoptic diagram of the structure of the book.

XVII

XVIII

Preface

4. Some References on Theoretical and Applied Related Topics Here is a short list of references for related topics on analysis on stratified Lie groups and applications.5 Alexopoulos [Ale02], Altafini [Alt99], Bellaïche and Risler [BR96], Birindelli, Capuzzo Dolcetta and Cutrì [BCC97], Bahri [Bah04,Bah03], Barletta [Bar03], Barletta and Dragomir [BD04], Barletta, Dragomir and Urakawa [BDU01], Birindelli, Capuzzo Dolcetta and Cutrì [BCC98], Brandolini, Rigoli and Setti [BRS98], Capogna [Cap99], Capogna and Cowling [CC69], Capogna and Garofalo [CG98,CG03,CG06], Capogna, Garofalo and Nhieu [CGN00,CGN02], Capuzzo Dolcetta [CD98], Chandresekhar [Cha43], Citti [Cit98], Citti, Lanconelli and Montanari [CLM02], Citti, Manfredini and Sarti [CMS04], Citti and Montanari [CM00], Citti, Pascucci and Polidoro [CPP01], Citti and Sarti [CS06], Citti and Tomassini [CT04] Cowling, De Mari, Korányi and Reimann [CDKR02], Cowling and Reimann [CR03], Danielli, Garofalo, Nhieu and Tournier [DGNT04], Danielli, Garofalo and Salsa [DGS03], Dragomir [Dra01], Franchi, Gutiérrez and van Nguyen [FGvN05], Franchi, Serapioni and Serra Cassano [FSS03a,FSS03b,FSS01], Gamara [Gam01], Gamara and Yacoub [GY01], Garofalo and Lanconelli [GL92], Garofalo and Tournier [GT06], Garofalo and Vassilev [GV00], Golé and Karidi [GK95], Gutiérrez and Lanconelli [GL03], Gutiérrez and Montanari [GM04a,GM04b], Heinonen [Hei95b], Heinonen and Holopainen [HH97], Heinonen and Koskela [HK98], Huisken and Klingenberg [HK99], Jerison and Lee [JL87,JL88,JL89], Juutinen, Lu, Manfredi and Stroffolini [JLMS07], Korányi and Reimann [KR90,KR95], Lanconelli [Lan03], Lanconelli, Pascucci and Polidoro [LPP02], Lanconelli and Uguzzoni [LU00], Lu, Manfredi and Stroffolini [LMS04], Lu and Wei [LW97], Malchiodi and Uguzzoni [MU02], Manfredi and Stroffolini [MS02], Montanari [Mo01], Montanari and Lascialfari [ML01], Montgomery [Mon02], Montgomery, Shapiro and Stolin [MSS97], Monti and Morbidelli [MM05], Monti and Rickly [MR05], Monti and Serra Cassano [MSC01], Petitot and Tondut [PT99], Reimann [Rei01a,Rei01b], Slodkowski and Tomassini [ST91], Stein [Ste81], Uguzzoni [Ugu00], Varopoulos, Saloff-Coste and Coulhon [VSC92]. 5 The list is alphabetically ordered and the grouping of the references in different lines is

only meant for typographical readability.

5. Acknowledgments

XIX

5. Acknowledgments The authors would like to thank Chiara Cinti and Andrea Tommasoli for careful reading some chapters of the manuscript. We would also like to express our gratitude to Italo Capuzzo Dolcetta, Cristian E. Gutiérrez, Guozhen Lu and Juan J. Manfredi for the kind encouraging appreciation of our work. It is also a pleasure to thank the Springer-Verlag staff for the kind collaboration, in particular Dr. Catriona M. Byrne, Dr. Marina Reizakis and Dr. Susanne Denskus. Some topics presented in this book have partially appeared in the following papers by joint collaborations of the authors (and of one of us with C. Cinti) [Bon04, BC04,BC05,BL01,BL02,BL03,BL07,BLU02,BLU03,BU04a,BU04b,BU05a]. Bologna, Italy April 2007 Andrea Bonfiglioli Ermanno Lanconelli Francesco Uguzzoni

Contents

Part I Elements of Analysis of Stratified Groups 1

Stratified Groups and Sub-Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Vector Fields in RN : Exponential Maps and Lie Algebras . . . . . . . . . 1.1.1 Vector Fields in RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Integral Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Exponentials of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Lie Brackets of Vector Fields in RN . . . . . . . . . . . . . . . . . . . . . 1.2 Lie Groups on RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The Lie Algebra of a Lie Group on RN . . . . . . . . . . . . . . . . . . 1.2.2 The Jacobian Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 The (Jacobian) Total Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 The Exponential Map of a Lie Group on RN . . . . . . . . . . . . . 1.3 Homogeneous Lie Groups on RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 δλ -homogeneous Functions and Differential Operators . . . . . 1.3.2 The Composition Law of a Homogeneous Lie Group . . . . . . 1.3.3 The Lie Algebra of a Homogeneous Lie Group on RN . . . . . 1.3.4 The Exponential Map of a Homogeneous Lie Group . . . . . . . 1.4 Homogeneous Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The Sub-Laplacians on a Homogeneous Carnot Group . . . . . . . . . . . 1.5.1 The Horizontal L-gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Exercises of Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 3 6 8 10 13 13 19 22 23 31 32 38 44 48 56 62 68 73

2

Abstract Lie Groups and Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.1 Abstract Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.1.1 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.1.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.1.3 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.1.4 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.1.5 Commutators. ϕ-relatedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.1.6 Abstract Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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2.1.7 Left Invariant Vector Fields and the Lie Algebra . . . . . . . . . . 2.1.8 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.9 The Exponential Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Some Properties of the Stratification of a Carnot Group . . . . 2.2.2 Some General Results on Nilpotent Lie Groups . . . . . . . . . . . 2.2.3 Abstract and Homogeneous Carnot Groups . . . . . . . . . . . . . . 2.2.4 More Properties of the Lie Algebra . . . . . . . . . . . . . . . . . . . . . 2.2.5 Sub-Laplacians of a Stratified Group . . . . . . . . . . . . . . . . . . . . 2.3 Exercises of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 112 116 121 125 128 130 138 144 147

3

Carnot Groups of Step Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Heisenberg–Weyl Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Homogeneous Carnot Groups of Step Two . . . . . . . . . . . . . . . . . . . . . 3.3 Free Step-two Homogeneous Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The Exponential Map of a Step-two Homogeneous Group . . . . . . . . 3.6 Prototype Groups of Heisenberg Type . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 H-groups (in the Sense of Métivier) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Exercises of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 155 158 163 165 166 169 173 177

4

Examples of Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 A Primer of Examples of Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Euclidean Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Carnot Groups with Homogeneous Dimension Q ≤ 3 . . . . . 4.1.3 B-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 K-type Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Sum of Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 From a Set of Vector Fields to a Stratified Group . . . . . . . . . . . . . . . . 4.3 Further Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 The Vector Fields ∂1 , ∂2 + x1 ∂3 . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Classical and Kohn Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Bony-type Sub-Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Kolmogorov-type Sub-Laplacians . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Sub-Laplacians Arising in Control Theory . . . . . . . . . . . . . . . 4.3.6 Filiform Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Fields not Satisfying One of the Hypotheses (H0), (H1), (H2) . . . . . 4.4.1 Fields not Satisfying Hypothesis (H0) . . . . . . . . . . . . . . . . . . . 4.4.2 Fields not Satisfying Hypothesis (H1) . . . . . . . . . . . . . . . . . . . 4.4.3 Fields not Satisfying Hypothesis (H2) . . . . . . . . . . . . . . . . . . . 4.5 Exercises of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183 183 183 184 184 186 190 191 198 198 200 202 204 205 207 210 210 212 215 215

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5

XXIII

The Fundamental Solution for a Sub-Laplacian and Applications . . . . 5.1 Homogeneous Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Control Distances or Carnot–Carathéodory Distances . . . . . . . . . . . . 5.3 The Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 The Fundamental Solution in the Abstract Setting . . . . . . . . . 5.4 L-gauges and L-radial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Gauge Functions and Surface Mean Value Theorem . . . . . . . . . . . . . . 5.6 Superposition of Average Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Harnack Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Liouville-type Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Asymptotic Liouville-type Theorems . . . . . . . . . . . . . . . . . . . . 5.9 Sobolev–Stein Embedding Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Analytic Hypoellipticity of Sub-Laplacians . . . . . . . . . . . . . . . . . . . . . 5.11 Harmonic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 An Integral Representation Formula for Γ . . . . . . . . . . . . . . . . . . . . . . 5.13 Appendix A. Maximum Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13.1 A Decomposition Theorem for L-harmonic Functions . . . . . 5.14 Appendix B. The Improved Pseudo-triangle Inequality . . . . . . . . . . . 5.15 Appendix C. Existence of Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . 5.16 Exercises of Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227 229 232 236 244 246 251 257 262 269 274 276 280 287 291 293 303 306 309 319

Part II Elements of Potential Theory for Sub-Laplacians 6

Abstract Harmonic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Sheafs of Functions. Harmonic Sheafs . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Harmonic Measures and Hyperharmonic Functions . . . . . . . . 6.2.2 Directed Families of Functions . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Harmonic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Directed Families of Harmonic and Hyperharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 B-hyperharmonic Functions. Minimum Principle . . . . . . . . . . . . . . . . 6.5 Subharmonic and Superharmonic Functions. Perron Families . . . . . . 6.6 Harmonic Majorants and Minorants . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 The Perron–Wiener–Brelot Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 S-harmonic Spaces: Wiener Resolutivity Theorem . . . . . . . . . . . . . . 6.8.1 Appendix: The Stone–Weierstrass Theorem . . . . . . . . . . . . . . 6.9 H-harmonic Measures for Relatively Compact Open Sets . . . . . . . . . 6.10 S∗ -harmonic Spaces: Bouligand’s Theorem . . . . . . . . . . . . . . . . . . . . 6.11 Reduced Functions and Balayage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Exercises of Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

337 338 340 341 342 345 347 348 353 358 359 363 366 367 370 375 378

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7

The L-harmonic Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The L-harmonic Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Some Basic Definitions and Selecta of Properties . . . . . . . . . . . . . . . . 7.3 Exercises of Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

381 381 388 392

8

L-subharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Sub-mean Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Some Characterizations of L-subharmonic Functions . . . . . . . . . . . . . 8.3 Continuous Convex Functions on G . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Exercises of Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

397 397 401 411 422

9

Representation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 L-Green Function for L-regular Domains . . . . . . . . . . . . . . . . . . . . . . 9.2 L-Green Function for General Domains . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Potentials of Radon Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Potentials Related to the Average Operators . . . . . . . . . . . . . . 9.4 Riesz Representation Theorems for L-subharmonic Functions . . . . . 9.5 The Poisson–Jensen Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Bounded-above L-subharmonic Functions in G . . . . . . . . . . . . . . . . . 9.7 Smoothing of L-subharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . 9.8 Isolated Singularities—Bôcher-type Theorems . . . . . . . . . . . . . . . . . . 9.8.1 An Application of Bôcher’s Theorem . . . . . . . . . . . . . . . . . . . 9.9 Exercises of Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

425 425 427 432 435 441 445 451 455 458 462 463

10 Maximum Principle on Unbounded Domains . . . . . . . . . . . . . . . . . . . . . . 10.1 MP Sets and L-thinness at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 q-sets and the Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 The Maximum Principle on Unbounded Domains . . . . . . . . . . . . . . . 10.4 The Proof of Lemma 10.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Exercises of Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

473 473 477 482 483 487

11 L-capacity, L-polar Sets and Applications . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Continuity Principle for L-potentials . . . . . . . . . . . . . . . . . . . . . . . 11.2 L-polar Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The Maria–Frostman Domination Principle . . . . . . . . . . . . . . . . . . . . . 11.4 L-energy and L-equilibrium Potentials . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 L-balayage and L-capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 The Fundamental Convergence Theorem . . . . . . . . . . . . . . . . . . . . . . . 11.7 The Extended Poisson–Jensen Formula . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Further Results. A Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9 Further Reading and the Quasi-continuity Property . . . . . . . . . . . . . . 11.10 Exercises of Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

489 489 491 494 497 500 510 514 519 527 533

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12 L-thinness and L-fine Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 The L-fine Topology: A More Intrinsic Tool . . . . . . . . . . . . . . . . . . . . 12.2 L-thinness at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 L-thinness and L-regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Functions Peaking at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 L-thinness and L-regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Wiener’s Criterion for Sub-Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 A Technical Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Wiener’s Criterion for L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Exercises of Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

537 537 538 542 542 544 547 547 550 553

13 d-Hausdorff Measure and L-capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 d-Hausdorff Measure and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 d-Hausdorff Measure and L-capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 New Phenomena Concerning the d-Hausdorff Dimension . . . . . . . . . 13.4 Exercises of Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

557 557 561 569 572

Part III Further Topics on Carnot Groups 14 Some Remarks on Free Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Free Lie Algebras and Free Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . 14.2 A Canonical Way to Construct Free Carnot Groups . . . . . . . . . . . . . . 14.2.1 The Campbell–Hausdorff Composition . . . . . . . . . . . . . . . . 14.2.2 A Canonical Way to Construct Free Carnot Groups . . . . . . . . 14.3 Exercises of Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

577 577 584 584 586 589

15 More on the Campbell–Hausdorff Formula . . . . . . . . . . . . . . . . . . . . . . . 15.1 The Campbell–Hausdorff Formula for Stratified Fields . . . . . . . . . . . 15.2 The Campbell–Hausdorff Formula for Formal Power Series–1 . . . . . 15.3 The Campbell–Hausdorff Formula for Formal Power Series–2 . . . . . 15.4 The Campbell–Hausdorff Formula for Smooth Vector Fields . . . . . . 15.5 Exercises of Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

593 593 599 605 610 616

16 Families of Diffeomorphic Sub-Laplacians ........................ 16.1 An Isomorphism Turning i,j ai,j Xi Xj into ΔG . . . . . . . . . . . . . . . 16.2 Examples and Counter-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Canonical or Non-canonical? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Further Reading: An Application to PDE’s . . . . . . . . . . . . . . . . . . . . . 16.5 Exercises of Chapter 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

621 622 628 637 641 644 645

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17 Lifting of Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Lifting to Free Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 An Example of Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 An Example of Application to PDE’s . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Folland’s Lifting of Homogeneous Vector Fields . . . . . . . . . . . . . . . . 17.4.1 The Hypotheses on the Vector Fields . . . . . . . . . . . . . . . . . . . . 17.5 Exercises of Chapter 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

649 649 659 661 666 669 676

18 Groups of Heisenberg Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Heisenberg-type Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 A Direct Characterization of H-type Groups . . . . . . . . . . . . . . . . . . . . 18.3 The Fundamental Solution on H-type Groups . . . . . . . . . . . . . . . . . . . 18.4 H-type Groups of Iwasawa-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5 The H-inversion and the H-Kelvin Transform . . . . . . . . . . . . . . . . . . . 18.6 Exercises of Chapter 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

681 681 686 695 702 704 709

19 The Carathéodory–Chow–Rashevsky Theorem . . . . . . . . . . . . . . . . . . . . 19.1 The Carathéodory–Chow–Rashevsky Theorem for Stratified Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 An Application of Carathéodory–Chow–Rashevsky Theorem . . . . . . 19.3 Exercises of Chapter 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

715

20 Taylor Formula on Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1 Polynomials and Derivatives on Homogeneous Carnot Groups . . . . . 20.1.1 Polynomial Functions on G . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1.2 Derivatives on G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Taylor Polynomials on Homogeneous Carnot Groups . . . . . . . . . . . . 20.3 Taylor Formula on Homogeneous Carnot Groups . . . . . . . . . . . . . . . . 20.3.1 Stratified Taylor Formula with Peano Remainder . . . . . . . . . . 20.3.2 Stratified Taylor Formula with Integral Remainder . . . . . . . . 20.4 Exercises of Chapter 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

733 734 734 736 741 746 751 754 766

715 727 730

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 Index of the Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795

Part I

Elements of Analysis of Stratified Groups

1 Stratified Groups and Sub-Laplacians

In this first chapter, we introduce the main notation and the basic definitions concerning with vector fields in RN : algebras of vector fields, exponentials of smooth vector fields, Lie brackets. Then, we introduce the main geometric structure investigated throughout the book: the homogeneous Carnot groups. To this end, we first study Lie groups G on RN and the Lie algebra of their left-invariant vector fields. Subsequently, we equip G with a homogeneous structure by the datum of a well-behaved group of dilations {δλ }λ>0 on G. The composition of G thus takes a transparent form, allowing us to study homogeneous Lie groups on RN by very direct methods. In particular, it will be a simple exercise of calculus to verify that the relevant exponential and logarithmic maps are global polynomial diffeomorphisms, a result which will throughout account for a very useful tool. Finally, we introduce the notion of homogeneous Carnot group and of sub-Laplacian. A wide number of explicit examples of homogeneous Carnot groups will be given in Chapters 3 and 4, after (in Chapter 2) we have analyzed the due relationship between abstract Carnot groups and the homogeneous ones. Indeed, despite our notions of Lie group on RN , homogeneous and homogeneous-Carnot group are dependent on a fixed system of coordinates for the group (thus being non-intrinsic notions), every abstract Carnot group is, as we shall see, isomorphic via the exponential map to its Lie algebra, which is a homogeneous Carnot group. This basic fact provides another motivation for this introductory chapter.

1.1 Vector Fields in RN . Exponential Maps. Lie Algebras of Vector Fields 1.1.1 Vector Fields in RN Given N ∈ N, we set, as usual, RN = {(x1 , . . . , xN ) : x1 , . . . , xN ∈ R}. We use any of the notation ∂ ∂j , ∂xj , , ∂/∂ xj ∂xj

4

1 Stratified Groups and Sub-Laplacians

to indicate the partial derivative operator with respect to the j -th coordinate of RN . Let Ω ⊆ RN be an open (and non-empty) set. Given an N-tuple of scalar functions a1 , . . . , aN , aj : Ω → R, j ∈ {1, . . . , N }, the first order linear differential operator X=

N

(1.1)

aj ∂j

j =1

will be called a vector field on Ω with component functions (or simply, components) a1 , . . . , aN . If O ⊆ Ω is an open set and f : O → R is a differentiable function, we denote by Xf the function on O defined by Xf (x) =

N

aj (x) ∂j f (x),

x ∈ O.

j =1

Occasionally, we shall also use the notation Xf when f : O → Rm is a vector-valued function, to mean the component-wise action of X. More precisely,1 ⎛ ⎞ ⎛ ⎞ f1 (x) Xf1 (x) .. ⎠. if f (x) = ⎝ ... ⎠ , we set Xf (x) = ⎝ . fm (x)

Xfm (x)

1 We warn the reader that points in RN will be usually denoted as N -tuples x

=

(x1 , . . . , xN ). When this does not lead to confusion, the column-vector notation ⎛ ⎞ x1 ⎜ . ⎟ x = ⎝ .. ⎠ xN will also be allowed. For example, this last notation will be sometimes used (with no risk of misunderstanding) for vector-valued functions f (x) = (f1 (x), . . . , fN (x))T , e.g. I (x) = (x1 , . . . , xN )T will always denote (this time as a rule) the identity map in RN . However, there are cases in which we shall keep the notation well distinguished: namely, if f : RN → R is a differentiable function, its gradient ∇f = (∂1 f, . . . , ∂N f ) will always be written as a row-vector. On the other side, we shall always use the column-vector notation when vectors in RN appear in matrix calculation. For example, if x, y ∈ RN , x T y will denote the row×column product ⎞ ⎛ y1 N ⎜ . ⎟ xj yj . x T y = (x1 , . . . , xN ) ⎝ .. ⎠ = j =1 yN

1.1 Vector Fields in RN : Exponential Maps and Lie Algebras

5

(Typically, this notation will be used when f = I is the identity map of RN , i.e. I (x) = (x1 , . . . , xN )T ; see (1.2) below.) Furthermore, given a differentiable function f : O → Rm , we shall denote by Jf (x), x ∈ O the Jacobian matrix of f at x. Let C ∞ (O, R) (for brevity, C ∞ (O)) be the set of smooth (i.e. infinitely-differentiable) real-valued functions. If the components aj ’s of X are smooth, we shall call X a smooth vector field and we shall often consider X as an operator acting on smooth functions, f → Xf. X : C ∞ (O) → C ∞ (O), We shall prevalently deal with smooth vector fields. We shall denote by T (RN ) the set of all smooth vector fields in RN . Equipped with the natural operations, T (RN ) is a vector space over R. We adopt the following (non-conventional) notation: I will denote the identity map on RN and, if X is the vector field in (1.1), then ⎛ ⎞ a1 . XI := ⎝ .. ⎠ (1.2) aN will be the column vector of the components of X. This notation is obviously consistent with our definition of the action of X on a vector-valued function.2 Thus, XI may also be regarded as a smooth map from RN to itself (that is what some authors call a “vector field” on RN ). Often, many authors identify X and XI . Instead, in order to avoid any confusion between a smooth vector field as a function belonging to C ∞ (RN , RN ) and a smooth vector field as a differential operator from C ∞ (RN ) to itself,3 we prefer to use the different notation XI and X as described in (1.2) and (1.1), respectively. By consistency of notation, we may write Xf = (∇f ) · XI, where

∇ = (∂1 , . . . , ∂N )

is the gradient operator in RN , f is any real-valued smooth function on RN and · denotes the row×column product. For example, for the following two vector fields on R3 (whose points are denoted by x = (x1 , x2 , x3 )) X2 = ∂x2 − 2 x1 ∂x3 , (1.3a) X1 = ∂x1 + 2 x2 ∂x3 ,

we have X1 I (x) =

1 0 2 x2

,

X2 I (x) =

0 1 −2 x1

.

2 Indeed, since I = (I , . . . , I ) with I (x) = x , we have X(I ) = a . N j j j j 1 3 Or even from C ∞ (RN , RN ) to itself!

(1.3b)

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1 Stratified Groups and Sub-Laplacians

1.1.2 Integral Curves A path γ : D → RN , D being an interval of R, will be said an integral curve of the smooth vector field X if γ˙ (t) = XI (γ (t)) for every t ∈ D. If X is a smooth vector field, then, for every x ∈ RN , the Cauchy problem γ˙ = XI (γ ), γ (0) = x

(1.4)

has a unique solution γX (·, x) : D(X, x) → RN . We agree to denote by D(X, x) the greatest open interval of R on which γX (·, x) exists. For example, if X1 is as in (1.3a), the solution to the relevant problem (1.4) is obtained by solving the following system of ODE’s (we write γ = (γ1 , γ2 , γ3 )) γ˙1 (t) = 1, γ˙2 (t) = 0, γ˙3 (t) = 2 γ2 (t), γ1 (0) = x1 ,

γ2 (0) = x2 ,

γ3 (0) = x3 ,

i.e. after simple computations, we obtain D(X1 , x) = R and γX1 (t, x) = (x1 + t, x2 , x3 + 2 x2 t).

(1.5)

Since X is smooth, t → γX (t, x) is a C ∞ function whose n-th Taylor expansion in a neighborhood of t = 0 is given by t2 tn γX (t, x) = x + t X (1) I (x) + X (2) I (x) + · · · + X (n) I (x) 2! n! t 1 (t − s)n X (n+1) I (γX (s, x)) ds. + n! 0

(1.6)

Hereafter, for k ∈ N, we denote by X (k) the vector field X (k) =

N (X k−1 aj )∂xj , j =1

being X 0 = I (the identity map) and X h , h ≥ 1, the h-th order iterated of X, i.e. X h := X · · ◦ X .

◦ · h times

In other words, we have

1.1 Vector Fields in RN : Exponential Maps and Lie Algebras

7

⎞ ⎞ ⎛ ⎛ ⎞ X(a1 ) X(X(a1 )) a1 . . .. ⎠,.... X (1) I = XI = ⎝ .. ⎠ , X (2) I = ⎝ .. ⎠ , X (3) I = ⎝ . aN X(aN ) X(X(aN )) ⎛

We remark that X h is a differential operator of order at most h, whereas X (h) is a differential operator of order at most 1. To check (1.6) we use (1.4). Writing γ (t) instead of γX (t, x), (1.4) gives (d/dt)|t=0 γ (t) = XI (x) and

γ (0) = x,

d d2 γ (t) = (XI )(γ (t)) = JXI (γ (0)) · γ˙ (0) = JXI (x) · XI (x) dt t=0 dt 2 t=0 ⎛ ⎞ ⎛ ⎞ ∇a1 (x) · XI (x) Xa1 (x) .. .. ⎠=⎝ ⎠ = X (2) I (x). =⎝ . . ∇aN (x) · XI (x)

XaN (x)

By iterating this argument, we obtain dk (k) γ (0) := k γ (t) = X (k) I (x), dt t=0

k ≥ 2.

Replacing this identity in the Taylor formula γ (t) = x +

n tk k=1

k!

γ

(k)

1 (0) + n!

t

(t − s)n γ (n+1) (s) ds,

0

we obtain (1.6). Since the identity map I is linear and since the first order part of X h coincides with X (h) , one has X (h) I ≡ X h I. Thus formula (1.6) can be rewritten as t2 tn γX (t, x) = x + t XI (x) + X 2 I (x) + · · · + X n I (x) 2! n! t 1 (t − s)n X n+1 I (γX (s, x)) ds. + n! 0 Example 1.1.1. For example, if X1 is as in (1.3a), since

1 0 (1) (2) X1 I = , X1 I = 0 = X1(k) I 0 2 x2 0 we have γX1 (t, x) = x + t X1 I (x) = as we directly found in (1.5).

x1 x2 x3

+t

1 0 2 x2

=

(1.7)

∀ k ≥ 3,

x1 + t x2 x3 + 2 x2 t

,

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1 Stratified Groups and Sub-Laplacians

1.1.3 Exponentials of Vector Fields The expansion in (1.7) suggests to use an “exponential-type” notation. For this (more than notational) reason, we give the following definition. Definition 1.1.2 (Exponential of a vector field). Let X be a smooth vector field on RN . Following all the above notation, we set exp(tX)(x) := γX (t, x),

(1.8)

where γX (·, x) is the solution of (1.4). This definition makes sense4 for every X ∈ T (RN ), for every x ∈ RN and every t ∈ D(X, x). (See Fig. 1.1.)

Fig. 1.1. Figure of Definition 1.1.2

Then, being X smooth, for every n ∈ N, we have the expansion exp(tX)(x) =

n tk k=0

+

k!

1 n!

X k I (x)

t

(t − s)n X n+1 I exp(sX)(x) ds.

(1.9)

0

In particular, for n = 1, exp(tX)(x) = x + t XI (x) +

t

(t − s) X 2 I exp(sX)(x) ds.

(1.10)

0

If we define U := {(t, x) ∈ R × RN | x ∈ RN , t ∈ D(X, x)}, from the basic theory of ordinary differential equations (see, e.g. [Har82]) we know that U is open and the map 4 Definition 1.1.2 is well-posed also when X has only Lipschitz-continuous component func-

tions, but we shall prevalently deal, as already stated, with smooth vector fields.

1.1 Vector Fields in RN : Exponential Maps and Lie Algebras

9

U (t, x) → exp(tX)(x) ∈ RN is smooth. Moreover, from the unique solvability of the Cauchy problem related to smooth vector fields we get: t ∈ D(−X, x) iff −t ∈ D(X, x) and exp(−tX)(x) := exp((−t)X)(x) = exp(t (−X))(x), exp(−tX) exp(tX)(x) = x, exp((t + τ )X)(x) = exp(tX) exp(τ X)(x) , exp((tτ )X)(x) = exp(t (τ X))(x),

(1.11a) (1.11b) (1.11c) (1.11d)

when all the terms are defined. If D(X, x) = R, identities (1.11a)–(1.11d) hold true for every t, τ ∈ R. See also the note.5 Remark 1.1.3 (Pyramid-shaped vector fields). For our aims, the vector fields of the following type N X= aj (x1 , . . . , xj −1 ) ∂xj (1.12) j =1

will play a crucial rôle. In (1.12), the function aj only depends on the variables x1 , . . . , xj −1 , and we agree to let aj (x1 , . . . , xj −1 ) = constant when j = 1. Roughly speaking, such a remarkable kind of vector field is “pyramid”-shaped, ⎛ ⎞ a1 ⎜ ⎟ a2 (x1 ) ⎜ ⎟ ⎜ ⎟ (x , x ) a 3 1 2 ⎜ ⎟ X = ⎜ a4 (x1 , x2 , x3 ) ⎟ . ⎜ ⎟ ⎜ ⎟ .. ⎝ ⎠ . aN (x1 , . . . , xN −1 ) For example, the fields in (1.3b) have this form. For any smooth vector field X of the form (1.12), the map (x, t) → exp(tX)(x) is well defined for every x ∈ RN and t ∈ R. Indeed, if γ = (γ1 , . . . , γN ) is the solution to the Cauchy problem γ˙ = XI (γ ), γ (0) = x, x = (x1 , . . . , xN ), then γ˙1 = a1 and γ˙j = aj (γ1 , . . . , γj −1 ) for j = 2, . . . , N . As a consequence, t aj (γ1 (x, s), . . . , γj −1 (x, s)) ds, γ1 (x, t) = x1 + ta1 , γj (x, t) = xj + 0

5 Strictly speaking, according to the very definition of exp(tX)(x) := γ (t, x), one may X

observe that t and X should be kept separate in notation. Though, we explicitly remark that identity (1.11d) justifies the notation “tX” in exp(tX)(x).

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1 Stratified Groups and Sub-Laplacians

and γj (x, t) is defined for every x ∈ RN and t ∈ R. Moreover, γ1 (·, t) only depends on x1 , whereas for j = 2, . . . , N , γj (·, t) only depends on x1 , . . . , xj . Let us put A1 (t) = ta1 and, for j = 2, . . . , N, t Aj (x, t) = Aj (x1 , . . . , xj −1 , t) := aj (γ1 (x, s), . . . , γj −1 (x, s)) ds. 0

Then, for every x ∈ RN , t ∈ R, ⎛

⎞

x1 + A1 (t) x2 + A2 (x1 , t) .. .

⎜ ⎜ exp(tX)(x) = ⎜ ⎝

⎟ ⎟ ⎟, ⎠

(1.13)

xN + AN (x1 , . . . , xN −1 , t) and the map x → exp(tX)(x) is a global diffeomorphism of RN onto RN for every fixed t ∈ R. Its inverse map y → L(y, t) is given by y → L(y, t) = exp(−tX)(y).

(1.14)

This last statement follows from identity (1.11b).

Remark 1.1.4. Let us consider a smooth function u : RN → R and the vector field in (1.1). Then Xu(x) = lim

t→0

u(exp(tX)(x)) − u(x) t

∀ x ∈ RN .

(1.15)

Indeed, since exp(tX)(x) = x + tXI (x) + O(t 2 ), the limit on the right-hand side of (1.15) is equal to the following one: lim

t→0

u(x + tXI (x)) − u(x) = ∇u(x) · XI (x) = Xu(x). t

1.1.4 Lie Brackets of Vector Fields in RN Given two smooth vector fields X and Y in RN , we define the Lie-bracket [X, Y ] as follows [X, Y ] := XY − Y X. This definition is only seemingly deceitful, for it writes [X, Y ] (which is a first order differential operator) as a difference Nof two second order differential operators. a ∂ and Y = Indeed, if X = N j j j =1 j =1 bj ∂j , a direct computation shows that the Lie bracket [X, Y ] is the vector field [X, Y ] =

N (Xbj − Y aj )∂j . j =1

1.1 Vector Fields in RN : Exponential Maps and Lie Algebras

11

As a consequence, ⎛

⎞ ⎛ ⎞ Xb1 Y a1 . . [X, Y ]I = ⎝ .. ⎠ − ⎝ .. ⎠ = JY I · XI − JXI · Y I. XbN Y aN

(1.16)

For example, if X1 , X2 are as in (1.3b) (page 5), we have [X1 , X2 ] = (X1 (−2x1 ) − X2 (2x2 )) ∂x3 = −4 ∂x3 . It is quite trivial to check that (X, Y ) → [X, Y ] is a bilinear map on the vector space T (RN ) satisfying the Jacobi identity [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0 for every X, Y, Z ∈ T (RN ). We shall refer to T (RN ) (equipped with the above Lie-bracket) as the Lie algebra of the vector fields on RN . Any sub-algebra a of T (RN ) will be called a Lie algebra of vector fields. More explicitly, a is a Lie algebra of vector fields if a is a vector subspace of T (RN ) closed with respect to [ , ], i.e. [X, Y ] ∈ a for every X, Y ∈ a. We now introduce some other notation on the algebras of vector fields. Given a set of vector fields Z1 , . . . , Zm ∈ T (RN ) and a multi-index J = (j1 , . . . , jk ) ∈ {1, . . . , m}k , we set ZJ := [Zj1 , . . . [Zjk−1 , Zjk ] . . .]. We say that ZJ is a commutator of length (or height) k of Z1 , . . . , Zm . If J = j1 , we also say that ZJ := Zj1 is a commutator of length 1 of Z1 , . . . , Zm . A commutator of the form ZJ will also be called nested, in order to emphasize its difference from, e.g. a commutator of the form [[Z1 , Z2 ], [Z3 , Z4 ]]. What is striking is that this last commutator is a linear combination of nested ones,6 as we prove in Proposition 1.1.7. First, we give a definition. Definition 1.1.5 (The Lie algebra generated by a set). If U is any subset of T (RN ), we denote by Lie{U } the least sub-algebra of T (RN ) containing U , i.e. Lie{U } := h, where h is a sub-algebra of T (RN ) with U ⊆ h. We define

rank Lie{U }(x) := dimR ZI (x) | Z ∈ Lie{U } .

6 Namely,

[[Z1 , Z2 ], [Z3 , Z4 ]] = −[Z3 , [Z4 , [Z1 , Z2 ]]] + [Z4 , [Z3 , [Z1 , Z2 ]]].

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1 Stratified Groups and Sub-Laplacians

Example 1.1.6. Let X1 and X2 be as in (1.3b) (page 5). Since [X1 , X2 ] = −4∂x3 and since any commutator involving X1 , X2 more than twice is identically zero, then Lie{X1 , X2 } = span{X1 , X2 , [X1 , X2 ]}, and rank(Lie{X1 , X2 }(x)) = 3

for every x ∈ R3

(the well-known Hörmander condition).

The following result holds. Proposition 1.1.7 (Nested commutators). Let U ⊆ T (RN ) be any set of smooth vector fields on RN . We set U1 := span{U },

Un := span{[u, v] | u ∈ U, v ∈ Un−1 },

n ≥ 2.

Then we have Lie{U } = span{Un | n ∈ N}. Moreover, [u, v] ∈ Ui+j

for every u ∈ Ui , v ∈ Uj .

We explicitly remark that, from the very definition of Un , the vector fields in Un are linear combination of nested brackets, i.e. brackets of the type [u1 [u2 [u3 [· · · [un−1 , un ] · · ·]]]] with u1 , . . . , un ∈ U . The above proposition then states that every element of Lie{U } is a linear combination of nested brackets. To show the idea behind the proof, let us take u1 , u2 , v1 , v2 ∈ U and prove that [[u1 , u2 ], [v1 , v2 ]] is a linear combination of nested brackets. By the Jacobi identity [X, [Y, Z]] = −[Y, [Z, X]] − [Z, [X, Y ]], one has [[u1 , u2 ], [ v1 , v2 ]] = −[v1 , [v2 , [u1 , u2 ]]] − [v2 , [[u1 , u2 ], v1 ]]

X

Y

Z

= −[v1 , [v2 , [u1 , u2 ]]] + [v2 , [v1 , [u1 , u2 ]]] ∈ U4 . Proof (of Proposition 1.1.7). We set U ∗ := span{Un | n ∈ N}. Obviously, U ∗ contains U and is contained in any algebra of vector fields which contains U . Hence, we are left to prove that U ∗ is closed under the bracket operation. Obviously, it is enough to show that, for any i, j ∈ N and for any u1 , . . . , ui , v1 , . . . , vj ∈ U , we have [u1 [u2 [· · · [ui−1 , ui ] · · ·]]]; [v1 [v2 [· · · [vj −1 , vj ] · · ·]]] ∈ Ui+j . We argue by induction on k := i + j ≥ 2. For k = 2 and 3, the assertion is obvious. Let us now suppose the thesis holds for every i + j ≤ k with k ≥ 4, and prove it also holds when i + j = k + 1. We can suppose, by skew-symmetry, j ≥ 3. Exploiting repeatedly the induction hypothesis and the Jacobi identity, we have

1.2 Lie Groups on RN

13

u; [v1 [v2 [· · · [vj −1 , vj ] · · ·]]] = −[v1 , [[v2 , [v3 , · · ·]], u]] − [[v2 , [v3 , · · ·]], [u, v1 ]]

length k

= {element of Uk+1 } − [[v1 , u], [v2 , [v3 , · · ·]]] = {element of Uk+1 } + [v2 , [[v3 , · · ·], [v1 , u]]] + [[v3 , · · ·], [[v1 , u]v2 ]]

length k

= {element of Uk+1 } + [[v2 , [v1 , u]], [v3 , · · ·]] (after finitely many steps) = {element of Uk+1 } + (−1)j −1 [[vj −i , [vj −2 , · · · [v1 , u]]], vj ] = {element of Uk+1 } + (−1)j [vj , [vj −i , [vj −2 , · · · [v1 , u]]]] ∈ Uk+1 . This ends the proof.

Corollary 1.1.8. Let Z1 , . . . , Zm ∈ T (RN ) be fixed. Then Lie{Z1 , . . . , Zm } = span ZJ | with J = (j1 , . . . , jk ) ∈ {1, . . . , m}k , k ∈ N . This (non-trivial) fact (direct consequence of Proposition 1.1.7) will be used throughout the next sections, often without mention. The following notation will be used when dealing with “stratified” (or “graded”) Lie algebras. If V1 , V2 are subsets of T (RN ), we denote [V1 , V2 ] := span [v1 , v2 ] | vi ∈ Vi , i = 1, , 2 . (1.17)

1.2 Lie Groups on RN 1.2.1 The Lie Algebra of a Lie Group on RN We first recall a well-known definition. Definition 1.2.1 (Lie group on RN ). Let ◦ be a given group law on RN , and suppose that the map RN × RN (x, y) → y −1 ◦ x ∈ RN is smooth. Then G := (RN , ◦) is called a Lie group on RN . Convention. For the simplicity of notation, we shall assume that the origin 0 of RN is the identity of G. This assumption is not restrictive. Indeed, if e ∈ G is the identity of G (a Lie group on RN ), we can consider new coordinates on RN given by the C ∞ -diffeomorphism defined by T (x) = x − e. Thus, we obtain a new Lie group on = (RN , ∗), where RN , G y ∗ y = (y + e) ◦ (y + e) − e,

14

1 Stratified Groups and Sub-Laplacians

are isomorphic Lie groups via the with identity T (e) = 0. Obviously, G and G diffeomorphism T . Furthermore, we shall see that every homogeneous Lie group on RN (see Section 1.3) has identity element equal to the origin of RN . Finally, the main subject of this book, i.e. Carnot groups, are naturally isomorphic to a homogeneous Lie group on RN . This justifies our convention and motivates the study of Lie groups on RN , i.e. roughly speaking, Lie groups with a global chart. Fixed α ∈ G, we denote by τα (x) := α ◦ x the left-translation by α on G. A (smooth) vector field X on RN is called left-invariant on G if X(ϕ ◦ τα ) = (Xϕ) ◦ τα

(1.18)

for every α ∈ G and for every smooth function ϕ : RN → R. We denote by g the set of the left-invariant vector fields on G. It is quite obvious to recognize that for every X, Y ∈ g and for every λ, μ ∈ R, we have λX + μY ∈ g and [X, Y ] ∈ g.

(1.19)

Then, g is a Lie algebra of vector fields, sub-algebra of T (RN ). It will be called the Lie algebra of G. Example 1.2.2. The map (x1 , x2 , x3 ) ◦ (y1 , y2 , y3 ) = x1 + y1 , x2 + y2 , x3 + y3 + 2 (x2 y1 − x1 y2 ) endows R3 with a structure of Lie group. In several next examples, we shall refer to H1 = (R3 , ◦) as the Heisenberg–Weyl group on R3 . It is a direct computation to show that the vector fields X1 = ∂x1 + 2 x2 ∂x3 ,

X2 = ∂x2 − 2 x1 ∂x3

(1.20)

are left invariant w.r.t. ◦. Consequently, X1 , X2 , [X1 , X2 ] ∈ h1 , say, the Lie algebra of H1 (this notation is not standard). We shall show that, precisely, h1 =

span{X1 , X2 , [X1 , X2 ]} = Lie{X1 , X2 }. From the theorem of differentiation of composite functions, we easily get the following characterization of left-invariant vector fields on G. Proposition 1.2.3 (Characterization of g. I). Let G be a Lie group on RN , and let g be the Lie algebra of G. The (smooth) vector field X belongs to g if and only if (XI )(α ◦ x) = Jτα (x) · (XI )(x)

∀ α, x ∈ G.

(1.21)

As usual, Jτα (x) denotes the Jacobian matrix at the point x of the map τα . Proof. For every smooth function ϕ on RN , we have (X(ϕ ◦ τα ))(x) = ∇(ϕ ◦ τα )(x) · XI (x) = (∇ϕ)(τα (x)) · Jτα (x) · XI (x) and

1.2 Lie Groups on RN

15

(Xϕ)(τα (x)) = (∇ϕ)(τα (x)) · XI (τα (x)). Then X ∈ g if and only if (∇ϕ)(τα (x)) · Jτα (x) · XI (x) = (∇ϕ)(τα (x)) · XI (τα (x))

(1.22)

N ∞ ∞ for Nevery α, x ∈ R and for every ϕ ∈ C (C , R). By choosing ϕ(x) = j =1 hj xj with hj ∈ R for 1 ≤ j ≤ N, (1.22) gives

hT · Jτα (x) · XI (x) = hT · XI (τα (x)) for every h ∈ RN , which obviously implies (1.21).

Interchanging α with x in (1.21), we obtain (XI )(x ◦ α) = Jτx (α) · (XI )(α) for all α, x ∈ G, so that, when α = 0, (XI )(x) = Jτx (0) · (XI )(0)

∀ x ∈ G.

(1.23)

This identity says that a left-invariant vector field on G is completely determined by its value at the origin (and by the Jacobian matrix at the origin of the left-translation). The following result shows that (1.23) characterizes the vector fields in g. Proposition 1.2.4. Let G be a Lie group on RN , and let g be the Lie algebra of G. Let η be a fixed vector of RN , and define the (component functions of the) vector field X as follows (1.24) XI (x) := Jτx (0) · η, x ∈ RN . Then X ∈ g. Proof. Definition (1.24) gives XI (α ◦ x) = Jτα◦x (0) · η,

α, x ∈ RN .

(1.25)

On the other hand, since the composition law on G is associative, we have τα◦x = τα ◦ τx , so that Jτα◦x (0) = Jτα (x) · Jτx (0). Replacing this identity in (1.25), we get XI (α ◦ x) = Jτα (x) · Jτx (0) · η, which implies, by (1.24), XI (α ◦ x) = Jτα (x) · XI (x). Then, by Proposition 1.2.3, X ∈ g. This ends the proof.

Corollary 1.2.5 (Characterization of g. II). Let G be a Lie group on RN , and let g be the Lie algebra of G. The vector field X belongs to g iff (XI )(x) = Jτx (0) · (XI )(0)

∀ x ∈ G.

(1.26)

16

1 Stratified Groups and Sub-Laplacians

Proof. If X satisfies (1.26), then, by Proposition 1.2.4 (setting η := XI (0)), we have X ∈ g. Vice versa, if X ∈ g, then we already showed that (1.26) follows from (1.21) of Proposition 1.2.3.

Example 1.2.6. If G = H1 (see Example 1.2.2), we have

1 0 0 Jτx (0) = 0 1 0 . 2x2 −2x1 1 For example, for X1 = ∂x1 + 2 x2 ∂x3 , we recognize that, for every x ∈ H1 ,

1 1 1 0 0 (X1 I )(x) = = 0 0 1 0 · 0 = Jτx (0) · (XI )(0). 2 x2 0 2x2 −2x1 1 The same obviously holds, e.g. for the fields X2 = ∂x2 − 2 x1 ∂x3 and [X1 , X2 ] = −4∂x3 . From Proposition 1.2.3 and identity (1.23) it follows that g is a vector space of dimension N. Indeed, the following proposition holds. Proposition 1.2.7 (Characterization of g RN . III). Let G be a Lie group on RN , and let g be the Lie algebra of G. The map J : RN → g,

η → J (η)

with J (η) defined by J (η)I (x) = Jτx (0) · η

(1.27)

is an isomorphism of vector spaces. In particular, dim g = N. Proof. We first observe that J is well defined since, by Proposition 1.2.4, J (η) ∈ g for every η ∈ RN . Moreover, by identity (1.23), J (RN ) = g. The linearity of J is obvious. Then it remains to prove that J is injective. Suppose J (η) = 0. This means that Jτx (0)·η = 0 for every x ∈ RN . In particular Jτ0 (0) · η = 0. On the other hand, since the left-translation τ0 is the identity map,

Jτ0 (0) · η = η. Then η = 0, and J is one-to-one. Example 1.2.8. The Lie algebra h1 of G = H1 (see Example 1.2.2 for the notation) is given by span{X1 , X2 , [X1 , X2 ]}. Indeed, X1 , X2 , [X1 , X2 ] are three linearly independent left-invariant vector fields and dim(h1 ) = 3, as stated in Proposition 1.2.7. Again using the same proposition, we could also argue as follows: X1 , X2 , [X1 , X2 ] are the vector fields obtained by multiplying Jτx (0) respectively times the basis of R3 (1, 0, 0)T ,

(0, 1, 0)T ,

(0, 0, −4)T .

In what follows, the next remarks will be very useful.

1.2 Lie Groups on RN

17

Remark 1.2.9. Let X ∈ g, and denote by η the value of XI at x = 0, i.e. η = XI (0). Then, by identity (1.23), XI (x) = Jτx (0) · η. As a consequence, for every smooth function ϕ on RN , d d ϕ(x ◦ (tη)) = ϕ(τx (tη)) dt t=0 dt t=0 = ∇ϕ(x) · Jτx (0) · η = ∇ϕ(x) · XI (x) = (Xϕ)(x). Then (Xϕ)(x) =

d ϕ(x ◦ (tη)), dt t=0

η = XI (0).

(1.28)

Identity (1.28) characterizes the left-invariant vector fields on G. This follows from the next remark. Remark 1.2.10. Let X be a vector field on RN . Assume that there exists η ∈ RN such that, for every ϕ ∈ C ∞ (RN , R), d (Xϕ)(x) = ϕ(x ◦ (tη)) ∀ x ∈ RN . (1.29) dt t=0 Then η = XI (0) and X ∈ g. Indeed, by taking ϕ(x) = xj = Ij (x) and x = 0 in (1.29), one gets d (XI )j (0) = (tη)j = ηj , dt t=0 i.e. XI (0) = η. Then (1.29) and the associativity of ◦ imply d d (Xϕ)(α ◦ x) = ϕ((α ◦ x) ◦ (tη)) = (ϕ ◦ τα )(x ◦ (tη)) dt t=0 dt t=0 = X(ϕ ◦ τα )(x) for every α, x ∈ G. Then X is left-invariant on G.

Collecting together the above remarks, we have proved the following result. Proposition 1.2.11 (Characterization of g. IV). Let G be a Lie group on RN , and let g be the Lie algebra of G. The vector field X belongs to g iff there exists η ∈ RN such that, for every ϕ ∈ C ∞ (RN , R), d (1.30) (Xϕ)(x) = ϕ(x ◦ (tη)) ∀ x ∈ RN . dt t=0 In this case η = XI (0).

18

1 Stratified Groups and Sub-Laplacians

Remark 1.2.12. For every x ∈ RN and X ∈ g, the following expansion holds exp(tX)(x) = x ◦ (tη) + o(t) as t → 0,

η = XI (0).

(1.31)

Indeed, since XI (x) = Jτx (0) · η, x ◦ tη = τx (tη) = τx (0) + tJτx (0) · η + o(t) = x + tXI (x) + o(t). Then (1.31) follows from (1.10), see page 8. We remark that two vector fields can be linearly independent in T (RN ) without being linearly independent at every point. Take, for example, ∂x1 and x1 ∂x2 in R2 . Moreover, two vector fields can be linearly dependent at every point without being linearly dependent in T (RN ). Take, for example, ∂x1 and x1 ∂x1 in R2 . The following result shows that neither of the previous situations can occur for left-invariant vector fields on a Lie group G on RN . Indeed, given a family of vector fields X1 , . . . , Xm ∈ g, the rank of the subset of RN spanned by {X1 I (x), . . . , Xm I (x)} is independent of x. More precisely, we have the following result. Proposition 1.2.13 (Constant rank). Let G be a Lie group on RN , and let g be the Lie algebra of G. Let X1 , . . . , Xm ∈ g. Then the following statements are equivalent: (i) (ii) (iii) (iv)

X1 , . . . , Xm are linearly independent (in g); X1 I (0), . . . , Xm I (0) are linearly independent (in RN ); ∃ x0 ∈ RN : X1 I (x0 ), . . . , Xm I (x0 ) are linearly independent (in RN ); X1 I (x), . . . , Xm I (x) are linearly independent (in RN ) for all x ∈ RN .

Proof. We first recall that, by identity (1.23), Xj I (x) = Jτx (0) · ηj , for every x ∈

RN .

with ηj = Xj I (0),

On the other hand, since τx −1 ◦ τx = I , Jτx −1 (x) · Jτx (0) = IN .

Hence Jτx (0) is non-singular for every x ∈ RN . Then (ii), (iii) and (iv) are equivalent. The equivalence between (i) and (ii) follows from Proposition 1.2.7. Indeed, with the notation of that proposition, for every j ∈ {1, . . . , m}, Xj = J (ηj ) with

ηj = Xj I (0), and J is an isomorphism of RN onto g. Example 1.2.14. The vector fields on R2 defined by X1 = ∂x1 ,

X2 = x1 ∂x2

do satisfy the so-called Hörmander condition rank(Lie{X1 , X2 }(x)) = 2

for every x ∈ R2 .

However, since X1 and X2 are independent as vector fields but X1 I (0), X2 I (0) are dependent as vectors of R2 , X1 and X2 are not left-invariant with respect to any group law on R2 .

1.2 Lie Groups on RN

19

1.2.2 The Jacobian Basis From Proposition 1.2.7 it follows that any basis of g is the image via J of a basis of RN . A natural definition is thus in order. Definition 1.2.15 (Jacobian basis). Let G be a Lie group on RN , and let g be the Lie algebra of G. If {e1 , . . . , eN } is the canonical basis7 of RN and J is the map defined in Proposition 1.2.7, we call {Z1 , . . . , ZN },

Zj := J (ej )

the Jacobian basis of g. (Note. We warn the reader that the notion of Jacobian basis is strictly related to the fact that, presently, G is a Lie group on RN , and we are making reference to the fixed Cartesian coordinates on RN . Hence, the Jacobian basis is not well-posed on general Lie groups. Moreover, if we perform a change of coordinates on RN , even a linear one, the Jacobian basis changes. Despite this “non-invariant”, non-coordinatefree nature of the Jacobian basis, the reader will soon recognize its usefulness.) From the very definition of J we obtain Zj I (x) = Jτx (0) · ej = j -th column of Jτx (0) so that, since Jτx (0) = IN ,

∀ x ∈ RN ,

(1.32)

Zj I (0) = ej .

From Remark 1.2.10 we also have d ∂ (Zj ϕ)(x) = ϕ(x ◦ tej ) = ϕ(x ◦ y) dt t=0 ∂yj y=0

(1.33)

for every ϕ ∈ C ∞ (RN ) and every x ∈ G. Consequently, the Jacobian basis {Z1 , . . . , ZN } of g is given by the N column of the Jacobian matrix Jτx (0) (whence the name). Moreover, Zj |0 = ∂/∂xj |0 and (Zj ϕ)(x) = (∂/∂yj )|y=0 ϕ(x ◦ y)

∀ ϕ ∈ C ∞ (RN ), x ∈ G.

(1.34)

Summing up the above results, we have the following equivalent characterizations of the Jacobian basis. 7 Id est, for every j ∈ {1, . . . , N },

1 , . . . , 0)T . ej = (0, . . . , j

20

1 Stratified Groups and Sub-Laplacians

Proposition 1.2.16 (Jacobian basis). Let G be a Lie group on RN , and let g be the Lie algebra of G. Let j ∈ {1, . . . , N } be fixed. Then there exists one and only one vector field in g, say Zj , characterized by any of the following equivalent conditions: 1. Zj |0 = (∂/∂xj )|0 , i.e. (Zj ϕ)(0) =

∂ϕ (0) ∂ xj

for every ϕ ∈ C ∞ (RN , R);

2. for every ϕ ∈ C ∞ (RN , R), it holds ∂ ϕ(x ◦ y) (Zj ϕ)(x) = ∂ yj y=0

for every x ∈ G;

3. if ej denotes the j -th vector of the canonical basis of RN , then Zj I (0) = ej ; 4. the column vector of the component functions of Zj is Zj I (x) = Jτx (0) · ej = j -th column of Jτx (0); 5. for every x ∈ G, we have d (Zj ϕ)(x) = ϕ(x ◦ (tej )) dt t=0

for every ϕ ∈ C ∞ (RN , R).

The system of vector fields Z := {Z1 , . . . , ZN } is a basis of g, the Jacobian basis. The coordinates of X ∈ g w.r.t. Z are, orderly, the entries of the column vector XI (0). (For the proof of the last statement, see Remark 1.2.20.) In the sequel, when we need to endow g with a differentiable structure, we shall consider the vector space structure of g, making g (in a natural way) a differentiable manifold. Although the choice of a basis for g is completely immaterial, we shall prevalently fix a system of coordinates on g by choosing the Jacobian basis, then identifying g with RN in a fixed way. Example 1.2.17 (The Jacobian basis of H1 ). The Jacobian basis for the Lie algebra of H1 (see Example 1.2.2) is given by Z1 = ∂x1 + 2 x2 ∂x3 ,

Z2 = ∂x2 − 2 x1 ∂x3 ,

Z3 = ∂x3 ,

since, in this case, the Jacobian matrix at 0 of the left-translation is

1 0 0 Jτx (0) =

0 1 0 . 2x2 −2x1 1

1.2 Lie Groups on RN

21

Example 1.2.18 (A non-polynomial nilpotent Lie group on R3 ). It is a simple exercise to verify that the following operation (x1 , x2 , x3 ) ◦ (y1 , y2 , y3 ) := (arcsinh(sinh(x1 ) + sinh(y1 )), x2 + y2 + sinh(x1 )y3 , x3 + y3 ) endows R3 with a Lie group structure (G, ◦). The Lie algebra g of G is spanned by the vector fields, coefficient-vectors are given by the columns of the Jacobian matrix 1

0 cosh(x1 ) 0 Jτx (0) = 0 1 sinh(x1 ) 0 0 1 (the Jacobian basis), i.e. g = span{Z1 , Z2 , Z3 }, where 1 ∂x cosh(x1 ) 1 Z2 = ∂x2 Z3 = ∂x3 + sinh(x1 ) ∂x2 .

Z1 =

Note that G is nilpotent (of step two). Example 1.2.19 (A non-polynomial non-nilpotent Lie group on R2 ). It is a simple exercise to verify that the following operation on R2 (x1 , x2 ) ◦ (y1 , y2 ) = (x1 + y1 , y2 + x2 ey1 ) defines a Lie group structure, and the Jacobian basis is Z1 = ∂x1 + x2 ∂x2 , Z2 = ∂x2 . Hence, the relevant Lie algebra is not nilpotent, for [Z2 , Z1 ] = Z2 , so that, inductively, [· · · [[[Z2 , Z1 ], Z1 ], Z1 ] · · · Z1 ] = Z2

for all k ∈ N.

k times

Remark 1.2.20. Let us consider the map π : g → RN ,

X → π(X) := XI (0).

(1.35)

From the very definition of π, the following fact follows: if X ∈ g and we write η := π(X), then we have Jτx (0) · η = Jτx (0) · XI (0) = (XI )(x),

(1.36)

the last equality following from (1.23). Let now J be the map introduced in Proposition 1.2.7. Comparing (1.36) to (1.27), we recognize that J (π(X)) = J (η) = X,

π(J (η)) = η

∀ X ∈ g, η ∈ RN .

Thus π is the inverse map of J . We explicitly remark that π is the linear map which assigns, to every vector field X in g, the N-tuple η in RN of the coordinates of X with respect to the Jacobian basis. The coordinate of X with respect to this basis is simply given by XI (0) (see also Fig. 1.2).

22

1 Stratified Groups and Sub-Laplacians

Fig. 1.2. Figure of Remark 1.2.20

1.2.3 The (Jacobian) Total Gradient Let G = (RN , ◦) be a Lie group on RN , and let Z1 , . . . , ZN be the Jacobian basis8 of the Lie algebra g of G. For any differentiable function u defined on an open set Ω ⊆ RN , we consider a sort of “intrinsic” gradient of u given by (Z1 u, . . . , ZN u) (in the sequel, we shall call it (Jacobian) total gradient). Then it follows from (1.32) that (Z1 u(x), . . . , ZN u(x)) = ∇u(x) · Jτx (0)

∀ x ∈ Ω.

(1.37)

On the other hand, since Jτx (0) is non-singular and its inverse is given by Jτx −1 (0), we can write the Euclidean gradient of u in terms of its total gradient in the following way ∀ x ∈ Ω. (1.38) ∇u(x) = (Z1 u(x), . . . , ZN u(x)) · Jτx −1 (0) From (1.38) we immediately obtain the following result. We shall follow the notation of Remark 1.2.3. Proposition 1.2.21. Let G be a Lie group on RN , and let Z1 , . . . , ZN be the relevant Jacobian basis (or any basis for g). Let Ω ⊆ RN be an open and connected set. A function u ∈ C 1 (Ω, R) is constant in Ω if and only if its total gradient (Z1 u, . . . , ZN u) vanishes identically on Ω. (Note. A significant improvement of this result will be available in the stratified setting. See, e.g. Proposition 1.5.6). Proof. From (1.37) and (1.38) it follows that the total gradient of u vanishes at x ∈ Ω if and only if ∇u(x) = 0.

8 In this section, we consider the Jacobian basis for the sake of brevity. In fact, Z , . . . , Z N 1

can be replaced by any basis X1 , . . . , XN of g. Indeed, note that in this case there exists a N × N non-singular constant matrix M such that (X1 I (x) · · · XN I (x)) = M · (Z1 I (x) · · · ZN I (x)).

1.2 Lie Groups on RN

23

Example 1.2.22. When G = H1 , it indeed holds (Z1 u, Z2 u, Z3 u) = (∂x1 u + 2 x2 ∂x3 u, ∂x2 u − 2 x1 ∂x3 u, ∂x3 u)

1 0 0 = ∂x1 u, ∂x2 u, ∂x3 u · 0 1 0 = ∇u · Jτx (0), 2x2 −2x1 1 and, vice versa, (Z1 u, Z2 u, Z3 u) · Jτx −1 (0) = (Z1 u, Z2 u, Z3 u) ·

1 0 −2x2

0 1 2x1

0 0 1

= ∇u.

1.2.4 The Exponential Map of a Lie Group on RN The next lemma will be useful to define the notion of Exponential map from g to G, one of the most important tools in the Lie group theory. Lemma 1.2.23 (Completeness of g). Let (G, ◦) be a Lie group on RN , and let g be its Lie algebra. Let X ∈ g, and let γ : [t0 , t0 + T ] → RN be an integral curve of X. Then: (i) α ◦ γ is an integral curve of X for every α ∈ G. (ii) γ can be continued to an integral curve of X on the interval [t0 − T , t0 + 2T ]. Proof. (i) For every t ∈ [t0 , t0 + T ], we have (by (1.21)) d d (α ◦ γ (t)) = (τα (γ (t))) = Jτα (γ (t)) · γ˙ (t) dt dt = Jτα (γ (t)) · XI (γ (t)) = X(α ◦ γ (t)). (ii) Define Γ : [t0 − T , t0 + 2T ] → RN as follows: ⎧ −1 ⎪ ⎨ γ (t0 ) ◦ (γ (t0 + T )) ◦ γ (t + T ) if t0 − T ≤ t ≤ t0 , if t0 ≤ t ≤ t0 + T , Γ (t) := γ (t), ⎪ ⎩ −1 γ (t0 + T ) ◦ (γ (t0 )) ◦ γ (t − T ) if t0 + T ≤ t ≤ t0 + 2T . Then, by (i), Γ is an integral curve of X and, obviously, Γ |[t0 ,t0 +T ] ≡ γ .

From assertion (ii) of this Lemma we immediately obtain the following important statement: For every X ∈ g, the map (x, t) → exp(tX)(x) is well-defined for every x ∈ RN and every t ∈ R. The next corollary easily follows from the assertion (i) of Lemma 1.2.23.

24

1 Stratified Groups and Sub-Laplacians

Corollary 1.2.24. Let (G, ◦) be a Lie group on RN , and let g be its Lie algebra. Let X ∈ g and x, y ∈ G. Then x ◦ exp(tX)(y) = exp(tX)(x ◦ y)

(1.39)

for every t ∈ R. In particular, for y = 0, exp(tX)(x) = x ◦ exp(tX)(0). Proof. By Lemma 1.2.23-(i), t → x ◦ exp(tX)(y) is an integral curve of X. Moreover, (x ◦ exp(tX)(y))|t=0 = x ◦ y. Then (1.39) follows.

Definition 1.2.25 (Exponential map). Let G be a Lie group on RN , and let g be its Lie algebra. The exponential map of the Lie group G is defined by Exp : g → G,

Exp (X) = exp(1 · X)(0).

More explicitly, Exp (X) is the value at the time t = 1 of the path γ (t) solution to γ˙ (t) = XI (γ (t)), γ (0) = 0. (See Fig. 1.3.) From Corollary 1.2.24 and identity (1.11b) (with τ = −t) we get Exp (−X) ◦ Exp (X) = 0. Indeed, Exp (−X) ◦ Exp (X) = Exp (−X) ◦ exp(X)(0) = exp(X)(Exp (−X)) = exp(X)(exp(−X)(0)) = 0. Then we have

(Exp (X))−1 = Exp (−X).

Fig. 1.3. Figure of Definition 1.2.25

We give an explicit example of exponential map.

(1.40)

1.2 Lie Groups on RN

25

Example 1.2.26 (The Exp map on H1 ). Let us consider once again the Heisenberg– Weyl group H1 on R3 . In Example 1.2.8, we showed that a basis for its Lie algebra h1 is given by X1 , X2 , X3 , where X1 = ∂x1 + 2 x2 ∂x3 , X2 = ∂x2 − 2 x1 ∂x3 and X3 = [X1 , X2 ] = −4∂x3 . Let us construct the exponential map. We set, for ξ ∈ R3 , ξ · X := ξ1 X1 + ξ2 X2 + ξ3 X3

1 0 0 ξ1 = ξ1 + ξ2 + ξ3 0 = . ξ2 0 1 −4ξ3 + 2ξ1 x2 − 2ξ2 x1 2 x2 −2 x1 −4 By Definition 1.1.2, for fixed x ∈ H1 , we have exp(ξ · X)(x) = γ (1), where γ (s) = (γ1 (s), γ2 (s), γ3 (s)) is the solution to γ˙ (s) = (ξ · X)I (γ (s)) = (ξ1 , ξ2 , −4ξ3 + 2ξ1 γ2 (s) − 2 ξ2 γ1 (s)), γ (0) = x. Solving the above system of ODE’s, one gets ⎛

⎞ x1 + ξ1 ⎠. x 2 + ξ2 exp(ξ · X)(x) = ⎝ x 3 − 4 ξ3 + 2 ξ1 x 2 − 2 ξ2 x 1

As a consequence, by Definition 1.2.25, we obtain ⎛

⎞ ξ1 Exp (ξ · X) = exp(ξ · W )(0) = ⎝ ξ2 ⎠ , −4 ξ3 so that Exp is globally invertible and its inverse map is given by ⎞ ⎛ y1 Log (y) := (Exp )−1 (y) = ⎝ y2 ⎠ · X. − 14 y3 For example, we have ⎛

⎞ −ξ1 −1 Exp (−ξ · X) = ⎝ −ξ2 ⎠ = −Exp (ξ · X) = Exp (−ξ · X) , +4 ξ3 since the inverse of x in H1 coincides with −x. This fact tests, in this simple example, the validity of (1.40).

Remark 1.2.27 (Local invertibility of Exp ). Let {X1 , . . . , XN } be a basis of g. Then, for every X ∈ g, X=

N j =1

ξj X j

for a suitable ξ = (ξ1 , . . . , ξN ) ∈ RN ,

26

1 Stratified Groups and Sub-Laplacians

Fig. 1.4. The relation between the group G, its algebra g and RN

so that

Exp (X) = exp

N

ξj Xj (0).

j =1

From the classical theory of ODE’s we know that the map N

ξj Xj (0) (ξ1 , . . . , ξN ) → exp j =1

is smooth. Then we can say that the map g X → Exp (X) ∈ G is smooth. From the Taylor expansion (1.10) (page 8) we get Exp (X) =

N

ξj ηj + O(|ξ |2 ),

as |ξ | → 0,

j =1

where ηj = Xj I (0). Denote by E the matrix whose column vectors are η1 , . . . , ηN . Then JExp (0) = E

(we fix on g coordinates related to the Xj ’s).

In particular, if {X1 , . . . , XN } = {Z1 , . . . , ZN } is the Jacobian basis of g, then JExp (0) = IN

(we fix on g coordinates related to the Zj ’s).

(1.41)

1.2 Lie Groups on RN

27

As a consequence, Exp is a diffeomorphism from a neighborhood of 0 ∈ g onto a neighborhood of 0 ∈ G. Where defined,9 we denote by Log the inverse map of Exp. Example 1.2.28. With the notation of Example 1.2.26, we recall that the Jacobian basis for h1 is given by (see Example 1.2.17) Z1 = X1 ,

Z2 = X2 ,

1 Z3 = − X3 . 4

Hence, by the computations in Example 1.2.26, we have 1 Exp (ξ1 Z1 + ξ2 Z2 + ξ3 Z3 ) = Exp ξ1 X1 + ξ2 X2 − ξ3 X3 4

! = (ξ1 , ξ2 , ξ3 ), (1.42)

and (1.41) is readily verified.

The next proposition is an easy consequence of Corollary 1.2.24 and shows an important link between the composition law in G and the exponential map. Proposition 1.2.29 (Exponentiation and composition). Let (G, ◦) be a Lie group on RN , and let g be its Lie algebra. Let x, y ∈ G. Assume Log (y) is defined. Then x ◦ y = exp(Log (y))(x).

(1.43)

Proof. Let X = Log (y). This means that y = Exp (X) = exp(X)(0). Then, by Corollary 1.2.24, we infer x ◦ y = x ◦ exp(X)(0) = exp(X)(x). This is precisely (1.43), and the proof is complete.

By writing y = Exp (X) in (1.43), we obtain x ◦ Exp (X) = exp(X)(x)

for every X ∈ g and every x ∈ G.

(1.44)

We give two examples of this proposition, the first is very simple, the second one a little more elaborated. Example 1.2.30 (of (1.43) for H1 ). Let us consider once again the Heisenberg–Weyl group H1 on R3 . Proceeding with the computations in Example 1.2.26, we have 9 We shall see in Chapter 2 that, in many important situations, such as for connected and

simply connected nilpotent Lie groups, Exp is globally invertible. Any Carnot group is a connected and simply connected nilpotent Lie group.

28

1 Stratified Groups and Sub-Laplacians

Fig. 1.5. Figure of Proposition 1.2.29

! ! 1 exp(Log (y))(x) = exp y1 , y2 , − y3 · X (x) 4 ⎛ ⎞ x1 + y1 ⎠ = x ◦ y, x2 + y2 =⎝ x3 + y3 + 2 y1 x2 − 2 y2 x1 which tests, in this simple example, the validity of (1.43).

Example 1.2.31. Let us consider once again the Lie group G introduced in Example 1.2.18. The relevant Jacobian basis is {Z1 , Z2 , Z3 }, where Z1 =

1 ∂x , cosh(x1 ) 1

Z2 = ∂x2 ,

Z3 = ∂x3 + sinh(x1 ) ∂x2 .

Given ξ = (ξ1 , ξ2 , ξ3 ) ∈ R3 and x ∈ G, we set ξ · Z = Z))(x) = γ (t) is the solution to

3

i=1 ξi

Zi , so that exp(t (ξ ·

γ˙ (t) = ξ1 Z1 I (γ (t)) + ξ2 Z2 I (γ (t)) + ξ3 Z3 I (γ (t)), i.e. more explicitly (writing γ (t) = (γ1 (t), γ2 (t), γ3 (t))), ⎧ −1 ⎪ ⎨ γ˙1 (t) = ξ1 (cosh(γ1 (t))) , γ1 (0) = x1 , γ˙2 (t) = ξ2 + ξ3 sinh(γ1 (t)), γ2 (0) = x2 , ⎪ ⎩ γ˙3 (t) = ξ3 , γ3 (0) = x3 . A direct computation gives ⎧ ⎪ ⎨ γ1 (t) = arcsinh(sinh(x1 ) + ξ1 t), 2 γ2 (t) = x2 + (ξ2 + ξ3 sinh(x1 ))t + ξ1 ξ3 t2 , ⎪ ⎩ γ3 (t) = x3 + ξ3 t.

γ (0) = x,

1.2 Lie Groups on RN

29

In particular, Exp (ξ · Z) = exp(1 (ξ · Z))(0), i.e. ! 1 Exp (ξ · Z) = arcsinh(ξ1 ), ξ2 + ξ1 ξ3 , ξ3 , 2 so that

! 1 Log (y1 , y2 , y3 ) = sinh(y1 ), y2 − sinh(y1 ) y3 , y3 · Z. 2

Collecting the above facts together, we get ! ! 1 sinh(y1 ) y3 , y3 · Z (x) 2 arcsinh(sinh(x1 ) + sinh(y1 )) = x2 + (y2 − 12 sinh(y1 ) y3 + y3 sinh(x1 )) + sinh(y1 ) y3 x3 + y3

arcsinh(sinh(x1 ) + sinh(y1 )) = = x ◦ y, x2 + y2 + y3 sinh(x1 ) x3 + y3

exp(Log (y))(x) = exp

sinh(y1 ), y2 −

1 2

as stated in (1.43) of Proposition 1.2.29.

Remark 1.2.32 (The composition of G induces a composition on g via Exp: The Campbell–Hausdorff operation). Suppose that Exp : g → G and Log : G → g are globally defined C ∞ maps, inverse to each other. We then define on g the operation X Y := Log (Exp (X) ◦ Exp (Y )), X, Y ∈ g. (1.45) It is immediately seen that defines a Lie group structure on g and Exp : (g, ) → (G, ◦) is a Lie-group isomorphism. Indeed, this last fact is obvious from the very definition of , whereas the associativity of on g follows immediately from the associativity of ◦ on G. One of the most striking facts about Lie algebras and Lie groups is that (under suitable hypotheses) the operation on g is well-posed and can be expressed in a somewhat “universal” way as a sum of iterated Lie-brackets of X and Y (see (2.43) and Theorem 2.2.13, page 129). For example, the first few terms are XY =X+Y +

1 1 1 [X, Y ] + [X, [X, Y ]] − [Y, [X, Y ]] + · · · . 2 12 12

(1.46)

We shall deal extensively on the composition law throughout the book. We give an example of when G is the Heisenberg–Weyl group on R3 .

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1 Stratified Groups and Sub-Laplacians

Example 1.2.33 (The operation for H1 ). The reader is invited to compare this example to the diagram in Fig. 1.6. We use the notation and computations of Example 1.2.26. When G = H1 , we saw that Exp is globally invertible. Hence (1.45) makes sense. We fix X ∈ h1 . If Z1 , Z2 , Z3 is the Jacobian basis for h1 , and we set, for brevity, ξ := XI (0), we have (see Remark 1.2.20) X = ξ1 Z1 + ξ2 Z2 + ξ3 Z3 =: ξ · Z. Analogously, if Y ∈ h1 , we set η := Y I (0), so that Y = η · Z. Thus, we derive Log Exp (X) ◦ Exp (Y ) = Log Exp (ξ · Z) ◦ Exp (η · Z) = (see (1.42))

Log (ξ ◦ η)

= Log (ξ1 + η1 , ξ2 + η2 , ξ3 + η3 + 2 η1 ξ2 − 2 η2 ξ1 ) (again from (1.42)) = (ξ1 + η1 , ξ2 + η2 , ξ3 + η3 + 2 η1 ξ2 − 2 η2 ξ1 ) · Z = (ξ1 + η1 ) Z1 + (ξ2 + η2 ) Z2 + (ξ3 + η3 + 2 η1 ξ2 − 2 η2 ξ1 ) Z3 .

(1.47)

On the other hand, we consider (1.46), truncated to the commutators of length two (sine h1 is nilpotent of step two!), and we explicitly write down X Y in our case, thus obtaining 1 [ξ · Z, η · Z] 2 = ξ1 Z1 + ξ2 Z2 + ξ3 Z3 + η1 Z1 + η2 Z2 + η3 Z3

(ξ · Z) (η · Z) = ξ · Z + η · Z +

1 [ξ1 Z1 + ξ2 Z2 + ξ3 Z3 , η1 Z1 + η2 Z2 + η3 Z3 ] 2 (here we use [Z1 , Z2 ] = −4 Z3 , [Z1 , Z3 ] = [Z2 , Z3 ] = 0) +

= (ξ1 + η) Z1 + (ξ2 + η2 ) Z2 + (ξ3 + η3 ) Z3 1 (−4 ξ1 η2 + 4 ξ2 η1 ) Z3 + 2 = (ξ1 + η) Z1 + (ξ2 + η2 ) Z2 + (ξ3 + η3 + 2 η1 ξ2 − 2 η2 ξ1 ) Z3 , which equals the last term in (1.47). As a consequence, we have proved that in this case it holds 1 Log Exp (X) ◦ Exp (Y ) = X + Y + [X, Y ]. 2 For the group considered in Examples 1.2.18 and 1.2.31, the reader is invited to test a similar formula.

1.3 Homogeneous Lie Groups on RN

31

1.3 Homogeneous Lie Groups on RN We begin by giving the definition of homogeneous Lie group (see also E.M. Stein [Ste81]). Definition 1.3.1 (Homogeneous Lie group (on RN )). Let G = (RN , ◦) be a Lie group on RN (according to Definition 1.2.1). We say that G is a homogeneous (Lie) group (on RN ) if the following property holds: (H.1)

There exists an N-tuple of real numbers σ = (σ1 , . . . , σN ), with 1 ≤ σ1 ≤ · · · ≤ σN , such that the “dilation” δλ : RN → RN ,

δλ (x1 , . . . , xN ) := (λσ1 x1 , . . . , λσN xN )

is an automorphism of the group G for every λ > 0. We shall denote by G = (RN , ◦, δλ ) the datum of a homogeneous Lie group on RN with composition law ◦ and dilation group {δλ }λ>0 . (Note. A note similar to that given after Definition 1.2.15 of Jacobian basis applies to the notion of homogeneous Lie group: the notion of homogeneous Lie group is not coordinate-free and strongly depends on the choice of a fixed system of coordinates on RN . Nonetheless, the reader will soon recognize the suitability of this notion.) The family of dilations {δλ }λ>0 forms a one-parameter group of automorphisms of G whose identity is δ1 = I, the identity map of RN . Indeed, we have δr s (x) = δr δs (x)

∀ x ∈ G, r, s > 0.

Moreover, (δλ )−1 = δλ−1 . In the sequel, {δλ }λ>0 will be referred to as the dilation group (or group of dilations) of G. From (H.1) it follows that δλ (x ◦ y) = (δλ x) ◦ (δλ y)

∀ x, y ∈ G

(1.48)

and, if e denotes the identity of G, δλ (e) = e for every λ > 0. This obviously implies that e = 0. This is consistent with our previous assumption that the origin is the identity of G. For example, the Heisenberg–Weyl group H1 (see Example 1.2.2, page 14) is a homogeneous Lie group if R3 is equipped with the dilations δλ (x1 , x2 , x3 ) = (λx1 , λx2 , λ2 x3 ). Remark 1.3.2. Suppose G = (RN , ◦) is a Lie group on RN such that there exists an N-tuple of positive real numbers σ = (σ1 , . . . , σN ) such that δλ : RN → RN ,

dλ (x1 , . . . , xN ) := (λσ1 x1 , . . . , λσN xN )

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1 Stratified Groups and Sub-Laplacians

is an automorphism of the group G for every λ > 0. Then, modulo a permutation of the variables of RN , it is always not restrictive to suppose that σ1 ≤ · · · ≤ σN . Obviously, this permutation of the coordinates does not alter neither (the new permuted) G being a Lie group on RN nor the (relevant permuted) dilation δλ satisfying (1.48). Moreover, there exists a group of dilations δλ on G such that σ1 σN δλ (x1 , . . . , xN ) = (λ x1 , . . . , λ xN )

with 1 = σ1 ≤ · · · ≤ σN . Indeed, it suffices to take (once the σj ’s have been ordered increasingly) σj := σj /σ1 for every j = 1, . . . , N . Indeed, with this choice, we have δλ ≡ dλ1/σ1 , and δλ (x ◦ y) = δλ (x) ◦ δλ (y) follows from (1.48), restated for dλ , with λ replaced by λ1/σ1 . 1.3.1 δλ -homogeneous Functions and Differential Operators Before we continue the analysis of homogeneous Lie groups, we show some basic properties of homogeneous functions and homogeneous differential operators with respect to the family {δλ }λ . In this subsection, no group law is required on RN . Here, we only suppose that it is given on RN a family of maps δλ of the form δλ : RN → RN ,

δλ (x1 , . . . , xN ) := (λσ1 x1 , . . . , λσN xN ),

(1.49)

with fixed positive real numbers σ1 , . . . , σN . We set σ := (σ1 , . . . , σN ). A real function a defined on RN is called δλ -homogeneous of degree m ∈ R if a does not vanish identically and, for every x ∈ RN and λ > 0, it holds a(δλ (x)) = λm a(x). A non-identically-vanishing linear differential operator X is called δλ -homogeneous of degree m ∈ R if, for every ϕ ∈ C ∞ (RN ), x ∈ RN and λ > 0, it holds X(ϕ(δλ (x))) = λm (Xϕ)(δλ (x)). Let a be a smooth δλ -homogeneous function of degree m ∈ R and X be a linear differential operator δλ -homogeneous of degree n ∈ R. Then Xa is a δλ homogeneous function of degree m − n (unless Xa ≡ 0). Indeed, for every x ∈ RN and λ > 0, we have λn (Xa)(δλ (x)) = X(a(δλ (x))) = X(λm a(x)) = λm (Xa)(x). Given a multi-index α ∈ (N ∪ {0})N , α = (α1 , . . . , αN ), we define the δλ -length (or δλ -height) of α as

1.3 Homogeneous Lie Groups on RN

|α|σ = α, σ =

N

αi σi .

33

(1.50)

i=1

G-length of a multi-index. G -degree). When G = (RN , ◦, δλ ) is Definition 1.3.3 (G a homogeneous Lie group on RN with its given group of dilations {δλ }λ , we shall use the notation |α|G for the relevant δλ -length as defined in (1.50). In this case, we shall refer to |α|G as the G-length (or G-height) of α. Moreover, if p : G → R is a polynomial function (the sum below is intended to be finite) p(x) = cα x α , cα ∈ R, α

we say that degG (p) := max{|α|G : cα = 0} is the G-degree or δλ -(homogeneous) degree of p. Since x → xj and ∂/∂xj , j ∈ {1, . . . , N }, are obviously δλ -homogeneous of degree σj , the function x → x α and the differential operator D α are both δλ homogeneous of degree |α|σ . If a is a continuous function, δλ -homogeneous of degree m and a(x0 ) = 0 for some x0 ∈ RN , then m ≥ 0. Indeed, from a(δλ (x0 )) = λm a(x0 ) we get a(0) a(δλ (x0 )) = . λ→0 a(x0 ) a(x0 )

lim λm = lim

λ→0

Moreover, the continuous and δλ -homogeneous of degree 0 functions are precisely the constant (non-zero) functions. Indeed, a(x) = a(δλ (x)) = lim a(δλ (x)) = a(0). λ→0+

Let us now consider a smooth and δλ -homogeneous of degree m ∈ R function a and a multi-index α. Assume that D α a is not identically zero. Then, since D α a is smooth and δλ -homogeneous of degree m − |α|σ , it has to be m − |α|σ ≥ 0, i.e. |α|σ ≤ m. This result can be restated as follows: D α a ≡ 0 ∀ α such that |α|σ > m. Thus a is a polynomial function. Let a(x) = α∈A aα x α , where A is a finite set of multi-indices and aα ∈ R for every α ∈ A. Since a is δλ -homogeneous of degree m, we have λm aα x α = λm a(x) = a(δλ (x)) = aα λ|α|σ x α . α∈A

Hence

λm a

α

=

λ|α|σ

α∈A

aα for every λ > 0, so that |α|σ = m if aα = 0. Then

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1 Stratified Groups and Sub-Laplacians

a(x) =

aα x α .

(1.51)

|α|σ =m

It is quite obvious that every polynomial function of the form (1.51) is δλ -homogeneous of degree m. Thus, we have proved the following proposition. Proposition 1.3.4 (Smooth δλ -homogeneous functions). Let δλ be as in (1.49). Suppose that a ∈ C ∞ (RN , R). Then a is δλ -homogeneous of degree m ∈ R if and only if a is a polynomial function of the form (1.51) with some aα = 0. As a consequence, the set of the degrees of the smooth δλ -homogeneous (non-vanishing) functions is precisely the set of the nonnegative real numbers A = {|α|σ : α ∈ (N ∪ {0})N }, with |α|σ = 0 if and only if a is constant. From the proposition above one easily obtains the following characterization of the smooth δλ -homogeneous vector fields. Proposition 1.3.5 (Smooth δλ -homogeneous vector fields). Let δλ be as in (1.49). Let X be a smooth non-vanishing vector field on RN , X=

N

aj (x) ∂xj .

j =1

Then X is δλ -homogeneous of degree n ∈ R if and only if aj is a polynomial function δλ -homogeneous of degree σj − n (unless aj ≡ 0). Hence, the degree of δλ homogeneity of X belongs to the set of real (possibly negative) numbers Aj = {σj − |α|σ : α ∈ (N ∪ {0})N }, whenever j is such that aj is not identically zero. Proof. A direct computation shows the “if” part of the proposition. Vice versa, if X(ϕ ◦ δλ ) = λn (X ϕ) ◦ δλ , the choice ϕ(x) = xj yields λσj aj (x) = λn aj (δλ (x)), whence aj is a (smooth) δλ -homogeneous function of degree σj − n. By Proposi tion 1.3.4, aj is a polynomial function. For example, the differential operators X1 = ∂x1 + 2 x2 ∂x3 ,

X2 = ∂x2 − 2 x1 ∂x3

(1.52)

on R3 are δλ -homogeneous of degree one with respect to the dilation δλ (x1 , x2 , x3 ) = (λx1 , λx2 , λ2 x3 ). Also, the vector fields x13 X1 = x13 ∂x1 + 2 x13 x2 ∂x3 and x2 X2 = x2 ∂x1 − 2 x1 x2 ∂x3 are respectively δλ -homogeneous of degrees −2 and 0 w.r.t. the same dilation.

1.3 Homogeneous Lie Groups on RN

35

Corollary 1.3.6. Let δλ be as in (1.49). Let X be a smooth non-vanishing vector field. Then X is δλ -homogeneous of degree n ∈ R iff δλ XI (x) = λn XI (δλ (x)). Proof. Let X = N j =1 aj ∂xj . By Proposition 1.3.5, X is δλ -homogeneous of degree n iff aj (δλ (x)) = λσj −n aj (x) for any j ∈ {1, . . . , N }. This is equivalent to T T δλ (XI (x)) = δλ a1 (x), . . . , aN (x) = λσ1 a1 (x), . . . , λσN aN (x) T = λn a1 (δλ (x)), . . . , aN (δλ (x)) = λn XI (δλ (x)). This ends the proof.

As a straightforward consequence, we have the following simple fact. Remark 1.3.7. Let δλ be as in (1.49). Let X = 0 be a smooth vector field on RN of the form N aj (x) ∂xj . X= j =1

If X is δλ -homogeneous of degree n ∈ R, then, for every aj non-identically zero, we must have n ≤ σj . As a consequence, it has to be n ≤ σN (i.e. the set of the δλ homogeneous degrees of the smooth vector fields is bounded above by the maximum exponent of the dilation). Hence, X has the form aj (x) ∂/∂xj . X= j ≤N, σj ≥n

Suppose now n > 0. Since aj is a polynomial function of degree σj − n and n > 0, then aj does not depend on xj , . . . , xN , aj (x) = aj (x1 , . . . , xj −1 ) (we agree to let aj (x1 , . . . , xj −1 ) = constant when j = 1). We already highlighted in the previous section the importance of these “pyramid”-shaped vector fields (see Remark 1.1.3). From this remark the next proposition straightforwardly follows. N Proposition 1.3.8. Let δλ be as in (1.49). Let X = j =1 aj (x) ∂xj be a smooth -homogeneous of positive degree. Then its adjoint operator X ∗ = vector field δ λ N ∗ − j =1 ∂j (aj ·) satisfies X = −X and X 2 = div(A · ∇ T ), where A is the square matrix (ai aj )i,j ≤N . Finally, X has null divergence.

(1.53)

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1 Stratified Groups and Sub-Laplacians

Proof. By the previous remark, the coefficient aj does not depend on xj . Then, for every smooth function ϕ, X∗ ϕ = −

N

∂j (aj ϕ) = −

j =1

N

aj ∂j ϕ = −Xϕ.

j =1

Moreover, it holds X = 2

N

ai ∂i (aj ∂j ) =

i,j =1

N i=1

∂i

N

ai aj ∂j

= div(A · ∇ T ),

j =1

where A is as in the statement of the proposition. Finally, div(XI ) =

0, since aj is independent of xj .

N

j =1 ∂j (aj )

=

Vector fields with different degree of δλ -homogeneity are linearly independent if they do not vanish at the origin. Indeed, the following proposition holds. Proposition 1.3.9. Let δλ be as in (1.49). Let X1 , . . . , Xk ∈ T (RN ) be δλ -homogeneous vector fields of degree n1 , . . . , nk , respectively. If ni = nj for i = j and if Xj I (0) = 0 for every j ∈ {1, . . . , k}, then X1 , . . . , Xk are linearly independent. Proof. Let c1 , . . . , ck ∈ R be such that kj =1 cj Xj = 0. Then, for every smooth function ϕ, 0=

k

cj Xj (ϕ(δλ x)) =

j =1

cj λnj (Xj ϕ)(δλ x)

∀ x ∈ RN .

j =1

If we take ϕ(x) = h, x = 0=

k

k

N

j =1 hj

xj , this identity at x = 0 gives

cj λnj ηj , h

∀ h ∈ RN ,

∀ λ > 0,

j =1

where ηj = Xj I (0). Equivalently, " 0=

k

# cj λnj ηj , h .

j =1

Due to the arbitrariness of h ∈ RN , this gives that (since ni = nj if i = j )

k

j =1 cj

λnj ηj = 0 for all λ > 0, so

cj ηj = 0 for any j ∈ {1, . . . , k}. This implies cj = 0, since, for every j = 1, . . . , k, ηj = 0 by the hypothesis.

1.3 Homogeneous Lie Groups on RN

37

The following simple proposition will be useful in the sequel. Proposition 1.3.10. Let δλ be as in (1.49). Let X1 , X2 be δλ -homogeneous vector fields of degree n1 , n2 , respectively. Then [X1 , X2 ] is δλ -homogeneous of degree n1 + n2 (unless X1 and X2 commute). As a consequence, if n1 , n2 are both positive, then every commutator of X1 , X2 containing k1 times X1 and k2 times X2 vanish identically whenever k1 n1 + k2 n 2 > σ N . Proof. It suffices to note that, for every smooth function ϕ on RN , one has (X1 X2 )(ϕ(δλ (x))) = λn2 X1 ((X2 ϕ)(δλ (x))) = λn2 +n1 (X1 X2 )(ϕ(δλ (x))). This proves the first part of the assertion, since [X1 , X2 ] = X1 X2 − X2 X1 (and [X1 , X2 ] ≡ 0 iff X1 and X2 commute). Finally, let X be a commutator of X1 , X2 containing k1 times X1 and k2 times X2 . By the first part of this proof, it follows inductively that X is δλ -homogeneous of degree k1 n1 + k2 n2 (unless X ≡ 0). By Remark 1.3.7, we know that if a smooth vector field is δλ -homogeneous of degree n ∈ R, then n ≤ σN . This ends the proof.

For example, the differential operators X1 = ∂x1 + 2x2 ∂x3 , X2 = ∂x2 − 2x1 ∂x3 on the Heisenberg–Weyl group H1 are homogeneous of degree one with respect to the dilation δλ (x1 , x2 , x3 ) = (λx1 , λx2 , λ2 x3 ), and [X1 , X2 ] = −4∂x3 is indeed δλ homogeneous of degree two. Moreover, any commutator of X1 , X2 of length ≥ 3 vanish identically, as stated in the last part of Proposition 1.3.10. Corollary 1.3.11. Let G = (RN , ◦, δλ ) be a homogeneous Lie group on RN , and let g be the Lie algebra of G. Let X1 , . . . , Xk ∈ g be non-identically vanishing and δλ -homogeneous of degrees n1 , . . . , nk , respectively. If ni = nj for i = j , then X1 , . . . , Xk are linearly independent. Proof. Since Xj I (x) = Jτx (0)·Xj I (0) for every x ∈ RN , and Xj is non-identically vanishing, then Xj I (0) = 0 for any j ∈ {1, . . . , k}. Hence the assertion follows from the previous proposition.

Proposition 1.3.12 (Nilpotence of homogeneous Lie groups on RN ). Let G = (RN , ◦, δλ ) be a homogeneous Lie group on RN , and let g be the Lie algebra of G. Then G is nilpotent of step ≤ σN , i.e. every commutator of vector fields in g containing more than σN terms vanishes identically. Moreover, if Zj is the j -th element of the Jacobian basis of g, Zj is δλ homogeneous of degree σj . Proof. Let Zj be the j -th element of the Jacobian basis of g. By Proposition 1.2.16-(5), for every ϕ ∈ C ∞ (RN , R), we have d d Zj (ϕ(δλ (x))) = (ϕ(δλ (x ◦ (tej )))) = (ϕ(δλ (x) ◦ δλ (tej ))) dt t=0 dt t=0 d = λσj (ϕ(δλ (x) ◦ (r ej ))) = λσj (Zj ϕ)(δλ (x)), dr r=0

38

1 Stratified Groups and Sub-Laplacians

i.e. Zj is δλ -homogeneous of degree σj . This proves the last part of the assertion. Since (Z1 , . . . , ZN ) is a linear basis for g, the above arguments show that every X ∈ g is a linear combination of δλ -homogeneous smooth vector fields of degrees at least σ1 ≥ 1. The first part of the assertion hence follows from Proposition 1.3.10.

1.3.2 The Composition Law of a Homogeneous Lie Group By using the elementary properties of the homogeneous functions showed in the previous section, we shall obtain a structure theorem for the composition law in a homogeneous Lie group (RN , ◦, δλ ). We first prove two lemmas. Lemma 1.3.13. Let δλ be as in (1.49). Let P : RN × RN → R be a smooth nonvanishing function such that P (δλ (x), δλ (y)) = λσj P (x, y)

∀ x, y ∈ RN , ∀ λ > 0,

for some j such that 1 ≤ j ≤ N . Assume also that P (x, 0) = xj ,

P (0, y) = yj .

(1.54)

Then P (x, y) = x1 + y1 if j = 1 and, if j ≥ 2, (x1 , . . . , xj −1 , y1 , . . . , yj −1 ), P (x, y) = xj + yj + P is a polynomial, the sum of mixed monomials in x1 , . . . , xj −1 , y1 , . . . , yj −1 . where P (δλ (x), δλ (y)) = λσj P (x, y). Finally, P (x, y) only depends on the xk ’s Moreover, P and yk ’s with σk < σj . Proof. By Proposition 1.3.4, P is a polynomial function of the following type: P (x, y) = cα,β x α y β , cα,β ∈ R. |α|σ +|β|σ =σj

On the other hand, by (1.54), xj = P (x, 0) =

cα,0 x α

|α|σ =σj

and yj = P (0, y) =

c0,β y α .

|β|σ =σj

Then P (x, y) = xj + yj +

cα,β x α y β .

(1.55)

|α|σ +|β|σ =σj , α,β=0

We can complete the proof by noticing that the condition |α|σ + |β|σ = σj , α, β = 0 is empty when j = 1, whereas it implies α = (α1 , . . . , αj −1 , 0, . . . , 0), β = (β1 , . . . , βj −1 , 0, . . . , 0) when j ≥ 2. As for the last assertion of the lemma, being α, β = 0 in the sum in the right-hand side of (1.55), the sum itself may depend only on the α’s and β’s with |α|σ , |β|σ <

σj , hence, on the xk ’s and yk ’s with σk < σj .

1.3 Homogeneous Lie Groups on RN

39

Lemma 1.3.14. Let δλ be as in (1.49). Let Q : RN × RN → R be a smooth function such that Q(δλ (x), δλ (y)) = λm Q(x, y) where m ≥ 0. Then x →

∀ x, y ∈ RN , ∀ λ > 0,

∂Q (x, 0) ∂ yj

is δλ -homogeneous of degree m − σj (unless it vanishes identically). Proof. By Proposition 1.3.13, Q is a polynomial of the following type Q(x, y) = cα,β x α y β . |α|σ +|β|σ =m

Then, denoting by ej the j -th element of the canonical basis of RN , we have ∂Q (x, y) = ∂ yj

cα,β βj x α y β−ej ,

|α|σ +|β|σ =m

so that, since |ej |σ = σj , ∂Q (x, 0) = ∂ yj

cα,β x α .

|α|σ =m−σj ,β=ej

This completes the proof.

Now, we are in the position to prove the previously mentioned structure theorem for the composition law of a homogeneous Lie group on RN . Theorem 1.3.15 (Composition of a homogeneous Lie group on RN ). Let (RN , ◦, δλ ) be a homogeneous Lie group on RN . Then ◦ has polynomial component functions. Furthermore, we have (x ◦ y)1 = x1 + y1 ,

(x ◦ y)j = xj + yj + Qj (x, y),

2 ≤ j ≤ N,

and the following facts hold: 1. Qj only depends on x1 , . . . , xj −1 and y1 , . . . , yj −1 ; 2. Qj is a sum of mixed monomials in x, y; 3. Qj (δλ x, δλ y) = λσj Qj (x, y). More precisely, Qj (x, y) only depends on the xk ’s and yk ’s with σk < σj . Proof. Let j ∈ {1, . . . , N }, and define Pj : RN × RN → R, Since δλ is an automorphism of G, we have

Pj (x, y) = (x ◦ y)j .

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1 Stratified Groups and Sub-Laplacians

Pj (δλ (x), δλ (y)) = (δλ (x ◦ y))j = λσj (x ◦ y)j = λσj Pj (x, y). Moreover, since x ◦ 0 = x, 0 ◦ y = y, we have Pj (x, 0) = xj ,

Pj (0, y) = yj .

Then the proof follows from Lemma 1.3.13.

For example, the Lie group G = (R3 , ◦) considered in Example 1.2.18 is not homogeneous with respect to any dilation on R3 , since its composition x ◦ y = (arcsinh(sinh(x1 ) + sinh(y1 )), x2 + y2 + sinh(x1 )y3 , x3 + y3 ) does not fulfill the requirements of Theorem 1.3.15. Corollary 1.3.16 (Inversion of a homogeneous Lie group on RN ). Let G = (RN , ◦, δλ ) be a homogeneous Lie group on RN . Let j ∈ {1, . . . , N }. For every y ∈ G, we have (y −1 )j = −yj + qj (y), where qj (y) is a polynomial function in y, δλ -homogeneous of degree σj , only depending on the yk ’s with σk < σj . Proof. We let p : RN → R,

p(y) = y −1 .

For every y ∈ G, we have 0 = δλ (y −1 ◦ y) = δλ (y −1 ) ◦ δλ (y), whence δλ (y −1 ) = (δλ (y))−1 , i.e. δλ (p(y)) = p(δλ (y))

∀ y ∈ RN .

(1.56)

If j ∈ {1, . . . , N } is fixed and pj is the j -th component function of p, (1.56) means that pj is δλ -homogeneous of degree σj . As a consequence, since the inversion is a smooth map (by definition of Lie group), we can apply Proposition 1.3.4 and infer that pj is a polynomial function, δλ -homogeneous of degree σj . Now, we exploit the explicit form of the composition in Theorem 1.3.15. If x ◦ y = 0, then we have ()

xj = −yj

whenever σj = 1.

Hence, if σj = 2, we have xj = −yj + Qj (x, y), where Qj only depends on the xk ’s and yk ’s with σk = 1. As a consequence of (), we infer xj = −yj + qj (y)

whenever σj = 2,

where qj only depends on the yk ’s with σk = 1. An inductive argument now proves that xj = −yj + qj (y), where qj only depends on the yk ’s with σk < σj , and the proof is complete.

1.3 Homogeneous Lie Groups on RN

41

Example 1.3.17. Let us consider on R4 the composition law ⎞ ⎛ x 1 + y1 ⎟ ⎜ x2 + y2 ⎟. x◦y =⎜ ⎠ ⎝ x3 + y3 + y1 x2 x4 + y4 + y1 x22 + 2 x2 y3 Then ◦ equips R4 with a homogeneous Lie group structure with dilations δλ (x1 , x2 , x3 , x4 ) = (λx1 , λx2 , λ2 x3 , λ3 x4 ). Notice that the inversion on this group is given by ⎞ ⎛ −y1 ⎟ ⎜ −y2 ⎟. y −1 = ⎜ ⎠ ⎝ −y3 + y1 y2 −y4 − y1 y22 + 2 y2 y3 Corollary 1.3.18. Let G = (RN , ◦, δλ ) be a homogeneous Lie group on RN . Let j ∈ {1, . . . , N }. For every x, y ∈ G, we have (j ) (y −1 ◦ x)j = xj − yj + Pk (x, y)(xk − yk ), k:σk <σj (j )

where Pk (x, y) is a polynomial function in x and y only depending on the xk ’s and yk ’s with σk < σj . Proof. We fix j ∈ {1, . . . , N } and y ∈ RN , and we let fy : RN → R,

fy (x) := (y −1 ◦ x)j .

By Theorem 1.3.15, fy is a polynomial function on RN . Hence, Taylor’s formula centered at ξ ∈ RN gives fy (x) =

(D α fy )(ξ ) α!

α

(x − ξ )α ,

(1.57)

where the sum is finite. Choose ξ = y and note that the summand with α = 0 in (1.57) is fy (ξ ) = fy (y) = (y −1 ◦ y)j = 0. Consequently, (1.57) gives (y −1 ◦ x)j =

(D α fy )(y) α=0

α!

(x − y)α .

(1.58)

Note that any summand in (1.58) contains at least one factor of the type xk − yk (since α = 0). On the other hand, by Theorem 1.3.15, we have (y −1 ◦ x)j = (y −1 )j + xj + Qj (y −1 , x),

(1.59)

42

1 Stratified Groups and Sub-Laplacians

where Qj (x, y) only depends on the xk ’s and yk ’s with σk < σj . Then, by Corollary 1.3.16, (1.59) gives (y −1 ◦ x)j = xj − yj + qj (y) + Qj (y −1 , x),

(1.60)

where qj (y) is a polynomial function in y, only depending on the yk ’s with σk < σj . Note that Qj (y −1 , x) depends on the xk ’s with σk < σj and the (y −1 )k ’s with σk < σj . Exploiting once again Corollary 1.3.16, this proves that Qj (y −1 , x) depends on the xk ’s and the yk ’s with σk < σj . As a consequence, (1.60) proves that j (x, y), (y −1 ◦ x)j = xj − yj + Q

(1.61)

j (x, y) is a polynomial function only depending on the xk ’s and yk ’s with where Q σk < σj . By collecting together (1.58) and (1.61), we immediately get the assertion of the corollary.

For example, on the Heisenberg–Weyl group H1 , we have y −1 ◦ x = (x1 − y1 , x2 − y2 , x3 − y3 + 2 (−y2 x1 + y1 x2 )) = (x1 − y1 , x2 − y2 , x3 − y3 − 2 y2 (x1 − y1 ) + 2 y1 (x2 − y2 )). Also, the map x ◦ y = (x1 + y1 , x2 + y2 , x3 + y3 + x1 y2 ) endows

R3

with a homogeneous Lie group structure. Since the inversion is y −1 = (−y1 , −y2 , −y3 + y1 y2 ),

we have y −1 ◦ x = (x1 − y1 , x2 − y2 , x3 − y3 + 2 (−y2 x1 + y1 x2 )) = (x1 − y1 , x2 − y2 , x3 − y3 − y1 (x2 − y2 )). Note that y −1 ◦ x contains also non-mixed monomials of degree > 1, y −1 ◦ x = (x1 − y1 , x2 − y2 , x3 − y3 − y1 x2 + y1 y2 ). The following result describes in a very explicit way the Jacobian matrix at 0 of the left-translation τx on a homogeneous Lie group on RN . Corollary 1.3.19 (The Jacobian basis of a homogeneous Lie group). Let G = (RN , ◦, δλ ) be a homogeneous Lie group on RN . Then we have ⎛ ⎞ 1 0 ··· 0 .. ⎟ ⎜ (1) .. ⎜ a2 . 1 .⎟ ⎜ ⎟, (1.62) Jτx (0) = ⎜ . ⎟ . . .. .. ⎝ .. 0⎠ (1) (N −1) · · · aN 1 aN

1.3 Homogeneous Lie Groups on RN

43

(j )

where ai is a polynomial function δλ -homogeneous of degree σi − σj . As a consequence, if we let Zj = ∂xj +

N

(j )

ai

∂xi

for 1 ≤ j ≤ N − 1 and ZN = ∂xN ,

i=j +1

then Zj is a left-invariant vector field δλ -homogeneous of degree σj . Moreover, Jτx (0) = (Z1 I (x) · · · ZN I (x)). In other words, the Jacobian basis Z1 , . . . , ZN for the Lie algebra g of G is formed by δλ -homogeneous vector fields of degree σ1 , . . . , σN , respectively. Proof. By Theorem 1.3.15, the Jacobian matrix Jτx (0) takes the form (1.62) with (j )

ai (x) =

∂ Qi (x, 0). ∂ yj

(j )

Then, by Lemma 1.3.14, ai (x) is a polynomial function, δλ -homogeneous of degree σi − σj . This proves the first part of the corollary. The second one follows from Proposition 1.3.12.

Example 1.3.20. In Example 1.2.6, we showed that the Jacobian matrix of the left translation on H1 is

1 0 0 Jτx (0) = 0 1 0 . 2x2 −2x1 1 We recognize that the three columns of this matrix give raise to the Jacobian basis Z1 = ∂x1 + 2 x2 ∂x3 , Z2 = ∂x2 − 2 x1 ∂x3 and Z3 = ∂x3 and these vector fields are homogeneous of degree, respectively, 1, 1, 2 with respect to δλ (x1 , x2 , x3 ) =

(λx1 , λx2 , λ2 x3 ). The structure Theorem 1.3.15 of the composition law of (RN , ◦, δλ ) implies that the Lebesgue measure on RN is invariant under left and right translations on G. Indeed, by Theorem 1.3.15, the Jacobian matrices of the functions x → α ◦ x and x → x ◦ α have the following lower triangular form ⎛ ⎞ 1 0 ··· 0 ⎜ .⎟ ⎜ 1 . . . .. ⎟ ⎜ ⎟. ⎜ . . ⎟ . . . . ⎝ . . . 0⎠ ··· 1 Then, we have proved the following proposition. Proposition 1.3.21 (Haar measure on a homogeneous Lie group). Let G be a homogeneous Lie group on RN . Then the Lebesgue measure on RN is invariant with respect to the left and the right translations on G.

44

1 Stratified Groups and Sub-Laplacians

The above proposition is also restated as: the Lebesgue measure on RN is the Haar measure for G. If we denote by |E| the Lebesgue measure of a measurable set E ⊆ RN , we then have |α ◦ E| = |E| = |E ◦ α| ∀ α ∈ G. We also have that the Lebesgue measure is homogeneous with respect to the dilations {δλ }λ>0 . More precisely, as a trivial computation shows, |δλ (E)| = λQ |E|, where Q=

N

σj .

(1.63)

j =1

The positive number Q is called the homogeneous dimension of the group G = (RN , ◦, δλ ). For example, in the case of the Heisenberg–Weyl group H1 , where τα is given by τα (x) = (α1 + x1 , α2 + x2 , α3 + x3 + 2 (α2 x1 − α1 x2 )), and δλ (x1 , x2 , x3 ) = (λx1 , λx2 , λ2 x3 ), we have

1 0 0 λ 0 0 Jδλ (x) = 0 λ 0 , Jτα (x) = 0 1 0 , 2α2 −2α1 1 0 0 λ2 so that, for every α, x ∈ H1 and every λ > 0, we have det Jτα (x) = 1,

det Jδλ (x) = λ4 = λQ ,

since the homogeneous dimension of H1 is Q = 1 + 1 + 2 = 4. 1.3.3 The Lie Algebra of a Homogeneous Lie Group on RN The following remark holds. Remark 1.3.22. Let G be a homogeneous Lie group on RN with Lie algebra g. From Corollary 1.3.19 we easily obtain the splitting of g as a direct sum of linear spaces spanned by vector fields of constant degree of δλ -homogeneity. More precisely, let us recall that the exponents σj ’s in the dilation δλ of G (see (H.1) in Definition 1.3.1) satisfy σ1 ≤ · · · ≤ σN and can then be grouped together to produce real and natural numbers, respectively, say n1 , . . . , nr

and

N1 , . . . , Nr ,

such that n1 < n2 < · · · < nr ,

N1 + N2 + · · · + Nr = N,

1.3 Homogeneous Lie Groups on RN

defined by ⎧ n1 = σj ⎪ ⎪ ⎪ ⎨ n2 = σj .. ⎪ . ⎪ ⎪ ⎩ nr = σj

45

for 1 ≤ j ≤ N1 , for N1 < j ≤ N1 + N2 , for N1 + · · · + Nr−1 < j ≤ N1 + · · · + Nr−1 + Nr .

Let Z1 , . . . , ZN be the Jacobian basis of g. Define g1 = span{Zj | 1 ≤ j ≤ N1 }

and, for i = 2, . . . , r,

gi = span{Zj | N1 + · · · + Ni−1 < j ≤ N1 + · · · + Ni−1 + Ni }. By Corollary 1.3.19, the generators Zj ’s of gi are δλ -homogeneous vector fields of degree ni , 1 ≤ i ≤ r. Moreover, we obviously have g = g1 ⊕ · · · ⊕ gr . We also explicitly notice that, by Proposition 1.3.9, a vector field X ∈ g is δλ homogeneous of degree n iff, for a suitable i ∈ {1, . . . , r}, n = ni and X ∈ gi . In the next section, we shall deal with homogeneous groups in which ni = i for 1 ≤ i ≤ r, and the layer (or slice) gi , i ∈ {1, . . . , r}, is precisely generated by the commutators of length i of the vector fields in g1 . Example 1.3.23. The usual additive group (R3 , +) is a homogeneous Lie group if equipped with the dilation δλ (x1 , x2 , x3 ) = (λ2 x1 , λπ x2 , λ4 x3 ). The decomposition of the Lie algebra as in Remark 1.3.22 is span{∂x1 } ⊕ span{∂x2 } ⊕ span{∂x3 }. Moreover, R4 is a homogeneous Lie group if equipped with the group law ⎛ ⎞ x 1 + y1 ⎜ ⎟ x2 + y2 ⎟ x◦y =⎜ ⎝ x3 + y3 + 2 y1 x2 − 2 y2 x1 ⎠ x4 + y4 and the dilation δλ (x1 , x2 , x3 , x4 ) = (λx1 , λx2 , λ2 x3 , λ2 x4 ). The decomposition of the Lie algebra as in Remark 1.3.22 is g1 ⊕ g2 = span{X1 , X2 } ⊕ span{∂x3 , ∂x4 }, where X1 = ∂x1 + 2 x2 ∂x3 , X2 = ∂x2 − 2 x1 ∂x3 . Note that (see the notation in (1.17))

46

1 Stratified Groups and Sub-Laplacians

[g1 , g1 ] g2 . Observe that the above (R4 , ◦) is isomorphic to the homogeneous Lie group (R4 , ∗) with the composition law ⎞ ⎛ ξ1 + η1 ⎟ ⎜ ξ 2 + η2 ⎟ ξ ∗η =⎜ ⎠ ⎝ ξ 3 + η3 ξ 4 + η 4 + 2 η1 ξ 2 − 2 η 2 ξ 1 and the new group of dilations δλ (ξ1 , ξ2 , ξ3 , ξ4 ) = (λξ1 , λξ2 , λξ3 , λ2 x4 ). The decomposition of the Lie algebra as in Remark 1.3.22 is g1 ⊕ g2 = span{Z1 , Z2 , ∂ξ3 } ⊕ span{∂ξ4 }, where Z1 = ∂ξ1 + 2 ξ2 ∂ξ4 , Z2 = ∂ξ2 − 2 ξ1 ∂ξ4 . Note that this time we have [g1 , g1 ] = g2 . Definition 1.3.24 (Dilations on the Lie algebra of a homogeneous Lie group). Let G = (RN , ◦, δλ ) be a homogeneous Lie group on RN with Lie algebra g and dilation δλ (x1 , . . . , xN ) = (λσ1 x1 , . . . , λσN xN ). We define a group of dilations on g (which we still denote by δλ ) as follows: δλ is the (only) linear (auto)morphism of g mapping the j -th element Zj of the Jacobian basis for g into λσj Zj . In other words, if X ∈ g is written w.r.t. the Jacobian basis Z1 , . . . , ZN as X=

N

we then have δλ (X) =

cj Zj ,

j =1

N

cj λσj Zj .

j =1

We immediately recognize that, if π : g → RN is the map defined by π(X) = XI (0) (see also Remark 1.2.20), it holds π(δλ (X)) = δλ (π(X))

∀ X ∈ g.

(1.64)

Indeed, we have δλ (π(X)) = δλ π = δλ

N j =1

N

= δλ

cj Zj

N

cj π(Zj )

j =1

cj (Zj )I (0) = δλ (c1 , . . . , cN )

j =1

= (λσ1 c1 , . . . , λσN cN )

1.3 Homogeneous Lie Groups on RN

47

and, on the other hand, π(δλ (X)) = π δλ

N

=π

cj Zj

j =1

=

N

N

σj

cj λ Zj

j =1

cj λσj π(Zj ) = (λσ1 c1 , . . . , λσN cN ).

j =1

The following simple and very useful fact holds. Proposition 1.3.25. Let G be a homogeneous Lie group on RN with Lie algebra g. The dilation on g introduced in Definition 1.3.24 is a Lie algebra automorphism of g, i.e. (1.65) δλ ([X, Y ]) = [δλ (X), δλ (Y )] ∀ X, Y ∈ g. Proof. First we remark that, for every i, j ∈ {1, . . . , N }, δλ ([Zi , Zj ]) = λσi +σj [Zi , Zj ].

(1.66)

Indeed, since Zi and Zj are δλ -homogeneous of degrees σi and σj , respectively, then [Zi , Zj ] is a δλ -homogeneous vector field of degree σi + σj (see Proposition 1.3.10). This implies that, if we express [Zi , Zj ] w.r.t. the Jacobian basis [Zi , Zj ] =

N

ck Zk ,

k=1

then the sum runs over the k’s such that σk = σi + σj

(1.67)

(here we use, as a crucial tool, Corollary 1.3.11). Consequently, N

N δλ ([Zi , Zj ]) = δλ ck Zk = ck λσk Zk k=1

= λσi +σj

(by (1.67))

k=1

N

ck Zk = λσi +σj [Zi , Zj ].

k=1

This proves (1.66). N Let now X = N i=1 xi Zi and Y = j =1 yj Zj . Then we have $ δλ ([X, Y ]) = δλ = δλ

N i=1

N i,j =1

xi Zi ,

N j =1

%

yj Zj

xi yj [Zi , Zj ]

48

1 Stratified Groups and Sub-Laplacians N

=

xi yj δλ ([Zi , Zj ])

(see (1.66))

i,j =1 N

=

xi yj λσi +σj ([Zi , Zj ])

i,j =1

=

$N

σi

ci λ Zi ,

N

% σj

cj λ Zj

= [δλ (X), δλ (Y )].

j =1

i=1

This completes the proof.

1.3.4 The Exponential Map of a Homogeneous Lie Group Let G = (RN , ◦, δλ ) be a homogeneous Lie group on RN with Lie algebra g. The exponential map on g has some remarkable properties, due to the homogeneous structure of G. We prove such properties in what follows. Let Z1 , . . . , ZN be the Jacobian basis of g. By Corollary 1.3.19, Zj is δλ homogeneous of degree σj and takes the form Zj =

N

(j )

ak (x1 , . . . , xk−1 )∂xk ,

(1.68)

k=j (j )

(j )

where ak is a polynomial function δλ -homogeneous of degree σk −σj and aj ≡ 1. We now consider on g the dilation group introduced in Definition 1.3.24, i.e. with abuse of notation (soon justified) N

N δλ ξj Zj := λσj ξj Zj . (1.69) δλ : g −→ g, j =1

j =1

Remark 1.3.26 (Consistency of the dilations on g and G). The dilation (1.69) is consistent with the one in G. More precisely, if Z ∈ g then, for every λ > 0, it holds δλ (ZI (x)) = (δλ Z)I (δλ (x))

∀ x ∈ G.

(1.70)

We first check this identity in the case Z = Zj , j = 1, . . . , N. Since Zj is δλ -homogeneous of degree σj , by Corollary 1.3.6, we have δλ (Zj I (x)) = λσj (Zj I )(δλ (x)), so that (see (1.69)) Then, given Z =

δλ (Zj I (x)) = (δλ Zj )I (δλ (x)).

N

j =1 ξj

δλ (ZI (x)) =

Zj ∈ g, we have (since δλ is linear on g)

N

ξj δλ (Zj I (x)) =

j =1

=

N j =1

N

ξj (δλ Zj )I (δλ (x))

j =1

ξj (δλ Zj ) I (δλ (x)) = (δλ Z)I (δλ (x)).

1.3 Homogeneous Lie Groups on RN

49

From the previous remark, we easily obtain the following lemma. Lemma 1.3.27. Let G = (RN , ◦, δλ ) be a homogeneous Lie group on RN with Lie algebra g. Denote also by δλ the dilation (1.69) on g. Let γ : [0, T ] → RN be an integral curve of Z with Z ∈ g. Then Γ := δλ (γ ) is an integral curve of δλ (Z). Proof. Identity (1.70) gives Γ˙ = δλ (γ˙ ) = δλ (ZI (γ )) = (δλ Z)I (δλ (γ )) = (δλ Z)I (Γ ). This ends the proof.

We are now in the position to prove the following important theorem. Theorem 1.3.28 (Exponential map of a homogeneous Lie group). Let G = (RN , ◦, δλ ) be a homogeneous Lie group with Lie algebra g. Then Exp : g → G and Log : G → g are globally defined diffeomorphisms with polynomial component functions (provided g is equipped with its vector space structure and any fixed system of linear coordinates). Moreover, denote also by δλ the dilation on g defined in (1.69). Then, for every Z ∈ g and x ∈ G, it holds Exp δλ (Z) = δλ (Exp (Z)) and Log (δλ (x)) = δλ (Log (x)). (1.71) Proof. Let Z ∈ g, Z =

N

Z=

j =1 ξj

N k=1

Zj . From (1.68) we obtain

k

(j ) ξj ak (x1 , . . . , xk−1 )

∂xk .

(1.72)

j =1

Then the system of ODE’s defining Exp (Z) is “pyramid”-shaped, and the first part of the theorem follows from Remark 1.1.3. In order to prove the first identity in (1.71), we consider the solution γ to the Cauchy problem γ˙ = ZI (γ ),

γ (0) = 0.

By the very definition of Exp (Z), we have γ (1) = Exp (Z). Let us put Γ = δλ (γ ). By Lemma 1.3.27, Γ is an integral curve of δλ (Z). Moreover, Γ (0) = δλ (γ (0)) = δλ (0) = 0. Then Γ (1) = Exp (δλ (Z)), so that Exp (δλ (Z)) = Γ (1) = δλ (γ (1)) = δλ (Exp (Z)). This proves the first identity in (1.71). The second one is trivially equivalent to the first one.

50

1 Stratified Groups and Sub-Laplacians

The first part of Theorem 1.3.28 together with (1.40) and Proposition 1.2.29 (page 27) give the following corollary. Corollary 1.3.29. For every x, y ∈ G, we have and x −1 = Exp (−Log (x)).

x ◦ y = exp(Log (y))(x)

Remark 1.3.30 (Exp and Log preserve the mass). If Z is the vector field (1.72), then ZI (x) =

(1) ξ1 , ξ2 + ξ1 a2 (x1 ), . . . , ξN

+

N −1

(j ) aN (x1 , . . . , xN −1 )

.

j =1

This implies (see (1.13), page 10) Exp (Z) = exp(Z)(0) = ξ1 , ξ2 + B2 (ξ1 ), . . . , ξN + BN (ξ1 , . . . , ξN −1 ) , (1.73) where the Bj ’s are suitable polynomial functions. Then the Jacobian matrix of the map RN (ξ1 , . . . , ξN ) → Exp (ξ1 Z1 + · · · + ξN ZN ) ∈ RN takes the following form

⎛

1

0

⎜ 1 ⎜ ⎜ . . ⎝ .. .. ···

⎞ ··· 0 . .. . .. ⎟ ⎟ ⎟. .. . 0⎠ 1

(1.74)

Thus, with respect to the Jacobian basis of g and the canonical basis of G ≡ RN , Exp preserves the Lebesgue measure. The same property holds for the map Log , since Log = (Exp )−1 . NEquivalently, if Z1 , . . . , ZN is the Jacobian basis for g and, as usual, ξ · Z = j =1 ξj Zj , then Exp and Log have the following remarkable forms ⎛ ⎜ Exp (ξ · Z) = ⎜ ⎝

⎞

ξ1 ξ2 + B2 (ξ1 ) .. .

⎟ ⎟ ⎠

(1.75a)

⎟ ⎟ · Z, ⎠

(1.75b)

ξN + BN (ξ1 , . . . , ξN −1 ) and

⎛ ⎜ Log (x) = ⎜ ⎝

x1 x2 + C2 (x1 ) .. .

⎞

xN + BN (x1 , . . . , xN −1 ) where the Bi ’s and Ci ’s are polynomial functions (δλ -homogeneous of degree σi ) completely determined by the composition law on G.

1.3 Homogeneous Lie Groups on RN

51

For example, the above results are readily verified for H1 , since in that case (as we proved in Example 1.2.28, page 27) Exp is represented by the identity matrix (if the algebra of H1 is equipped with the Jacobian basis). A little more elaborated example is given below. Example 1.3.31. Let us consider on R4 the following composition law (we denote the points of R4 by x = (x1 , x2 , x3 , x4 )): ⎛ ⎞ x 1 + y1 ⎜ ⎟ x2 + y2 ⎟. x◦y =⎜ ⎝ ⎠ x3 + y3 + y1 x2 2 x4 + y4 + y1 x2 + 2 x2 y3 It is readily verified that ◦ equips R4 with a homogeneous Lie group structure, provided that the dilation group is given by δλ (x1 , x2 , x3 , x4 ) := (λx1 , λx2 , λ2 x3 , λ3 x4 ). Let us construct the exponential map. To begin with, the first two vector fields X1 , X2 of the related Jacobian basis can be found as follows (see (1.33)): for every ϕ ∈ C ∞ (R4 ) and every x ∈ R4 , we have ∂ ∂ (X1 ϕ)(x) = ϕ(x ◦ y), (X2 ϕ)(x) = ϕ(x ◦ y), ∂y1 y=0 ∂y2 y=0 so that the chain rule straightforwardly gives X1 = ∂x1 + x2 ∂x3 + x22 ∂x4 ,

X2 = ∂x2 .

A direct computation shows that [X1 , X2 ] = −∂3 − 2 x2 ∂4 ,

[X1 , [X1 , X2 ]] = 0,

[X2 , [X1 , X2 ]] = −2 ∂4 ,

whereas all commutators of length > 3 vanish. We now remark that X1 ,

X2 ,

[X1 , X2 ]

and [X2 , [X1 , X2 ]]

satisfy the following properties: – they are left-invariant w.r.t. ◦ (as iterated commutators of left-invariant vector fields, see also (1.19)); – they are linearly independent vector fields10 ; – they form a basis of g (since dim(g) = 4, see Proposition 1.2.7). 10 For example, we can notice that their respective evaluations at 0

⎛1⎞

⎛0⎞

⎜0⎟ ⎝ ⎠, 0 0

⎜1⎟ ⎝ ⎠, 0 0

⎛ 0 ⎞ ⎜ 0 ⎟ ⎝ ⎠, −1 0

⎛ 0 ⎞ ⎜ 0 ⎟ ⎝ ⎠ 0 −2

are linearly independent vectors of R4 (and use Proposition 1.2.13).

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1 Stratified Groups and Sub-Laplacians

We now set W1 := X1 , W2 := X2 , W3 := [X1 , X2 ], W4 := [X2 , [X1 , X2 ]], and, for ξ ∈ R4 , we also let ξ · W := ξ1 W1 + ξ2 W2 + ξ3 W3 + ξ4 W4 . By the definition in (1.8), we have exp(ξ · W )(x) = γ (1), where γ (s) solves γ˙ (s) = (ξ · W )I (γ (s)) = (ξ1 , ξ2 , ξ1 γ2 − ξ3 , ξ1 γ22 − 2 ξ3 γ2 − 2ξ4 ), γ (0) = x. Solving the above system of ODE’s, one gets ⎛

⎞ x1 + ξ1 ⎟ ⎜ x 2 + ξ2 ⎟. exp(ξ · W )(x) = ⎜ 1 ⎠ ⎝ x 3 − ξ3 + 2 ξ1 ξ2 + ξ1 x 2 1 2 2 x4 − 2ξ4 + 3 ξ1 ξ2 − ξ2 ξ3 + ξ1 x2 + ξ1 ξ2 x2 − 2ξ3 x2

As a consequence, by Definition 1.2.25, we obtain ⎛

⎞ ξ1 ⎜ ⎟ ξ2 ⎟, Exp (ξ · W ) = exp(ξ · W )(0) = ⎜ 1 ⎝ ⎠ −ξ3 + 2 ξ1 ξ2 1 2 −2ξ4 + 3 ξ1 ξ2 − ξ2 ξ3

so that the inverse map of Exp is given by ⎛

⎞ x1 ⎜ ⎟ x2 ⎟ · W. Log (x) = ⎜ 1 ⎝ ⎠ −x3 + 2 x1 x2 1 1 1 2 − 2 x4 − 12 x1 x2 + 2 x2 x3

One can now directly check11 the validity of Theorem 1.3.28, Corollary 1.3.29 and Remark 1.3.30. We explicitly remark the vectors Wi ’s do not form the Jacobian basis for g, which, instead, is given by Z1 = W1 ,

Z2 = W2 ,

Z3 = −W3 ,

1 Z4 = − W4 . 2

Hence, if we write ξ · Z := ξ1 Z1 + ξ2 Z2 + ξ3 Z3 + ξ4 Z4 , we see that ! 1 (ξ, ξ2 , ξ3 , ξ4 ) · Z = ξ, ξ2 , −ξ3 , − ξ4 · Z, 2 11 For example, we see that the inverse of x ∈ G is given by

⎞ −x1 & ' ⎜ ⎟ −x2 ⎟. x −1 = Exp −Log (x) = ⎜ ⎠ ⎝ −x3 + x1 x2 −x4 + 2 x2 x3 − x1 x22 ⎛

1.3 Homogeneous Lie Groups on RN

53

whence, with respect to the Jacobian coordinates, the exponential and the logarithmic maps are respectively given by ⎞ ⎛ ξ1 ⎟ ⎜ ξ2 ⎟, Exp (ξ · Z) = ⎜ ⎠ ⎝ ξ3 + 12 ξ1 ξ2 ξ4 + 13 ξ1 ξ22 + ξ2 ξ3 ⎛ ⎞ x1 ⎜ ⎟ x2 ⎟ · Z. Log (x) = ⎜ 1 ⎝ ⎠ x3 − 2 x1 x2 1 2 x4 + 6 x1 x2 − x2 x3 Compare this to (1.74).

Theorem 1.3.28 has many important consequences. We collect some of them in the following remark. Remark 1.3.32. From Theorem 1.3.28 we infer, in particular, that Exp : g → G and Log : G → g are globally defined

C∞

maps. Hence, by Remark 1.2.32, the operation on g

X Y := Log (Exp (X) ◦ Exp (Y )),

X, Y ∈ g,

(1.76)

defines a Lie group structure isomorphic to (G, ◦). We consider on g the dilation (still denoted by δλ ) introduced in Definition 1.3.24. We claim that δλ is a Lie group automorphism of (g, ), i.e.

δλ (X Y ) = (δλ (X)) (δλ (Y ))

∀ X, Y ∈ g.

(1.77)

Roughly speaking, (g, , δλ ) is a homogeneous Lie group too. To prove the claim, we notice that δλ (X Y ) = δλ {Log (Exp (X) ◦ Exp (Y ))} (see (1.71)) = Log {δλ (Exp (X) ◦ Exp (Y ))} = Log {(δλ (Exp (X))) ◦ (δλ (Exp (Y )))} (see (1.71)) = Log {(Exp (δλ (X))) ◦ (Exp (δλ (Y )))} = (δλ (X)) (δλ (Y )). This proves our claim.12 12 We explicitly remark that, if we already knew that is defined by a “universal” composition

of iterated Lie brackets, we could derive (1.77) from (1.65) in Proposition 1.3.25. Indeed, ! 1 1 [X, [X, Y ]] + · · · δλ (X Y ) = δλ X + Y + [X, Y ] + 2 12 1 1 [δλ (X), [δλ (X), δλ (Y )]] + · · · = δλ (X) + δλ (Y ) + [δλ (X), δλ (Y )] + 2 12 = (δλ (X)) (δλ (Y )), the second identity following by a repeated application of (1.65).

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1 Stratified Groups and Sub-Laplacians

We now identify g with RN taking coordinates with respect to the Jacobian basis. In other words, we consider the map (see Remark 1.2.20) π : g → RN ,

X → π(X) := XI (0).

Again, we transfer the Lie group structure of (g, ) into a Lie group (RN , ∗) in the natural way, by setting ξ ∗ η := π(π −1 (ξ ) π −1 (η)),

ξ, η ∈ RN .

(1.78)

As a consequence, (RN , ∗) is isomorphic to (g, ) and hence to (G, ◦). We finally consider on RN the same dilation δλ defined on G (this makes sense, since the underlying manifold for G is RN too). We claim that (RN , ∗, δλ ) is a homogeneous Lie group. In other words, we have to show that δλ is a Lie group automorphism of (RN , ∗). This follows from the argument below, δλ (ξ ∗ η) (see (1.64)) (see (1.77)) (see (1.64))

= δλ {π(π −1 (ξ ) π −1 (η))} = π{δλ (π −1 (ξ ) π −1 (η))} = π{(δλ (π −1 (ξ ))) (δλ (π −1 (η)))} = π{(π −1 (δλ (ξ ))) (π −1 (δλ (η)))} = (δλ (ξ )) ∗ (δλ (η)).

We can summarize the above remarked facts as follows. Given a homogeneous Lie group G = (RN , ◦, δλ ), we can consider a somewhat “more canonical” homogeneous Lie group on RN C-H(G) := (RN , ∗, δλ ) (which we may call “of Campbell–Hausdorff type”) obtained by the natural identification of the Lie algebra of G (equipped with the Campbell–Hausdorff composition law in (1.76)) to RN (via coordinates w.r.t. the Jacobian basis). (See also the Fig. 1.6.) Example 1.3.33. Let us consider the homogeneous Lie group on R3 with the dilation δλ (x) = δλ (x1 , x2 , x3 ) = (λx1 , λx2 , λ2 x3 ) and the composition law defined by ⎛

⎞ x1 + y1 ⎠. x2 + y2 x◦y =⎝ x3 + y3 + x1 y2

The Jacobian basis for g (the Lie algebra of G) is

1.3 Homogeneous Lie Groups on RN

55

Fig. 1.6. Figure related to Remark 1.3.32

Z1 = ∂1 ,

Z2 = ∂2 + x1 ∂3 ,

Z3 = [Z1 , Z2 ] = ∂3 . For ξ = (ξ1 , ξ2 , ξ3 ) ∈ R3 , we fix the notation ξ · Z := 3i=1 ξi Zi ∈ g. It is easy to show that the exponential and logarithmic maps are given by ⎞ ⎛ ⎞ ⎛ ξ1 x1 ⎠ , Log (x) = ⎝ ⎠ · Z. ξ2 x2 Exp (ξ · Z) = ⎝ x1 x2 ξ1 ξ2 x3 − 2 ξ3 + 2 Hence, the map ∗ considered in (1.78) is given by ⎛ ⎞ ξ 1 + η1 ⎠. ξ 2 + η2 ξ ∗η =⎝ 1 ξ3 + η3 + 2 (ξ1 η2 − ξ2 η1 ) The Jacobian basis related to C-H(G) = (R3 , ∗, δλ ) is (1 = ∂1 − ξ2 ∂3 , Z (2 = ∂2 + ξ1 ∂3 , Z (3 = [Z (1 , Z (2 ] = ∂3 . Z 2 2 Now, it is interesting to see what happens if we iterate this “C-G” process. It is easy to see that, if we consider once again the group obtained from C-H(G) in the same way (i.e. C-H(C-H(G))), we obtain nothing else than C-H(G) itself (a rigorous formulation of this fact will be given in Proposition 2.2.24). We remark that G and C-H(G) are isomorphic and the canonical sub-Laplacian of G ΔG = {(∂/∂x1 )}2 + {(∂/∂x2 ) + x1 (∂/∂x3 )}2 is “equivalent” to the canonical sub-Laplacian of C-H(G) )2 )2 ξ2 ξ1 ΔC-H(G) = (∂/∂ξ1 ) − (∂/∂x3 ) + (∂/∂ξ2 ) + (∂/∂x3 ) . 2 2

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1.4 Homogeneous Carnot Groups We now enter into the core of the chapter by introducing the central definition of this book. Our definition here of Carnot group will be properly compared to the classical one in Section 2.2, page 121. Definition 1.4.1 (Homogeneous Carnot group). We say that a Lie group on RN , G = (RN , ◦), is a (homogeneous) Carnot group or a homogeneous stratified group, if the following properties hold: (C.1) RN can be split as RN = RN1 × · · · × RNr , and the dilation δλ : RN → RN δλ (x) = δλ (x (1) , . . . , x (r) ) = (λx (1) , λ2 x (2) , . . . , λr x (r) ),

x (i) ∈ RNi ,

is an automorphism of the group G for every λ > 0. Then (RN , ◦, δλ ) is a homogeneous Lie group on RN , according to Definition 1.3.1. Moreover, the following condition holds: (C.2) If N1 is as above, let Z1 , . . . , ZN1 be the left invariant vector fields on G such that Zj (0) = ∂/∂xj |0 for j = 1, . . . , N1 . Then13 rank(Lie{Z1 , . . . , ZN1 }(x)) = N

for every x ∈ RN .

If (C.1) and (C.2) are satisfied, we shall say that the triple G = (RN , ◦, δλ ) is a homogeneous Carnot group. We also say that G has step r and N1 generators. The vector fields Z1 , . . . , ZN1 will be called the (Jacobian) generators of G, whereas any basis for span{Z1 , . . . , ZN1 } is called a system of generators of G. (Note. As already remarked for Lie groups on RN and homogeneous ones, the notion of homogeneous Carnot group is not coordinate-free. This fact will not distract us from recognizing its importance.) In the sequel, we use the following notation to denote the points of G x = (x1 , . . . , xN ) = (x (1) , . . . , x (r) ) with

(i) ) ∈ R Ni , x (i) = (x1(i) , . . . , xN i

i = 1, . . . , r.

(1.79a) (1.79b)

Furthermore, we shall denote by g the Lie algebra of G. Remark 1.4.2 (Equivalent definition of homogeneous Carnot group). An equivalent definition of homogeneous Carnot group can be given: Suppose G = (RN , ◦) is a Lie group on RN , and there exist positive real numbers τ1 ≤ · · · ≤ τN such that dλ (x) = (λτ1 x1 , . . . , λτN xN ) is a Lie group morphism of G for every λ > 0. Let g be the Lie algebra of G, and let g1 be the linear subspace 13 See the notation in Definition 1.1.5.

1.4 Homogeneous Carnot Groups

57

of g of the left-invariant vector fields which are dλ -homogeneous of degree τ1 . If g1 Lie-generates14 the whole g, then G is a homogeneous Carnot group according to Definition 1.4.1. Precisely, G has step r := τN /τ1 , it has m := dim(g1 ) generators, and it is a homogeneous Lie group with respect to the dilation δλ = dλ1/τ1 . Also, set σj := τj /τ1 , then {σ1 , σ2 , . . . , σN } are consecutive integers starting from 1 up to r. A sketch of the proof is in order. As we observed in Remark 1.3.2, δλ is a morphism of (G, ◦), i.e. G = (RN , ◦, δλ ) is a homogeneous Lie group on RN . Obviously, X ∈ g1 if and only if X is δλ homogeneous of degree 1. Let ν be the maximum of the integers k’s such that σk = 1. Let us denote by {Z1 , . . . , ZN } the Jacobian basis related to G and observe that (by Proposition 1.3.12), for every j ≤ N , Zj is δλ -homogeneous of degree σj . We claim that () ν = dim(g1 ) =: m, and {Z1 , . . . , Zm } is a basis for g1 . Indeed, let X ∈ g1 . Then X = ξ1 Z1 + · · · + ξN ZN for suitable scalars ξj ’s. Since X is δλ -homogeneous of degree 1, by Corollary 1.3.11 and the definition of ν, it holds ξj = 0 for every j > ν. Hence, g1 is spanned by {Z1 , . . . , Zν } whence (this system of vectors being linearly independent) the claimed () holds. By the assumption Lie(g1 ) = g and (), it follows ()

Lie(Z1 , . . . , Zm ) = g.

For every j ∈ N, j ≥ 2, let us set (see the notation in (1.17)) gj := [g1 , gj −1 ]. By Proposition 1.3.12, gj = {0} for every j > r := σN . Also, by Proposition 1.3.10, any X ∈ gj is δλ -homogeneous of degree j . Let now j ∈ {m + 1, . . . , N } be fixed. Then, by (), Zj is a linear combination of nested commutators of Z1 , . . . , Zm . But any such commutator is δλ -homogeneous of an integer degree in 1, . . . , r. This proves that σj (the δλ -homogeneous degree of Zj ) is integer and (again from Corollary 1.3.11) σj ∈ {1, . . . , r}. As a consequence, we have the splitting of RN , as requested in (C.1) of Definition 1.4.1, with N1 = m. Finally, let us prove that (C.2) holds too. This is obvious thanks to (), since (see the notation in Definition 1.1.5) rank(g(x)) ≥ rank Z1 I (x), . . . , ZN I (x) = rank Z1 I (0), . . . , ZN I (0) = N for every x ∈ G (see Proposition 1.2.13).

Example 1.4.3. The Heisenberg–Weyl group H1 is a Carnot group of step two and two generators. Indeed, it is a homogeneous Lie group (with dilations δλ (x1 , x2 , x3 ) = (λx1 , λx2 , λ2 x3 )). Moreover (since the first two vector fields of the Jacobian basis are Z1 = ∂x1 + 2x2 ∂x3 and Z2 = ∂x2 − 2x1 ∂x3 ), we have rank(Lie{Z1 , Z2 }(x)) = 3 for every x ∈ R3 , 14 This means that Lie(g ) = g, see the notation in Proposition 1.1.7. 1

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1 Stratified Groups and Sub-Laplacians

as we proved in Example 1.1.6. Thus, the above properties (C.1) and (C.2) are fulfilled. We now give an example of a homogeneous Lie group which is not a Carnot group. Let us consider the following composition law on R2 (x1 , x2 ) ◦ (y1 , y2 ) = (x1 + y1 , x2 + y2 + x1 y1 ). It can be readily verified that G = (R2 , ◦) is a Lie group (here (x1 , x2 )−1 = (−x1 , −x2 + x12 )). Moreover, G is a homogeneous group, if equipped with the dilation δλ (x1 , x2 ) := (λx1 , λ2 x2 ). Hence (C.1) is satisfied. However, (C.2) is not. Indeed, if Z1 = ∂x1 + x1 ∂x2 is the first vector field of the Jacobian basis, we have rank(Lie{Z1 }(x)) = 1 = 2 for every x ∈ R2 . Hence G is not a homogeneous Carnot group. Finally, let us remark that the triple (R2 , +, δλ ) is a homogeneous Carnot group if δλ (x1 , x2 ) = (λx1 , λx2 ), whereas if δλ (x1 , x2 ) = (λx1 , λ2 x2 ), (R2 , +, δλ ) is a homogeneous Lie group but not a Carnot one.

From properties (C.1) and (C.2) of Definition 1.4.1 and the results on the homogeneous Lie groups showed in Section 1.3 we immediately get the assertions contained in the following remarks. Remark 1.4.4. Let (RN , ◦, δλ ) be a homogeneous Carnot group. Then ◦ has polynomial component functions. Moreover, following the notation in (1.79a) and denoting x ◦ y by ((x ◦ y)(1) , . . . , (x ◦ y)(r) ), we have (x ◦ y)(1) = x (1) + y (1) ,

(x ◦ y)(i) = x (i) + y (i) + Q(i) (x, y),

2 ≤ i ≤ r,

where 1. Q(i) only depends on x (1) , . . . , x (i−1) and y (1) , . . . , y (i−1) ; 2. the component functions of Q(i) are sums of mixed monomials in x, y; 3. Q(i) (δλ x, δλ y) = λi Q(i) (x, y). Remark 1.4.5. Let (RN , ◦, δλ ) be a homogeneous Carnot group. Then we have ⎛ ⎞ IN1 0 ··· 0 .. ⎟ ⎜ (1) .. ⎜ J2 (x) IN2 . . ⎟ ⎜ ⎟, (1.80) Jτx (0) = ⎜ . ⎟ . . .. .. ⎝ .. 0 ⎠ (1) (r−1) (x) INr Jr (x) · · · Jr (i)

where In is the n × n identity matrix, whereas Jj (x) is a Nj × Ni matrix whose entries are δλ -homogeneous polynomials of degree j − i. In particular, if we let Jτx (0) = Z (1) (x) · · · Z (r) (x) , where Z (i) (x) is a N × Ni matrix, then the column vectors of Z (i) (x) define δλ homogeneous vector fields of degree i: those of the relevant Jacobian basis.

1.4 Homogeneous Carnot Groups

59

Remark 1.4.6. Let G = (RN , ◦, δλ ) be a homogeneous Carnot group with Lie algebra g. Let Z1 , . . . , ZN be the Jacobian basis of g, i.e. Zj ∈ g and Zj (0) = ∂xj |0 ,

j = 1, . . . , N.

With a notation consistent with (1.79a) and (1.79b), we shall also denote the Jacobian basis by (1) (1) (r) (r) Z1 , . . . , ZN1 ; . . . ; Z1 , . . . , ZNr . Obviously, Zj(1) = Zj for 1 ≤ j ≤ N1 . By Corollary 1.3.19, Zj(i) is δλ -homogeneous of degree i and takes the form (i) Zj

=

(i) ∂/∂xj

Nh r (i,h) (h) + aj,k (x (1) , . . . , x (h−i) ) ∂/∂xk ,

(1.81)

h=i+1 k=1 (i,h)

where aj,k

is a δλ -homogeneous polynomial function of degree h − i. In par(1)

(1)

ticular, the Jacobian generators of G, i.e. the vector fields Z1 , . . . , ZN1 are δλ homogeneous of degree 1.

Remark 1.4.7. With the notation of the above remark, the Lie algebra g is generated by Z1 , . . . ZN1 , (1.82) g = Lie{Z1 , . . . ZN1 }. Indeed, the inclusion Lie{Z1 , . . . ZN1 } ⊆ g is obvious. Since dim(g) = N, in order to show the opposite inclusion, it is enough to prove that dim(Lie{Z1 , . . . ZN1 }) = N. By condition (C.2), there exists X1 , . . . , XN ∈ Lie{Z1 , . . . ZN1 } such that X1 I (0), . . . , XN I (0) are linearly independent vectors in RN . Then (by Proposition 1.2.13) X1 , . . . , XN are linearly independent in g. Hence N ≥ dim(Lie{Z1 , . . . ZN1 }) ≥ N, and this ends the proof.

Remark 1.4.8 (Stratification of the algebra of a homogeneous Carnot group). Let the notation of Remark 1.4.6 be employed. Let us denote by W (k) the vector space spanned by the commutators of length k of Z1 , . . . , ZN1 , W (k) := span ZJ | J ∈ {1, . . . , N1 }k . Obviously, W (k) ⊆ g, and every Z ∈ W (k) is δλ -homogeneous of degree k. Then, by Corollary 1.3.11 and Proposition 1.3.12, W (k) = {0} if k > r, while (k)

(k)

W (k) ⊆ span{Z1 , . . . , ZNk }

if 2 ≤ k ≤ r.

(1.83)

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1 Stratified Groups and Sub-Laplacians

Then, if we agree to let (1)

(1)

W (1) = span{Z1 , . . . , ZN1 } = span{Z1 , . . . , ZN1 }, we have dim(W (k) ) ≤ Nk

for any k ∈ {1, . . . , r}.

(1.84)

On the other hand, by Proposition 1.1.7, (1)

(1)

span{W (1) , . . . , W (r) } = Lie{Z1 , . . . , ZN1 }. Thus, by Remark 1.4.7, g = span{W (1) , . . . , W (r) }, so that, since W (h) ∩ W (k) = {0} if h = k (see Corollary 1.3.11), we have g = W (1) ⊕ W (2) ⊕ · · · ⊕ W (r) . As a consequence, dim(g) = On the other hand, dim(g) = N =

r

r

dim(W (k) ).

k=1

k=1 Nk .

dim(W (k) ) = Nk

Then, by (1.84),

for any k ∈ {1, . . . , r},

and, by (1.83), (k)

(k)

W (k) = span{Z1 , . . . , ZNk }

if 1 ≤ k ≤ r.

We also have [W (1) , W (i−1) ] = W (i)

for 2 ≤ k ≤ r

(1.85a)

and [W (1) , W (r) ] = {0}. Indeed, let us put V1 :=

W (1)

(1.85b)

and

Vi := [V1 , Vi−1 ]

for i = 2, . . . , r.

By the definition of W (k) and Proposition 1.1.7, Vi ⊆ W (i) for i = 2, . . . , r. Then dim(Vi ) ≤ dim(W (i) ) = Ni . On the other hand, by Proposition 1.3.12, [V1 , Vr ] = {0}, and, by Proposition 1.1.7, (1)

(1)

g = Lie{Z1 , . . . , ZN1 } = span{V1 , V2 , . . . , Vr }.

Then N = ri=1 dim(Vi ) ≤ ri=1 Ni = N . This implies dim(Vi ) = Ni for every i ∈ {1, . . . , r}. As a consequence, Vi = W (i) for every i ∈ {1, . . . , r}, and (1.85a) and (1.85b) hold.

1.4 Homogeneous Carnot Groups

61

Summing up, we have proved the “stratification” of the Lie algebra g, i.e. the decomposition g = W (1) ⊕ W (2) ⊕ · · · ⊕ W (r) with [W (1) , W (i−1) ] = W (i) [W (1) , W (r) ] = {0}, where

(k)

for 2 ≤ k ≤ r,

(k)

W (k) = span{Z1 , . . . , ZNk }

if 1 ≤ k ≤ r.

Remark 1.4.9. Following all the notation and definitions in Remark 1.3.32, if G = (RN , ◦, δλ ) is a homogeneous Carnot group, then the Lie group C-H(G) := (RN , ∗, δλ ) obtained by the natural identification of the algebra of G to RN is a homogeneous Carnot group too. The proof of this fact is left to the reader as an exercise. Remark 1.4.10 (Stratified change of basis on a homogeneous Carnot group). Let (RN , ◦, δλ ) be a homogeneous Carnot group according to the Definition 1.4.1. As usual, we denote the points of G by x = (x (1) , . . . , x (r) )

with x (i) ∈ RNi

and the dilation group by δλ (x) = (λx (1) , . . . , λr x (r) ). Let C (1) , . . . , C (r) be r fixed non-singular matrices with C (i) of dimension Ni × Ni for every i = 1, . . . , r. We denote by C the N ×N matrix having C (1) , . . . , C (r) as diagonal blocks and 0’s elsewhere, i.e. ⎛ (1) ⎞ C ··· 0 . .. ⎠ .. C = ⎝ .. . . . (r) 0 ··· C Finally, we denote again by C the relevant linear change of basis on RN , i.e. the linear map C : RN → RN , C(x) = C · x. We define on RN a new composition law ∗ obtained by writing ◦ in the new coordinates defined by ξ = C(x). More precisely, we have ξ ∗ η := C (C −1 (ξ )) ◦ (C −1 (η)) ∀ ξ, η ∈ RN . (1.86) We claim that H = (RN , ∗, δλ ) is a homogeneous Carnot group isomorphic to G = (RN , ◦, δλ ). The proof of this (not obvious) assertion is left as an exercise (see also Section 16.3 of Chapter 16, page 637).

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1.5 The Sub-Laplacians on a Homogeneous Carnot Group Definition 1.5.1 (Sub-Laplacian on a homogeneous Carnot group). If Z1 , . . . , ZN1 are the Jacobian generators of the homogeneous Carnot group G = (RN , ◦, δλ ), the second order differential operator ΔG =

N1

Zj2

(1.87)

j =1

is called the canonical sub-Laplacian on G. Any operator L=

N1

Yj2

(1.88)

j =1

where Y1 , . . . , YN1 is a basis of span{Z1 , . . . , ZN1 }, is simply called a sub-Laplacian on G. The vector valued operator ∇G = (Z1 , . . . , ZN1 )

(1.89)

will be called the canonical (or horizontal) G-gradient. Finally, if L is as in (1.88), the notation ∇L = (Y1 , . . . , YN1 ) will be used to denote the L-gradient (or horizontal L-gradient). Example 1.5.2. The canonical sub-Laplacian of the Heisenberg–Weyl group H1 is ΔH1 = {∂x1 + 2 x2 ∂x3 }2 + {∂x2 − 2 x1 ∂x3 }2 = (∂x1 )2 + (∂x2 )2 + 4(x12 + x22 ) (∂x3 )2 + 4 x2 ∂x1 ,x3 − 4 x1 ∂x2 ,x3 . A (non-canonical) sub-Laplacian on H1 is, for example, L = {(∂x1 + 2 x2 ∂x3 ) − (∂x2 − 2 x1 ∂x3 )}2 + {∂x2 − 2 x1 ∂x3 }2 = (∂x1 )2 + 2(∂x2 )2 + 4(x12 + (x1 + x2 )2 ) (∂x3 )2 − 2∂x1 ,x2 + 4(x1 + x2 ) ∂x1 ,x3 − 4(x1 + (x1 + x2 )) ∂x2 ,x3 . The following one is a (non-canonical!) sub-Laplacian on the classical additive group (R2 , +) L = {2 ∂x1 − 5 ∂x2 }2 + {−∂x1 + 3 ∂x2 }2 = 5(∂x1 )2 + 34(∂x2 )2 − 26 ∂x1 ,x2 . It is not difficult to verify that R4 equipped with the operation ⎛ ⎞ x 1 + y1 ⎜ ⎟ x2 + y2 ⎟ x◦y =⎜ 1 ⎝ ⎠ x3 + y3 + 2 (y2 x1 − y1 x2 ) 1 1 x4 + y4 + 2 (y3 x1 − y1 x3 ) + 12 (x1 − y1 )(y2 x1 − y1 x2 )

1.5 The Sub-Laplacians on a Homogeneous Carnot Group

63

and the dilation δλ (x1 , x2 , x3 , x4 ) := (λx1 , λx2 , λ2 x3 , λ3 x4 ) is a homogeneous Carnot group, say G, whose canonical sub-Laplacian is )2 1 1 1 ΔG = ∂1 − x2 ∂3 − x3 ∂4 − x1 x2 ∂4 2 2 12 )2 1 1 2 + ∂2 + x1 ∂3 + x1 ∂4 2 12 1 = ∂11 + ∂22 + (x12 + x22 )∂33 + x1 ∂23 − x2 ∂13 4 ! ! ! ! x2 x12 2 1 x1 x2 2 x1 x2 ∂44 + 1 ∂24 − x3 + ∂14 + + x3 + 4 6 6 6 6 ! !! x12 1 x1 x2 1 x1 + x2 x3 + ∂34 + x2 ∂4 . + 2 6 6 6 We notice that ΔG also contains a first order (underlined) partial differential opera tor.15 We would like to list some basic properties of the sub-Laplacians, straightforward (1) (1) consequences of the properties of the vector fields Z1 , . . . , ZN1 . In what follows 1 2 L= N j =1 Yj will denote any sub-Laplacian on G. (A0) L is hypoelliptic, i.e. every distributional solution to Lu = f is of class C ∞ whenever f is of class C ∞ (see Section 5.10, page 280, for further comments). This follows from the celebrated Hörmander hypoellipticity theorem [Hor67, Theorem 1.1], recalled in the Preface (Theorem 1, page VIII) and the fact that, 1 2 if L = N j =1 Yj , then the following rank-condition holds rank Lie {Y1 , . . . , YN1 }(x) = N

∀ x ∈ RN .

This is an obvious consequence of hypothesis (C.2) in Definition 1.4.1, page 56. (A1) L is invariant with respect to the left translations on G, i.e. for every fixed α ∈ G, L(u(α ◦ x)) = (Lu)(α ◦ x)

for every x ∈ G and every u ∈ C ∞ (RN ).

This holds since the Yj ’s are left-translation invariant on G. (A2) L is δλ -homogeneous of degree two, i.e. for every fixed λ > 0, L(u(δλ (x))) = λ2 (Lu)(δλ (x))

for every x ∈ G and every u ∈ C ∞ (RN ).

This holds since the Yj ’s are δλ -homogeneous of degree one, see Remark 1.4.6. 15 This cannot happen for the sub-Laplacians on groups of step two, provided the inverse

map on the group is −x, i.e. any sub-Laplacian on a step-two homogeneous Carnot group (whose inverse map is −x) contains only second order coordinate partial derivatives (see Section 3.2).

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1 Stratified Groups and Sub-Laplacians

(A3) L can be written as

L = div(A(x)∇ T ),

where div denotes the divergence operator in N × N symmetric matrix

(1.90a)

RN ,

∇ = (∂1 , . . . , ∂N ), A is the

A(x) = σ (x) σ (x)T

(1.90b)

and σ (x) is the N × N1 matrix whose columns are Y1 I (x), . . . , YN1 I (x). In other words, A(x) is the Gram matrix of the system of vectors {Y1 I (x), . . . , YN1 I (x)}; hence, since these vectors are linearly independent for every x ∈ G, the rank of A(x) is N1 for every x ∈ G. Now, (1.90a) is a consequence of the following computation N

N1 N1 N Yk2 = (Yk I )i (x) ∂i (Yk I )j (x)∂j L= k=1

=

N

∂i

N1

*

(Yk I )i (x) (Yk I )j (x) ∂j ,

j =1

i=1

j =1

k=1 i=1 N

k=1

since (Yk I )i (x) does not depend on xi . Hence, this proves that L = div(A(x)∇ T ) with N

1 A(x) = (Yk I )i (x) (Yk I )j (x) = σ (x) σ (x)T . i,j =1,...,N

k=1

The matrix A takes the following block form A=

A1,1 A2,1

A1,2 A2,2

! (1.91)

,

where Ai,j stands for a mi ×mj matrix with polynomial entries, with m1 = N1 and m2 = N − N1 . Furthermore, A1,1 is constant and non-singular. Indeed, for 1 a suitable non-singular matrix B = (bj,k )N j,k=1 , we have Yj =

N1

bj,k Zk ,

j = 1, . . . , N1 .

(1.92)

k=1

On the other hand, see (1.81), Zk = ∂k +

N i=N1 +1

(k)

(k)

ai ∂i = ∂k +

N

(k)

∂i (ai ·),

(1.93)

i=N1 +1

where the ai ’s are suitable polynomial functions independent of xi . Replacing (1.93) in (1.92) and squaring, we obtain

1.5 The Sub-Laplacians on a Homogeneous Carnot Group

L=

N1

Yj2 =

j =1

N1

ah,k ∂h,k +

65

∂h (ah,k ∂k ),

h,k≤N, h∨k>N1

h,k=1

where A1,1 = (ah,k )h,k≤N1 = B T B is a constant N1 × N1 matrix. Moreover, when h ∨ k := max{h, k} > N1 , then ah,k is a suitable polynomial function. We notice that if L = ΔG , then B = IN1 , so that A1,1 = IN1 . The above computations also give the expression of L with respect to the usual coordinate partial derivatives, L=

N1 k=1

Yk2 =

N

ai,j (x)∂i,j +

i,j =1

N

bj (x) ∂j ,

j =1

where ai,j (x) =

N1 (Yk I )i (x) (Yk I )j (x),

bj (x) =

k=1

N1

Yk ((Yk I )j (x)).

k=1

Analogous formulas hold for general sum of squares of vector fields (see Ex. 4 at the end of the chapter). (A4) If x ∈ G is fixed and A(x) is the matrix in (1.90a), then the quadratic form in ξ ∈ RN qL (x, ξ ) := A(x)ξ, ξ is called the characteristic form of L. We have qL (x, ξ ) =

N1 Yj I (x), ξ 2 , j =1

so that qL (x, ·) is obtained by formally replacing in L the coordinate derivatives ∂1 , . . . , ∂N by ξ1 , . . . , ξN . This can be easily seen from (1.90b), for we have qL (x, ξ ) = A(x) ξ, ξ = σ (x) σ (x)T ξ, ξ = σ (x)T ξ, σ (x)T ξ = |σ (x)T ξ |2 =

N1 Yj I (x), ξ 2 . j =1

Then qL (x, ξ ) ≥ 0 for every x ∈ G and every ξ ∈ RN , i.e. A(x) is positive semi-definite for every x ∈ G. Moreover, qL (x, ξ ) = 0 iff Yj I (x), ξ = 0 for every j ∈ {1, . . . , N1 }. Hence, for a fixed x ∈ G, the set N (x) of vectors ξ ’s which annihilate the quadratic form related to A(x) is a linear space given by ⊥ ⊥ N (x) := {ξ ∈ RN | qL (x, ξ ) = 0} = Y1 I (x) ∩ · · · ∩ YN1 I (x) . (1.94)

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1 Stratified Groups and Sub-Laplacians

We recall that, by Proposition 1.2.13, since Y1 , . . . , YN1 are linearly independent in g, then Y1 I (x), . . . , YN1 I (x) are linearly independent in RN for every fixed x. Thus, if N1 < N , that is if r ≥ 2, for every x ∈ G there exists ξ ∈ RN \ {0} such that qL (x, ξ ) = 0. More precisely, the set of isotropic vectors for the quadratic form related to A(x) is a linear subspace of RN of dimension N − N1 , equal to the kernel16 of the matrix A(x). This means that if r ≥ 2, then L is not elliptic at any point of G. On the other hand, if N1 = N (that is the step r of G is 1) the block A1,1 in (1.91) has dimension N × N, while the other blocks disappear. Then L is a constant coefficient operator of the form L = N i,j =1 ai,j ∂i,j with A = (ai,j )i,j symmetric and strictly positive definite. Thus, we can summarize these results as follows: The sub-Laplacian L is a second order differential operator in divergence form with polynomial coefficients. The characteristic form of L is positive semi-definite. If the step of G is ≥ 2, then L is not elliptic at any point of G. If the step of G is 1, then L is an elliptic operator with constant coefficients. Example 1.5.3. The canonical sub-Laplacian of the Heisenberg–Weyl group H1 has been written in Example 1.5.2: in that case the characteristic form is q(x, ξ ) = (ξ1 )2 + (ξ2 )2 + 4(x12 + x22 )(ξ3 )2 + 4x2 ξ1 ξ3 − 4x1 ξ2 ξ3 = (ξ1 + 2x2 ξ3 )2 + (ξ2 − 2x1 ξ3 )2 . Hence q(x, ξ ) = 0 if and only if (1, 0, 2 x2 ), ξ = (0, 1, −2 x1 ), ξ = 0, 16 Indeed, given a N × N real symmetric matrix A, we denote by Isotr(A) the set of the

isotropic vectors w.r.t. the quadratic form related to A, i.e. Isotr(A) := {ξ ∈ RN | Aξ, ξ = 0}. Obviously, it holds Ker(A) ⊆ Isotr(A). In general, the reverse inclusion does not necessarily hold (as Isotr(A) is not necessarily a vector space!) as the following example shows: when ! 1 0 A := , 0 −1 we have Ker(A) = {(0, 0)}, whereas Isotr(A) = span{(1, 1)} ∪ span{(1, −1)}. However, if A is positive semi-definite, then Ker(A) = Isotr(A). Indeed, let R be a real, symmetric matrix such that A = R 2 . If ξ ∈ Isotr(A), it holds 0 = Aξ, ξ = R 2 ξ, ξ = R T Rξ, ξ = Rξ, Rξ = R ξ 2 . Hence R ξ = 0, so that A ξ = R 2 ξ = RRξ = 0. This gives ξ ∈ Ker(A).

1.5 The Sub-Laplacians on a Homogeneous Carnot Group

67

so that, for any fixed x ∈ H1 , the set N (x) of vectors ξ ’s which annihilate the relevant quadratic form is (see (1.94)) N(x) = (1, 0, 2 x2 )⊥ ∩ (0, 1, −2 x1 )⊥ = span{(−2x2 , 2x1 , 1)}. Therefore, it is always one-dimensional. As we showed in (A4), N (x) can also be found as N (x) = Ker(A(x)), where

! 1 0 1 0 2x2 T A(x) = σ (x) σ (x) = · 0 1 0 1 −2x1 2x2 −2x1

1 0 2x2 = . 0 1 −2x1 2x2 −2x1 4(x12 + x22 ) This kernel is one-dimensional, as A(x) has rank 2 for every x ∈ H1 . (A5) The sub-Laplacian L is the second order partial differential operator related to the Dirichlet form u →

|∇L u|2 dx.

More precisely, let Ω ⊆ RN be an open set, and consider the functional 1 C (Ω, R) u → J (u) = 2 ∞

Ω

|∇L u|2 dx,

|∇L u|2 =

N1

(Yj u)2 .

j =1

Denoting by , the inner product in RN1 , we have J (u + h) − J (u) = ∇L u, ∇L h dx + J (h) Ω

C0∞ (Ω, R).

We call critical point of J any function u ∈ for every h ∈ C ∞ (Ω, R) such that ∇L u, ∇L h dx = 0 ∀ h ∈ C0∞ (Ω, R). Ω

Then, given u ∈ C ∞ (Ω, R), we have u is a critical point of J if and only if Lu = 0 in Ω. Indeed, since Yj∗ = −Yj , an integration by parts gives Ω

∇L u, ∇L h dx =

N1

Yj uYj h dx = −

j =1 Ω

=−

(Lu)h dx Ω

for every u ∈ C ∞ (Ω, R) and h ∈ C0∞ (Ω, R).

N1 j =1 Ω

(Yj2 u)h dx

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1 Stratified Groups and Sub-Laplacians

1.5.1 The Horizontal L-gradient We end this section with some useful results on the horizontal L-gradient. N1 2 Proposition 1.5.4. Let L = j =1 Xj be a sub-Laplacian on the homogeneous ∞ Carnot group G. Let u ∈ C (G, R) be such that Xj u is a polynomial function of G-degree not exceeding m for every j = 1, . . . , N1 . Then u is a polynomial function of G-degree not exceeding m + 1. Proof. Let Z1 , . . . , ZN be the Jacobian basis of g, the Lie algebra of G. Since the Xj ’s have polynomial coefficients and g = Lie{X1 , . . . , XN1 }, pk := Zk u is a polynomial function, k = 1, . . . , N . Moreover, if we denote δλ (x) = δλ (x1 , . . . , xN ) := (λσ1 x1 , . . . , λσN xN ) with 1 = σ1 ≤ · · · ≤ σN = r, keeping in mind the stratification of g, one easily recognizes that degG pk ≤ m + 1 − σk , k = 1, . . . , N. Now, by using (1.38) and (1.62) of Section 1.1, we see that ∂xj u(x) =

N

pk (x) ak (x −1 ), (j )

j = 1, . . . , N,

k=j

where y → ak (y) is δλ -homogeneous of degree σk − σj . Then, since δλ (x −1 ) = (δλ (x))−1 and the map x → x −1 = Exp (−Log x) has polynomial components (see (j ) Theorem 1.3.28 and Corollary 1.3.29), the function x → ak (x −1 ) is a polynomial δλ -homogeneous of degree σk − σj . It follows that (j )

x → ∇u(x), x =

N

xj ∂xj u(x)

j =1

is a polynomial function of G-degree not exceeding m + 1. Therefore, also 1 1 d dt u(tx) dt = ∇u(tx), tx u(x) − u(0) = t 0 dt 0 is a polynomial function of G-degree not exceeding m + 1.

In order to state the next corollary, we introduce a new notation. Let β = (i1 , . . . , ik ) be a multi-index with components in the set {1, . . . , N1 }. We set X β := Xi1 ◦ · · · ◦ Xik and |β| = k. Corollary 1.5.5. Let the hypotheses of Proposition 1.5.4 hold. Let u ∈ C ∞ (G, R) be such that X β u = 0 ∀ β : |β| = m for a suitable integer m ≥ 1. Then u is a polynomial function on G of G-degree not exceeding m − 1.

1.5 The Sub-Laplacians on a Homogeneous Carnot Group

69

Proof. We argue by induction on m. By Proposition 1.5.6, the assertion holds if m = 1. Suppose it holds for m = p, and let us prove that it holds for m = p + 1. Now, if X β u = 0 for any multi-index β with |β| = p + 1, then X γ (Xj u) = 0 ∀ γ : |γ | = p and for every j ∈ {1, . . . , N1 }. Hence, by the induction assumption, Xj u is a polynomial function of G-degree not exceeding p − 1 for every j ∈ {1, . . . , N1 }. From Proposition 1.5.4 it follows that u is a polynomial function of G-degree not exceeding p. This completes the proof.

Proposition 1.5.6. Let Ω be an open and connected subset of the homogeneous Carnot group G. Let L be any sub-Laplacian on G. Then a function u ∈ C 1 (Ω, R) is constant in Ω if and only if the relevant horizontal L-gradient ∇L u vanishes identically on Ω. Proof. It is obviously non-restrictive to suppose that L = ΔG . Suppose Z1 u, . . . , ZN1 u vanish identically on Ω. Since the Lie algebra of G is given by Lie{Z1 , . . . ZN1 } (see (1.82)), then for every vector field Zj of the Jacobian basis, we have Zj u ≡ 0. We end by applying Proposition 1.2.21.

Example 1.5.7. Our proof of Proposition 1.5.6, despite its simplicity, conceals a deep geometric argument, which can be applied in more general situations (namely, for vector fields satisfying Hörmander’s condition). We describe the underlying geometric idea by an explicit example, leaving to the reader the task to generalize it in more general cases. (See also Chapter 19, where we study the so-called Carathéodory– Chow–Rashevsky connectivity theorem.) Consider the Heisenberg–Weyl group H1 on R3 . The Jacobian generators of its Lie algebra are the vector fields X1 = ∂1 + 2x2 ∂3 ,

X2 = ∂2 − 2x1 ∂3 .

Proposition 1.5.6 then states that if a C 1 -function u satisfies X1 u = 0 and X2 u = 0 in a domain Ω ⊆ R3 ,

(1.95)

then u is constant in Ω. For the sake of brevity, rather than applying some (simple) connectedness argument, we may suppose Ω = R3 . The basic idea is the following one. Suppose that two points x, y ∈ R3 can be joined by an integral curve γ of one of the fields ±X1 or ±X2 , then u(x) = u(y). Indeed, suppose that γ : [0, T ] → R3 is such that γ (0) = x, γ (T ) = y and γ˙ (s) = XI (γ (s))

for all s ∈ [0, T ],

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1 Stratified Groups and Sub-Laplacians

where X is one of the fields ±X1 or ±X2 . Then, taking into account (1.95), we have

T

u(y) − u(x) = 0

T

=

T d (u(γ (s))) ds = (∇u)(γ (s)), γ˙ (s) ds = ds 0 T + , (Xu)(γ (s)) ds = 0, (∇u)(γ (s)), XI (γ (s)) ds =

0

0

whence u(x) = u(y). As a consequence, we can prove that u is constant, if we show that any couple of points in H1 can be joined by a finite sequence of paths which are integral curves of ±X1 and ±X2 .

(1.96)

We shall refer to (1.96) by saying that H1 is (X1 , X2 )-connected. The (X1 , X2 )-connectedness has a deep motivation, namely the fact that X1 ,

X2 ,

[X1 , X2 ] are linearly independent.17

(1.97)

In its turn, the fact that (1.97) implies the (X1 , X2 )-connectedness has a profound motivation too, mainly based on the so-called Campbell–Hausdorff formula, as we shall explain at the end of this section. Before entering into the details of the Campbell–Hausdorff formula, we show how simple the argument is in the case of H1 . Indeed, let us fix a point P0 := (x1 , x2 , x3 ) ∈ H1 , and let us consider the path γ , integral curve of X1 starting from this point. As we showed in (1.5), we have γ (s) = (x1 + s, x2 , x3 + 2 x2 s).

(1.98)

We denote this point by P1 . We then proceed along the integral curve of X2 starting from P1 : at the time s, we arrive to the following point (as a simple calculation shows) P2 = (x1 + s, x2 + s, x3 + 2x2 s − 2(x1 + s)s). Moreover, we proceed along the integral curve of −X1 starting from P2 : at the time s, we arrive to the following point (it is enough to have in mind (1.98) and replace s with −s) P3 = (x1 , x2 + s, x3 − 2(x1 + s)s − 2s 2 ). Finally, we proceed along the integral curve of −X2 starting from P3 : at the time s, we arrive to the following point (again, it is enough to notice that the integral curve of −X2 at time s coincides with the integral curve of X2 at time −s) 17 When X , X are arbitrary smooth vector fields in R3 , the result 1 2

(X1 , X2 , [X1 , X2 ] linearly independent) ⇒ (R3 is [X1 , X2 ]-connected), is known as Carathéodory’s theorem. For a more general version of this result in RN (but under some further assumptions on the fields Xi ’s) we refer the reader to Chapter 19.

1.5 The Sub-Laplacians on a Homogeneous Carnot Group

71

P4 = (x1 , x2 , x3 − 4s 2 ). An analogous calculation shows that, if we start from (x1 , x2 , x3 ) and proceed along the integral curves of, respectively, X2 , X1 , −X2 , −X1 , we arrive to 4 = (x1 , x2 , x3 + 4s 2 ). P 4 , this shows that we can join any two points having the Being s arbitrary in P4 and P same x1 , x2 -coordinates and the third one arbitrarily given. We now start from (x1 , x2 , x3 ) and, along an integral curve of X1 , at the time s we arrive to (x1 + s, x2 , x3 + 2x2 s). By the preceding argument, keeping fixed the first two coordinates, we can vary the third one, in order to arrive to the point (x1 + s, x2 , x3 ) (after finitely many integral curves of ±X1 , ±X2 ). Being s arbitrary, this shows that we can join any two points having the same x2 , x3 coordinates and the first one arbitrarily given. Finally, an obvious analogous argument shows that we can join any two points having the same x1 , x3 coordinates and the second one arbitrarily given. All these facts together prove (1.96). To end the section, we describe the reason why the validity of (1.97) implies the (X1 , X2 )-connectedness. For instance, let us denote by Γ (s) the point we obtain if (as we did above) we follow paths of X1 , X2 , −X1 , −X2 for a time s, this is the same as saying that we follow paths of sX1 , sX2 , −sX1 , −sX2 for unit time. More explicitly, Γ (s) = exp(−sX2 )(exp(−sX1 )(exp(sX2 )(exp(sX1 )(x)))). Very simple arguments (showed in the proof of Lemma 5.13.18, page 301) show that (see precisely (5.117), page 302) Γ (s) − x = [X1 , X2 ]I (x). s→0 s2 lim

(1.99)

Roughly speaking, (1.99) ensures that, by following suitable integral curves of ±X1 , ±X2 , we can arrive as close as we want to the endpoints of the integral curves of the commutator [X1 , X2 ]. So, if X1 , X2 , [X1 , X2 ] span R3 at every point, it is intuitively evident that we can connect any two points by suitable integral curves of ±X1 , ±X2 . This argument becomes completely apparent if we make use of the so-called Campbell–Hausdorff formula. For two vector fields X1 , X2 generating an algebra nilpotent of step two, the Campbell–Hausdorff formula states that18 ! 1 exp(X2 )(exp(X1 )(x)) = exp X1 + X2 + [X1 , X2 ] (x). (1.100) 2 18 Further details on the Campbell–Hausdorff formula can be found in Definition 2.2.11,

Theorem 2.2.13 (page 129), in Lemma 4.2.4 (page 194) and, mostly, in Theorem 15.1.1 (page 595).

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1 Stratified Groups and Sub-Laplacians

We now apply three times formula (1.100), in order to simplify the above Γ (s). The following calculation then applies: Γ (s) = exp(−sX2 )(exp(−sX1 )(exp(sX2 )(exp(sX1 )(x)))) ! !! s2 = exp(−sX2 ) exp(−sX1 ) exp sX1 + sX2 + [X1 , X2 ] (x) 2 ! ! s2 s2 = exp(−sX2 ) exp sX1 + sX2 + [X1 , X2 ] − sX1 − [X2 , X1 ] (x) 2 2 = exp(−sX2 )(exp(sX2 + s 2 [X1 , X2 ])(x)) = exp(sX2 + s 2 [X1 , X2 ] − sX2 )(x) = exp(s 2 [X1 , X2 ])(x). This says that, following suitable integral curves of ±X1 , ±X2 , we can arrive wherever the integral curves of the commutator [X1 , X2 ] do arrive! So, again, if X1 , X2 , [X1 , X2 ] span R3 at every point, we can obviously connect any two points by suitable integral curves of ±X1 , ±X2 . We explicitly remark that the assumption that X1 , X2 generate an algebra nilpotent of step two has made the above calculation very transparent. For general smooth vector fields, the Campbell–Hausdorff formula leads to a formula with a remainder term, exp(−sX2 )(exp(−sX1 )(exp(sX2 )(exp(sX1 )(x)))) = exp(s 2 [X1 , X2 ] + Ox (s 3 ))(x).

Bibliographical Notes. “Carnot” groups seem to owe their name after an paper by C. Carathéodory [Car09] (related to a mathematical model of thermodynamics) dated 1909. The same denomination was then used in the school of M. Gromov [Gro96] and it is nowadays commonly used. The definition of stratified group given in this chapter is seemingly different from the classical one by G.B. Folland [Fol75] and by G.B. Folland & E.M. Stein [FS82] (see also M. Gromov [Gro96], P. Pansu [Pan89]). Indeed, here we focused on stratified groups having an underlying homogeneous structure. Nonetheless, we shall prove in Chapter 2 that any (abstract) stratified group is canonically isomorphic to a homogeneous one, so that our definition here is nonrestrictive and seems to be more operative and easier to deal with. A direct approach to homogeneous Carnot groups can also be found in E.M. Stein [Ste81] or in N.T. Varopoulos, L. Saloff-Coste, T. Coulhon [VSC92]; see also P. Hajlasz and P. Koskela [HK00].

1.6 Exercises of Chapter 1

73

1.6 Exercises of Chapter 1 Ex. 1) Verify that the following operation (x1 , x2 , x3 ) ◦ (y1 , y2 , y3 ) := (arcsinh(sinh(x1 ) + sinh(y1 )), x2 + y2 + sinh(x1 )y3 , x3 + y3 ) endows R3 with a Lie group structure. Ex. 2) a) Verify that, for every fixed α ∈ R, the following operation ⎛ x1 + y1 , ⎜ x2 + y2 , ⎜ 1 ⎜ + y + x 3 3 2 (x1 y2 − x2 y1 ), ⎜ 1 ⎝ x4 + y4 + 2 (x1 y3 − x3 y1 ) + α2 (x2 y3 − x3 y2 ) 1 α (x1 − y1 ) (x1 y2 − x2 y1 ) + 12 (x2 − y2 ) (x1 y2 − x2 y1 ) + 12

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

defines on R4 a homogeneous Carnot group G. b) Verify that the Jacobian basis for the algebra g of the above G is ! 1 1 1 x3 + x2 (x1 + α x2 ) ∂4 , Z1 = ∂1 − x2 ∂3 − 2 2 12 ! 1 1 1 Z2 = ∂2 + x1 ∂3 + − α x3 + x1 (x1 + α x2 ) ∂4 , 2 2 12 ! 1 1 x1 + α x2 ∂4 , Z3 = [Z1 , Z2 ] = ∂3 + 2 2 Z4 = [Z1 , [Z1 , Z2 ]] = [Z1 , Z3 ] = ∂4 . Verify that the only other non-trivial commutator identity is [Z2 , [Z1 , Z2 ]] = α [Z1 , [Z1 , Z2 ]]. c) Verify that the exponential map for G (written w.r.t. the Jacobian basis) is the “identity map” in the following sense: if ξ = (ξ1 , ξ2 , ξ3 , ξ4 ) ∈ R4 , and ξ · Z ∈ g denotes the vector field 4i=1 ξi Zi , then Exp (ξ · Z) = ξ ∈ G. Ex. 3) With reference to what we proved in Example 1.2.33 (page 30), prove that, for the group considered in Examples 1.2.18 and 1.2.31 (pages 21 and 28) it holds 1 Log Exp (X) ◦ Exp (Y ) = X + Y + [X, Y ]. 2 Ex. 4) In this exercise, we provide some compact formulas for general sum of squares of vector fields. Suppose we are assigned m vector fields of class at least C 1 on RN ,

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1 Stratified Groups and Sub-Laplacians

⎛α Xk =

N

⎞

⎜ α2,k ⎟ ⎟ Xk I = ⎜ ⎝ .. ⎠ , . αN,k

i.e. as usual,

αj,k ∂j ,

1,k

j =1

k ≤ m.

Let S be the N × m matrix whose k-th column vector is given by Xk I , ⎛α

1,1

S := ⎝ ... αN,1

···

α1,k .. .

α1,m ⎞ .. ⎠ = (α ) j,k j =1,...,N . .

···

αN,k

αN,m

k=1,...,m

With this notation, for every u ∈ C 1 (RN , R), we have ∇u · S = (X1 u, . . . , Xm u) =: ∇X u, where ∇ = (∂1 , . . . , ∂N ) is the usual gradient operator, and ∇X := (X1 , . . . , Xm ) is the “intrinsic gradient” related to the family {X1 , . . . , Xm }. We also define the N × N symmetric matrix m

T αi,k αj,k =: (ai,j )i,j ≤N . A := S · S = i,j =1,...,N

k=1

For every u ∈ C 1 (RN , R), it holds |∇X u|2 :=

m (Xk u)2 = ∇X u, ∇X u = ∇u · S, ∇u · S k=1

= (∇u · S) · (∇u · S)T = ∇u · (S · S T ) · (∇u)T = ∇u · A · (∇u)T , i.e. |∇X u|2 =

N

ai,j ∂i u ∂j u.

i,j =1

Show that, for every k = 1, . . . , m, it holds

N N N 2 αi,k αj,k ∂i,j + αi,k (∂i αj,k ) ∂j . Xk2 = i,j =1

j =1

i=1

Derive that, in the coordinate form, the sum of squares related to the vector 2 fields Xk ’s, L := m k=1 Xk , is given by

1.6 Exercises of Chapter 1

L=

N

2 ai,j ∂i,j +

i,j =1

N

75

bj ∂j ,

j =1

with ai,j =

m

αi,k αj,k ,

k=1

bj =

m N

αi,k (∂i αj,k ) =

k=1 i=1

m

Xk αj,k .

k=1

We note that bj is a sort of “X-divergence” of the j -th row of the matrix S. Besides, show that N

N

N

N 2 ∂i αi,k αj,k ∂j − ∂i αi,k · αj,k ∂j Xk = i=1

=

N i=1

∂i

j =1 N

αi,k αj,k ∂j

j =1

i=1

− div(Xk I ) · Xk .

j =1

Deduce that L can be written in the following equivalent ways:

N m

m N N N L= ∂i αi,k αj,k ∂j − αj,k (∂i αi,k ) ∂j i=1

j =1 k=1

i=1

j =1

j =1

k=1 i=1

N

m N N = ∂i ai,j ∂j − αj,k div(Xk I ) ∂j = div(A · ∇ T ) −

j =1 m k=1

k=1

div(Xk I ) · Xk

⎞ # div(X1 I ) . T T ⎠,∇ .. = div(A · ∇ ) − S · ⎝ div(Xm I ) T = div(A · ∇ ) − (div(X1 I ), . . . , div(Xm I )), ∇X . "

⎛

In particular, the sum of squares L is in divergence form if and only if ⎞ ⎛ div(X1 I ) m N .. ⎠ ≡ 0. ∀ j ≤ N, αj,k (∂i αi,k ) = 0, i.e. S · ⎝ . k=1 i=1 div(Xm I ) (1.101) For example, re-derive that any sub-Laplacian L on a homogeneous Carnot group N is in divergence form, because in this case it holds div(Xk I ) = i=1 ∂i αi,k = 0, for αi,k does not depend on xi . We explicitly remark that, in the case of non-homogeneous Carnot groups (which will be introduced in the

76

1 Stratified Groups and Sub-Laplacians

next chapter), this is not necessarily true, as the following example shows: R3 equipped with the composition x ◦ y = (arcsinh(sinh(x1 ) + sinh(y1 )), x2 + y2 + sinh(x1 )y3 , x3 + y3 ) is a non-homogeneous Carnot group; the first two vector fields of the relevant Jacobian basis are X1 = (cosh(x1 ))−1 ∂1 ,

X2 = ∂3 + sinh(x1 ) ∂2 ,

so that the canonical sub-Laplacian is not a divergence form operator, for (1.101) is not satisfied, being, for j = 1, α1,1 div(X1 I ) = − sinh(x1 )/(cosh(x1 ))3 . Ex. 5) Prove the assertions made in Remark 1.4.9 (page 61). Ex. 6) Prove the following simple formulas of calculus for vector fields. Here, 2 N N u, f, g ∈ C 2 (RN , R), α ∈ C 2 (R, R), ϕ = (ϕ 1 , . . . , ϕ2 N ) ∈ C (R , R ), X is a sum of squares X = ∇(·) · XI is a vector field on RN , L = m j =1 j of vector fields on RN , ∇L = (X1 , . . . , Xm ) is the intrinsic gradient related to L. • X(f g) = (Xf ) g + f (Xg), • ∇L (f g) = (∇L f ) g + f (∇L g), • X 2 (f g) = (X 2 f ) g + 2 (Xf ) (Xg) + f (X 2 g), • L(f g) = (Lf ) g + 2 ∇L f, ∇L g + f (Lg), • X(α(u)) = α (u) Xu, • ∇L (α(u)) = α (u) ∇L u, • X 2 (α(u)) = α (u) (Xu)2 + α (u) X 2 u, |∇L u|2 + α (u) Lu, • L(α(u)) = α (u) N • X(u(ϕ(x))) = j =1 uj (ϕ(x)) Xϕj (x) = (∇u)(ϕ(x)), Xϕ(x), • ∇L (u(ϕ(x))) = N j =1 (∂j u)(ϕ(x)) ∇L ϕj (x); if we use the usual columnvector notation ⎞ ⎛ ϕ1 (x) . ϕ(x) = ⎝ .. ⎠ ϕN (x) for vectors in RN , the row-vector notation for the gradients ∇u = (∂1 u, . . . , ∂N u),

∇L u = (X1 u, . . . , Xm u),

and we introduce the X-Jacobian matrix ⎛ ⎞ ⎛ ⎞ X1 ϕ1 (x) · · · Xm ϕ1 (x) ∇L ϕ1 (x) ∂ϕ .. .. .. ⎠=⎝ ⎠ (x) := ⎝ . ··· . . ∂X X1 ϕN (x) · · · Xm ϕN (x) ∇L ϕN (x) (which is a N × m matrix) which can also be denoted by

1.6 Exercises of Chapter 1

77

∇L ϕ(x) = (X1 ϕ(x) · · · Xm ϕ(x)) ⎛ ⎛ ⎛ ⎞ ⎞⎞ ϕ1 (x) ϕ1 (x) = ⎝X1 ⎝ ... ⎠ · · · Xm ⎝ ... ⎠⎠ , ϕN (x)

ϕN (x)

then last formula can be rewritten as ∇L (u(ϕ(x))) = (∇u)(ϕ(x)) ·

∂ϕ (x), ∂X

or equivalently ∇L (u(ϕ(x))) = (∇u)(ϕ(x)) · ∇L ϕ(x), which is resemblant to the classical chain rule ∇(u(ϕ(x))) = (∇u)(ϕ(x)) · Jϕ (x), •

prove that X 2 (u(ϕ(x))) =

N

(∂i,j u)(ϕ(x))Xϕi (x) Xϕj (x)

i,j =1

+

N (∂j u)(ϕ(x))X 2 ϕj (x), j =1

or equivalently, following the previous notation (here Hessu (x) is the usual Hessian matrix of u at x), X 2 (u(ϕ(x))) = (Xϕ(x))T · Hessu (ϕ(x)) · (Xϕ(x)) + (∇u)(ϕ(x)) · X 2 ϕ(x), •

deduce from the previous formulas that L(u(ϕ(x))) =

N

+ , (∂i,j u)(ϕ(x)) ∇L ϕi (x), ∇L ϕj (x)

i,j =1

+

N (∂j u)(ϕ(x))Lϕj (x), j =1

which can also be rewritten as L(u(ϕ(x))) =

m (Xk ϕ(x))T · Hessu (ϕ(x)) · (Xk ϕ(x)) k=1

+ (∇u)(ϕ(x)) · Lϕ(x)

78

1 Stratified Groups and Sub-Laplacians

or, introducing the Gram matrix of the system of N vectors in Rm {∇L ϕ1 (x), . . . , ∇L ϕN (x)}, i.e. the N × N symmetric matrix !T ∂ϕ ∂ϕ Gϕ (x) := (x) · (x) ∂X ∂X ⎛ ⎞ ∇L ϕ1 (x) . ⎠ · ∇L ϕ1 (x) T · · · ∇L ϕN (x) T .. = ⎝ ∇L ϕN (x) ⎛ ⎞ ∇L ϕ1 (x), ∇L ϕ1 (x) · · · ∇L ϕ1 (x), ∇L ϕN (x) .. .. .. ⎠, = ⎝ . . . ∇L ϕN (x), ∇L ϕ1 (x)

···

∇L ϕN (x), ∇L ϕN (x)

the above formula can be rewritten as L(u(ϕ(x))) = trace(Hessu (ϕ(x)) · Gϕ (x)) + (∇u)(ϕ(x)) · Lϕ(x). Ex. 7) Let X be a smooth vector field on some open set Ω ⊆ RN , and let x ∈ Ω. Let γ be the solution to the system of ODE’s γ˙ (t) = XI (γ (t)),

γ (0) = x,

defined on the open (maximal) interval D(X, x) ⊆ R. Denote (momentarily) γ (t) by E(X, x, t). Let λ ∈ R be arbitrary. Prove the homogeneity relation E(X, x, λ t) = E(λ X, x, t)

for all t such that λ t ∈ D(X, x). (1.102)

Let now t0 ∈ D(X, x) be fixed. Take λ = t0 and t = 1 in (1.102) and derive that (for the arbitrariness of t0 ) E(X, x, t) = E(t X, x, 1)

for all t ∈ D(X, x).

(1.103)

Throughout the following exercise, we shall adopt our usual notation exp(tX)(x) to denote E(X, x, t). Equation (1.103) says that the notation is not improper. Ex. 8) In this exercise, we give a generalization of formula (1.7) (page 7). Let Ω ⊆ RN be an open set, and let X be a smooth vector field on Ω. Let also f be a smooth function on Ω. Finally, we fix x ∈ Ω. Then the function t → f (exp(tX)(x)) is C ∞ near t = 0, and its Taylor expansion at 0 is given by f (exp(tX)(x)) =

n tk

(X k f )(x) k! k=0 1 t (t − s)n X n+1 f (exp(sX)(x)) ds. + n! 0

(1.104)

1.6 Exercises of Chapter 1

79

Indeed, prove by induction that it holds dk (f (exp(tX)(x))) = (X k f )(exp(tX)(x)). dt k Derive from it the (very useful) formula dk (f (exp(tX)(x))) = (X k f )(x). dt k t=0 Derive from (1.44) that if G = (RN , ◦) is a Lie group on RN and X is leftinvariant on G, then dk (f (x ◦ Exp (tX))) = (X k f )(x). dt k t=0 Ex. 9) Let Ω ⊆ RN be an open set, and let X be a smooth vector field on Ω. Let us consider the (autonomous) equation γ˙ = XI (γ ). From general results on the existence and uniqueness of solution of ODE’s (see, e.g. [Har82]) we know that, for every fixed compact set K ⊂ Ω, there exists δ = δ(X, K) > 0 such that the solution γ (t, x) to γ˙ = XI (γ ), γ (0) = x exists for every t ∈ ]−δ, δ[ and every initial value x ∈ K. From the uniqueness of the solution (and the autonomous nature of the equation) derive that γ (t, γ (s, x)) = γ (t + s, x),

|t| + |s| < δ,

x ∈ K.

(1.105)

Deduce from this fact that, for every fixed t0 ∈ ]−δ, δ[, the mapping K x → γ (t0 , x) is injective. (Hint: If γ (t0 , x) = γ (t0 , y), then γ (t +t0 , x) = γ (t, γ (t0 , x)) = γ (t, γ (t0 , y)) = γ (t + t0 , y), and evaluate at t = −t0 .) If O is an open set such that O ⊂ Ω is compact, prove that (whenever |t0 | < δ(X, O)) the map O x → γ (t0 , x) is a C ∞ -diffeomorphism onto its image. (Hint: Again thanks to (1.105), the inverse map is given by y → γ (−t0 , x).) Ex. 10) Let Y1 , . . . , Ym be smooth vector fields on an open set Ω ⊆ RN . Let ξ = (ξ1 , . . . , ξm ) ∈ Rm . Since the dependence on ξ of ξ1 Y1 I + · · · + ξm Ym I is smooth, by general results on ODE’s (see, e.g. [Har82]), given a compact set K ⊂ Ω, we infer the existence of ε > 0 such that the solution γ to γ˙ =

m

ξj Yj I (γ ),

γ (0) = x

j =1

exists for every x ∈ K, every ξ satisfying |ξ | < ε and every t ∈ ]−ε, ε[ (here |ξ | denotes any fixed norm on Rm ). Moreover, the dependence of γ on (x, ξ, t) is smooth. Using (1.103) (and the notation therein), observe that

80

1 Stratified Groups and Sub-Laplacians

Fig. 1.7. The flow of the vector field X

E

! ξj Yj , x, t

=E

j

! for all t ∈ ]−ε, ε[

tξj Yj , x, 1

j

and derive that the map (u1 , . . . , um ) → Θx (u) := exp

uj Yj (x)

j

is well-defined (and smooth) on the open ball B = {u ∈ Rm : |u| < ε 2 } (for any x ∈ K). Prove that the Jacobian matrix of Θx at u = 0 is the matrix whose j -th column vector is Yj I (x) or, equivalently, that the differential d0 Θx sends the tangent vector (∂/∂uj )|0 ∈ T0 (B) to Yj |x ∈ Tx (Ω). Hint: Use the following fact ! ! ∂ ∂ 2 I (x) + exp uj Yj (x) = uj Yj I (x) + Ou→0 (|u| ) ∂uj 0 ∂uj 0 j

j

= Yj I (x). As a corollary, deduce that if m = N and Y1 I (x), . . . , YN I (x) are linearly independent, then Θx is a C ∞ -diffeomorphism of a neighborhood of 0 onto a neighborhood of x. Finally, generalize (1.104) of Exercise 8 (we follow the notation therein) and obtain the following Taylor expansion at u = 0 f

exp

m j =1

uj Yj

k m ∞ 1 uj Yj f (x). (x) ∼ k! k=0

j =1

(1.106)

1.6 Exercises of Chapter 1

Hint: Set

F (u) := f

exp

m

81

and G(t, u) := F (tu).

uj Yj (x)

j =1

Hence G(t, u) = f (exp(tX)(x)), where X = m j =1 uj Yj . Use Ex. 8 to derive that

k m ∂ k G(t, u) = uj Yj f (x), ∂t k t=0 j =1

and the Taylor expansion of F at u = 0 follows, as usual, from that of G at t = 0. Ex. 11) (Rellich–Pohozaev identities). We introduce the following notation. If Ω ⊂ RN is a domain with boundary regular enough, we denote by ν the outer unit normal to ∂Ω and by dσ the Hausdorff (N − 1)-dimensional measure on ∂Ω. Provide the details for the following proposition. Proposition Let G be an arbitrary homogeneous Carnot group, and 1.6.1. 2 be a sub-Laplacian on G. Let Ω be a bounded domain X let L = m i=1 i in G, regular for the divergence theorem. Finally, let Z be a vector field of class C 1 (G). Then, for every ϕ ∈ C 2 (Ω), we have m 2 Xi ϕ Xi I, ν Zϕ dσ − ZI, ν |∇L ϕ|2 dσ ∂Ω i=1

∂Ω

m =2 Xi ϕ [Xi , Z]ϕ + 2 Lϕ Zϕ − |∇L ϕ|2 div(ZI ). (1.107) Ω i=1

Ω

Ω

Proof. By applying two times the divergence theorem (and recalling that Xi∗ = −Xi ), we obtain 2 |∇L ϕ| ZI, ν dσ = div(|∇L ϕ|2 ZI ) ∂Ω Ω |∇L ϕ|2 div(ZI ) + Z(|∇L ϕ|2 ) = Ω Ω = |∇L ϕ|2 div(ZI ) + 2 Z(∇L ϕ), ∇L ϕ Ω Ω m 2 |∇L ϕ| div(ZI ) + 2 [ZI, Xi ]ϕ Xi ϕ = Ω

+2

Ω

Ω i=1

Xi (Zϕ) Xi ϕ

Ω i=1

=

m

|∇L ϕ|2 div(ZI ) + 2

m

+2

∂Ω i=1

This ends the proof.

m [ZI, Xi ]ϕ Xi ϕ Ω i=1

Zϕ Xi ϕ Xi I, ν dσ − 2

Zϕ Lϕ. Ω

82

1 Stratified Groups and Sub-Laplacians

We now specify the integral identity (1.107) when Z is given by the socalled generator of the translations. Let ◦ be the group law on the Carnot group G, and fix z0 ∈ G. We denote by Z z0 the following vector field on G d z0 Z I (z) = ((hz0 ) ◦ z). (1.108) dh h=0 Recalling that, for every i = 1, . . . , m, Xi (z) = (d/dh)h=0 (z ◦ (hei )) (here ei is the i-th versor of the canonical basis of RN ), derive that the bracket [Xi , Z z0 ] vanishes identically. Hint: ◦ is associative and d d f ((sz0 ) ◦ (z ◦ (hei ))) [Xi , Z z0 ]f (z) = dh h=0 ds s=0 d d − f (((sz0 ) ◦ z) ◦ (hei )). ds s=0 dh h=0 Furthermore, the divergence of the vector field Z z0 vanishes identically. Indeed, we recall that ◦ has the form z ◦ ζ = ((z ◦ ζ )(1) , . . . , (z ◦ ζ )(r) ), where (z ◦ζ )(1) = z(1) +ζ (1) ,

(z ◦ζ )(j ) = z(j ) +ζ (j ) +Q(j ) (z, ζ ),

2 ≤ j ≤ r,

Q(j ) being a function with values in RNj and whose components are mixed polynomials in z and ζ such that Q(j ) (δλ z, δλ ζ ) = λj Q(j ) (z, ζ ). We then recognize that the components of Z z0 (z) in the j -th layer have the following form (j )

(1)

(j −1)

(j )

(Z z0 (z))(j ) = z0 + z0 , q1 (z) + · · · + z0

(j )

, qj −1 (z),

(j )

where qi is a RNi -valued function whose components are polynomials δλ -homogeneous of degree j − i. In particular, (Z z0 (z))(j ) does not depend on zj , whence div(Z z0 ) = 0. From Proposition 1.6.1 and the above remarks, the next result immediately follows. Proposition 1.6.2. Let G be a homogeneous Carnot group, and let L = m 2 be a sub-Laplacian on G. Let Ω be a bounded domain in G, regX i=1 i ular for the divergence theorem. Finally, for a fixed z0 ∈ G, let Z z0 be the vector field defined in (1.108). Then, for every ϕ ∈ C 2 (Ω), we have 2

m z0 Xi ϕ Xi I, ν Z ϕ dσ −

∂Ω i=1

∂Ω

Lϕ Z z0 ϕ.

=2 Ω

Z z0 I, ν |∇L ϕ|2 dσ

1.6 Exercises of Chapter 1

83

Ex. 12) (Maps commuting with a sum of squares). a) Consider a second order differential operator N

L :=

2 ai,j (x) ∂i,j +

i,j =1

N

bi (x) ∂i

i=1

with coefficients ai,j and bi in C 2 (RN ) such that ai,j = aj,i for all i and j . Consider ψ ∈ C 2 (RN , RN ). Prove that the necessary and sufficient conditions in order to have L(u ◦ ψ) = (L u) ◦ ψ

∀ u ∈ C 2 (RN )

are the following ones: ⎧ bk (ψ) = Lψk ∀ k = 1, . . . , N, ⎪ ⎪ ⎨ N ⎪ ai,j (ψ) = ar,s ∂r ψi ∂s ψj ∀ i, j = 1, . . . , N. ⎪ ⎩

(1.109)

(1.110)

r,s=1

(Hint: Take u(x) = xk and then u(x) := xi xj .) If Jψ denotes the usual Jacobian matrix of ψ and ⎞ ⎛ b1 (x) ⎟ ⎜ A(x) := ai,j (x) 1≤i,j ≤N , B(x) := ⎝ ... ⎠ , bN (x) so that

L = trace A(x) · Hess + ∇, B(x),

then system (1.110) can be rewritten as B(ψ(x)) = Lψ(x) A(ψ(x)) = Jψ (x) · A(x) · (Jψ (x))T .

(1.111)

p b) Suppose furthermore that L is a sum of squares L = k=1 Xk2 with (k) Xk = N i=1 σi ∂i , so that p

p

N N (k) (k) (k) 2 σi σj ∂i,j + Xk σi L= ∂i . i,j =1

k=1

i=1

k=1

Prove that the necessary and sufficient conditions for (1.109) to hold are ⎧ p ⎪ (k) ⎪ ⎪ (Xk σi )(ψ) = Lψi ∀ i = 1, . . . , N, ⎪ ⎨ k=1

p ⎪ ⎪ (k) (k) ⎪ ⎪ σi (ψ) σj (ψ) = ∇L ψi , ∇L ψj ⎩

∀ i, j = 1, . . . , N,

k=1

(1.112)

84

1 Stratified Groups and Sub-Laplacians

where ∇L u := (X1 u, . . . , Xp u). Set, as usual, ⎞ σ1(k) ⎟ ⎜ Xk I := ⎝ ... ⎠ . ⎛

(k)

σN Considering the N × p matrix (j )

S(x) := (σi (x))i≤N, j ≤p = (X1 I (x) · · · Xp I (x)), we have

A(x) =

p

(k) (k) σi σj

= S(x) · (S(x))T , i,j ≤N

k=1

so that Jψ (x) · A(x) · (Jψ (x))T = (Jψ (x) · S(x)) · (Jψ (x) · S(x))T . Moreover, ⎞ ∇L ψ 1 ⎟ ⎜ = ⎝ ... ⎠ . ∇L ψ N ⎛

Jψ (x) · S(x) = (Xj ψi (x))i≤N, j ≤p As a consequence,

(Jψ (x) · S(x)) · (Jψ (x) · S(x))T = (∇L ψi (x), ∇L ψj (x))i,j ≤N . Moreover, B(x) =

p k=1

(k)

Xk σi

i≤N

⎛ p ⎜ =⎝

k=1

p k=1

(k) ⎞ Xk σ1 ⎟ .. ⎠. .

Xk σN(k)

Finally, (1.112) becomes B(ψ(x)) = Lψ(x), A(ψ(x)) = (∇L ψi (x), ∇L ψj (x))i,j ≤N . Ex. 13) Consider the Lie group (and the notation) in Example 1.2.19 (page 21). Show that, for every ξ1 , ξ2 ∈ R, the integral curve of ξ1 Z1 + ξ2 Z2 starting at (0, 0) is (ξ1 t, (ξ2 eξ1 t − ξ2 )/ξ1 ) if ξ1 = 0, γ (t) = (0, ξ2 t) if ξ1 = 0.

1.6 Exercises of Chapter 1

85

Equivalently, considering the function f : R → R,

f (z) :=

ez − 1 if z = 0, z

f (0) := 1,

we have γ (t) = (ξ1 t, ξ2 t f (ξ1 t)). Derive that Exp : g → G,

exp(ξ1 Z1 + ξ2 Z2 ) = (ξ1 , ξ2 f (ξ1 )).

Prove that Exp is smooth (indeed, real analytic!) and globally invertible. Find the Log function. (Hint: The function f is analytic and invertible.)

2 Abstract Lie Groups and Carnot Groups

The aim of this chapter is to prove that, up to a canonical isomorphism, the classical definition of stratified group (or Carnot group) (see Definition 2.2.3) coincides with our definition of homogeneous Carnot group, as given in Chapter 1. To this aim, we begin by recalling some basic facts about abstract Lie groups, providing all the terminology and the main results about manifolds, tangent vectors, left-invariant vector fields, Lie algebras, homomorphisms, the exponential map. Our exposition in Section 2.1 is self-contained and is intended to provide the topics from differential geometry and Lie group theory, which are strictly necessary to read this book. In Section 2.2, we provide the cited equivalence between the two notions of Carnot group. This is accomplished by showing that the Lie algebra g of the abstract Carnot group G possesses a natural structure of homogeneous Carnot group. Indeed, the group operation on g is the one induced by that of G via the exponential map, and the dilations on g are modeled on its stratification. A central rôle here will be played by the Campbell–Hausdorff formula, that we assume in this chapter by recalling (without proofs) some of its abstract and very general properties. Nonetheless, we shall devote Chapter 15 to provide a self-contained investigation of such a remarkable formula in the significant case of homogeneous stratified groups.

2.1 Abstract Lie Groups 2.1.1 Differentiable Manifolds Let N ∈ N, and let us define, for i = 1, . . . , N , the coordinate projections on RN (whose points will be denoted by ξ = (ξ1 , . . . , ξN ) ∈ RN with ξ1 , . . . , ξN ∈ R) πi : RN −→ R,

πi (ξ ) := ξi .

Definition 2.1.1 (N -dimensional locally Euclidean space). An N -dimensional locally Euclidean space M is a Hausdorff topological space such that every point of M

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2 Abstract Lie Groups and Carnot Groups

has a neighborhood in M homeomorphic to an open subset of RN . If ϕ is a homeomorphism between a connected open set U ⊆ M and an open subset of RN , we say that ϕ : U → RN is a coordinate map, xi := πi ◦ ϕ : U → R is a coordinate function, and the pair (U, ϕ) (sometimes also denoted by (U, x1 , . . . , xN )) is a coordinate system or a chart. If m ∈ U and ϕ(m) = 0, we say that the coordinate system is centered at m. Definition 2.1.2 (Differentiable manifold). A C ∞ differentiable structure F on a locally Euclidean space M is a collection of coordinate systems {(Uα , ϕα ) : α ∈ A } with the following properties: • α∈A Uα = M; • ϕα ◦ ϕβ−1 is C ∞ for every α, β ∈ A (whenever it is defined); • F is maximal w.r.t. the second property in the sense that if (U, ϕ) is a coordinate system such that ϕ ◦ ϕα−1 and ϕα ◦ ϕ −1 are C ∞ for every α ∈ A, then (U, ϕ) ∈ F. An N -dimensional C ∞ differentiable manifold is a couple (M, F), where M is a second countable N -dimensional locally Euclidean space and F is a C ∞ differentiable structure. As usual, when we say “M is an N-dimensional C ∞ differentiable manifold”, we leave unsaid that M is equipped with the fixed datum of a C ∞ differentiable structure F on M. Let M and M be differentiable manifolds of dimension N and N , respectively, and let f : M −→ M . Then we say that f is C ∞ in m ∈ M if, for every (or, equivalently, for at least one) coordinate system (U, ϕ) of M and for every (or, equivalently, for at least one) coordinate system (U , ϕ ) of M such that m ∈ U and f (m) ∈ U , the function ϕ ◦ f ◦ ϕ −1 : ϕ(U ) ⊆ RN −→ ϕ (U ) ⊆ RN is C ∞ in a neighborhood of ϕ(m). Let now μ(t) be a C ∞ function defined on a real interval and with values in M. We say that μ(t) is a curve passing through m at the time t0 if μ(t0 ) = m. We say that a C ∞ real valued function f defined in a neighborhood of m ∈ M is horizontal in m if d f (μ(t)) =0 dt t=0 for every curve passing through m at the time t = 0. Remark 2.1.3. The following characterizations hold: (I). It is immediate to see that a real-valued function f is C ∞ in a neighborhood of m ∈ M if and only if there exist a coordinate system (U, ϕ) with m ∈ U and a

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(ordinary) C ∞ real-valued function f on (the open subset of RN ) ϕ(U ) such that f = f◦ ϕ on U . (For example, any coordinate map xi = πi ◦ ϕ is C ∞ on U for πi is smooth.) (II). Analogously, μ is a curve passing through m ∈ M at t0 if and only if there exist a coordinate system (U, ϕ) with m ∈ U and a (ordinary) C ∞ curve ν : ]a, b[ ⊆ R −→ ϕ(U ) ⊆ RN with t0 ∈]a, b[, ν(t0 ) = ϕ(m) and μ = ϕ −1 ◦ ν on ]a, b[. (III). Furthermore, we show that f is horizontal in m ∈ M if and only if there exist a coordinate system (U, ϕ) with m ∈ U and a C ∞ real-valued function f on ϕ(U ) such that f = f◦ ϕ on U and the ordinary gradient (∇ f)(ϕ(m)) is null. Indeed, suppose this fact holds. Let μ be any curve passing through m at t = 0. By (II), there exist a coordinate system (V , ψ) with m ∈ V and a C ∞ curve ν : ]a, b[ −→ ψ(U ) with 0 ∈ ]a, b[, ν(0) = ψ(m) and μ = ψ −1 ◦ ν on ]a, b[. Then, for ε > 0 sufficiently small (so that ν(t) ∈ ψ(U ∩ V ) for every t ∈ ]−ε, ε[), the composition f ◦ μ = f◦ ϕ ◦ ψ −1 ◦ ν is well defined and smooth on ]−ε, ε[ and, by the ordinary chain rule, it holds d f (μ(t)) = (∇ f) ϕ(ψ −1 (ν(0))) · Jϕ◦ψ −1 (ν(0)) · ν˙ (0) = 0, dt t=0 for (∇ f)(ϕ(ψ −1 (ν(0)))) = (∇ f)(ϕ(m)) = 0. This proves that f is horizontal in m. Vice versa, suppose f is horizontal in m. By (I), there exist a coordinate system (U, ϕ) with m ∈ U and a C ∞ real-valued function f on ϕ(U ) such that f = f◦ ϕ on U . We aim to prove that (∇ f)(ϕ(m)) = 0. Let h ∈ RN be any fixed vector. Let ε > 0 be small enough, so that, for every |t| < ε, we have ν(t) := ϕ(m) + t h ∈ ϕ(U ). Then, by (II), μ := ϕ −1 ◦ ν is a C ∞ curve passing through m at t = 0. Hence, being f horizontal in m, we have d d f (μ(t)) f ◦ ϕ ◦ ϕ −1 ◦ ν (t) = 0= dt d t t=0 t=0 = (∇ f)(ν(0)) · ν˙ (0) = (∇ f)(ϕ(m)), h . Consequently, due to the arbitrariness of h, this proves that (∇ f)(ϕ(m)) = 0. (IV). Finally, we prove that f is horizontal in m ∈ M if and only if for every coordinate system (V , ψ) with m ∈ V , the C ∞ real-valued function f = f ◦ ψ −1 on ψ(V ) satisfies

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(∇ f)(ψ(m)) = 0. The “if” part follows by (III). Vice versa, suppose f is horizontal in m ∈ M. By (III), there exist a coordinate system (U, ϕ) with m ∈ U and a C ∞ real-valued function g on ϕ(U ) such that f = g ◦ ϕ on U and (∇g)(ϕ(m)) = 0. As a consequence, (∇ f)(ψ(m)) = (∇(f ◦ ψ −1 ))(ψ(m)) = (∇(g ◦ ϕ ◦ ψ −1 ))(ψ(m)) = (∇g)(ϕ ◦ ψ −1 ◦ ψ(m)) · Jϕ◦ψ −1 (ψ(m)) = (∇g)(ϕ(m)) · Jϕ◦ψ −1 (ψ(m)) = 0, for (∇g)(ϕ(m)) = 0.

Example 2.1.4. (i). Any constant function on M is trivially horizontal at every point of M. The same is true for a function vanishing near m. (ii). Let m ∈ M, and fix any coordinate system (U, ϕ) with m ∈ U . Let i, j ∈ {1, . . . , N } be chosen. Consider the relevant coordinate functions xi , xj , i.e. xi = πi ◦ ϕ and xj = πj ◦ ϕ on U . Denote x 0 := ϕ(m). We show that f := (xi − xi0 ) (xj − xj0 ) is horizontal at m. By (III) of Remark 2.1.3, in order to prove that f is horizontal at m, it suffices to show that ∇(f ◦ ϕ −1 )(ϕ(m)) = 0. But this is obvious since, for every u ∈ ϕ(U ), (f ◦ ϕ −1 )(u) = (πi ◦ ϕ − xi0 ) (πj ◦ ϕ − xj0 ) (ϕ −1 (u)) = (ui − xi0 ) (uj − xj0 ), and trivially

∂ (ui − xi0 )(uj − xj0 ) = 0 ∂ uk x 0

for every k = 1, . . . , N . (iii). If f is horizontal at m ∈ M, f (m) = 0 and g is any smooth function in a neighborhood of m, then f g is horizontal at m ∈ M. Indeed, let (U, ϕ) be a coordinate system with m ∈ M. By (IV) of Remark 2.1.3, we need to show that ∇((f g) ◦ (ϕ −1 ))(ϕ(m)) = 0. This is obvious since ∇ (f g) ◦ (ϕ −1 ) = (g ◦ ϕ −1 ) ∇ f ◦ (ϕ −1 ) + (f ◦ ϕ −1 ) ∇ g ◦ (ϕ −1 ) and recalling that ∇(f ◦ ϕ −1 )(ϕ(m)) = 0 (for f is horizontal in m) and f (m) = 0 by hypothesis.

As we shall see in the proof of Proposition 2.1.7, the functions in (i), (ii), (iii) of Example 2.1.4 are “infinitesimally” the only functions worth considering.

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2.1.2 Tangent Vectors Definition 2.1.5 (Tangent vector, space and bundle). Let M be an N -dimensional C ∞ differentiable manifold. A tangent vector v at m ∈ M is a linear functional, defined on the collection of the real-valued functions C ∞ in some neighborhood of m, such that v(f ) = 0 whenever f is horizontal in m. We denote by Mm the set of the tangent vectors at m ∈ M, and we say that Mm is the tangent space to M at m. We finally set

T (M) := (2.1) {m} × Mm = (m, v) : m ∈ M, v ∈ Mm . m∈M

T (M) is called the tangent bundle to M. Remark 2.1.6. Suppose f is a real-valued C ∞ function defined in a neighborhood of m ∈ M. Hence, there exists a coordinate system (U, ϕ) such that m ∈ U , and f is defined on U . Consider the open set ϕ(U ) ⊆ RN . For n ∈ {1, 2, 3, 4}, denote by Bn the Euclidean ball centered at ϕ(m) with radius n ε, where ε > 0 is small enough, so that B4 is contained in ϕ(U ). Let χ be a smooth cut-off function on RN such that χ ≡ 1 on B1 and χ ≡ 0 on RN \ B2 . Then, consider the function on M χ(ϕ(n)) f (n) if n ∈ ϕ −1 (B3 ), f (n) := n ∈ M. 0 otherwise, It is easy to see that f ≡ f on the set ϕ −1 (B1 ), open neighborhood of n in M, and that ϕ is defined and smooth on M (note that χ ◦ ϕ is C ∞ on U by Remark 2.1.3-(I)). Since f − f vanishes in a neighborhood of m, by Example 2.1.4-(i), we have v(f − f ) = 0 for every tangent vector v at m. Hence, by linearity, v(f ) = v(f ). This shows that in the definition of vector field it is not restrictive to suppose that a tangent vector at a point is a linear functional defined on the collection of the real-valued functions C ∞ in M. We remark that Mm is a vector space with the usual operations of sum of functionals and multiplication of a functional times a scalar factor. Denoting by dim M the dimension N of the differentiable manifold M, we have the following result. (Note. The reader will recognize in the proof below that a tangent vector at a point is indeed a linear “differential” operator of the first order defined on the C ∞ functions on M.) Proposition 2.1.7. Let M be an N-dimensional C ∞ differentiable manifold. Then dim(Mm ) = N = dim M. Proof. Let v ∈ Mm . Let (U, ϕ) be a coordinate system such that m ∈ U . Let f be an arbitrary C ∞ function defined in a neighborhood of ϕ(m). We set

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v(f) := v(f ),

where f := f◦ ϕ.

By Remark 2.1.3-(I), v is meaningful, and it is easily seen that v is a linear functional defined on the functions C ∞ in a neighborhood of x 0 := ϕ(m). In particular, v (precisely, a suitable restriction of v) is linear on the vector space V of the real-valued linear functions a(x) defined on RN (where N := dim M; note that V = (RN )∗ , the usual dual space of RN ). Now, any linear functional on V is uniquely represented by a vector λ ∈ RN , i.e. v(a(u)) = a( λ). Let now f = f(u) be any function C ∞ in a neighborhood of x 0 . By Taylor expansion with the integral remainder, we have f(u) = f(x 0 ) + ∇ f(x 0 ), u − x 0 1 N

∂ 2 f 0 + x + t (u − x 0 ) dt (ui − xi0 )(uj − xj0 ) (1 − t) ∂ui ∂uj 0 i,j =1

=: f(x 0 ) − a(x 0 ) + a(u) + R(u) for every u near x 0 . By Example 2.1.4, we see that f(x 0 ) − a(x 0 ) and R ◦ ϕ are horizontal in m. Indeed, the summand in the (i, j )-sum defining R ◦ ϕ is the product of (ui − xi0 )(uj − xj0 ) ◦ ϕ = (xi − xi0 )(xj − xj0 ) (where xi , xj are the relevant coordinate functions), which is horizontal and vanishing in m, times the smooth function near m

1

U n → g(n) := 0

(1 − t)

∂ 2 f 0 x + t (ϕ(n) − x 0 ) dt. ∂ui ∂uj

We now set f = f◦ ϕ, so that f = f(x 0 ) − a(x 0 ) + a ◦ ϕ + R ◦ ϕ = a ◦ ϕ + {a horizontal function in m}. As a consequence of the definition of tangent vector, we infer v(f ) = v(a ◦ ϕ) = v (a ◦ ϕ) ◦ ϕ −1 λ. = v(a(u)) = ∇(f ◦ ϕ −1 )(x 0 ), Now, it is easy (and is left as an exercise) to prove that the map Mm v → λ ∈ RN is linear, injective and surjective. This completes the proof.

(2.2)

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Remark 2.1.8. It is easy to see that v ∈ Mm if and only if for every coordinate map (U, ϕ) with m ∈ U (or, equivalently, for at least one such coordinate map) there exists λ = λ(ϕ, m) ∈ RN such that v(f ) = ∇(f ◦ ϕ −1 )(ϕ(m)), λ for every f , real valued and C ∞ near m. The “only if” part follows from the proof of Proposition 2.1.7 (see in particular (2.2)). Suppose now that there exists (U, ϕ) and λ as above. Let f be horizontal in m ∈ M. We need to show that ∇(f ◦ ϕ −1 )(ϕ(m)), λ = 0. By Remark 2.1.3, the function f = f ◦ ϕ −1 satisfies (∇ f)(ϕ(m)) = 0. This definitely suffices for what we needed to prove.

Remark 2.1.9. Remark 2.1.8 furnishes another very useful characterization of tangent vectors (actually, an alternative definition frequently used in literature). If m ∈ M, then v ∈ Mm if and only if there exists a C ∞ curve on M passing through m at t0 such that d v(f ) = (f (μ(t))) for every f , real valued and C ∞ near m. d t t0 Indeed, if v is defined in this way, we have, for any given coordinate map (U, ϕ), m ∈ U, d d (f ◦ ϕ −1 ) ◦ (ϕ ◦ μ) (t) (f (μ(t))) = v(f ) = dt t d t t0 0 d = ∇(f ◦ ϕ −1 )(ϕ(m)), (ϕ ◦ μ)(t0 ) , dt whence v satisfies the condition in Remark 2.1.8 with λ=

d (ϕ ◦ μ)(t0 ). dt

Vice versa, suppose v ∈ Mm . Fix any coordinate map (U, ϕ) with m ∈ U . Then, again by Remark 2.1.8, there exists λ ∈ RN such that ( ) v(f ) = ∇(f ◦ ϕ −1 )(ϕ(m)), λ for every f , real valued and C ∞ near m. Consider the curve on M given by (see (II) in Remark 2.1.3) μ(t) := ϕ −1 (ϕ(m) + t λ), |t| < ε, with ε > 0 small enough. Obviously, μ passes through m at t = 0. Moreover,

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d d (f ◦ ϕ −1 ) ◦ (ϕ ◦ μ) (t) (f (μ(t))) = dt 0 dt 0 d = ∇(f ◦ ϕ −1 )(ϕ(m)), ϕ(m) + t λ (0) dt 0 −1 = ∇(f ◦ ϕ )(ϕ(m)), λ . Comparing to ( ), we have proved our assertion.

Definition 2.1.10 (Partial derivatives on M). Let M be an N -dimensional C ∞ differentiable manifold. Let (U, ϕ) be a coordinate system with coordinate functions x1 , . . . , xN (xi := πi ◦ ϕ), and let m ∈ U . For every i ∈ {1, . . . , N } we define a tangent vector, denoted ∂ ∈ Mm , ∂x i m

by setting

∂ ∂ (f ) := (f ◦ ϕ −1 )(ξ ) ∂ x i m ∂ ξi ϕ(m)

(2.3)

for every C ∞ function f defined in a neighborhood of m. We remark that f → ((∂/∂ xi )|m )(f ) actually defines an element of Mm (see Remark 2.1.8 with λ = the i-th element of the canonical basis of RN ). Note that the definition of (∂/∂ xi )|m is not coordinate-free: despite the notation, forgetful of ϕ, it depends on the coordinate map ϕ, as xi itself. Example 2.1.11. With the notation of the above definition, ∂ (xj ) = δi,j (of Kronecker). ∂ xi m Indeed, xj ◦ ϕ −1 = πj , so that (2.3) gives ∂ ∂ (x ) = (ξj ) = δi,j . j ∂ x i m ∂ ξi ϕ(m) Remark 2.1.12. (i). If v ∈ Mm , then (by collecting together Remark 2.1.8 and Example 2.1.11) we have the following suggesting formula: v=

N

i=1

∂ v(xi ) · , ∂ x i m

xi := πi ◦ ϕ,

and then (∂/∂ x1 )|m , . . . , (∂/∂ xN )|m is a basis for Mm . (ii). In particular, two tangent vectors v, w ∈ Mm coincide if and only if

(2.4)

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v(πi ◦ ϕ) = w(πi ◦ ϕ)

95

∀ i = 1, . . . , N,

for every (or, equivalently, for at least one) coordinate system (U, ϕ) such that m ∈ U. (iii). The above formula (2.4) allows to represent a tangent vector in a very explicit way, once a coordinate system has been fixed. Indeed, if ϕ : U −→ RN is a coordinate map, ϕi is the i-th component of ϕ, (ξ1 , . . . , ξN ) are the coordinates on RN , m ∈ M, f is a C ∞ function defined in U and v ∈ Mm , then we have v(f ) =

N

i=1

v(ϕi ) ·

∂(f ◦ ϕ −1 ) (ϕ(m)). ∂ ξi

(2.5)

2.1.3 Differentials Definition 2.1.13 (Differential at a point). Let ψ : M −→ M be a C ∞ map between two differentiable manifolds, and let m ∈ M. The differential of ψ at m is the linear map dm ψ : Mm −→ Mψ(m) defined as follows: if v ∈ Mm , dm ψ(v) is the tangent vector in Mψ(m) acting in the following way: if f is a C ∞ function in a neighborhood of ψ(m), we set dm ψ(v) (f ) := v(f ◦ ψ). (2.6)

Even if many authors use to write dψ instead of dm ψ, we shall keep the notation well distinguished, preserving “dψ” for a suitable further notion. Remark 2.1.14. We verify that (2.6) actually defines a tangent vector at ψ(m). Let f be a C ∞ function in a neighborhood of ψ(m), and let (U, ϕ), (U , ϕ ) be coordinate systems in M and M , respectively, with m ∈ U , ψ(m) ∈ U . By Remark 2.1.8, there exists λ ∈ RN such that v(g) = ∇(g ◦ ϕ −1 )(ϕ(m)), λ for every g, C ∞ near m. Hence, we have (we denote λ as a column vector) dm ψ(v) (f ) = v(f ◦ ψ) = v f ◦ (ϕ )−1 ◦ ϕ ◦ ψ ◦ ϕ −1 ◦ ϕ = ∇ f ◦ (ϕ )−1 ◦ ϕ ◦ ψ ◦ ϕ −1 (ϕ(m)), λ = ∇ f ◦ (ϕ )−1 ϕ (ψ(m)) · Jϕ ◦ψ◦ϕ −1 (ϕ(m)) · λ = ∇ f ◦ (ϕ )−1 ϕ (ψ(m)) , Jϕ ◦ψ◦ϕ −1 (ϕ(m)) · λ . This shows that dm ψ(v) (f ) = ∇ f ◦ (ϕ )−1 ϕ (ψ(m)) , λ ,

(2.7a)

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where

λ = Jϕ ◦ψ◦ϕ −1 (ϕ(m)) · λ.

(2.7b)

Again thanks to Remark 2.1.8, this proves that dm ψ(v) is a tangent vector at ψ(m).

Remark 2.1.15. We remark that, by (2.4), (2.7a) and (2.7b), if (U, x1 , . . . , xN ) and (U , y1 , . . . , yN ) are coordinate systems at m ∈ M and ψ(m) ∈ M , respectively, we have N ∂ ∂ ∂ = (y ◦ ψ) · . (2.8) dm ψ j ∂ x i m ∂ x i m ∂ yj ψ(m) j =1

Indeed, (2.8) follows from (2.7b) by taking f = πj ◦ ϕ , λ = the i-th element of the standard basis of RN and by recalling that yj ◦ ψ = πj ◦ ϕ ◦ ψ and ∂ (yj ◦ ψ) = ∂i (πj ◦ ϕ ◦ ψ ◦ ϕ −1 ) (ϕ(m)) = Jϕ ◦ψ◦ϕ −1 (ϕ(m)) . j,i ∂ xi m Remark 2.1.16 (Transformation of tangent vectors via a differential). In general, a vector field v at m transforms under the differential of ψ as follows: N

∂ dm ψ v(πi ◦ ϕ) · ∂ x i m i=1 N N

∂ −1 = v(πi ◦ ϕ) · ∂i πj ◦ ϕ ◦ ψ ◦ ϕ ∂ yj ψ(m) j =1

=

N

j =1

where

i=1

Jϕ ◦ψ◦ϕ −1 (ϕ(m)) · v(ϕ)

j

∂ , ∂ yj ψ(m)

(2.9)

⎛

⎞ v(π1 ◦ ϕ) ⎜ ⎟ .. v(ϕ) = ⎝ ⎠ . v(πN ◦ ϕ)

and (U, ϕ), (U , ϕ ) are, respectively, coordinate systems at m ∈ M, ψ(m) ∈ M (M, M are differentiable manifolds of dimensions N, N , respectively). Remark 2.1.17 (Differential of the composition and of the inverse). Let ψ : M −→ M , φ : M −→ M be C ∞ maps between differentiable manifolds M, M , M . Then it is easily seen that φ ◦ ψ is C ∞ , and we have dm (φ ◦ ψ) = dψ(m) φ ◦ dm ψ

∀ m ∈ M.

(2.10)

Furthermore, let ψ : M −→ M be a C ∞ map between two differentiable manifolds. Suppose ψ −1 : M −→ M is a C ∞ map too. In this case, we say that ψ is a diffeomorphism.

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It is immediate to observe that if ψ is a diffeomorphism and m ∈ M, then dm ψ is also invertible (for every m ∈ M) and dψ(m) (ψ −1 ) : Mψ(m) −→ Mm

is the inverse function of

. dm ψ : Mm −→ Mψ(m)

Definition 2.1.18 (dψ as a map on the tangent bundle). Let ψ : M → M be a C ∞ map between two differentiable manifolds M, M . We set dψ : T (M) → T (M ),

dψ(m, v) := (ψ(m), dm ψ(v)).

(2.11)

Note that, whereas dm ψ is a map from Mm to Mψ(m) (for any fixed m ∈ M), dψ is a map from T (M) to T (M ).

2.1.4 Vector Fields The following definition is one of the most important in differential geometry. Definition 2.1.19 (Vector field). Let Ω ⊆ M be an open subset of a differentiable manifold M. A vector field X on Ω is an application X : Ω −→ T (M) such that, X(m) = (m, v(m)) ∈ T (M)

∀ m ∈ Ω.

Equivalently, we have X(m) = (m, v(m)),

where v(m) ∈ Mm for every m ∈ Ω.

In order to avoid any confusion, we make explicit the following conventional identification of a vector field with its projection onto the second argument. This identification is frequently tacitly employed in literature. Convention–Notation. If T (M) is the tangent bundle of a differentiable manifold M, and, for every m ∈ M, v ∈ Mm , we set π(m, v) := v, then the following map is well posed on T (M): π : T (M) → Mm , (m, v) → v. m∈M

In the sequel, if X is a vector field on an open set Ω ⊆ M, we shall use the notation X(m) for the map X : Ω → T (M), m → X(m), whereas Xm will denote the map

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Ω→

Mm ,

m → Xm := (π ◦ X)(m).

m∈M

So, the above positions can be summarized as X(m) = (m, Xm )

for every m ∈ M.

(2.12)

Finally, if f is a C ∞ function on Ω and X is a vector field on Ω, we shall denote (with an abuse of notation) by X(f ) or shortly Xf the function on Ω whose value at m is Xm (f ), i.e. (2.13) Xf : Ω → R, (Xf )(m) := Xm (f ). Definition 2.1.20 (Smooth vector field). Let X be a vector field defined on a manifold M. We say that X is C ∞ (or smooth) if, for every open set Ω ⊆ M and every smooth real-valued function f on Ω, the function Xf as defined in (2.13) is smooth on Ω. Remark 2.1.21. It is straightforward to verify that X is a smooth vector field on M if and only if, for every coordinate system (U, x1 , . . . , xN ), the functions a1 , . . . , aN defined on U by N

∂ Xm = ai (m) · ∂ x i m i=1

C∞

(see (2.12) and (2.4)), are functions on U . Following the above notation, we have ai (m) = Xm (xi ),

where xi = πi ◦ ϕ.

Recalling (2.5), we see that, if (U, ϕ) is a coordinate system, a smooth vector field acts on a function f ∈ C ∞ (U ) in the following way: Xm (f ) =

N

ai (m) · ∂i (f ◦ ϕ −1 ) (ϕ(m))

i=1

=

N

Xm (πi ◦ ϕ) · ∂i (f ◦ ϕ −1 ) (ϕ(m)),

(2.14)

i=1

where a1 , . . . , aN ∈ C ∞ (U ) are fixed functions depending on X and on the coordinate map. This shows that a vector field X on M is smooth iff, for every coordinate system (U, ϕ), the functions m → Xm (π1 ◦ ϕ), . . . , Xm (πN ◦ ϕ) are C ∞ on U .

Remark 2.1.22 (Smooth vector fields as operators on C ∞ (M, R)). Let X be a smooth vector field on a differentiable manifold M. Besides a map from M to T (M), it is possible to identify X with the map

2.1 Abstract Lie Groups

X : C ∞ (M, R) → C ∞ (M, R),

99

f → Xf,

where (see (2.13)) Xf : M → R,

m → (Xf )(m) = Xm f.

We denote by X (M) the set of the smooth vector fields considered as linear operators (i.e. endomorphisms) on C ∞ (M, R). Note that X (M) is a vector space over R. The correspondence between X as a vector field on M and as an endomorphism on C ∞ (M, R) is faithful in the sense that, if X, Y are vector fields such that Xf ≡ Yf for every f ∈ C ∞ (M, R), then X = Y . (Indeed, take any m ∈ M, any coordinate system (U, ϕ) around m, any i ∈ {1, . . . , N } and choose f = πi ◦ ϕ. Then apply Remark 2.1.12-(ii).) Remark 2.1.23. Definition 2.1.13 defines dm ψ as a map from Mm to Mψ(m) . Moreover, in Definition 2.1.18, we introduced a natural map denoted by dψ between the tangent bundles T (M) and T (M ). It may be thought that a third natural map can be defined between X (M) and X (M ) by mapping X ∈ X (M) into the vector field Y such that Yψ(m) = dm ψ(Xm ). Unfortunately, in general, this defines a “vector field” only on the points of ψ(M) and not on the whole M . However, if ψ is a diffeomorphism, this can be done as we describe below.

Definition 2.1.24 (dψ as a map on X (M)). Suppose ψ : M → M is a C ∞ diffeomorphism of differentiable manifolds M, M . We set dψ : X (M) −→ X (M ),

X → dψ(X),

where, for every f ∈ C ∞ (M , R), {dψ(X)}m (f ) = dψ −1 (m ) ψ(Xψ −1 (m ) )(f )

∀ m ∈ M .

(2.15a)

Since ψ is onto, (2.15a) is equivalent to set {dψ(X)}ψ(m) = dm ψ(Xm ) for every m ∈ M.

(2.15b)

We say that X ∈ X (M) and the above defined dψ(X) ∈ X (M ) are ψ-related (see also Definition 2.1.34). We leave to the reader the verification that dψ(X) is indeed a smooth vector field on M according to the definition of X (M ) in Remark 2.1.22. The following example shows that the above mapping X → dψ(X) appears naturally when a “change of variable” occurs. Example 2.1.25 (Related vector fields via a diffeomorphism). We consider a simple but significant example. Let T : RN → RN be a C ∞ -diffeomorphism (i.e. an invertible map, smooth together with its inverse function) from RN onto itself. We consider on the domain of T a fixed system of Cartesian coordinates (x1 , . . . , xN ), and we equip the image of T (which still coincides with RN = T (RN )) with the new system of coordinates y defined by y = T (x). We use the following notation

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T : (RN , x) → (RN , y). on To every smooth vector field X on (RN , x), there corresponds a vector field X (RN , y) in a natural way, roughly speaking, by expressing X in the new coordinates y. Namely, given any f = f (y), f ∈ C ∞ ((RN , y), R), we set y (f ) := XT −1 (y) (f ◦ T ), X also written as or equivalently,

(y) := X(f ◦ T )(T −1 (y)), Xf )(T (x)) = X(f ◦ T )(x). (Xf

(2.16)

or that X and X are T -related. With reference to (2.16), we say that T turns X into X, is the representation1 of X in the new system of coordinates Roughly speaking, X defined by y = T (x). 1 A simple example is in order. Let R2 be equipped with coordinates x = (x , x ) and 1 2

consider the linear change of coordinates given by y = (y1 , y2 ) = T (x1 , x2 ) := (2x1 − x2 , −5x1 + 3x2 ). Following the above definition, the ordinary partial derivatives X1 := ∂x1 and X2 := ∂x2 1 and X 2 , respectively, where are turned into the operators X 1 f (y) = ∂ f (2x1 − x2 , −5x1 + 3x2 ) X ∂x1 −1 x=T

(y)

= 2∂y1 f (y1 , y2 ) − 5 ∂y2 f (y1 , y2 ), ∂ f (2x1 − x2 , −5x1 + 3x2 ) X2 f (y) = ∂x2 −1 x=T

(y)

= −∂y1 f (y1 , y2 ) + 3 ∂y2 f (y1 , y2 ). Since the Jacobian matrix of T equals JT (x1 , x2 ) =

2 −1 −5 3

,

2 are the vector fields whose component functions at y are respectively given 1 and X then X by 2 −1 1 · , JT (T −1 (y)) · X1 I (T −1 (y)) = −5 3 0 2 −1 0 · . JT (T −1 (y)) · X2 I (T −1 (y)) = −5 3 1 Hence ∂x1 is turned by T into 2 ∂y1 − 5 ∂y2 and ∂x2 is turned by T into − ∂y1 + 3 ∂y2 . Consequently, the ordinary Laplace operator Δ = (∂x1 )2 + (∂x2 )2 is turned by T into the following second order constant coefficient differential operator of elliptic type (which is a sub-Laplacian on (R2 , +)!)

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If we write, as usual, X = ∇ · XI , the chain rule then gives (y) = ∇(f ◦ T )(T −1 (y)) · XI (T −1 (y)) Xf = (∇y f )(y) · JT (T −1 (y)) · XI (T −1 (y)). In other words, for every y ∈ (RN , y), one has )(y) y = ∇y · (XI X

)(y) = JT (T −1 (y)) · XI (T −1 (y)), with (XI

(2.17a)

or equivalently, for every x ∈ (RN , x), )(T (x)) T (x) = ∇T (x) · (XI X

)(T (x)) = JT (x) · XI (x). with (XI

(2.17b)

Keeping in mind (2.9) of Remark 2.1.16, (2.17b) shows that T (x) = dx T (Xx ), X

(2.17c)

which gives a significant interpretation of the differential map (at least in the present case when T is a diffeomorphism): Given a vector field X, dx T (Xx ) is the tangent vector at T (x) which represents Xx with respect to the change of variable y = T (x).

In what follows, we introduce an important definition. The adjectives “regular” and “smooth” will always mean “of class C ∞ ”. Definition 2.1.26 (Tangent vector to a curve). Let μ : [a, b] → M be a regular curve. The tangent vector to the curve μ at time t is defined by d ∈ Mμ(t) . μ(t) ˙ := dt μ (2.18) d r r=t Hence, fixed t ∈ [a, b], if f is C ∞ near μ(t), we have d (f (μ(r))). μ(t)(f ˙ )= d r r=t Remark 2.1.27. Note that, as Remark 2.1.9 shows, any tangent vector at a point of M can be represented as the tangent vector to a certain curve at a suitable time. Definition 2.1.28 (Integral curve). Let X be a smooth vector field on the differentiable manifold M. A regular curve μ : [a, b] −→ M is called an integral curve of X if μ(t) ˙ = Xμ(t) for every t ∈ [a, b]. (2.19) = 5(∂y1 )2 + 34(∂y2 )2 − 26 ∂y1 ∂y2 . Δ is simply the ordinary Laplace operator expressed in a new system of In other words, Δ coordinates y in R2 . In Chapter 16, we show that every second order constant coefficient differential operator of elliptic type in RN is simply the ordinary Laplace operator expressed in a suitable new system of coordinates via a linear change of basis.

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More explicitly, (2.19) means that d (f (μ(r))) = X(f )(μ(t)) d r r=t for every smooth function f on M and every t ∈ [a, b]. We remark that if Xm = N i=1 ai (m)(∂/∂ xi )|m and (U, ϕ) is a coordinate system such that μ(t) ∈ U , then (2.19) is also equivalent to d

ϕi (μ(r)) = ai (μ(t)), d r r=t which becomes (setting γ (t) := ϕ(μ(t)) ∈ RN ) γ˙i (t) = (ai ◦ ϕ −1 )(γ (t)),

i = 1, . . . , N,

(2.20)

which is an ODE on RN . Equivalently, making more explicit the rôle of X, μ : [a, b] → M is an integral curve of X if and only if, whenever (U, ϕ) is a coordinate system and (t0 , t1 ) ⊂ [a, b] is such that μ(t) ∈ U for every t ∈ (t0 , t1 ), it holds μ(t) = ϕ −1 (γ (t)), where γ˙ (t) = Xϕ −1 (γ (t)) (ϕ)

∀ t ∈ (t0 , t1 )

and Xm (ϕ) = (Xm (π1 ◦ ϕ), . . . , Xm (πN ◦ ϕ)). Remark 2.1.29. By the above observation on the coordinate form of Definition 2.1.28 and by recalling the existence “in small” for smooth systems of ODE’s, we infer that, given any smooth vector field X on M and fixed any m ∈ M, there exists one and only one integral curve of X passing through m at time 0. Hence the following definition is well-posed. Definition 2.1.30 (Complete vector field). Let X be a smooth vector field on the differentiable manifold M. We say that X is complete if, for every m ∈ M, the integral curve μ of X such that μ(0) = m is defined on the whole R (i.e. its maximal interval of definition is R). 2.1.5 Commutators. ϕ-relatedness In the sequel, we denote by C ∞ (M, R) or, shortly, C ∞ (M) the set of the smooth real-valued functions defined on a differentiable manifold M. It is immediate to observe that if X is a smooth vector field on M and f ∈ C ∞ (M, R), we have Xf ∈ C ∞ (M, R). We explicitly recall that, here and in the sequel, we use the notation in (2.13): Xf : M → R, (Xf )(m) = Xm (f ). As a consequence, the following definition is well posed.

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Definition 2.1.31 (Commutators). Let X and Y be smooth vector fields on a differentiable manifold M. We define a vector field on M (called the commutator of X and Y ) in the following way: [X, Y ] : M → T (M),

[X, Y ](m) := (m, [X, Y ]m ),

where [X, Y ]m (f ) := Xm (Yf ) − Ym (Xf ) for every m ∈ M and every f ∈

(2.21)

C ∞ (M, R).

Definition 2.1.31 is well posed as it follows from (i) in the proposition below. Proposition 2.1.32. If X, Y and Z are smooth vector fields on M, we have: (i) [X, Y ] is a smooth vector field on M; (ii) [X, Y ]m = −[Y, X]m for every m ∈ M; (iii) [[X, Y ], Z]m + [[Y, Z], X]m + [[Z, X], Y ]m = 0 for every m ∈ M. Proof (Sketch). The verification that these facts hold true, can be performed via a computation in a coordinate map. This reduces the above identities (ii) and (iii) to identities between usual differential operators in RN . For instance, let us prove (i). Let m ∈ M be fixed and choose any coordinate map (U, ϕ) such that m ∈ M. According to Definition 2.1.20, we have Xm (f ) =

N

Xm (πi ◦ ϕ) ∂i (f ◦ ϕ −1 ) (ϕ(m)).

i=1

As a consequence, Xm (Yf ) =

N

i,j =1

=

N

∂ Yϕ −1 (ξ ) (πj ◦ ϕ) ∂ξj (f ◦ ϕ −1 )(ξ ) (ϕ(m)) Xm (πi ◦ ϕ) ∂ ξi Xm (πi ◦ ϕ) · ∂ξi Yϕ −1 (ξ ) (πj ◦ ϕ) ∂ξj (f ◦ ϕ −1 )(ξ )

i,j =1

+ Yϕ −1 (ξ ) (πj ◦ ϕ)

∂2 (f ◦ ϕ −1 )(ξ ) . ∂ ξi ξj ξ =ϕ(m)

By reversing the rôles of X and Y and then subtracting, we obtain (note, before the second equality sign, the cancellation of the second order terms) Xm (Yf ) − Ym (Xf ) =

N

i,j =1

Xm (πi ◦ ϕ) ∂ξi Yϕ −1 (ξ ) (πj ◦ ϕ) ∂ξj (f ◦ ϕ −1 )(ξ ) ξ =ϕ(m)

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−

N

i,j =1

+

N

i,j =1

−

N

i,j =1

=

N

j =1

Ym (πi ◦ ϕ) ∂ξi Xϕ −1 (ξ ) (πj ◦ ϕ) ∂ξj (f ◦ ϕ −1 )(ξ ) ξ =ϕ(m) 2 ∂ −1 Xm (πi ◦ ϕ) Ym (πj ◦ ϕ) (f ◦ ϕ ) (ϕ(m)) ∂ ξi ξj 2 ∂ Ym (πi ◦ ϕ) Xm (πj ◦ ϕ) (f ◦ ϕ −1 ) (ϕ(m)) ∂ ξi ξj

N

Xm (πi ◦ ϕ) ∂ξi Yϕ −1 (ξ ) (πj ◦ ϕ) (ϕ(m))

i=1

− Ym (πi ◦ ϕ) ∂ξi Xϕ −1 (ξ ) (πj ◦ ϕ) (ϕ(m)) ∂ξj (f ◦ ϕ −1 )(ϕ(m)). Thus, [X, Y ]m is actually a tangent vector at m (see, for instance, Remark 2.1.8), and, following the notation in Definition 2.1.20, we have proved that, in a coordinate system (U, ϕ) around m, [X, Y ]m (f ) =

N

cj (m) ∂j (f ◦ ϕ −1 )(ϕ(m))

with

j =1

cj (m) = [X, Y ]m (πj ◦ ϕ) = Xm Y (πj ◦ ϕ) − Ym X(πj ◦ ϕ)

∀ j ≤ N. (2.22)

We see that the maps cj ’s are smooth on U , so that, by Remark 2.1.21, [X, Y ] is a smooth vector field on M. Now, (iii) in the assertion can be proved with an analogous coordinate-computation, whereas (ii) is trivial.

Remark 2.1.33 (Commutators in X (M)). Consider the alternative definition of smooth vector field as an element of X (M), see Remark 2.1.33. The commutator operation rewrites as an operation on X (M) in the following way: Given X, Y ∈ X (M), we consider the operator on C ∞ (M, R) defined by [X, Y ] : C ∞ (M, R) → C ∞ (M, R),

f → [X, Y ]f,

where ([X, Y ]f )(m) := [X, Y ]m f = Xm (Yf ) − Ym (Xf ). Then, obviously, [X, Y ] ∈ X (M) is the operator on C ∞ (M, R) related to the (usual) vector field [X, Y ]. With this meaning of the commutation, Proposition 2.1.32 rewrites as: If X, Y and Z belong to X (M), we have: (i) [X, Y ] ∈ X (M); (ii) [X, Y ] = −[Y, X]; (iii) [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0.

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105

be a C ∞ function between two Definition 2.1.34 (ϕ-relatedness). Let ϕ : M → M on M are called ϕ-related differentiable manifolds. The vector fields X on M and X if we have ◦ ϕ. dϕ ◦ X = X (2.23a) (see Definition 2.1.18) and, Here, dϕ is intended as a map from T (M) to T (M) as usual, the vector fields are maps from the underlying differentiable manifolds to the relevant tangent bundles. Hence, (2.23a) is intended as an equality of functions Hence, ϕ-relatedness is equivalent to saying that the following from M to T (M). diagram is commutative ϕ

M

M X

X dϕ

T (M)

T (M).

Since, for every m ∈ M, we have (dϕ ◦ X)(m) = dϕ(m, Xm ) = (ϕ(m), dm ϕ(Xm )), whereas

◦ ϕ)(m) = X(ϕ(m)) ϕ(m) ), (X = (ϕ(m), X

then (2.23a) can be rewritten as the collection of identities ϕ(m) ∀ m ∈ M dm ϕ(Xm ) = X between tangent vectors at ϕ(m), i.e. between two functionals on Finally, condition (2.23a) is equivalent to the following ones: ϕ(m) (f ), Xm (f ◦ ϕ) = X

(2.23b) R). C ∞ (M,

)◦ϕ X(f ◦ ϕ) = (Xf

(2.23c) (2.23d)

for every m ∈ M and for every f smooth in a neighborhood of ϕ(m) in M. Remark 2.1.35. When ϕ is a diffeomorphism, for every X ∈ X (M), there always ∈ X (M) which is ϕ-related to X (see Remark 2.1.23), namely, exists a (unique) X = dϕ(X). with the notation of Definition 2.1.24, X With reference to (2.17c), we have an interpretation of ϕ-relatedness when ϕ : are ϕ-related if and only if X is the RN → RN is a diffeomorphism: X and X expression of X in the new coordinates defined by the new Cartesian coordinates y = ϕ(x).

Remark 2.1.36 (ϕ-relatedness in X (M)). Consider the alternative definition of vector field as in Remark 2.1.22. Then, the notion of ϕ-relatedness of vector fields rewrites R), ∈ X (M) are ϕ-related if, for every f ∈ C ∞ (M, as follows: X ∈ X (M) and X the functions X(f ◦ ϕ), (Xf ) ◦ ϕ on M do coincide, i.e. ϕ(m) f for every m ∈ M, Xm (f ◦ ϕ) = X or, equivalently, (2.23b) holds. The following simple result will be soon of a crucial importance.

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be a C ∞ Proposition 2.1.37 (ϕ-relatedness and commutators). Let ϕ : M → M function between two differentiable manifolds. Let X, Y be smooth vector fields on Y be smooth vector fields on M. If X is ϕ-related to X and Y is ϕ-related M and X, to Y , then [X, Y ] is ϕ-related to [X, Y ]. Proof. We have to prove Y ] ◦ ϕ, dϕ ◦ [X, Y ] = [X, and dϕ◦Y = Y ◦ϕ. To this end, let m ∈ M and f ∈ C ∞ (M). provided dϕ◦X = X◦ϕ By the equivalent restatement (2.23c) of ϕ-relatedness, we have to show that Y ]ϕ(m) (f ). ( ) dm ϕ [X, Y ]m (f ) = [X, By definition of dm ϕ, we have dm ϕ [X, Y ]m (f ) = [X, Y ]m (f ◦ ϕ) (see (2.21)) = Xm Y (f ◦ ϕ) − Ym X(f ◦ ϕ) f ) ◦ ϕ − Ym (Xf )◦ϕ (see (2.23d)) = Xm (Y (f ) − dm ϕ(Ym ) X(f ) = dm ϕ(Xm ) Y ϕ(m) Y (f ) − Y ϕ(m) X(f ) (see (2.23b)) = X Y ]ϕ(m) (f ). (see (2.21)) = [X, This gives ( ) thus ending the proof.

2.1.6 Abstract Lie Groups Definition 2.1.38 (Lie group). A Lie group G is a differentiable manifold G along with a group law ∗ : G × G −→ G such that the applications G × G (x, y) → x ∗ y ∈ G,

G x → x −1 ∈ G

are smooth.2 In the following, we shall always denote by e the identity of (G, ∗). Moreover, fixed σ ∈ G, we denote by τσ the left translation on G by σ , i.e. the map G x → τσ (x) := σ ∗ x ∈ G. In case, when more than only one composition law is involved, we may also write τσ∗ instead of τσ . 2 The notion of “smoothness” of a map from G × G should be properly defined. The prod-

uct of two differentiable manifolds is indeed endowed with a differentiable structure in a natural way. Here, we just remark that a function f : G × G → R is smooth if, for every couple of coordinate systems (U, ϕ) and (V , φ) on G, the function RN × RN ⊇ ϕ(U ) × φ(V ) (u, v) → f ϕ −1 (u) ∗ φ −1 (v) ∈ R is smooth. For more details see, e.g. [War83].

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107

Definition 2.1.39 (Lie algebra). A (real) Lie algebra is a real vector space g with a bilinear operation [·, ·] : g × g −→ g (called (Lie) bracket) such that, for every X, Y , Z ∈ g, we have: 1. (anti-commutativity) [X, Y ] = −[Y, X]; 2. (Jacobi identity) [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0. A very remarkable fact is that, given any Lie group, there exists a certain finitedimensional Lie algebra such that the group properties are reflected into properties of the algebra. For instance (as we shall see later on), any connected and simply connected Lie group is completely determined (up to isomorphism) by its Lie algebra. Therefore, the study of a Lie group is often reduced to the study of its Lie algebra. Remark 2.1.40. Straightforwardly adapting the proof of Proposition 1.1.7 in Section 1.1 (see page 12), we have: if X1 , . . . , Xm are elements of an (abstract) Lie algebra, then a system of generators of Lie{X1 , . . . , Xm } is given by the commutators XI := [Xi1 , [Xi2 , [Xi3 , . . . [Xik−1 , Xik ] . . .]]], where {i1 , i2 , . . . , ik } ⊆ {1, . . . , m} and I = (i1 , i2 , . . . , ik ), k ∈ N. Indeed, the proof of Proposition 1.1.7 is only based on anti-commutativity and the Jacobi identity. 2.1.7 Left Invariant Vector Fields and the Lie Algebra Definition 2.1.41 (Left invariant vector fields). Let G be a Lie group. A smooth vector field X on G is called left invariant if, for every σ ∈ G, X is τσ -related to itself, i.e. (2.24) dτσ ◦ X = X ◦ τσ . Here dτσ is intended as a map from T (G) to itself (as in Definition 2.1.18). As shown by the remarks after Definition 2.1.34, condition (2.24) is equivalent to the following one: ∀ x, σ ∈ G. (2.25a) (dx τσ )(Xx ) = Xσ ∗x Applying (2.25a) at the identity e, it follows immediately that if X is a left invariant vector field, we have ∀ σ ∈ G, (2.25b) de τσ (Xe ) = Xσ which proves that a left invariant vector field is determined by its action at the origin. Equality (2.25b) is actually equivalent3 to (2.25a). Moreover, (2.25a) can also be written as 3 Indeed, by applying (2.25b) with σ replaced by σ ∗ x, subsequently (2.10) and finally using

again (2.25b) with σ replaced by x, we have Xσ ∗x = de τσ ∗x (Xe ) = dx τσ de τx (Xe ) = dx τσ (Xx ).

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2 Abstract Lie Groups and Carnot Groups

Xx (f ◦ τσ ) = Xσ ∗x (f ) for every x, σ ∈ G and every f ∈ C ∞ (G, R), or again as (the most commonly used) Xx y → f (σ ∗ y) = (Xf )(σ ∗ x),

(2.25c)

(2.25d)

i.e. comparing it to (1.18) (page 14), the very analogue of the usual left-invariance. Before giving the following central Definition 2.1.42, we pause a moment in order to recall the multiple ways a smooth vector field can be thought of. A smooth vector field on G is a map X : G → T (G) such that, for every x ∈ G, it holds X(x) = (x, Xx ), where Xx ∈ Gx for every x ∈ G and such that, for every f ∈ C ∞ (G, R), the function x → Xx (f ) is smooth on G. A smooth vector field can be identified to the operator X : C ∞ (G, R) −→ C ∞ (G, R), f → Xf : G → R, x → Xx f. The set of the vector fields, as the above described operators, is denoted by X (G). Obviously, the set of the left invariant operators on G gives rise to a relevant subset in X (G), following the above identification. We are ready to give the following central definition. Definition 2.1.42 (Algebra of a Lie group). Let G be a Lie group. Then the subset of X (G) of the smooth left invariant vector fields on G is called the (Lie) algebra of G. It will be denoted by g. More precisely, following Remark 2.1.22, we henceforth identify a left invariant vector field X on G with the following operator X : C ∞ (G, R) → C ∞ (G, R) such that, for every f ∈ C ∞ (G, R), the function Xf on G is defined by (Xf )(x) := Xx f

∀ x ∈ G.

Hence, g is a (linear) set of endomorphisms on C ∞ (G, R), g ⊆ X (G). Note that, from the left invariance of X ∈ g, we have (Xf )(x) = X(f ◦ τx )(e)

∀ x ∈ G ∀ f ∈ C ∞ (G, R).

Along with the above definition of the algebra of a Lie group, there is a wide commonly used identification of g with Ge described in the following theorem. Theorem 2.1.43 (The Lie algebra of a Lie group). Let G be a Lie group and g be its algebra. Then we have:

2.1 Abstract Lie Groups

109

(i) g is a vector space, and the map α : g −→ Ge , X → α(X) := Xe is an isomorphism between g and the tangent space Ge to G at the identity e of G. As a consequence, dim g = dim Ge = dim G; (ii) The commutator of smooth left invariant vector fields (see also Remark 2.1.33) is a smooth left invariant vector field; (iii) g with the commutation operation is a Lie algebra. Proof. (i). It is evident that g (thought of as a subset of X (G)) is a vector space and that α is linear. Let us prove that α is injective. If α(X) = α(Y ), we have (see (2.25b)) Xσ = dτσ (Xe ) = dτσ (α(X)) = dτσ (α(Y )) = dτσ (Ye ) = Yσ , i.e. X = Y . Let us prove that α is surjective. If v ∈ Ge , we set Xσ := (de τσ )(v)

for every σ ∈ G.

Here, de τσ is the differential of the map τσ at the identity e of G. By definition, we have Xσ ∈ Gσ , i.e. σ → (σ, Xσ ) is a vector field on G. Moreover, it is not difficult4 to prove that X is smooth. On the other hand, X is left invariant since Xσ ∗x = de τσ ∗x (v) = (dx τσ ◦ de τx )(v) = dx τσ (Xx ), which gives (2.25a). Finally, we have 4 Indeed, let us prove that, for every f ∈ C ∞ (G, R), G σ → X f is smooth. Fix σ

σ0 ∈ G, and let (U, ϕ) be a coordinate system around σ0 . We have to prove that the map ϕ(U ) u → (Xf )(ϕ −1 (u)) is smooth. By definition, it holds (Xf )(ϕ −1 (u)) = Xϕ −1 (u) (f ) = (de τϕ −1 (u) )(v)(f ) = v(f ◦ τϕ −1 (u) ) = ().

By Remark 2.1.8, fixed a coordinate system (E, χ ) around e, there exists λ = λ(χ , e, v) ∈ RN such that ! " ! " () = λ, ∇ f ◦ τϕ −1 (u) ◦ χ −1 (χ (e)) = λ, ∇v f (ϕ −1 (u) ∗ χ −1 (v)) (χ (e)) . By the very definition of Lie group, ϕ(U ) × χ (E) (u, v) → f (ϕ −1 (u) ∗ χ −1 (v)) is smooth, so that ϕ(U ) u → ∇v f (ϕ −1 (u) ∗ χ −1 (v)) (χ (e)) is smooth, and this ends the proof that X is smooth.

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2 Abstract Lie Groups and Carnot Groups

α(X) = Xe = de τe (v) = v, i.e. α is surjective. Since α is an isomorphism of vector spaces, we have dim g = dim Ge = dim G (see Proposition 2.1.7). (ii). Let σ ∈ G be arbitrary. Let X and Y ∈ g, i.e. X is τσ -related to itself and Y is τσ -related to itself. From Proposition 2.1.37 it follows that [X, Y ] is τσ -related to itself, i.e. [X, Y ] ∈ g. (iii). The last statement of the theorem follows from Proposition 2.1.32 (see also Remark 2.1.33).

Incidentally, in the above proof, we have explicitly written the inverse map of the natural identification α : g −→ Ge , X → α(X) := Xe . Indeed, the inverse of α is given by α −1 : Ge −→ g, v → X, where Xσ = (de τσ )(v) for every σ ∈ G.

(2.26)

Example 2.1.44 (The Lie algebra of (R, +)). It is obvious that the Lie algebra r of the usual Euclidean Lie group (R, +) is d span , dr where

d : C ∞ (R, R) → C ∞ (R, R), f → f . dr With the usual formalism Xx for vector fields, this rewrites as d f = f (t) for every t ∈ R. d r t

Remark 2.1.45 (Left invariance in coordinates). We now make explicit the left invariance condition in terms of the coefficients ai of the coordinate form of a smooth vector field N

∂ ai (σ ) · . Xσ = ∂x i=1

i σ

We know that a smooth vector field X is left invariant on G if and only if (see (2.25b)) ( )

de τσ (Xe ) = Xσ

∀ σ ∈ G.

In turn (see Remark 2.1.21), fixed a coordinate system (E, χ) around the identity e, ( ) is equivalent to state that, for every σ ∈ G and for every (or, equivalently, for at least one) coordinate system (U, ϕ) around σ , it holds

2.1 Abstract Lie Groups N

111

∂ ∂ (f ◦ ϕ −1 )(ϕ(σ )) = Xe (χi ) · (f ◦ τσ ◦ χ −1 )(χ(e)) ∂ ui ∂ vi N

Xσ (ϕi ) ·

i=1

i=1

for every smooth function f around σ (or, equivalently, for the N smooth functions ϕ1 , . . . , ϕN ). Writing f = f ◦ ϕ −1 ◦ ϕ, on the right-hand side the above becomes N

Xσ (ϕi ) ·

i=1

=

N

j =1

=

N

j =1

=

N

Xe (χi )

i=1 N

i=1

N

j =1

∂ (f ◦ ϕ −1 )(ϕ(σ )) ∂ ui ∂ ∂ (ϕj ◦ τσ ◦ χ −1 )(χ(e)) · (f ◦ ϕ −1 )(ϕ(σ )) ∂ vi ∂ uj

∂ Jϕ◦τσ ◦χ −1 (χ(e)) Xe (χi ) · (f ◦ ϕ −1 )(ϕ(σ )) j,i ∂ uj

∂ Jϕ◦τσ ◦χ −1 (χ(e)) · Xe (χ) · (f ◦ ϕ −1 )(ϕ(σ )). j ∂ uj

Here, Xe (χ) = (Xe (χ1 ), . . . , Xe (χN ))T . Comparing the far left-hand and right-hand sides, we rewrite all as the matrix identity Jϕ◦τσ ◦χ −1 (χ(e)) · Xe (χ) = Xσ (ϕ).

(2.27)

Here, Xe (ϕ) = (Xe (ϕ1 ), . . . , Xe (ϕN ))T . As a consequence, we have proved that a smooth vector field X on the Lie group G is left invariant if and only if, fixed a coordinate system (E, χ) around the identity e, for every σ ∈ G, and for every (or, equivalently, for at least one) coordinate system (U, ϕ) around σ , it holds ∂ ai (σ ) , Xσ = ∂ x i σ i=1 ⎛ ⎞ ⎞ ⎛ a1 (σ ) Xe (χ1 ) ⎜ ⎟ ⎟ ⎜ .. where ⎝ ... ⎠ = Jϕ◦τσ ◦χ −1 (χ(e)) · ⎝ ⎠. . N

aN (σ )

(2.28)

Xe (χN )

Formula (2.28) is particularly useful for those Lie groups admitting a single global coordinate system defined on the whole group (such as Carnot groups or GLN (R), to give some example). Remark 2.1.46 (Bracket in the identity). Since the Lie algebra can be identified to Ge , it is interesting to analyze what form takes the bracket as seen as an operation on Ge . Let (E, χ) be a fixed coordinate system around the identity e ∈ G. We know that (see (2.14) and (2.22))

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2 Abstract Lie Groups and Carnot Groups

Xe f =

N

λj ∂j |χ(e) (f ◦ χ −1 ),

j =1

Ye f =

N

μj ∂j |χ(e) (f ◦ χ −1 ),

j =1

[X, Y ]e f =

N

νj ∂j |χ(e) (f ◦ χ −1 ),

j =1

where λj = Xe (χj ), μj = Ye (χj ), and νj = Xe (Y χj ) − Ye (Xχj )

λi ∂ui |χ(e) Yχ −1 (u) χj − μi ∂ui |χ(e) Xχ −1 (u) χj = i

i

(after a lengthy computation using also (2.28)) N

$ ∂ # −1 ∂ = χj χ (u) ∗ χ −1 (v) . (λi μk − μi λk ) ∂u ∂v i χ(e)

i,k=1

k χ(e)

2.1.8 Homomorphisms Definition 2.1.47 (Homomorphisms). Let (G, •) and (H, ∗) be Lie groups. A map ϕ : G −→ H is a homomorphism of Lie groups if it is C ∞ and if ϕ(x • y) = ϕ(x) ∗ ϕ(y)

∀ x, y ∈ G.

A map ϕ is an isomorphism of Lie groups if it is a homomorphism of Lie groups and a diffeomorphism of differentiable manifolds. An isomorphism of G onto itself is called an automorphism of G. Let (g, [·, ·]1 ) and (h, [·, ·]2 ) be Lie algebras. A map ϕ : g −→ h is a homomorphism of Lie algebras if it is linear and if ∀ X, Y ∈ g. ϕ [X, Y ]1 = [ϕ(X), ϕ(Y )]2 A map ϕ is an isomorphism of Lie algebras if it is a bijective homomorphism of Lie algebras. An isomorphism of g onto itself is called an automorphism of g. Example 2.1.48. Suppose (G, •) is a Lie group and M is a differentiable manifold. Suppose T : G → M is a C ∞ -diffeomorphism. We can consider on M a Lie group structure naturally induced by • via T . Precisely, we equip M with the following composition law (2.29) M × M (x, y) → x ∗ y := T T −1 (x) • T −1 (y) ∈ M. It is an easy exercise to verify that ∗ defines on M a Lie group structure, T : (G, •) → (M, ∗) is a Lie-group isomorphism, the identity of (M, ∗) is T (eG ) (eG being the identity of (G, •)) and, finally, the inverse of x in M with respect to ∗ is given by

2.1 Abstract Lie Groups

x −1 = T

113

−1 . T −1 (x)

Now, let X be a •-left invariant vector field on G. We showed in Example 2.1.25 how on M simply setting (see, for instance (2.17c)) to relate to X a vector field X y = dT −1 (y) T (XT −1 (y) ) X

for every y ∈ M.

(2.30)

are T -related according to Definition 2.1.34. This obviously ensures that X and X Moreover, we claim that is ∗-left invariant. X (2.31) This is equivalent to σ ∗y (f ) y (f (σ ∗ ·)) = X X

∀ y, σ ∈ M

∀ f ∈ C ∞ (M, R).

Now, this last equality follows from the following calculation y (f (σ ∗ ·)) = X

(see (2.30)) = dT −1 (y) T (XT −1 (y) ) (f (σ ∗ ·)) = (XT −1 (y) ) f (σ ∗ T (·))

(see (2.29)) = (XT −1 (y) ) f T (T −1 (σ ) • (·))

(X is •-left inv.) = (XT −1 (σ )•T −1 (y) ) f T (·)

(see (2.29)) = (XT −1 (σ ∗y) ) f T (·) = dT −1 (σ ∗y) T (XT −1 (σ ∗y) ) (f ) σ ∗y (f ). (see (2.30)) = X Roughly speaking, this gives the following fact. Let T : RN → RN be a C ∞ -diffeomorphism defining on RN a change of variable y = T (x). Suppose the domain (RN , x) is equipped with a group law •, and define on the image (RN , y) the induced group law ∗ as in (2.29). Then, if X is a which expresses X left-invariant vector field w.r.t. • on (RN , x), the vector field X in the y-coordinates is a left-invariant vector field w.r.t. ∗ on (RN , y).

Let ϕ : G −→ H be a homomorphism of Lie groups. Since ϕ sends the identity of G to the identity of H, the differential de ϕ of ϕ is a linear map from Ge to He (for brevity we denote by e the identity element both in G and in H). Now, by means of the natural identification between Ge and the algebra g of G and between He and the algebra h of H (see Theorem 2.1.43), de ϕ thus induces a natural linear map between g and h, which we denote by dϕ. In other words, the following definition naturally arises. Definition 2.1.49 (Differential of a homomorphism). If ϕ : G −→ H is a homomorphism of Lie groups, we define dϕ : g −→ h

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2 Abstract Lie Groups and Carnot Groups

as the linear map such that, for every X ∈ g, dϕ(X) is the only element of h satisfying {dϕ(X)}eH = deG ϕ(XeG ) (∈ HeH ).

(2.32)

At this point, some confusion in the notation may occur. Indeed, Definition 2.1.49 defines the differential of a Lie-group homomorphism ϕ : G → H as a map from the Lie-algebra g to the Lie-algebra h; but we already defined (see Definition 2.1.18) dϕ as a map between the relevant tangent bundles, when ϕ is smooth, and dϕ (see Definition 2.1.24) as a map between the smooth vector fields as operators, when ϕ is a diffeomorphism. The following theorem shows that the three definitions are consistent (see also Remark 2.1.51). Theorem 2.1.50. Let G and H be Lie groups with associated algebras g and h, respectively. Let ϕ : G −→ H be a homomorphism of Lie groups. Then: (i) for every X ∈ g, we have that X and dϕ(X) are ϕ-related, i.e.

∀ x ∈ G; dϕ(X) ϕ(x) = dx ϕ(Xx )

(2.33)

Condition (2.33) characterizes dϕ(X), i.e. dϕ(X) is the only left invariant vector field Y ∈ h such that Yϕ(x) = dx ϕ(Xx ) for every x ∈ G; (ii) dϕ : g −→ h is a homomorphism of Lie algebras. := dϕ(X). We have to prove that (see ReProof. Let X ∈ g be arbitrary. Let X mark 2.1.36 and (2.23b)) ( )

ϕ(σ ) = dσ ϕ(Xσ ) X

∀ σ ∈ G.

H Let σ ∈ G. We shall use the notation τσG and τϕ(σ ) for the left translations on G and on H and the notation eG , eH for the identity elements in G and H, respectively. Moreover, the group laws on G and H will be respectively denoted by ◦G and ◦H . Since ϕ is a Lie group homomorphism, we have H H G G (τϕ(σ ) ◦ ϕ)(·) = ϕ(σ ) ◦ ϕ(·) = ϕ(σ ◦ ·) = (ϕ ◦ τσ )(·),

i.e. the following identity of maps on G holds H G τϕ(σ ) ◦ ϕ ≡ ϕ ◦ τσ .

and X are left invariant) As a consequence, we have (also recall that X ϕ(σ ) = X H (see (2.25b)) = deH τϕ(σ ) (XeH ) H H (see (2.32)) = deH τϕ(σ ) (deG ϕ(XeG )) = deG (τϕ(σ ) ◦ ϕ)(XeG )

(see (2.34)) = deG (ϕ ◦ τσG )(XeG ) = dσ ϕ(deG τσG (XeG )) (see (2.25b)) = dσ ϕ(Xσ ). This is precisely ( ), and (i) is proved.

(2.34)

2.1 Abstract Lie Groups

115

Finally, we have to prove dϕ([X, Y ]) = [dϕ(X), dϕ(Y )]

∀ X, Y ∈ g.

By means of the first part (i) of the assertion, we have that X (resp. Y ) is ϕrelated to dϕ(X) (resp. dϕ(Y )). Thus, by Proposition 2.1.37, [X, Y ] is ϕ-related to [dϕ(X), dϕ(Y )], i.e. dx ϕ([X, Y ]x ) = [dϕ(X), dϕ(Y )]ϕ(x)

∀ x ∈ G.

In particular, for x = eG , we have deG ϕ([X, Y ]eG ) = [dϕ(X), dϕ(Y )]eH . Now, by definition, dϕ([X, Y ]) is the unique vector field in h whose value in eH is deG ϕ([X, Y ]eG ), then dϕ([X, Y ]) = [dϕ(X), dϕ(Y )], and the theorem is completely proved.

Remark 2.1.51 (Consistency of the notation “dϕ”). Let G and H be Lie groups with associated algebras g and h, respectively. Let ϕ : G → H be a homomorphism of Lie groups. In Definition 2.1.18, dϕ was defined as the map dϕ : T (G) → T (H),

dϕ(x, Xx ) = (ϕ(x), Yϕ(x) ),

where Yϕ(x) = dx ϕ(Xx ), Hence, by Theorem 2.1.50, dϕ, as defined in Definition 2.1.49, is just the “projection” of the previously defined dϕ on the second component. In case ϕ is also an isomorphism, we have defined dϕ in Definition 2.1.24. A comparison of (2.15b) and (2.33) shows that dϕ and dϕ of Definition 2.1.49 actually coincide.

Remark 2.1.52. (i) From Theorem 2.1.50 it immediately follows that if ϕ : G −→ H is a Lie group isomorphism then dϕ : g −→ h is a Lie algebra isomorphism. (ii) Let us now suppose we are given two Lie groups on RN as defined in Chapter 1: (G, ◦) and (H, ∗). Let the associated Lie-algebras be g and h, respectively. Suppose it is given a Lie group isomorphism ϕ from G to H. As usual, the identity elements in G and H are supposed to be the origin 0 of RN . Then dϕ sends the ◦Jacobian basis of g in a basis of h that in 0 coincides with the column vectors of the matrix Jϕ (0). Indeed, if Z1 , . . . , ZN is the Jacobian basis related to (G, ◦) and if f ∈ C ∞ (H, R), we have {dϕ(Zk )}0 (f ) = {d0 ϕ((Zk )0 )}(f ) = {Zk }0 (f ◦ ϕ) = ∂xk |0 (f ◦ ϕ) =

N

(∂ξj f )(0) · (∂xk ϕj )(0). j =1

In particular, dϕ sends the ◦-Jacobian basis of g in the ∗-Jacobian basis of h if and only if Jϕ (0) is the identity matrix.

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2 Abstract Lie Groups and Carnot Groups

(iii) Let F, G, H be three Lie groups with Lie algebras f, g, h. Let ϕ : F → G,

ψ :G→H

be Lie-group homomorphisms. Then d(ψ ◦ ϕ) = dψ ◦ dϕ. Indeed, for every X ∈ f, d(ψ ◦ ϕ)(X) is the only element of h such that ( ) d(ψ ◦ ϕ)(X) e = deF (ψ ◦ ϕ)(XeF ). H

But also (dψ ◦ dϕ)(X) when applied on eH coincides with the right-hand side of ( ). Indeed, (dψ ◦ dϕ)(X) e = dψ dϕ(X) e = deG ψ dϕ(X) e H H G = deG ψ deF ϕ(XeF ) = deF (ψ ◦ ϕ)(XeF ).

2.1.9 The Exponential Map We begin with a simple but crucial result on Lie groups, following from an ODE’s result. We recall that, according to Definition 2.1.30, a smooth vector field X on a Lie group G is complete if, for every x ∈ G, the integral curve μ of X such that μ(0) = x is defined on the whole R. Proposition 2.1.53 (Completeness of the left invariant vector fields). The left invariant vector fields on a Lie group G are complete. Proof. Let X ∈ g be fixed, g being the Lie algebra of G. By simple prolongation results for ODE’s, it is enough to prove that there exists ε = ε(X) > 0 such that, for every x ∈ G, the integral curve μx of X such that μx (0) = x is defined on [−ε, ε]. Let e be the identity of G. Let ε > 0 be such that the integral curve μe of X with μe (0) = e is defined on [−ε, ε]. Then, it is an immediate consequence of the left invariance of X to derive that the map [−ε, ε] t → (τx ◦ μe )(t) =: ν(t) ∈ G coincides with the above μx . Indeed, ν(0) = τx (μe (0)) = τx (e) = x and (see the Definition 2.1.26 for ν˙ (t)) d d = dt (τx ◦ μe ) ν˙ (t) = dt ν d r r=t d r r=t d = dμe (t) τx ◦ dt μe d r r=t = dμe (t) τx (μ˙e (t)) = dμe (t) τx (Xμe (t) ) (see (2.24)) = X(τx ◦μe )(t) = Xν(t) . This ends the proof.

2.1 Abstract Lie Groups

117

Definition 2.1.54 (The exponential curve expX (t)). Let G be a Lie group with Lie algebra g. Let X ∈ g be fixed. By Proposition 2.1.53, the integral curve μ(t) of X passing through the identity of G when t = 0 is defined on the whole R. We set expX (t) := μ(t). By the very Definition 2.1.28 of integral curve, we have ⎧ ⎨ expX (0)= eG , d expX : R → G with = XexpX (t) ⎩ dt expX d r r=t

∀ t ∈ R.

(2.35)

In terms of functionals on C ∞ (G, R), (2.35) can be written more explicitly as

d f (expX (r)) = XexpX (t) (f ) ∀ f ∈ C ∞ (G, R). (2.36a) dr r=t In particular, when t = 0,

d f (expX (r)) = Xe (f ) dr r=0

∀ f ∈ C ∞ (G, R).

Again from (2.35) with t = 0 we infer d = Xe . d0 expX dt 0

(2.36b)

(2.36c)

Remark 2.1.55 (The exponential curve as a homomorphism). It is a simple exercise on ODE’s (see also Ex. 9 at the end of Chapter 1) to verify that expX : (R, +) −→ (G, ∗) is a Lie group homomorphism. In other words, it holds expX (r + s) = expX (r) ∗ expX (s)

for every r, s ∈ R.

Hence, Definition 2.1.49 can be applied. The differential of expX is the following map (here r is the Lie algebra of (R, +)) d expX : r −→ g

is such that λ

d → λ X dt

∀ λ ∈ R,

d d expX = X. (2.37) dt Indeed, by definition of the differential of a homomorphism, d expX (d/dt) is the unique vector field of g such that d d d expX . = d0 expX dt dt e 0 i.e. it holds

The above right-hand side equals Xe thanks to (2.36c), whence (2.37).

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2 Abstract Lie Groups and Carnot Groups

For future reference, we collect some other useful formulas for expX (t), immediate consequence of the facts proved above. Theorem 2.1.56. Let (G, ∗) be a Lie group with algebra g. Let X ∈ g. Then: (i) expX (r + s) = expX (r) ∗ expX (s) for every r, s ∈ R; (ii) expX (−t) = (expX (t))−1 for every t ∈ R; (iii) expX (0) = e; (iv) R t → expX (t) ∈ G is a smooth curve; (v) expX (t) is the unique integral curve of X passing through the identity at time zero, so that, for every x ∈ G, t → x ∗ (expX (t)) is the unique integral curve of X passing through x at time zero. Note that (i) in the above theorem jointly with the left invariance of X and (2.36c) gives back5 (2.36b). We are ready to give the fundamental definition. Definition 2.1.57 (Exponential map). Let (G, ∗) be a Lie group with Lie algebra g. Following the notation in Definition 2.1.54, we set Exp : g −→ G, X → Exp(X) := expX (1). Exp is called the exponential map (related to the Lie group G). The following results hold. Proposition 2.1.58. Let (G, ∗) be a Lie group with Lie algebra g. For every X ∈ g, we have (i) Exp(t X) = expX (t) for every t ∈ R; (ii) Exp((r + s)X) = Exp(r X) ∗ Exp(sX) for every r, s ∈ R; (iii) Exp(−tX) = (Exp(tX))−1 , for every t ∈ R. 5 Indeed, a direct computation gives

d dr

# $ f (expX (r)) =

r=t

=

d dr

d dr

r=0

# $ f (expX (r + t))

# $ f (expX (t) ∗ expX (r)) =

r=0

= d0 expX

d dr

d dr

# $ (f ◦ τexpX (t) )(expX (r))

r=0

(f ◦ τexpX (t) ) = Xe (f ◦ τexpX (t) ) = XexpX (t) (f ). r=0

Here we used (2.36c) and (2.25c).

2.1 Abstract Lie Groups

119

Proof. Fix t ∈ R and consider the curve s → μ(s) := expX (s t). We have μ(0) = expX (0) = e. By Theorem 2.1.56, we have μ(s) ˙ = t XexpX (s t) = t Xμ(s) . Thus μ is the integral curve of tX, and, again by Theorem 2.1.56, we have μ(s) = exptX (s), i.e. expX (s t) = exptX (s). For s = 1, we have expX (t) = exptX (1) = Exp (t X). Therefore, this yields Exp ((r + s) X) = expX (r + s) = expX (r) ∗ expX (s) = Exp (r X) ∗ Exp (s X), and moreover, −1 −1 = Exp (t X) . Exp (−t X) = expX (−t) = expX (t) The proposition is thus completely proved.

Theorem 2.1.59. Let G and H be Lie groups with associated algebras g and h. We denote by ExpG and ExpH the exponential maps related to G and to H, respectively. Finally, let ϕ : G −→ H be a Lie group homomorphism. Then the following diagram is commutative: G

ϕ

ExpG

g

H ExpH

dϕ

h.

Proof. Let X ∈ g. We have to prove that Exp H (dϕ(X)) = ϕ Exp G (X) .

(2.38)

We set for brevity x := Exp H (dϕ(X)). By definition, we have x = expH dϕ(X) (1), where expH dϕ(X) (t) is the unique integral curve of dϕ(X) on H passing through the identity eH of H at time zero (see Theorem 2.1.56). We set y := ϕ(Exp G (X)), i.e. y = ϕ(expG X (1)), where expG X (t) is the unique integral curve of X on G passing through the identity eG of G at time zero. We want to show that G expH dϕ(X) (t) = ϕ(expX (t))

∀ t ∈ R.

When t = 1, this gives x = y, which is the claimed (2.38).

(2.39)

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2 Abstract Lie Groups and Carnot Groups

To this end, we show that μ(t) := ϕ(expG X (t)) is an integral curve of dϕ(X) on H passing through eH at time zero. By uniqueness, this will yield μ(t) = expH dϕ(X) (t), i.e. (2.39). Since ϕ is a homomorphism, we have μ(0) = ϕ(expG X (0)) = ϕ(eG ) = eH . Finally, since ϕ and expG X are Lie-group homomorphisms, μ is a homomorphism. Thus, we have d d d G = dμ = d ϕ ◦ exp μ(t) ˙ = dt μ X d r r=t dt dt μ(t) μ(t) d G = dϕ ◦ dexpX = dϕ(X) μ(t) . dt μ(t) In the first equality, we used (2.18); in the second, (2.33); in the third, we exploited the very definition of μ; the fourth follows from Remark 2.1.52-(iii); in the fifth equality, we used (2.37). This ends the proof.

Remark 2.1.60 (Some notable commutative diagrams). Theorem 2.1.59 has some important consequences: suppose that Exp G and Exp H are diffeomorphisms with inverse functions Log G and Log H , respectively. Let us analyze the following commutative diagram: G

ϕ

Exp G

g

H Log H

dϕ

h.

We deduce that, given the Lie group homomorphism ϕ, the map Log H ◦ ϕ ◦ Exp G : g −→ h coincides with the differential of ϕ, which is a Lie algebra homomorphism (in particular, a linear map!). Under the same hypotheses on Exp G and Exp H , suppose we are given a Lie algebra homomorphism ψ : g −→ h. Let us consider the map Exp H ◦ ψ ◦ Log G : G −→ H. If such a map is a Lie group homomorphism,6 then the differential of this map coincides with ψ. Indeed, the following diagram is commutative G

Exp H ◦ψ◦Log G

Log G

g

H Exp H

ψ

h.

6 We shall see that Exp ◦ ψ ◦ Log is always a Lie group homomorphism whenever G and H G

H are connected and simply connected nilpotent Lie groups.

2.2 Carnot Groups

121

Finally, given a Lie group homomorphism ϕ : G −→ H, by the commutative diagram G

ϕ

H

Log G

g

Exp H dϕ

h

the map Exp H ◦ dϕ ◦ Log G : G −→ H coincides with ϕ and is thus a Lie group homomorphism. (For other related topics7 not employed in this book, see, e.g. [War83].)

2.2 Carnot Groups We give two definitions of Carnot groups: the first one (which we already introduced in Chapter 1, Section 1.4, page 56) is the most convenient for our purposes and it seems very natural in an Analysis context; the second one is the classical one from Lie group theory. Then, we compare the two definitions showing that, up to isomorphism, they are equivalent. For reader’s convenience, we recall the definition of homogeneous Carnot group. Definition 2.2.1 (Homogeneous Carnot group). Let RN be equipped with a Lie group structure by the composition law ◦. Let also RN be equipped with a homogeneous group structure by a family {δλ }λ>0 of automorphisms of (RN , ◦) (called dilations) of the following form δλ (x (1) , x (2) , . . . , x (r) ) = (λx (1) , λ2 x (2) , . . . , λr x (r) ).

(2.40)

Here x (i) ∈ RNi for i = 1, . . . , r and N1 + N2 + · · · + Nr = N. Let g be the algebra of the group (RN , ◦). For i = 1, . . . , N1 , let Zi be the (unique) vector field of g agreeing with ∂/∂ xi at the origin. If the following assumption holds (H1) then G =

the Lie algebra generated by Z1 , . . . , ZN1 is the whole g,

(RN , ◦, δλ )

is called a homogeneous Carnot group.

7 For example, the following results hold.

Theorem 2.1.61. Let G and H be Lie groups with related algebras g and h. 1. Suppose that G is connected. Let ϕ, ψ : G −→ H be Lie group homomorphisms. Then ϕ and ψ coincide if and only if the Lie algebra homomorphisms dϕ and dψ from g to h coincide. 2. Suppose that G is simply connected. Let ψ : g −→ h be a Lie algebra homomorphism. Then there exists a unique Lie group homomorphism ϕ : G −→ H such that dϕ = ψ. 3. Suppose that G and H are simply connected. If g and h are isomorphic Lie algebras, G and H are isomorphic Lie groups.

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2 Abstract Lie Groups and Carnot Groups

Given a Lie algebra h, for any two subsets V , W of h, we set

[V , W ] = span [v, w] v ∈ V , w ∈ W . Remark 2.2.2. Let m ∈ N, and let X1 , . . . , Xm ∈ h. Then, for every k ∈ N, any Lie bracket of height k of X1 , . . . , Xm is a linear combination of nested brackets of the form (see Proposition 1.1.7, page 12) [Xi1 , [Xi2 , . . . [Xik−1 , Xik ] . . .]],

i1 , i2 , . . . , ik ∈ {1, . . . , m}.

Definition 2.2.3 (Stratified Lie group). A stratified group (or Carnot group) H is a connected and simply connected Lie group whose Lie algebra h admits a stratification, i.e. a direct sum decomposition [V1 , Vi−1 ] = Vi if 2 ≤ i ≤ r, (2.41) h = V1 ⊕ V2 ⊕ · · · ⊕ Vr such that [V1 , Vr ] = {0}. Convention. In the rest of the chapter, we shall use the term “stratified group” to refer to a Carnot group according to the above abstract, classical Definition 2.2.3. We shall instead use the prefix “homogeneous” for Carnot groups as in Definition 2.2.1. From Chapter 3 onwards, after we have proved that (see Theorem 2.2.18) every stratified group is isomorphic to a homogeneous Carnot group, we shall use the term “Carnot” group almost indifferently. Yet, whenever the homogeneous structure will have to be stressed, or whenever it will be important to distinguish stratified versus homogeneous Carnot groups, we shall re-invoke the prefix homogeneous. Remark 2.2.4. Our Definition 2.2.1 is very natural to deal with in an analytic context. However, its only inconvenient property is that it is not invariant under isomorphism of Lie groups, since the homogeneity property heavily depends on the choice of coordinates on the underlying manifold RN . For example, consider the following group law in R3 : x ◦ y = arcsinh(sinh(x1 ) + sinh(y1 )), x2 + y2 + sinh(x1 )y3 , x3 + y3 . It is an easy exercise to prove that (R3 , ◦) is a homogeneous Carnot group8 according to Definition 2.2.3: indeed, the stratification is given by span{(cosh(x1 ))−1 ∂x1 , ∂x3 + sinh(x1 )∂x2 } ⊕ span{∂x2 }. It is also evident that (R3 , ◦) is not a homogeneous group (if it were so, the composition law should have polynomial component functions!, see Theorem 1.3.15). However, (R3 , ◦) is isomorphic to the homogeneous Carnot group H1 = (R3 , ∗, δλ ), being 8 We remark that

2 2 (cosh(x1 ))−1 ∂1 + ∂3 + sinh(x1 )∂2

is the canonical sub-Laplacian on this group.

2.2 Carnot Groups

123

x ∗ y = (x1 + y1 , x2 + y2 , x3 + y3 + 2x2 y1 − 2x1 y2 ), and δλ (x) := (λx1 , λx2 , λ2 x3 ). The main goal of this section is to prove that in every equivalence class of isomorphic stratified groups there is one group which is homogeneous, according to Definition 2.2.1. Example 2.2.5. Consider the open subset of R3 π π × R. Ω := (0, ∞) × − , 2 2 Then Ω is equipped with a structure of (non-homogeneous) Carnot group by the composition (ξ = (ξ1 , ξ2 , ξ3 ) ∈ Ω and analogously for η ∈ Ω) ⎛ ⎞ ξ1 η1 ⎠. arctan(ξ1 + η1 + tan ξ2 + tan η2 − ξ1 η1 ) ξ ∗η =⎝ ξ3 + η3 + 2(ξ1 ln η1 + tan ξ2 ln η1 − η1 ln ξ1 − tan η2 ln ξ1 ) This can be seen, for example, by remarking that G = (Ω, ∗) is isomorphic to the Heisenberg–Weyl group H1 (see Example 1.2.2) via the isomorphism ϕ : Ω → H1 ,

ϕ(ξ1 , ξ2 , ξ3 ) = (ln ξ1 , ξ1 + tan ξ2 , ξ3 ).

Indeed, recalling that the composition law on H1 is given by (x1 , x2 , x3 ) ◦ (y1 , y2 , y3 ) = x1 + y1 , x2 + y2 , x3 + y3 + 2 (x2 y1 − x1 y2 ) , and the inverse map of ϕ is ϕ −1 : H1 → Ω, we notice that

ϕ −1 (x1 , x2 , x3 ) = exp(x1 ), arctan(x2 − exp(x1 )), x3 , ξ ∗ η = ϕ −1 (ϕ(ξ ) ◦ ϕ(η))

∀ ξ, η ∈ G.

This suffices to prove that G is a Carnot group. Equivalently, we show the stratification condition for g, the algebra of G: first we find the ∗-left-invariant vector fields which coincide with the partial derivatives ∂ξ1 , ∂ξ2 , ∂ξ3 at the identity of G, say π −1 eG = ϕ (0, 0, 0) = 1, − , 0 . 4 These vector fields Z1 , Z2 , Z3 can be found via the following formula (here f ∈ C ∞ (Ω, R)) ∂ (Zi f )(ξ ) = (f (ξ ∗ η)) ∀ ξ ∈ G. ∂ ηi η=eG A direct computation then shows that

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2 Abstract Lie Groups and Carnot Groups

Z1 = ξ1 ∂ξ1 +

1 − ξ1 ∂ξ + (2ξ1 + 2 tan ξ2 − 2 ln ξ1 ) ∂ξ3 , 1 + tan2 (ξ2 ) 2

2 ∂ξ − 4 ln ξ1 ∂ξ3 , 1 + tan2 (ξ2 ) 2 Z3 = ∂ξ3 . Z2 =

Equivalently, the Zi ’s are the vector fields whose column-vector of the coefficients is given by the columns of the Jacobian matrix of the map τξ (η) = ξ ∗ η at eG , ⎛ Jτξ (eG ) = ⎝

0

ξ1 1−ξ1 1+tan2 (ξ2 ) 2ξ1 + 2 tan ξ2 − 2 ln ξ1

2 1+tan2 (ξ2 ) −4 ln ξ1

⎞ 0 0⎠. 1

As a consequence, the stratification of g is given by g = span{Z1 , Z2 } ⊕ span{[Z1 , Z2 ]}, and all commutators of Z1 , Z2 with length > 2 vanish identically. Indeed, another direct computation gives [Z1 , Z2 ] = −8 ∂ξ3 . We can now directly check the validity of Theorem 2.1.50-(2), i.e. that dϕ is an algebra-isomorphism from g to h1 , the algebra of H1 . To this end, we remark that dϕ is the map that to a vector field Z ∈ g assigns the vector field in h1 whose vector of the coefficients (at x ∈ H1 ) is given by (dϕ(Z))I (x) = Jϕ (ϕ −1 (x)) · (ZI )(ϕ −1 (x)). Now, we have ⎛

⎞ ξ −1 0 0 Jϕ (ξ ) = ⎝ 1 1 + tan2 (ξ2 ) 0 ⎠ , 0 0 1 ⎛ −x ⎞ e 1 0 0 1 + (x2 − ex1 )2 0 ⎠ . Jϕ (ϕ −1 (x)) = ⎝ 1 0 0 1 Consequently, it holds ⎛

e−x1 ⎝ 1 (dϕ(Z1 ))I (x) = 0

0 1 + (x2 − ex1 )2

0

⎞ ⎛ ⎞ ⎞⎛ ex1 1 0 x1 1−e ⎠. 1 0 ⎠ ⎝ 1+(x −ex1 )2 ⎠ = ⎝ 2 2x2 − 2x1 1 2x2 − 2x1

This means that dϕ(Z1 ) = ∂x1 + ∂x2 + (2x2 − 2x1 ) ∂x3 = X1 + X2 ,

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125

where, as usual, X1 = ∂x1 + 2x2 ∂x3 , X2 = ∂x2 − 2x1 ∂x3 are the generators of the algebra of H1 . Analogously, we have ⎞ ⎛ ⎞ ⎛ −x ⎞⎛ 0 e 1 0 0 0 (dϕ(Z2 ))I (x) = ⎝ 1 1 + (x2 − ex1 )2 0 ⎠ ⎝ 1+(x22−ex1 )2 ⎠ = ⎝ 2 ⎠ , −4x1 0 0 1 −4x1 i.e. dϕ(Z2 ) = 2∂x2 − 4x1 ∂x3 = 2X2 . Moreover, ⎛

e−x1 (dϕ([Z1 , Z2 ]))I (x) = ⎝ 1 0

0 1 + (x2 − ex1 )2 0

⎞⎛ ⎞ ⎛ ⎞ 0 0 0 0⎠⎝ 0 ⎠ = ⎝ 0 ⎠, −8 −8 1

i.e. dϕ([Z1 , Z2 ]) = −8 ∂x3 = 2[X1 , X2 ]. Finally, we can straightforwardly check the algebra-isomorphism condition for dϕ (see Theorem 2.1.50-(2)), namely dϕ([Z1 , Z2 ]) = 2[X1 , X2 ] = [X1 + X2 , 2X2 ] = [dϕ(Z1 ), dϕ(Z2 )]. This completes our example.

2.2.1 Some Properties of the Stratification of a Carnot Group If H is a Carnot group, its Lie algebra admits at least a stratification, but it can have more than one. For example, if H = H1 is the Heisenberg–Weyl group on R3 (see Example 1.2.2 and its notation), its Lie algebra admits the stratifications span{X1 , X2 } ⊕ span{[X1 , X2 ]}, span{X1 − 3 [X1 , X2 ], X2 } ⊕ span{[X1 , X2 ]}, span{X1 + X2 , 3X1 + [X1 , X2 ]} ⊕ span{[X1 , X2 ]}. Definition 2.2.6 (Basis adapted to the stratification). Let H be a Carnot group. Let V = (V1 , . . . , Vr ) be a fixed stratification of the Lie algebra h of H as in (2.41). We say that a basis B of h is adapted to V if (1) (1) (r) (r) (2.42) B = E 1 , . . . , E N1 ; . . . ; E 1 , . . . , E Nr , where, for i = 1, . . . , r, we have Ni := dim Vi , and (i) (i) E1 , . . . , ENi is a basis for Vi . Obviously, every Carnot group admits an adapted basis to any of its stratifications.

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Definition 2.2.7 (Step and number of generators). Let H be a stratified group. Let (V1 , . . . , Vr ) be any stratification of the algebra of H, as in Definition 2.2.3. We say that H as step (of nilpotency) r and has m generators, where m := dim(V1 ). The following proposition shows that the above definitions are well-posed, i.e. they do not depend on the particular stratification of H. Proposition 2.2.8. Let H be a stratified group. Suppose that (V1 , . . . , Vr ) and r ) be any two stratifications of the algebra of H, as in Definition 2.2.3. 1 , . . . , V (V i ) for every i = 1, . . . , r. Moreover, the algebra of Then r = r and dim(Vi ) = dim(V H is a nilpotent Lie algebra of step r. Hence, the natural number Q :=

r

idim(Vi )

i=1

depends only on the stratified nature of H and not on the particular stratification. Q is called the homogeneous dimension of H. Proof. From the very Definition 2.2.3, we see that if (V1 , . . . , Vr ) is a stratification of h, the algebra of H, then h is a nilpotent9 Lie algebra of step r. Hence r depends only on h and not on the stratification. i ) for every i = 1, . . . , r is not trivial and follows The fact that dim(Vi ) = dim(V from Lemma 2.2.9 below.

Lemma 2.2.9 (The two-stratification lemma). Let H be a stratified group with Lie algebra h. Suppose V := (V1 , . . . , Vr ) and W := (W1 , . . . , Wr ) are two stratifications of h. Then, for every couple of bases V and W of h respectively adapted to the stratifications V and W , the transition matrix between the two bases is non-singular and has the block-triangular form ⎞ ⎛ (1) M 0 ··· 0 ⎜ .. ⎟ . ⎜ M (2) . . . ⎟ ⎟, ⎜ ⎟ ⎜ .. .. .. ⎝ . . . 0 ⎠ ··· M (r) 9 We recall that, given an abstract Lie algebra (g, [·, ·]), g is called nilpotent of step r, r ∈ N,

if for every X1 , . . . , Xr+1 ∈ g,

[X1 , [X2 , . . . [Xr , Xr+1 ] . . . ]] = 0, and there exist Y1 , . . . , Yr ∈ g such that [Y1 , [Y2 , . . . [Yr−1 , Yr ] . . . ]] = 0.

2.2 Carnot Groups

127

where, for every i = 1, . . . , r, the block M (i) is a Ni × Ni non-singular matrix (Ni being the common value of dim(Vi ) = dim(Wi )). Proof. Let the notation in the assertion be fixed. We also set, for every i = 1, . . . , r, Ni = dim(Vi ) and Mi = dim(Wi ). From V1 ⊆ W1 ⊕ · · · ⊕ Wr and from the stratification condition for V and for W , we infer that Vi = [V1 , · · · [V1 , V1 ] · · ·] )* + ( i times / , . ⊆ {W1 ⊕ · · ·}, · · · {W1 ⊕ · · ·}, {W1 ⊕ · · ·} · · · ⊆ Wi ⊕ Wi+1 ⊕ · · · ⊕ Wr . Hence (the second column obtained by reversing the rôles of V and W ), ⎧ V1 ⊆ W1 ⊕ W2 ⊕ · · · ⊕ Wr , W1 ⊆ V1 ⊕ V2 ⊕ · · · ⊕ Vr , ⎪ ⎪ ⎪ ⎪ .. ⎨ .. . . ⎪ ⎪ Wr−1 ⊆ Vr−1 ⊕ Vr , Vr−1 ⊆ Wr−1 ⊕ Wr , ⎪ ⎪ ⎩ Vr ⊆ Wr , Wr ⊆ Vr . All this proves that

( )

⎧ W1 ⊕ W2 ⊕ · · · ⊕ Wr = V1 ⊕ V2 ⊕ · · · ⊕ Vr , ⎪ ⎪ ⎪ ⎪ . ⎨. . ⎪ ⎪ Wr−1 ⊕ Wr = Vr−1 ⊕ Vr , ⎪ ⎪ ⎩ Wr = Vr .

In particular (recalling that dim(A ⊕ B) = dim(A) + dim(B)), this yields ⎧ M 1 + M2 + · · · + Mr = N 1 + N 2 + · · · + N r , ⎪ ⎪ ⎪ ⎪ ⎨ .. . ⎪ ⎪ Mr−1 + Mr = Nr−1 + Nr , ⎪ ⎪ ⎩ Mr = N r , whence, Ni = Mi , i.e. dim(Vi ) = dim(Wi ) for every i = 1, . . . , r. From ( ) it immediately follows the block-form for the matrix of the change of basis between V and W.

The following proposition shows that “to be a Carnot group” is an invariant under isomorphism of Lie groups. Proposition 2.2.10. Let H be a stratified group. Suppose G is a Lie group isomorphic to H. Then G is a stratified group too. Moreover, H and G have the same step, the same number of generators and even the dimensions of the layers of the relevant

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2 Abstract Lie Groups and Carnot Groups

stratifications are preserved. Also, H and G have the same homogeneous dimension Q. More precisely, suppose ϕ : H → G is a Lie group isomorphism and that (V1 , . . . , Vr ) is a stratification of h, the algebra of H, as in Definition 2.2.3. Then, if g is the algebra of G, a stratification for g is given by (dϕ(V1 ), . . . , dϕ(Vr )), where dϕ is the differential of ϕ (see Definition 2.1.49) which is an isomorphism of Lie algebras (and of vector spaces). Proof. We follow the notation in the assertion. Set Wi := dϕ(Vi ) for every i = 1, . . . , r. The proposition will be proved if we demonstrate that g = W1 ⊕ W2 ⊕ · · · ⊕ Wr

and

[W1 , Wi−1 ] = Wi [W1 , Wr ] = {0}.

if 2 ≤ i ≤ r,

Now, from the linearity and the invertibility of dϕ, it holds g = dϕ(h) = dϕ(V1 ⊕ · · · ⊕ Vr ) = dϕ(V1 ) ⊕ · · · ⊕ dϕ(Vr ) = W1 ⊕ · · · ⊕ Wr . Finally, dϕ being a Lie algebra homomorphism (see Theorem 2.1.50-(ii)) it is easy to see that [W1 , Wi−1 ] equals dϕ(Vi ) = Wi , 2 ≤ i ≤ r, [dϕ(V1 ), dϕ(Vi−1 )] = d [V1 , Vi−1 ] = dϕ({0}) = {0}. This ends the proof.

2.2.2 The Campbell–Hausdorff Formula and Some General Results on Nilpotent Lie Groups Before proving that, up to isomorphism, Carnot groups and homogeneous Carnot groups provide equivalent notions, we recall some results about the so-called Campbell–Hausdorff formula. Definition 2.2.11. Let h be a nilpotent Lie algebra. For X, Y ∈ h, we set10 X Y :=

(−1)n+1 n≥1

n

pi +qi ≥1

(ad X)p1 (ad Y )q1 · · · (ad X)pn (ad Y )qn −1 Y ( nj=1 (pj + qj )) p1 ! q1 ! · · · pn ! qn !

1≤i≤n

= X+Y +

1 [X, Y ] + 2

10 We use the notation (ad A)B = [A, B]. Moreover, if q = 0, the term in the sum (2.43) is n by convention · · · (ad X)pn−1 (ad Y )qn−1 (ad X)pn −1 X. Clearly, if qn > 1, or qn = 0 and pn > 1, the term is zero.

2.2 Carnot Groups

1 1 [X, [X, Y ]] − [Y, [X, Y ]] 12 12 1 1 [Y, [X, [X, Y ]]] − [X, [Y, [X, Y ]]] − 48 48 + {brackets of height ≥ 5}.

129

+

(2.43)

The sum over n actually runs on {1, . . . , r}, where r < ∞ is the step of nilpotency of h. The same is true for the sum over the pi ’s and qi ’s, for which it is left unsaid that ni=1 (pi + qi ) ≤ r. We shall refer to the operation defined in (2.43) as the Campbell–Hausdorff operation on h. The inner sum in (2.43) can be reordered so that the brackets appear with increasing height (here N0 = N ∪ {0}), XY =

r

(−1)n+1 n=1

×

n

r

h=1

(p1 ,q1 ),...,(pn ,qn )∈ N0 ×N0 (p1 ,q1 ),...,(pn ,qn )=(0,0) (p1 +q1 )+··· +(pn +qn )=h

×

(adX)p1 (ad Y )q1 · · · (adX)pn (ad Y )qn −1 Y . ( nj=1 (pj + qj )) p1 ! q1 ! · · · pn ! qn !

Remark 2.2.12. Since h is nilpotent, (2.43) is a finite sum, and defines a binary operation in h. A striking fact is that this operation is actually associative! Indeed, much more holds: (h, ) is a Lie group. We explicitly remark that the inverse of X ∈ h w.r.t. is simply −X. We shall prove this fact in Corollary 2.2.15 below, by making use of two abstract remarkable results whose proofs are out of our scope here (we refer the reader to the monographs [CG90] and [Var84] for more references). First of all we recall the following result (see [CG90, p. 13]), which also gives the well known Campbell–Hausdorff formula (2.44). Theorem 2.2.13 (Corwin and Greenleaf [CG90, Theorem 1.2.1]). Let (H, ∗) be a connected and simply connected Lie group. Suppose that the Lie algebra h of H is nilpotent. Then defines a Lie group structure on h (h being equipped with the manifold structure resulting from its finite-dimensional vector space structure) and Exp : (h, ) → (H, ∗) is a group-isomorphism. In particular, we have Exp (X) ∗ Exp (Y ) = Exp (X Y )

∀ X, Y ∈ h.

(2.44)

We now recall the third fundamental theorem of Lie (see [Var84, Theorem 3.15.1]). This is a very deep result in Lie group theory.

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2 Abstract Lie Groups and Carnot Groups

Theorem 2.2.14 (The third fundamental theorem of Lie). Let h be a finite-dimensional Lie algebra. Then there exists a connected and simply connected Lie group whose Lie algebra is isomorphic to h. Collecting the above two theorems, we obtain the following result. Corollary 2.2.15. Let h be a finite-dimensional nilpotent Lie algebra. Then defines a Lie group structure on h. Moreover, the Lie algebra associated to the Lie group (h, ) is isomorphic to the algebra h. Proof. Since h is a finite-dimensional Lie algebra, by Theorem 2.2.14 there exists a connected and simply connected Lie group (say, G) whose Lie algebra g is isomorphic to h. Let ϕ : g → h be a Lie algebra isomorphism. Since h is nilpotent by hypothesis, so is g (since g is isomorphic to h). Hence G satisfies the hypotheses of Theorem 2.2.13. As a consequence, g equipped with the operation defined in (2.43) is a Lie group. We denote by g such a composition law on the Lie algebra g. Analogously, we denote by h a similar operation on h. Now, the isomorphism ϕ between the two Lie algebras g and h transfers the operation g into h . More precisely, for every X, Y ∈ g, we have ϕ(X g Y ) 1 1 [X, [X, Y ]g ]g + · · · = ϕ X + Y + [X, Y ]g + 2 12 (since ϕ is a Lie algebra homomorphism) 1 1 [ϕ(X), [ϕ(X), ϕ(Y )]h ]h + · · · = ϕ(X) + ϕ(Y ) + [ϕ(X), ϕ(Y )]h + 2 12 = ϕ(X) h ϕ(Y ). (2.45) We explicitly remark that in the last equality we used, as a crucial tool, the “universal” way in which is defined on a Lie algebra. In particular, the associativity of g on g directly implies the associativity of h on h. Thus, the first part of the assertion of the corollary is proved. The second part follows from Ex. 2 at the end of the chapter.

Remark 2.2.16. If h and g are finite-dimensional nilpotent Lie algebras and ϕ : h → g is an algebra-homomorphism, then ϕ is also a group-homomorphism between (h, ) and (g, ). This directly follows from (2.45). 2.2.3 Abstract and Homogeneous Carnot Groups We now aim to prove that, up to isomorphism, the definitions of classical and homogeneous Carnot group are equivalent. To begin with, we prove the following simple fact:

2.2 Carnot Groups

131

Proposition 2.2.17 ((Homogeneous ⇒ stratified) Carnot). A homogeneous Carnot group is a stratified group. Proof. Let G = (RN , ◦, δλ ) be a homogeneous Carnot group. Clearly, G is connected and simply connected. Let g be the algebra of G. (i) For i = 1, . . . , r and j = 1, . . . , Ni , let Zj be the vector field of g agreeing (i)

with ∂/∂ xj at the origin. We set (i)

(i)

Vi := span{Z1 , . . . , ZNi }. Remark 1.4.8 proves that (V1 , . . . , Vr ) is a stratification of g, as in Definition 2.2.3. This ends the proof.

We are now in the position to prove the main result of this section. The proof is a detailed argument of what is commonly used (without comments) in literature, i.e. the identification of the group with its algebra. isom.

Theorem 2.2.18 ((Stratified ⇒ homogeneous) Carnot). Let H be a stratified group, according to Definition 2.2.3. Then there exists a homogeneous Carnot group H∗ (according to our Definition 2.2.1) which is isomorphic to H. We can choose as H∗ the Lie algebra h of H (identified to RN by a suitable choice of an adapted basis of h) equipped with the composition law defined by the Campbell–Hausdorff operation (2.43) in Definition 2.2.11. In this case, a groupisomorphism from H∗ to H is the exponential map Exp : (h, ) → (H, ∗). Proof. Let (H, ∗) be as in Definition 2.2.3. Let h be the algebra of H. Let h = V1 ⊕ · · · ⊕ Vr be a fixed stratification of h as in (2.41). By Proposition 2.2.8, h is nilpotent of step r. Then Theorem 2.2.13 yields that Exp : (h, ) → (H, ∗) is a Lie-group isomorphism,

(2.46)

where is as in (2.43). We now prove that (a coordinate version of) (h, ) is a homogeneous Carnot group according to Definition 2.2.1. We fix a basis for h adapted to its stratification (see Definition 2.2.6): for i = (i) ) be a basis for Vi . Then consider 1, . . . , r, set Ni := dim Vi , and let (E1(i) , . . . , EN i the basis for h given by (1) (1) (r) (r) E = E 1 , . . . , E N1 ; . . . ; E 1 , . . . , E Nr . By means of this basis, we fix a coordinate system on h, and we identify h with RN , where N := N1 + · · · + Nr . More precisely, we consider the map11 11 See also the map introduced in (1.35) of Remark 1.2.20, page 21. Incidentally, we notice

that (h, πE ) is a coordinate system for the whole h, determining its differentiable structure. Obviously, along with πE , we have other coordinate maps (for example those arising by a different choice of a linear basis of h, adapted or not) and the differentiable structure of h does not depend on E.

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Fig. 2.1. Turning a stratified Carnot group into a homogeneous one

πE : h → RN ,

E · ξ :=

Ni r

(i)

(i)

ξj Ej → (ξ (1) , . . . , ξ (r) ),

i=1 j =1 (i)

(i)

where ξ (i) = (ξ1 , . . . , ξNi ) ∈ RNi for every i = 1, . . . , r. Next, we set Ψ := Exp ◦ (πE )−1 : RN → H,

Ψ (ξ ) = (Exp (E · ξ )).

(See also Fig. 2.1.) Notice that, more explicitly, r N i

(i) (i) Ψ (ξ ) = Exp ξj E j

∀ ξ ∈ RN .

(2.47)

with the composition law E defined by ξ E η := Ψ −1 Ψ (ξ ) ∗ Ψ (η) , ξ, η ∈ RN ,

(2.48)

i=1 j =1

Finally, we equip

RN

We define a family of dilations {Δλ }λ>0 on the Lie algebra h as follows: r r

Δλ Xi := λi Xi , where Xi ∈ Vi . Δλ : h → h, i=1

(2.49a)

i=1

Obviously, Δλ is a vector-space automorphism of h.

(2.49b)

(Note that N, the Ni ’s and the form of Δλ do not depend neither on the choice of the stratification of h nor on the adapted basis; see, for instance, Lemma 2.2.9.) Obviously, Δλ turns into a family of dilations {δλ }λ>0 on RN via Ψ by setting δλ := πE ◦ Δλ ◦ πE−1 . We claim that H∗ := (RN , E , δλ ) is a homogeneous Carnot group (of step r and N1 generators) isomorphic to (H, ∗) via the Lie group isomorphism Ψ . To prove the claim, we split the proof in steps.

(2.49c)

2.2 Carnot Groups

133

(I). By the very definition of E and Ψ , we have Ψ (ξ E η) = Ψ (ξ ) ∗ Ψ (η)

∀ ξ, η ∈ RN ,

(2.50a)

which, in turn, is equivalent to (exploit (2.46) and recall that Ψ = Exp ◦ πE−1 ) πE−1 (ξ E η) = πE−1 (ξ ) πE−1 (η)

∀ ξ, η ∈ RN ,

(2.50b)

∀ X, Y ∈ h.

(2.50c)

or, equivalently, πE (X Y ) = πE (X) E πE (Y ) Now, we recognize that (2.50a) and (2.50b) mean that (RN , E ),

(h, ),

(H, ∗)

are isomorphic Lie groups via the Lie-group isomorphisms πE−1

Exp

(RN , E ) −→ (h, ) −→ (H, ∗). In particular, Ψ = Exp ◦ πE−1 : (RN , E ) → (H, ∗) is a Lie-group isomorphism.

(2.51)

(II). We now investigate the dilation δλ . The stratified notation hE·ξ =

Ni r

(i)

(i)

ξj E j

i=1 j =1

for an arbitrary vector of h and the fact that πE (E · ξ ) = ξ

(2.52)

suggests the notation RN ξ = (ξ (1) , . . . , ξ (r) ) for the points in RN . We claim that, with the above notation, δλ introduced in (2.49c) has the form in (2.40), i.e. δλ (ξ (1) , ξ (2) , . . . , ξ (r) ) = (λξ (1) , λ2 ξ (2) , . . . , λr ξ (r) ). Indeed, δλ (ξ ) = (see (2.49c)) πE ◦ Δλ ◦ πE−1 (ξ ) (see (2.52)) = πE Δλ (E · ξ ) r N i

(i) (i) ξj E j = πE Δλ i=1 j =1

(2.53)

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2 Abstract Lie Groups and Carnot Groups

(see (2.49b))

= πE

(see (2.49a))

= πE

Ni r

(i) (i) ξj Δλ (Ej )

i=1 j =1 Ni r

(i) (i) ξj λi E j

i=1 j =1

(see (2.52))

= πE E · λξ (1) , . . . , λr ξ (r) = λξ (1) , . . . , λr ξ (r) .

Next, we proceed by showing that Δλ is an automorphism of the Lie-group (h, ), i.e. (2.54) Δλ (X Y ) = Δλ (X) Δλ (Y ) ∀ X, Y ∈ h, ∀ λ > 0. Recalling Remark 2.2.16, it is enough to prove that Δλ [X, Y ] = [Δλ (X), Δλ (Y )] for every X, Y ∈ h. If X = ri=1 Xi and Y = ri=1 Yi , where Xi , Yi ∈ Vi , we have [Xi , Yj ] ∈ Vi+j (by the stratification condition), whence r r

Δλ [Xi , Yj ] = λi+j [Xi , Yj ] Δλ [X, Y ] =

=

i,j =1 r

[λi Xi , λj Yj ] =

i,j =1

i,j =1 r

[Δλ (Xi ), Δλ (Yj )] = [Δλ (X), Δλ (Y )].

i,j =1

Now, a joint application of (2.50b), (2.50c) and (2.54) prove that δλ is a Lie-group automorphism of (RN , E ), i.e. δλ (ξ E η) = δλ (ξ ) E δλ (η)

∀ ξ, η ∈ RN ,

∀ λ > 0.

(III). Thus, H∗ := (RN , E , δλ ) is a homogeneous Lie group on RN , as defined in Definition 1.3.1, page 31. Let now h∗ be the Lie algebra of H∗ . Dealing with a Lie group on RN (and the fixed Cartesian coordinates ξ ’s on RN ), the Jacobian basis related to the composition E is well-posed. We denote by (1) (1) (r) (r) Z = Z1 , . . . , ZN1 ; . . . ; Z1 , . . . , ZNr this Jacobian basis, i.e. Zk is the vector field in h∗ agreeing at the origin with (i) ∂/∂ ξk . The proof is complete if we show that the Lie algebra generated by Z1 , . . . , ZN1 coincides with the whole h∗ . (i)

To this end, we first observe that, thanks to (2.51), d Ψ : h∗ → h is an algebraisomorphism (see Theorem 2.1.50). Furthermore, from Remark (2.1.52)-(iii), we have

2.2 Carnot Groups

135

dΨ = (d Exp ) ◦ d(πE−1 ) . (1)

(1)

Moreover, since E1 , . . . , EN1 is a system of Lie-generators for h (by the very definition of stratification!), it is enough to prove that (i)

(i)

d Ψ (Zk ) = Ek

for every i = 1, . . . , r and every k = 1, . . . , Ni .

(2.55)

In order to prove (2.55), we recall that a left-invariant vector field is determined by its value at the identity. Hence, (2.55) will follow if we show that (i) (i) d Ψ (Zk ) e = (Ek )e . For every f ∈ C ∞ (H, R), we have (i) (i) (i) d Ψ (Zk ) e (f ) = d0 Ψ (Zk )0 (f ) = (Zk )0 (f ◦ Ψ ) r N i

(i) (i) (i) ξj E j = (∂/∂ ξk )|ξ =0 f Exp d (i) = f Exp (t Ek ) dt t=0 d = f expE (i) (t) k dt =

i=1 j =1

t=0 (i) (Ek )e (f ).

In the first equality, we used the very definition of the differential of a homomorphism (see Definition 2.1.49); in the second one, we used the definition of the dif(i) ferential at a point; in the third equality, we exploited the very definition of Zk and that of Ψ (see (2.47)); the fourth equality is a triviality from calculus; the fifth and the sixth equalities are the core of the computation: they follow, respectively, from Proposition 2.1.58-(1) and from (2.36b). The theorem is thus completely proved.

Remark 2.2.19. From Theorem 2.2.18 and Remark 2.2.12, it follows that any stratified group is isomorphic to a homogeneous Carnot group in which the group inversion law is simply given by x −1 = −x. This is commonly assumed in great part of the literature involving Carnot groups, without further comments. See Proposition 2.2.22 for more details. Remark 2.2.20 (Change of adapted basis). We give some more information on Theorem 2.2.18. Let H be a fixed stratified group with Lie algebra h. Let h = V1 ⊕· · ·⊕Vr be a given stratification of h, as in (2.41). We know that r and every Ni := dim(Vi ) (for i = 1, . . . , r) are invariants of the stratified group. So is N := N1 + · · · + Nr (= dim(h)). We fix any arbitrary basis E for h adapted to its stratification. We use the notation in the proof of Theorem 2.2.18. We therein demonstrated that RN is a homogeneous Carnot group isomorphic to (H, ∗) if RN is equipped with the composition law E defined by (2.48) and the dilation (2.53).

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2 Abstract Lie Groups and Carnot Groups

The map Ψ is a Lie-group isomorphism from (RN , E ) to (H, ∗). Another way to write (2.48) is obviously Exp E · (ξ E η) = (Exp (E · ξ )) ∗ (Exp (E · η)). (2.56a) Suppose now that we choose another basis E (adapted to the same stratification), with analogous notation as for E. RN can be equipped with an another composition E characterized by · ( · · Exp E ξ E η) = (Exp (E ξ )) ∗ (Exp (E η)). (2.56b) (i)

For every i ∈ {1, . . . , r} and every j ∈ {1, . . . , Ni }, there exist scalars ch,j ’s such that (i) (i) (i) (i) E 1 + · · · + cN E Ni . Ej(i) = c1,j i ,j Let us introduce the notation (i) C (i) := ch,j 1≤h≤N , 1≤j ≤N i

i

and C to denote the N × N matrix whose diagonal blocks are C (1) , . . . , C (r) , ⎛ (1) ⎞ C ··· 0 . . . .. .. ⎠ , C = ⎝ .. 0 · · · C (r) and, finally, let us use once again the notation C to denote the linear map C : RN → RN ,

x → Cx.

We note that if a vector field in h has coordinates ξ ∈ RN with respect to the basis i.e. E, then12 it has coordinates Cξ with respect to the basis E, · (Cξ ) E·ξ =E

for every ξ ∈ RN .

12 Indeed, it holds

E·ξ =

Ni r

(i) (i) ξj Ej i=1 j =1

⎛ ⎞ Ni Ni Ni Ni r r

(i) (i) (i) (i) (i) (i) ⎝ = ξj ch,j Eh = ch,j ξj ⎠ E h i=1 j =1

=

h=1

i=1 h=1

j =1

Ni Ni r r

(i) (i) (i) = C (i) · ξ (i) E (C · ξ )h E h h i=1 h=1

· (Cξ ). =E

h

i=1 h=1

(2.57)

2.2 Carnot Groups

137

We now claim that the same happens for the relevant group structures G := (RN , E ),

:= (RN , ), G E

Roughly i.e. the linear map C : RN → RN is a group isomorphism from G to G. speaking, the same change of basis in h from the basis E to the basis E acts as a Lie-group isomorphism from (RN , E ) to (RN , E). Indeed, from (2.56a), (2.56b) and (2.57) we have Exp E · (ξ E η) = (Exp (E · ξ )) ∗ (Exp (E · η)) · (Cξ ) ∗ Exp E · (Cη) = Exp E · ((Cξ ) (Cη)) = Exp E E

= Exp E · C −1 (Cξ ) E (Cη) , which implies

ξ E η = C −1 (Cξ ) E (Cη) ,

(2.58a)

i.e. C(ξ E η) = (Cξ ) E (Cη)

∀ ξ, η ∈ RN .

(2.58b)

Now, (2.58b) is equivalent to say that C : (RN , E ) → (RN , E),

x → Cξ

is a Lie-group isomorphism.

Example 2.2.21 (From “stratified” to “homogeneous”). We consider once again the Lie group introduced in Examples 1.2.18 and 1.2.31. To be consistent with the notation in Theorem 2.2.18, we denote this group by H. We explicitly remark that H is a Carnot group which is not homogeneous (w.r.t. the coordinates initially assigned to it). We find an explicit isomorphism of Lie groups turning H into a homogeneous Carnot group: the existence of such an isomorphism is indeed ensured by Theorem 2.2.18, whereas an explicit way to construct it comes from the very proof of Theorem 2.2.18 (see also Remark 2.2.20). We have H = (R3 , ∗), where x ∗ y = arcsinh(sinh(x1 ) + sinh(y1 )), x2 + y2 + sinh(x1 )y3 , x3 + y3 . (1)

(1)

A stratification of the algebra h of H is V1 ⊕ V2 , where V1 = span{E1 , E2 }, (2) V2 = span{E1 }, being (1)

E1 =

1 ∂x , cosh(x1 ) 1

(1)

E2 = ∂x3 + sinh(x1 ) ∂x2 ;

The relevant dilation on h is given by

(2)

E1 = ∂x2 .

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2 Abstract Lie Groups and Carnot Groups

δλ : h −→ h, δλ (ξ1 E1(1)

+ ξ2 E2(1)

+ ξ3 E1(2) ) := ξ1 λ E1(1) + ξ2 λ E2(1) + ξ3 λ2 E1(2) .

Since h is a nilpotent algebra of step two, the Campbell–Hausdorff operation on h is 1 X Y = X + Y + [X, Y ]. 2 (1)

(1)

(2)

Now, since [E1 , E2 ] = E1 , a direct computation gives (ξ1 E1(1) + ξ2 E2(1) + ξ3 E1(2) ) (η1 E1(1) + η2 E2(1) + η3 E1(2) ) 1 (1) (1) (2) = (ξ1 + η1 ) E1 + (ξ2 + η2 ) E2 + ξ3 + η3 + (ξ1 η2 − ξ2 η1 ) E1 . 2 (1)

(1)

(2)

Via the choice of the basis E = {E1 , E2 , E1 }, the Lie group (h, ) can be identified to H∗ = (R3 , E ), where 1 ξ E η = ξ1 + η1 , ξ2 + η2 , ξ3 + η3 + (ξ1 η2 − ξ2 η1 ) . 2 The Lie group isomorphism between (H∗ , E ) and (H, ∗) is the “exponential-type map” H∗ (ξ1 , ξ2 , ξ3 ) → Exp (ξ1 E1 + ξ2 E2 + ξ3 E1 ) ∈ H. (1)

(1)

(2)

Now, from the computation13 in Example 1.2.31 (page 28) we derive that this map is given by 1 (ξ1 , ξ2 , ξ3 ) → Ψ (ξ1 , ξ2 , ξ3 ) = arcsinh(ξ1 ), ξ3 + ξ1 ξ2 , ξ2 . 2 The map Ψ is the isomorphism of Lie groups turning the homogeneous Carnot group H∗ = (R3 , E , δλ ) (where δλ (ξ1 , ξ2 , ξ3 ) = (λξ1 , λξ2 , λ2 ξ3 )) into the nonhomogeneous Carnot group H. We recognize that x ∗ y = Ψ (Ψ −1 (x) E Ψ −1 (y))

∀ x, y ∈ H.

2.2.4 More Properties of the Lie Algebra of a Stratified Group In the proof of Theorem 2.2.18, we described how to identify a stratified group with a homogeneous Carnot group on RN . More precisely, given a stratified group H, the equivalence class of the Lie groups which are isomorphic to H contains at least one (in fact, infinite) homogeneous Carnot group H∗ on RN , according to Definition 2.2.1. Namely, H∗ is a “coordinate-copy” of (h, ), the Lie algebra of H equipped with the Campbell–Hausdorff operation. 13 We warn the reader that a slight change of notation here is needed, if compared to the

notation in Example 1.2.31, interchanging ξ2 and ξ3 .

2.2 Carnot Groups

139

Since, in the specialized literature, one often meets with phrases such as “it is not restrictive to suppose that. . . ” H has some distinguished properties, it is advisable to look more closely to the properties of (h, ). We furnish some of these properties in the following proposition which collect several already proved facts. Proposition 2.2.22. Let H be a stratified group with Lie algebra h and exponential map Exp H : h → H. Let also be the Campbell–Hausdorff operation on h defined in (2.43). Let V1 ⊕ · · · ⊕ Vr be a stratification of h, as in (2.41). Let E be any basis for h adapted to the stratification, as in Definition (2.2.6). Set N := dim(h), consider the map πE : h → RN , where, for every X ∈ h, πE (X) is the N -tuple of the coordinates of X w.r.t. E. Then the binary operation on RN defined by ∀ x, y ∈ RN x E y = πE πE−1 (x) πE−1 (y) has the following properties: (1) G := (RN , E ) is a Lie group on RN ; G is isomorphic to H via the map Ψ = Exp H ◦πE−1 and to (h, ) via πE , whence (G, E ) and (h, ) are stratified groups. (2) Let Z = {Z1 , . . . , ZN } be the Jacobian basis related to G; then, denoting the adapted basis by E = {E1 , . . . , EN }, we have dΨ (Zi ) = Ei

for every i = 1, . . . , N,

or, equivalently, Zi (f ◦ Ψ ) ≡ Ei (f ) ◦ Ψ on G ∞ C (H, R). Moreover, if g is the algebra of G, the exponential map

for every f ∈ Exp G : g → G is a linear map and it sends Zi in the i-th element of the standard basis of G ≡ RN , whence Exp G (x1 , . . . , xN )Z = (x1 , . . . , xN ), being (x1 , . . . , xN )Z = x1 Z1 + · · · + xN ZN . (3) The inversion on G is the Euclidean inversion −x. (4) For every i ∈ {1, . . . , N }, we have (x E y)i = xi + yi + Ri (x, y), where Ri (x, y) is a polynomial function depending on the xk ’s and yk ’s with k < i, and Ri (x, y) can be written as a sum of polynomials each containing a factor of the following type xh yk − xk yh

with h = k and h, k < i.

(5) Let Δλ be the linear map on h such that, for every i = 1, . . . , r, Δλ (X) = λi X

whenever X ∈ Vi .

Let δλ := πE ◦ Δλ ◦ πE−1 . Then (RN , E , δλ ) is a homogeneous Carnot group of the same step and number of generators as H.

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2 Abstract Lie Groups and Carnot Groups

Proof. (1). The proof of (1) is contained in the proof of Theorem 2.2.18. (2). In the proof of Theorem 2.2.18, we demonstrated that (see (2.55)) (i)

(i)

d Ψ (Zk ) = Ek

∀ i ∈ {1, . . . , r}, k ∈ {1, . . . , Ni } (1) (1) (r) (r) if E = E1 , . . . , EN1 ; . . . ; E1 , . . . , ENr (1) (1) (r) (r) and Z = Z1 , . . . , ZN1 ; . . . ; Z1 , . . . , ZNr

(2.59)

are, respectively, the usual notation for the adapted basis E and the notation for the Jacobian basis resulting from the coordinates induced by πE on RN . This proves the first part of (2). As to the first part, consider the following diagram (h∗ denotes the algebra of the Lie group (h, ) and Exp h : h∗ → h the related exponential map) (G, E ) Exp G

g

πE−1

(h, )

Exp H

(H, ∗)

dExp h d πE−1

h∗

Exp H d Exp H

h.

From the commutativity result in Theorem 2.1.59 it holds

Exp G = πE ◦ Exp h ◦ d πE−1 = πE ◦ (Exp H )−1 ◦ Exp H ◦ d Exp H ◦ d πE−1 = πE ◦ {d Exp H } ◦ d πE−1 ,

whence

Exp G = πE ◦ d Exp H ◦ d πE−1 .

(2.60)

Hence, Exp G is a linear map since πE , d Exp H and d πE−1 are linear. We now prove that (i) (i) Exp G (Zk ) = 0(1) , . . . , ek , . . . , 0(r) ∀ i ≤ r, k ≤ Ni ,

(2.61)

(i)

where ek denotes the k-th element of the standard basis of RNi (for k ∈ {1, . . . , Ni }). To this end, first notice that (i) (i) πE (Ek ) = 0(1) , . . . , ek , . . . , 0(r) ∀ i ≤ r, k ≤ Ni . (2.62) Then, by applying (2.60), we infer (i) (i) (i) Exp G (Zk ) = πE ◦ d Exp H ◦ d πE−1 (Zk ) = πE d Ψ (Zk ) (i) (i) = πE (Ek ) = 0(1) , . . . , ek , . . . , 0(r) . In the second equality, we invoked the definition of Ψ and Remark (2.1.52)-(iii), whereas in the third we exploited (2.59); finally, the last equality is (2.62).

2.2 Carnot Groups

141

Now, the linearity of Exp G and (2.61) produce Exp G (x1 , . . . , xN )Z r N i

(i) (i) xj Zj = Exp G i=1 j =1

=

Ni r

(i) (i) xj Exp G (Zj )

i=1 j =1

=

r

(1)

0 ,...,

xj(i) ej(i) , . . . , 0(r)

j =1

i=1

= (x

Ni

,...,x

(r)

(i) (1)

xj

i=1 j =1

(1)

=

Ni r

(i)

0 , . . . , ej , . . . , 0(r)

=

r

(1) 0 , . . . , x (i) , . . . , 0(r) i=1

).

(3). Since in (1) we proved that (G, E ) is isomorphic to (h, ) and πE provides a Lie-group isomorphism between them, which is also a linear map of vector spaces, we infer that, for any x ∈ G, −1 = πE − πE−1 (x) = −πE πE−1 (x) = −x. (x)−1 = πE πE−1 (x) Here we used the fact that the inversion in (h, ) is X → −X, for, by the Campbell– Hausdorff operation, we have X (−X) = X − X +

1 1 [X, −X] + [X, [X, −X]] + · · · = 0. 2 12

(4). To begin with, we know that, by definition, x E y = πE πE−1 (x) πE−1 (y) ∀ x, y ∈ RN . N Writing x = (x1 , . . . , xN ), we have X := πE−1 (x) = i=1 xi Ei . Analogously, N −1 Y := πE (y) = j =1 yj Ej . Let us consider the Campbell–Hausdorff operation X Y . Considering the form of the operation in (2.43), we see that, besides the summands N N N

xi E i + yj E j = (xi + yi )Ei , X+Y = i=1

j =1

i=1

any other summand has the following form: it is given by a rational number times ( ) (ad X)α1 (ad Y )β1 · · · (ad X)αn (ad Y )βn [X, Y ] , or an analogous term with [X, Y ] replaced by [Y, X]. Let us analyze the case [X, being analogous). Replacing X and Y (but in the final [X, Y ]) by NY ] (the other N x E and i i i=1 j =1 yj Ej , respectively, ( ) is a finite sum of summands of the following type:

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2 Abstract Lie Groups and Carnot Groups

(

)

p(x, y) [Ek1 , · · · [EkM , [X, Y ]]],

where p(x, y) is a polynomial, M ∈ N and i1 , . . . , iM ∈ {1, . . . , N }. Now, (

) is equal to p(x, y)[Ek1 , · · · [EkM , [X, Y ]]] =

N

p(x, y) xi yj [Ek1 , · · · [EkM , [Ei , Ej ]]]

i,j =1

=

{ · · ·} = { · · ·} + { · · ·} i=j

=

i<j

i>j

p(x, y) (xh yk − xk yh ) [Ek1 , · · · [EkM , [Eh , Ek ]]].

1≤h
In the last equality, we suitably renamed the dummy variables, and we exploited anti-commutativity [Ei , Ej ] = −[Ej , Ei ]. All these facts together prove that N

x E y = πE (X Y ) = πE (xi + yi )Ei + finite sum of terms like i=1

p(x, y) (xh yk − xk yh ) [Ek1 , · · · [EkM , [Eh , Ek ]]] . Finally, the linearity of πE proves that, for every i = 1, . . . , N , it holds (x E y)i = xi + yi + Ri (x, y), where Ri (x, y) is the sum of polynomials each containing a factor of the type xh yk − xk yh with h = k. The fact that Ri (x, y) does depend only on the xk ’s and yk ’s with k < i easily follows from (5) of the assertion and the results on homogeneous groups from Section 1.3.2. (5). The proof of (5) is contained in the proof of Theorem 2.2.18 (see also (1) and Proposition 2.2.8).

A natural question arises from the proof of Theorem 2.2.18. In that proof, we handled with a Lie group H, its Lie algebra h which (equipped with the Campbell– Hausdorff operation ) becomes a Lie group itself and, finally, we considered the Lie algebra h∗ of the Lie group (h, ) (roughly speaking, “the algebra of the algebra”). We now prove the very natural fact that h∗ is “essentially” h itself. Moreover, the exponential map Exp h : h∗ → h is essentially the “identity map”. Example 2.2.23. Before entering the details, we give an example. Let us consider the homogeneous Lie group (H, ∗) on R3 with the composition law ⎛ ⎞ x 1 + y1 ⎠. x2 + y2 x∗y =⎝ x3 + y3 + x1 y2

2.2 Carnot Groups

143

The Jacobian basis for h (the algebra of H) is Z1 = ∂1 ,

Z2 = ∂2 + x1 ∂3 ,

Z3 = [Z1 , Z2 ] = ∂3 .

Then a natural Lie group structure (h, ) is obtained by considering on h the Campbell–Hausdorff law and, at the same time, by identifying h to R3 via coordinates w.r.t. the Jacobian basis. This gives ⎛ ⎞ ξ1 + η1 ⎠. ξ 2 + η2 ξ η =⎝ ξ3 + η3 + 12 (ξ1 η2 − ξ2 η1 ) The next Jacobian basis related to (h, ) is 11 = ∂1 − ξ2 ∂3 , Z 2

12 = ∂2 + ξ1 ∂3 , Z 2

13 = [Z 11 , Z 12 ] = ∂3 . Z

Finally, consider the Lie algebra h∗ of (h, ). It is easily checked that the exponential map Exp h : h∗ → h i into Zi for i = 1, 2, 3. In other words, by an abuse of language, h∗ coincides maps Z with h.

A general formulation of the above facts is given by the following proposition, which highlights once more the relevance of the Jacobian basis. Notice that no stratification condition is supposed in the first part of the following statement. Proposition 2.2.24 (Algebra of the algebra). Let (H, ∗) be a connected and simply connected nilpotent Lie group with Lie algebra h. Let be the Campbell–Hausdorff operation on h defined in Definition 2.2.11, and let (h, ) be equipped with its natural Lie group structure (see Theorem 2.2.13). Finally, let h∗ denote the Lie algebra of the Lie group (h, ), and let Exp h : h∗ → h denote the related exponential map. Then, Exp h is a linear map of vector spaces. For example, if H is a homogeneous Carnot group, and we equip the above Lie algebras h and h∗ with the relevant Jacobian bases, then Exp h is the linear map related to the identity matrix. Proof. We follow the notation of the assertion, and we denote by Exp H : h → H the relevant exponential map. We know that the following diagram is commutative (see Theorem 2.1.59 jointly with Theorem 2.2.13): (h, )

Exp H

Exp h

h∗

(H, ∗) Exp H

d Exp H

h.

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2 Abstract Lie Groups and Carnot Groups

As a consequence, we have Exp h = (Exp H )−1 ◦ Exp H ◦ (d Exp H ) = d Exp H , so that Exp h is a linear map (here, the vector space structures on h and h∗ are considered). Let now (H, ∗) be a homogeneous Carnot group on RN . We consider on h the Jacobian basis Z1 , . . . , ZN (related to the operation ∗). We equip (h, ) with the natural system of coordinates given by h

N

ξj Zj → (ξ1 , . . . , ξN ) ∈ RN ,

j =1

thus identifying h to RN . On h ≡ RN , we have the Lie group law : we then consider ∗ of h∗ (related to the operation ). the Jacobian basis Z1∗ , . . . , ZN We have to show that d Exp H sends Zk∗ into Zk . To this end, given f ∈ C ∞ (H, R), we have (d Exp H (Zk∗ ))0 (f ) = (Zk∗ )0 (f ◦ Exp H ) N

d = (∂/∂ξk )|ξ =0 f Exp H ξj Zj f Exp (t Zk ) = (Zk )0 (f ). = dt 0 j =1

This ends the proof.

2.2.5 Sub-Laplacians of a Stratified Group We begin with a central definition. Definition 2.2.25 (Sub-Laplacian of a stratified group). Let H be a stratified group with Lie algebra h. The second order differential operator L=

m

Xj2

(2.63)

j =1

is referred to as a sub-Laplacian on H, if there exists a stratification for h h = V1 ⊕ · · · ⊕ Vr , as in Definition 2.2.3, such that X1 , . . . , Xm is a (linear) basis of V1 . With the above notation, we also say that L is a sub-Laplacian on H with related (or, generating) stratification (V1 , . . . , Vr ).

2.2 Carnot Groups

145

We shall treat a sub-Laplacian L as an operator on smooth functions on the open subsets of H. More precisely, if Ω ⊆ H is open and f ∈ C ∞ (Ω, R), by L(f ) we mean the function on Ω defined by Ω x →

m

(Xj )x Ω m → (Xj )m (f ) . j =1

Note that if L is as in (2.63), then m = dim(V1 ) is the number of generators of H 2 according to Definition 2.2.7. Observe also that if m j =1 Xj is a sub-Laplacian on H with related2 stratification (V1 , . . . , Vr ) and if {Y1 , . . . , Ym } is another basis of V1 , then m j =1 Yj is a sub-Laplacian on H. Since stratified groups and homogeneous Carnot’s are deeply related, a comparison of the above notion of sub-Laplacian and that of sub-Laplacian on a homogeneous Carnot group is in order. 2 Remark 2.2.26. Let H be a stratified group with an algebra h. Let L = m j =1 Xj be a sub-Laplacian on H, and let V = (V1 , . . . , Vr ) be the stratification of h related to L according to Definition 2.2.25. Let also E be a basis for h adapted to the stratification V and such that X1 , . . . , Xm are the first m elements of E. (The existence of such a basis E is evident.) By Proposition 2.2.22 (whose notation we presently follow), there exists a homogeneous Carnot group G on RN which is isomorphic to H. Let g be the Lie algebra of G, and let Z1 , . . . , ZN be the Jacobian basis of g (the Jacobian basis is well-posed for G is a homogeneous Carnot group on RN ). Let Ψ : G → H be the isomorphism, as in Proposition 2.2.22-(1). By Proposition 2.2.22-(2), we know, in particular, that dΨ (Zi ) = Xi

for every i = 1, . . . , m,

(2.64)

or, equivalently, Zi (f ◦ Ψ ) = Xi (f ) ◦ Ψ on G for every f ∈ C ∞ (H, R). Set ΔG :=

m

Zi2 (which is the canonical sub-Laplacian on G).

i=1

We infer that, for every f ∈ C ∞ (H, R), ΔG (f ◦ Ψ ) ≡ L(f ) ◦ Ψ, i.e. L(f ) ≡ ΔG (f ◦ Ψ ) ◦ Ψ −1 .

(2.65a)

Equivalently, since we have (with obvious meaning of the notation) C ∞ (H, R) ◦ Ψ ≡ C ∞ (G, R),

C ∞ (G, R) ◦ Ψ −1 ≡ C ∞ (H, R),

(2.65b)

for every given g ∈ C ∞ (G, R), if we set f := g ◦ Ψ −1 , (2.65a) rewrites as ΔG g ≡ L(g ◦ Ψ −1 ) ◦ Ψ ∀ g ∈ C ∞ (G, R). (2.65c) Roughly speaking, (2.65a) says that the sub-Laplacian L of H is turned by Ψ −1 into the canonical sub-Laplacian ΔG of G, or, equivalently, L and ΔG are Ψ -related.

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2 Abstract Lie Groups and Carnot Groups

This fact ensures that the theory of sub-Laplacians on a stratified group can be recast in its homogeneous-Carnot-group dual version. Moreover, let L1 and L2 be two sub-Laplacians on H with the same related stratification V . Suppose we perform the above identification of L1 with the canonical sub-Laplacian of G. What is L2 Ψ -related to on G? Obviously, to a (possibly noncanonical) sub-Laplacian on G. Indeed, let L1 =

m

Xi2 ,

L2 =

i=1

m

Yi2 .

i=1

We have that the Xi ’s and the Yi ’s form two bases for V1 . Hence, there exists a non-singular m × m matrix (ai,j ) such that Yi =

m

ai,j Xi

for every i = 1, . . . , m.

j =1

Thus we have (see (2.64)) i := dΨ −1 (Yi ) = Y

m

ai,j Zi ,

(2.66)

j =1

i ) = Yi ) so that (being dΨ (Y m m

2 2 i (f ◦ Ψ ) ≡ Yi (f ) ◦ Ψ Y j =1

j =1

2 for every f ∈ C ∞ (H, R). This proves the claimed fact that L2 = m j =1 Yi is Ψ m m 2 2 . In turn, j =1 Y is a sub-Laplacian on G, in force of (2.66), related to j =1 Y i i which grants that Y1 , . . . , Ym is a basis for span{Z1 , . . . , Zm }.

We introduce another central definition. Definition 2.2.27 (L-harmonic function on a stratified group). Let H be a stratified group, and let L be a fixed sub-Laplacian on H, according to Definition 2.2.25. Let Ω ⊆ H be an open set. A real-valued function f on Ω is called L-harmonic if and only if it holds f ∈ C ∞ (Ω, R)

and Lf = 0 on Ω.

(2.67)

We denote by HL (Ω) the vector space of the L-harmonic functions on Ω. m 2 Remark 2.2.28. (i) Let H be a stratified group, and let L = j =1 Xj be a subLaplacian on H with generating stratification V = (V1 , . . . , Vr ). Let E be any basis of the algebra of H adapted to the stratification V (but not necessarily containing X1 , . . . , Xm as its first elements).

2.3 Exercises of Chapter 2

147

Let Ψ = Exp H ◦ πE−1 be as in Proposition 2.2.22. Then, according to the results This means that in Remark 2.2.26, L is Ψ -related to a sub-Laplacian on G, say L. ◦ Ψ ) ◦ Ψ −1 , f = L(f◦ Ψ −1 ) ◦ Ψ Lf = L(f L (2.68) for every f smooth on H and every f smooth on G. := Ψ −1 (Ω) is an open (ii) Let now Ω = ∅ be an open subset of H. Then Ω subset of G. We claim that = HL (Ω) ◦ Ψ, HL(Ω)

◦ Ψ −1 . HL (Ω) = HL(Ω)

(2.69)

More precisely, f ∈ HL (Ω) f ∈ H (Ω) L

⇐⇒ ⇐⇒

f := f ◦ Ψ ∈ HL(Ω), f := f◦ Ψ −1 ∈ HL (Ω).

Indeed, (2.69) follows at once from (2.68). All the above facts ensure that the theory of the L-harmonic functions on (the open subsets of) a stratified group can be recast in its homogeneous-Carnot-group dual version. This is what it will be done in the foregoing chapters. Bibliographical Notes. In this chapter, we intended to provide only the topics of differential geometry and Lie group theory which are strictly necessary to read this book. The reader is referred to, e.g. M. Hausner, J.T. Schwartz [HS68], V.S. Varadarajan [Var84] and F.W. Warner [War83], for exhaustive introductions to the differential geometry and the theory of Lie groups and Lie algebras. See also L.J. Corwin and F.P. Greenleaf [CG90] for the theory of representation of nilpotent Lie groups.

2.3 Exercises of Chapter 2 Ex. 1) Following the lines of Example 2.2.21 (page 137), find an explicit isomorphism of Lie groups between a suitable homogeneous Carnot group and the (non-homogeneous) stratified group G = (Ω, ∗), where π π × (0, ∞) Ω := R × − , 2 2 and ξ ∗η =

ξ1 + η1 + 2(ξ3 ln η3 + tan ξ2 ln η3 − η3 ln ξ3 − tan η2 ln ξ3 ) arctan(ξ3 + η3 + tan ξ2 + tan η2 − ξ3 η3 ) ξ3 η3

.

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2 Abstract Lie Groups and Carnot Groups

Ex. 2) Let h be a nilpotent Lie algebra. Denote by the Campbell–Hausdorff multiplication on h (see Definition 2.2.11, page 128). Then (h, ) is a Lie group (see the first part of Corollary 2.2.15). Denote by h∗ the Lie algebra of this group. Prove that h and h∗ are isomorphic as Lie algebras. Hint: Let (G, ∗) be a Lie group whose Lie algebra g is isomorphic to h (see the third fundamental theorem of Lie, Theorem 2.2.14). Let ϕ : h → g be a Lie-algebra isomorphism. Denote by Exp G : g → G the exponential map related to G. Prove that Exp G ◦ ϕ is a Lie-group isomorphism from (h, ) to (G, ∗), using the universality of the operation . Derive that d(Exp G ◦ ϕ) : h∗ → g is a Lie-algebra isomorphism. Consequently, ϕ −1 ◦ d(Exp G ◦ ϕ) is a Lie-algebra isomorphism from h∗ to h. See also the following diagram ϕ

(g, )

Exp G

−→

−→

−→

(h, ) −→ ↑ h∗

−→ (G, ∗) ↑

d(Exp G ◦ ϕ)

g

ϕ −1

−→ h.

Ex. 3) Let h be a nilpotent Lie algebra. Denote by the Campbell–Hausdorff multiplication on h (see Definition 2.2.11, page 128). Let (G, ∗) be a Lie group isomorphic to the Lie group (h, ). Prove that the Lie algebra g of G is isomorphic to h as Lie algebras. (Hint: Use the preceding exercise: g is isomorphic to h∗ , which is isomorphic to h.) Ex. 4) Prove that the following is another equivalent definition of stratified group: (H∗) A stratified group is a connected and simply connected Lie group G whose Lie algebra g admits a (vector space) decomposition of the type g = V1 ⊕ · · · ⊕ Vr , where

[Vi , Vj ] ⊆ Vi+j

∀ i, j : i + j ≤ r,

[Vi , Vj ] = 0

∀ i, j : i + j > r,

(2.70)

and V1 generates (by iterated commutators) all g. In fact, in (2.70), it holds [Vi , Vj ] = Vi+j if i + j ≤ r. This is yet another equivalent definition of Carnot group, frequently adopted in literature. Hint: Use the following facts: If G is a stratified group according to Definition 2.2.3, then (setting Vi := {0} if i > r) it holds Vi = [V1 , · · · [V1 , V1 ]] )* + (

for every i ∈ N.

i times

In particular, V1 Lie-generates all the Vi ’s (whence it generates also g = V1 ⊕ · · · ⊕ Vr ). This also gives (using Proposition 1.1.7, page 12)

2.3 Exercises of Chapter 2

149

/ , [Vi , Vj ] = [V1 , · · · [V1 , V1 ]], [V1 , · · · [V1 , V1 ]] )* + ( )* + ( i times

j times

⊆ [V1 , · · · [V1 , V1 ]] = Vi+j . )* + ( i + j times

In particular, (2.70) holds. Vice versa, let G satisfy the above hypothesis (H∗). Set W1 := V1 and Wi := [W1 , Wi−1 ] = [V1 , · · · [V1 , V1 ]] )* + (

for i ≥ 2.

i times

Prove that condition (2.70) implies that Wi ⊆ Vi for every 1 ≤ i ≤ r and Wi = {0} for every i > r. Moreover, the second hypothesis in (H∗) (i.e. V1 Lie-generates g) ensures that g = W1 + · · · + Wr . Now, a simple linear algebra argument shows that the following conditions W1 + · · · + Wr = g = V1 ⊕ · · · ⊕ Vr ,

Wi ⊆ Vi

∀i ≤r

are sufficient to derive that Wi = Vi for every 1 ≤ i ≤ r. As a consequence, we have [V1 , Vj ] = [W1 , Wj ] = Wj +1 = Vj +1 whenever 1 + j ≤ r, and [V1 , Vj ] = [W1 , Wj ] = {0} whenever 1+j > r, so that G is a Carnot group according to Definition 2.2.3. Ex. 5) Let g be a Lie algebra. Consider the so-called lower central series of g, i.e. the sequence of subspaces defined by g(1) := g,

g(j ) := [g, g(j −1) ],

j ≥ 1.

In other words, for every j ≥ 2, we have g(j ) = [g, [g, · · · [g, g]]] . ( )* + j times

Prove the following facts: • It holds g(1) ⊇ g(2) ⊇ · · · ⊇ g(j ) ⊇ g(j +1) ⊇ · · · for every j ∈ N; • if there exists r ≥ 1 such that g(r) = g(r+1) , then g(r) = g(j ) for every j ≥ r; • g is nilpotent of step r iff g(r+1) = {0}, but g(r) = {0}, • g is nilpotent of step r iff g(1) g(2) · · · g(r) g(r+1) = {0}. Ex. 6) Prove the following fact: A (finite dimensional) nilpotent Lie algebra g of step two is necessarily stratified.

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2 Abstract Lie Groups and Carnot Groups

Indeed, let us set V2 = [g, g] and choose any V1 such that g = V1 ⊕ V2 : then it also holds [V1 , V1 ] = V2 and [V1 , V2 ] = {0}. Hint: for every X, Y ∈ g, write X = X1 + X2 and Y = Y1 + Y2 , where Xi , Yi ∈ Vi , and observe that [g, g] [X, Y ] = [X1 , Y1 ] ∈ [V1 , V1 ]. Ex. 7) Consider the Carnot group on R4 (whose points are denoted by (x, y) with x ∈ R, y = (y1 , y2 , y3 ) ∈ R3 ) with the composition law ⎛

⎞ x+ξ y 1 + η1 ⎜ ⎟ (x, y) ◦ (ξ, η) = ⎝ ⎠. y2 + η2 + 12 (xη1 − ξy1 ) 1 y3 + η3 + 12 (xη2 − ξy2 ) + 12 (x − ξ )(xη1 − ξy1 ) Its Lie algebra g is spanned by 1 1 1 y1 ∂y2 − y2 ∂y3 − xy1 ∂y3 , 2 2 12 1 1 2 x ∂y3 , Y1 = ∂y1 + x ∂y2 + 2 12 1 Y2 = ∂y2 + x ∂y3 , 2 Y3 = ∂y3 , X = ∂x −

and the following commutator relations hold: [X, Y1 ] = Y2 ,

[X, Y2 ] = Y3 ,

[X, Y3 ] = 0,

[Yi , Yj ] = 0,

i, j ∈ {1, 2, 3}. Prove that the following are three different stratifications of g: g = span{X, Y1 } ⊕ span{Y2 } ⊕ span{Y3 }, g = span{X, Y1 + Y2 } ⊕ span{Y2 + Y3 } ⊕ span{Y3 }, g = span{X, Y1 + Y2 + Y3 } ⊕ span{Y2 + Y3 } ⊕ span{Y3 }. Ex. 8) Let g be a (finite dimensional) stratified Lie algebra, i.e. suppose we have g = V (1) ⊕ V (2) ⊕ · · · ⊕ V (r) with (set V (i) := {0} whenever i > r) [V (1) , V (i) ] = V (i+1) for every i ∈ N. Prove the following facts: • •

•

g is nilpotent of step r; V (r) = g(r) , the r-th element of the lower central series for g (see Ex. 5); (1) (r) (Hint: for every X1 , · · · , Xr ∈ g, write Xi = Xi + · · · + Xi , where (1) (r) Xi ∈ V (1) , · · · , Xi ∈ V (r) , and observe that [X1 , · · · [Xr−1 , Xr ]] = (1) (1) (1) [X1 , · · · [Xr−1 , Xr ]].) (1) Suppose V ⊕ · · · ⊕ V (r) and W (1) ⊕ · · · ⊕ W (r) are two stratifications of g. Prove that

2.3 Exercises of Chapter 2

W (r) = V (r) , (r) W (r−1) = V (r−1) ⊕ Ur−1 (r−1)

151

(r)

(where Ur−1 is a subspace of V (r) ), (r)

W (r−2) = V (r−2) ⊕ Ur−2 ⊕ Ur−2 (r−1)

(r)

(where Ur−2 , Ur−2 are subspaces of V (r−1) , V (r) , respectively). In other words, we have W (r) = V (r) , W (r−1) = V (r−1) (modulo V (r) ), W (r−2) = V (r−2) (modulo V (r−1) + V (r) ), .. . (1) W = V (1) (modulo V (2) + · · · + V (r−1) + V (r) ). Precisely, prove the following facts: (i) dim(V (i) ) = dim(W (i) ) =: Ni for every i = 1, . . . , r. Choose two bases V and W of g adapted respectively to the stratifications with the V (i) ’s and with the W (i) ’s. Set V = {V1 , . . . , VN } and W = {W1 , . . . , WN } and prove the existence of non-singular matrices M (i) ’s of order Ni × Ni such that (with clear meaning of the notation) ⎛ (1) ⎞ M ··· ⎛ ⎛ ⎞ ⎞ W1 .. ⎟ V1 .. ⎜ (2) . . ⎟ ⎝ .. ⎠ 0 M ⎝ ... ⎠ = ⎜ . ⎜ . ⎟ . .. .. ⎝ .. ⎠ . . WN VN 0 ··· 0 M (r) (Hint: By the preceding part of the exercise, we have W (r) = V (r) = g(r) . Now consider the quotient g/g(r) . It holds (1)

(r−1)

(1)

(r−1)

W/W (r) ⊕ · · · ⊕ W/W (r) = g/g(r) = V/V (r) ⊕ · · · ⊕ V/V (r) , and these are stratifications of the Lie algebra g/g(r) too. Consequently, (r−1)

(r−1)

W/W (r) = V/V (r) . Proceed inductively. Alternatively, argue as in the two-stratification Lemma 2.2.9.) Ex. 9) Let g be a stratified Lie algebra, i.e. g = V (1) ⊕ V (2) ⊕ · · · ⊕ V (r) with (set V (i) := {0} whenever i > r) [V (1) , V (i) ] = V (i+1) for every i ∈ N. Prove that the lower central series of g (see Ex. 5 above) is given by g(k) =

r 2

V (i)

∀ k = 1, . . . , r,

g (k) = {0} ∀ k > r.

i=k

3r (i) arguing inductively as follows: (Hint: First prove that g(k) ⊆ i=k V 3r (2) (1) (r) (i) = [g, g] = [V + · · · + V , V (1) + · · · + V (r) ] ⊆ g i=2 V ;

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2 Abstract Lie Groups and Carnot Groups

3 3 3 g(3) = [g, [g, g]] ⊆ [ ri=1 V (i) , ri=2 V (i) ] ⊆ ri=3 V (i) , etc. Vice versa, it holds V (i) = span{[X1 , · · · [Xi−1 , Xi ] · · ·] : X1 , . . . , Xi ∈ V (1) } ⊆ [[g, · · · [g, g] · · ·]] = g(i) , ( )* + i times

then derive

3r i=k

V (i) = V (k) ⊕ · · · ⊕ V (r) ⊆ g(k) ⊕ · · · ⊕ g(r) = g(k) .)

Ex. 10) The natural identification between the Lie algebra g of Lie group G with the tangent space Ge to G at the identity must be “handled with care”, as the following example shows. Suppose G and H are Lie groups with Lie algebras g, h, respectively. We denote by e both the identity of G and that of H. Suppose F : G → H is a C ∞ -map such that F (e) = e. Obviously, we have de F : Ge → He . But Ge is identified with g and He is identified with h. Consequently, F induces a natural map, say f , between g and h. The following question arises: as in the case when F is a group homomorphism, does f represent pointwise the differential of F ? Id est (see (2.33)), does it hold (for every X ∈ g and every x ∈ G) ?

(f (X))F (x) = dx F (Xx ). The answer is in general negative. Indeed, with the above notation (see also (2.26)) the map f : g → h is defined in the following way: for every X ∈ g and every y ∈ H, we have (f (X))y = de τy (de F (Xe )), i.e. f (X) ∈ h is the vector field on H defined by H y → de (τy ◦ F )(Xe ). We ask whether ?

dx F (Xx ) = de (τF (x) ◦ F )(Xe )

(∗1)

∀ X ∈ g, x ∈ G.

Since X is left-invariant, it holds Xx = de τx (Xe ), so that dx F (Xx ) = dx F (de τx (Xe )) = de (F ◦ τx )Xe . As a consequence, (∗1) holds iff (recall that Ge = {Xe : X ∈ g} by Theorem 2.1.43-(1)) (∗2)

?

de (F ◦ τx ) = de (τF (x) ◦ F )

∀ x ∈ G.

(Note that this certainly holds if F is a group homomorphism!) In turn, this is equivalent to (∗3)

?

dx F = dx (τF (x) ◦ F ◦ τx −1 )

∀ x ∈ G.

2.3 Exercises of Chapter 2

Note that (∗3) is a “differential equation” dx F = de τF (x) ◦ de F ◦ dx τx −1 F (e) = e.

153

for every x ∈ G,

For example, prove that the above system holds for G = H = (RN , +) iff F is a linear map, i.e. F is an automorphism of the Lie group (RN , +). Consequently, the map F : (R, +) → (R, +), F (x) = x 2 furnishes a counterexample. Ex. 11) Suppose that G = (RN , ◦) and H = (Rn , ∗) are two Lie groups on RN and Rn , respectively. Let g and h denote the relevant Lie algebras. Finally, let ϕ : G → H be a homomorphism of Lie groups. If dϕ : g → h is the map defined in Definition 2.1.49, prove that it operates in the following way: for every X ∈ g, dϕ(X) is the left invariant vector field on H whose column vector of the coefficients at the point y ∈ Rn is given by dϕ(X) I (y) = Jτy ◦ϕ (0) · XI (0), or, equivalently, dϕ(X) I (y) = Jτy (0) · Jϕ (0) · XI (0). Observe that (since Jτ0 (0) is the identity matrix, and since the coordinates of a vector field Y w.r.t. the Jacobian basis are given by Y I (0)) the matrix representing the linear map dϕ with respect to the relevant Jacobian bases on g and h, respectively, is simply Jϕ (0). Ex. 12) Provide the ODE’s details for what is stated in the first paragraph of the proof of Proposition 2.1.53, page 116. Ex. 13) Suppose G = (RN , ◦) and H = (Rn , ∗) are two Lie groups on RN and Rn , respectively (according to the definition given in Chapter 1). Let g and h be the relevant Lie algebras. Let ϕ : G → H be a homomorphism of Lie groups. Show that the differential of ϕ, as defined in Definition 2.1.49, is the map dϕ : g → h such that, for every X ∈ g, dϕ(X) is the only vector field in h such that (2.71) dϕ(X)I (ϕ(x)) = Jϕ (x) (XI )(x). Ex. 14) The following is an example to the “change of adapted basis”, described in Remark 2.2.20. We follow the therein notation. Example 2.3.1 (Stratified groups of step two). Let (H, ∗) be a homogeneous Carnot group of step two on RN . We shall prove in Section 3.2, page 158, that if Z denotes the Jacobian basis related to H, then it holds

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2 Abstract Lie Groups and Carnot Groups

(ξ, τ ) Z (ξ , τ ) =

ξj + ξj (j = 1, . . . , m) τi + τi + 12 B (i) ξ, ξ (i = 1, . . . , n)

,

where m is the number of generators of H, n = N − m and if {X1 , . . . , Xm ; T1 , . . . , Tn } re-denotes the Jacobian basis, then B (i) is the m × m matrix (i) B (i) = bh,k 1≤h,k≤m defined by [Xk , Xh ] =

n

(i) k=1 bh,k Ti .

is another basis of the algebra of g adapted to its stratification, say If Z = {X 1 , . . . , X m ; T1 , . . . , Tn }, Z there exist two non-singular matrices U (of order m × m) and V (of order n × n) such that 1 I · · · X m I = X1 I · · · Xm I · U, X T1 I · · · Tn I = T1 I · · · Tn I · V . A simple computation shows that the new composition law Z is given by (using, for example, (2.58a)) ξj + ξj (j = 1, . . . , m) (ξ, τ ) Z (ξ , τ ) = , (i) ξ, ξ (i = 1, . . . , n) τi + τi + 12 B where, if V −1 = (wi,j )i,j ≤n , (i)

B

=U · T

n

wi,j B

j =1

The test of this fact is left to the reader.

(j )

· U.

3 Carnot Groups of Step Two

The aim of this chapter is to collect some results and many explicit examples of Carnot groups of step two. Some examples are well known in literature, some are new. To begin with, we present the most studied (and by far one of the most important) among Carnot groups, the Heisenberg–Weyl group. Then, we turn our attention to general homogeneous Carnot groups of step two and m generators, m ≥ 2. In particular, we show that they are naturally given with the data on Rm+n of n skewsymmetric matrices of order m. The set of examples that we provide here contains the free step-two homogeneous groups, the prototype groups of Heisenberg-type (which will be widely studied in Chapter 18) and the H-groups in the sense of Métivier.

3.1 The Heisenberg–Weyl Group Let us consider in Cn × R (whose points we denote by (z, t) with t ∈ R and z = (z1 , . . . , zn ) ∈ Cn ) the following composition law (z, t) ◦ (z , t ) = (z + z , t + t + 2 Im(z · z )).

(3.1)

In (3.1), we have set (i obviously denotes the imaginary unit) Im(x + iy) = y (x, y ∈ R), whereas z · z denotes the usual Hermitian inner product in Cn , z · z =

n (xj + iyj )(xj − iyj ). j =1

Hereafter we agree to identify Cn with R2n and to use the following notation1 to denote the points of Cn × R ≡ R2n+1 : 1 Someone may say that the notation (z, t) ≡ (x , y , . . . , x , y , t) would be more appron n 1 1

priate, but the other notation is so deeply entrenched that we have no choice but to go along with it.

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3 Carnot Groups of Step Two

(z, t) ≡ (x, y, t) = (x1 , . . . , xn , y1 , . . . , yn , t) with z = (z1 , . . . , zn ), zj = xj + iyj and xj , yj , t ∈ R. Then, the composition law ◦ can be explicitly written as (3.2) (x, y, t) ◦ (x , y , t ) = x + x , y + y , t + t + 2y, x − 2x, y , where ·, · denotes the usual inner product in Rn . It is quite easy to verify that (R2n+1 , ◦) is a Lie group whose identity is the origin and where the inverse is given by (z, t)−1 = (−z, −t). Let us now consider the dilations δλ : R2n+1 → R2n+1 ,

δλ (z, t) = (λz, λ2 t).

A trivial computation shows that δλ is an automorphism of (R2n+1 , ◦) for every λ > 0. Then Hn = (R2n+1 , ◦, δλ ) is a homogeneous group. It is called the Heisenberg–Weyl group in R2n+1 . For example, when n = 1, the Heisenberg–Weyl group H1 in R3 is equipped with the composition law (x, y, t) ◦ (x , y , t ) = (x + x , y + y , t + t + 2 (yx − xy )), while, when n = 2, the Heisenberg–Weyl group H2 in R5 is equipped with the composition law (x1 , x2 , y1 , y2 , t) ◦ (x1 , x2 , y1 , y2 , t )

= (x1 + x1 , x2 + x2 , y1 + y1 , y2 + y2 , t + t + 2 (y1 x1 + y2 x2 − x1 y1 − x2 y2 )).

The Jacobian matrix at the origin of the left translation τ(z,t) is the following block matrix ⎞ ⎛ In 0 0 0 ⎠, In Jτ(z,t) (0, 0) = ⎝ 0 2 y T −2 x T 1 where In denotes the n × n identity matrix, while 2 y T and −2 x T stand for the 1 × n matrices (2y1 · · · 2yn ) and (−2x1 · · · − 2xn ), respectively. Then, the Jacobian basis of hn , the Lie algebra of Hn , is given by Xj = ∂xj + 2yj ∂t ,

Yj = ∂yj − 2xj ∂t ,

j = 1, . . . , n,

T = ∂t .

Since [Xj , Yj ] = −4 ∂t , we have rank Lie{X1 , . . . , Xn , Y1 , . . . , Yn }(0, 0) = dim span{∂x1 , . . . , ∂xn , ∂y1 , . . . , ∂yn , −4∂t } = 2n + 1. This shows that Hn is a Carnot group with the following stratification2 hn = span{X1 , . . . , Xn , Y1 , . . . , Yn } ⊕ span{∂t }.

(3.3)

2 Obviously, there exist other possible stratifications, but the above one is the most frequently

adopted and the one we shall refer to.

3.1 The Heisenberg–Weyl Group

157

The step of (Hn , ◦) is r = 2 and its Jacobian generators are the vector fields Xj , Yj (j = 1, . . . , n). The canonical sub-Laplacian on Hn (also referred to as Kohn Laplacian) is then given by ΔHn =

n 2 Xj + Yj2 . j =1

An explicit formula for ΔHn can be found in Ex. 1, at the end of this chapter. Finally, we exhibit the explicit form of the exponential map for Hn . It is given by3 Exp ((ξ, η, τ ) · Z) = (ξ, η, τ ). (3.4) n Here we have set (ξ, η, τ ) · Z = j =1 (ξj Xj + ηj Yj ) + τ T . We now want to perform a change of variables in Hn , inspired by the change of basis in hn , which turns the Jacobian basis into the new basis Xj∗ := Xj ,

Yj∗ := Yj

(j = 1, . . . , n), T ∗ := [X1 , Y1 ] = −4 T . As above, we set (ξ, η, τ ) · Z ∗ = nj=1 (ξj Xj∗ + ηj Yj∗ ) + τ T ∗ . In other words, we have chosen another basis of hn adapted to the stratification (3.3). Then, what we aim to do (see the proof of Theorem 2.2.18, page 131) is to equip hn with a Lie-group structure isomorphic (via Exp ) to that of (Hn , ◦); we identify hn with R2n+1 via the cited basis Z ∗ : this amounts to equip R2n+1 with the composition ∗ such that Log (Exp ((ξ, η, τ ) · Z ∗ )) ◦ (Exp ((ξ , η , τ ) · Z ∗ )) = (ξ, η, τ ) ∗ (ξ , η , τ ) · Z ∗ . The explicit expression for ∗ is easily found,4 (ξ, η, τ ) ∗ (ξ , η , τ )

1 1 = ξ + ξ , η + η , τ + τ − η, ξ + ξ, η , 2 2

(3.5a)

or, equivalently, 1 (ζ, τ ) ∗ (ζ , τ ) = ζ + ζ , τ + τ + Bζ, ζ 2

with B=

0 In

−In 0

(3.5b)

.

The natural isomorphism ϕ : (R2n+1 , ∗) → (Hn , ◦) is given by ϕ(ξ, η, τ ) = Exp ((ξ, η, τ ) · Z ∗ ) = Exp ((ξ, η, −4τ ) · Z) = (ξ, η, −4τ ). 3 See (3.11a) in Section 3.5, page 167, and Remark 3.2.4, page 163. 4 For all the details, see Section 4.3.2, page 200.

158

3 Carnot Groups of Step Two

3.2 Homogeneous Carnot Groups of Step Two Let m, n ∈ N. Set RN := Rm × Rn and denote its points by z = (x, t) with x ∈ Rm and t ∈ Rn . Given an n-tuple B (1) , . . . , B (n) of m × m matrices with real entries, let

1 (3.6) (x, t) ◦ (ξ, τ ) = x + ξ, t + τ + Bx, ξ . 2 Here Bx, ξ denotes the n-tuple (1) B x, ξ , . . . , B (n) x, ξ

also written as

m

Bi,j xj ξi

i,j =1

and ·, · stands for the inner product in Rm . One can easily verify that (RN , ◦) is a Lie group whose identity is the origin and where the inverse is given by (x, t)−1 = −x, −t + Bx, x . We highlight that the inverse map is the usual −(x, t) if and only if, for every k = 1, . . . , n, it holds B (k) x, x = 0 ∀ x ∈ Rm , i.e. iff the matrices B (k) ’s are skew-symmetric. It is also quite easy to recognize that the dilation (3.7) δλ : RN → RN , δλ (x, t) = (λx, λ2 t) is an automorphism of (RN , ◦) for any λ > 0. Then G = (RN , ◦, δλ ) is a homogeneous Lie group. We explicitly remark that the composition law of any Lie group in Rm × Rn , homogeneous w.r.t. the dilations {δλ }λ as in (3.7), takes the form (3.6) (see Theorem 1.3.15, page 39). The Jacobian matrix at (0, 0) of the left translation τ(x,t) takes the following block form ⎛ ⎞ Im 0 ⎠, Jτ(x,t) (0, 0) = ⎝ 1 B x I n 2 (k)

where, if B (k) = (bi,j )i,j ≤m for k = 1, . . . , n, Bx denotes the matrix m

(k) bi,j xj . j =1

More explicitly, we have

k≤n, i≤m

⎛

⎜ ⎜ ⎜ ⎜ Jτ(x,t) (0, 0) = ⎜ ⎜ ⎜ ⎜ ⎝

Im

1 2

1 2

m

(1) j =1 b1,j

m

xj · · ·

.. .

(n) j =1 b1,j xj

··· ···

0m×n

1 2

1 2

m

(1) j =1 bm,j

m

xj

.. .

(n) j =1 bm,j

In xj

⎞ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎠

3.2 Homogeneous Carnot Groups of Step Two

159

Then the Jacobian basis of g, the Lie algebra of G, is given by m

n 1 (k) bi,l xl (∂/∂tk ) Xi = (∂/∂xi ) + 2 k=1

l=1

1 = (∂/∂xi ) + (Bx)i , ∇t , 2 Tk = ∂/∂tk , k = 1, . . . , n.

i = 1, . . . , m,

(3.8)

Here, we briefly denoted by (Bx)i the vector of Rn (1) (B x)i , . . . , (B (n) x)i , where (B (k) x)i is the i-th component of B (k) x. An easy computation shows that [Xj , Xi ] =

n n 1 (k) (k) (k) bi,j − bj,i ∂tk =: ci,j ∂tk . 2 k=1

k=1

(k)

We have denoted by C (k) = (ci,j )i,j ≤m the skew-symmetric part of B (k) , i.e. C (k) =

1 (k) B − (B (k) )T . 2

Let us now assume that C (1) , . . . , C (n) are linearly independent. This implies that the m2 × n matrix ⎛ ⎞ (1) (n) C1,1 ··· C1,1 ⎜ (n) ⎟ ⎜ C (1) ··· C1,2 ⎟ ⎜ ⎟ 1,2 ⎜ .. .. ⎟ ⎜ . ··· . ⎟ ⎜ ⎟ ⎜ (n) ⎟ ⎜ C (1) ··· C1,m ⎟ 1,m ⎜ ⎟ ⎜ (1) (n) ⎟ ⎜ C2,1 ⎟ ··· C2,1 ⎜ ⎟ ⎜ .. .. ⎟ ⎜ . ··· . ⎟ ⎜ ⎟ ⎜ (proceed analogously up to) ⎟ ⎜ ⎟ ⎜ .. .. ⎟ ⎜ . ··· . ⎟ ⎝ ⎠ (1) (n) ··· Cm,m Cm,m has rank equal to n. As a consequence, span{[Xj , Xi ] | i, j = 1, . . . , m} = span{∂t1 , . . . , ∂tn }. Therefore, rank Lie{X1 , . . . , Xm }(0, 0) = dim span{∂x1 , . . . , ∂xm , ∂t1 , . . . , ∂tn } = m + n. This shows that G is a Carnot group of step two and Jacobian generators X1 , . . . , Xm .

160

3 Carnot Groups of Step Two

We explicitly remark that the linear independence of the matrices C (1) , . . . , C (n) is also necessary for G to be a Carnot group. Then, we have proved the following proposition. Proposition 3.2.1 (Characterization. I). Every homogeneous Lie group G on RN , homogeneous with respect to the dilation δλ : RN → RN ,

δλ (x, t) = (λx, λ2 t)

(where x ∈ Rm , t ∈ Rn and N = m + n), is equipped with the composition law

1 (1) 1 (n) (x, t) ◦ (ξ, τ ) = x + ξ, t1 + τ1 + B x, ξ , . . . , tn + τn + B x, ξ 2 2 for n suitable m × m matrices B (1) , . . . , B (n) . Moreover, a characterization of homogeneous Carnot groups of step two and m generators is given by the above G = (Rm+n , ◦, δλ ), where the skew-symmetric parts of the B (k) ’s are linearly independent. We remark that the above arguments show that there exist Carnot groups of any dimension m ∈ N of the first layer and any dimension n ≤ m(m − 1)/2 of the second layer: it suffices to choose n linearly independent matrices B (1) , . . . , B (n) in the vector space of the skew-symmetric m × m matrices (which has dimension m(m − 1)/2) and then define the composition law as in (3.6). Finally, by means of the general results on stratified groups in Chapter 2, we obtain the following theorem. Theorem 3.2.2 (Characterization. II). Every N -dimensional (not necessarily homogeneous) stratified group of step two and m generators is naturally isomorphic to a homogeneous Carnot group (RN , ◦, δλ ) with Lie group law as in (3.6) for some m × m linearly independent skew-symmetric matrices B (k) ’s. The group of dilations is given by (3.7), and the inverse of x is −x. By (3.8), we can write explicitly the canonical sub-Laplacian of the Lie group G = (RN , ◦) with ◦ as in (3.6). It is given by ΔG = Δx +

n 1 (h) B x, B (k) x ∂th tk 4 h,k=1

+

m

B (k) x, ∇x ∂tk +

k=1

n 1 trace(B (k) ) ∂tk . 2 k=1

(3.9)

3.2 Homogeneous Carnot Groups of Step Two

161

Here, we denoted Δx =

m

∂xi ,xi

and ∇x = (∂x1 , . . . , ∂xm ).

i=1

We recognize that ΔG contains partial differential terms of second order only if trace(B (k) ) = 0 for every k = 1, . . . , n. This happens, for example, if the B (k) ’s are skew-symmetric, i.e. if the inverse map on G is x → −x. Example 3.2.3. Following all the above notation, let us take m = 3, n = 2 and ⎛ ⎞ ⎛ ⎞ 1 1 0 0 0 −1 B (2) = ⎝ 0 1 0 ⎠ . B (1) = ⎝ −1 0 0 ⎠ , 0 0 0 1 0 0 Then the composition law on R5 = R3 × R2 as in (3.6) becomes (denoting (x, t) = (x1 , x2 , x3 , t1 , t2 ) and analogously for (ξ, τ )) ⎛ ⎞ x 1 + ξ1 ⎜ ⎟ x 2 + ξ2 ⎜ ⎟ ⎜ ⎟, x + ξ 3 3 (x, t) ◦ (ξ, τ ) = ⎜ ⎟ 1 ⎝ t1 + τ1 + 2 (x1 ξ1 + ξ1 x2 − ξ2 x1 ) ⎠ t2 + τ 2 +

1 2

(x2 ξ2 − ξ1 x3 − ξ3 x1 )

and the dilation is δλ (x1 , x2 , x3 , t1 , t2 ) = (λx1 , λx2 , λx3 , λ2 t1 , λ2 t2 ). Then G = (R5 , ◦, δλ ) is a homogeneous Carnot group, for the skew-symmetric parts of B (1) and B (2) are linearly independent, ⎛ ⎞ ⎛ ⎞ 0 1 0 0 0 −1 1 (1) 1 B − (B (1) )T = ⎝ −1 0 0 ⎠ , B (2) − (B (2) )T = ⎝ 0 0 0 ⎠ . 2 2 0 0 0 1 0 0 In fact, we can compute the first three vector fields of the Jacobian basis and verify that they are Lie-generators for the whole Lie algebra, 1 1 X1 = ∂x1 + (x1 + x2 )∂t1 − x3 ∂t2 , 2 2 1 1 X2 = ∂x2 − x1 ∂t1 + x2 ∂t2 , 2 2 1 X3 = ∂x3 + x1 ∂t2 , 2 [X1 , X2 ] = −∂t1 , [X1 , X3 ] = ∂t2 , 1 [X2 , X3 ] = ∂t2 . 2

162

3 Carnot Groups of Step Two

The related canonical sub-Laplacian is ΔG = ∂x1 ,x1 + ∂x2 ,x2 + ∂x3 ,x3 1 (x1 + x2 )2 + (−x1 )2 ∂t1 ,t1 + (−x3 )2 + (x2 )2 + (x1 )2 ∂t2 ,t2 + 4 + 2 (x1 + x2 )(−x3 ) + (−x1 )(x2 ) ∂t1 ,t2 + {(x1 + x2 ) ∂x1 − x1 ∂x2 } ∂t1 + {−x3 ∂x1 + x2 ∂x2 + x1 ∂x3 } ∂t2 1 1 + ∂t1 + ∂t2 . 2 2 ΔG contains first order terms, for trace(B (1) ) = 0 = trace(B (2) ). On the contrary, if ⎛ ⎞ ⎛ ⎞ 1 1 0 0 −2 0 B (1) = ⎝ −1 0 0 ⎠ , B (2) = ⎝ 2 1 0 ⎠ , 0 0 0 0 0 0 then the composition law on R5 given by ⎛

⎞ x 1 + ξ1 ⎜ ⎟ x 2 + ξ2 ⎜ ⎟ ⎜ ⎟ x + ξ (x, t) ◦ (ξ, τ ) = ⎜ 3 3 ⎟ ⎝ t1 + τ1 + 1 (x1 ξ1 + ξ1 x2 − ξ2 x1 ) ⎠ 2 t2 + τ2 + 12 (x2 ξ2 − 2ξ1 x2 + 2ξ2 x1 )

does not define a homogeneous Carnot group, because the skew-symmetric parts of B (1) and B (2) are linearly dependent, ⎛ ⎞ ⎛ ⎞ 0 1 0 0 −2 0 1 (1) 1 B − (B (1) )T = ⎝ −1 0 0 ⎠ , B (2) − (B (2) )T = ⎝ 2 0 0 ⎠ . 2 2 0 0 0 0 0 0 In fact, the only admissible dilation would be δλ (x, t) = (λ x1 , λ x2 , λ x3 , λ2 t1 , λ2 t2 ), but the first three vector fields of the related Jacobian basis are not Lie-generators for the whole Lie algebra, since 1 X1 = ∂x1 + (x1 + x2 )∂t1 − x2 ∂t2 , 2

1 1 x2 + x1 ∂t2 , X2 = ∂x2 − x1 ∂t1 + 2 2 X3 = ∂x3 , [X1 , X2 ] = −∂t1 + 2∂t2 , [X1 , X3 ] = [X2 , X3 ] = 0.

3.3 Free Step-two Homogeneous Groups

163

Remark 3.2.4. The composition law on the Heisenberg–Weyl group Hk on R2k+1 , which is given by (x, y, t) ◦ (x , y , t ) = x + x , y + y , t + t + 2 (y, x − x, y ) , is of the form (3.6) with m = 2k, n = 1 and

0 B (1) = 4 −Ik

Ik 0

.

3.3 Free Step-two Homogeneous Groups In this section, following the notation of the previous one, we fix a particular set of matrices B (k) ’s and consider the relevant homogeneous Carnot group (Fm,2 , ), which will serve as prototype for what we shall call free Carnot group of step two and m generators. Throughout the section, m ≥ 2 is a fixed integer. Let i, j ∈ {1, . . . , m} be fixed with i > j , and let S (i,j ) be the m × m skewsymmetric matrix whose entries are −1 in the position (i, j ), +1 in the position (j, i) and 0 elsewhere. For example, if m = 3, we have ⎛ ⎞ ⎛ ⎞ 0 1 0 0 0 1 S (3,1) = ⎝ 0 0 0 ⎠ , S (2,1) = ⎝ −1 0 0 ⎠ , 0 0 0 −1 0 0 ⎛ ⎞ 0 0 0 S (3,2) = ⎝ 0 0 1 ⎠ . 0 −1 0 Then, we agree to denote by (Fm,2 , ) the Carnot group on RN associated to these m(m − 1)/2 matrices according to (3.6) of the previous section. We set n := m(m − 1)/2, N = m + n = m(m + 1)/2, I := {(i, j ) | 1 ≤ j < i ≤ m}. We observe that the set I has exactly n elements. In the sequel of this section, we shall use the following notation, different from the one used in the previous section: instead of using the notation t for the coordinate in the “second layer” of the group, we denote the points of Fm,2 by (x, γ ), where x = (x1 , . . . , xm ) ∈ Rm , γ ∈ Rn , and the coordinates of γ are denoted by γi,j

where (i, j ) ∈ I.

Here we have ordered I in an arbitrary (henceforth) fixed way. Then, the composition law is given by

xh + xh , h = 1, . . . , m . (x, γ ) (x , γ ) = + 1 (x x − x x ), (i, j ) ∈ I γi,j + γi,j j i 2 i j

164

3 Carnot Groups of Step Two

For example, when m = 3, we have ⎛

⎞ x1 + x1 ⎜ ⎟ x2 + x2 ⎜ ⎟ ⎜ ⎟ x3 + x3 ⎜ ⎟ (x, γ ) ◦ (x , γ ) = ⎜ γ2,1 + γ + 1 (x2 x − x1 x ) ⎟ . 2,1 1 2 ⎟ 2 ⎜ ⎜ + 1 (x x − x x ) ⎟ ⎝ γ3,1 + γ3,1 1 3 ⎠ 2 3 1 + 1 (x x − x x ) γ3,2 + γ3,2 2 3 2 3 2 By (3.8), we can compute the Jacobian basis Xh ,

h = 1, . . . , m,

Γi,j ,

(i, j ) ∈ I,

of fm,2 , the Lie algebra of Fm,2 : it holds m

(i,j ) 1 Xh = (∂/∂xh ) + Sh,l xl (∂/∂γi,j ) 2 1≤j
if h = 1, if 1 < h < m, if h = m,

(i, j ) ∈ I.

Moreover, for every (i, j ) ∈ I, we have the commutator identities (h,k) [Xj , Xi ] = Si,j ∂γh,k = ∂γj,i − ∂γi,j , 1≤k
whence we recognize that the algebra fm,2 is “the most non-Abelian as possible” (as it is allowed for an algebra with m generators and step two). This is the reason why we shall refer to (any algebra isomorphic to) fm,2 as a free Lie algebra with m generators and step two (see Chapter 14 for more details on free Lie algebras). For example, when m = 3, we have 1 X1 = ∂x1 + (x2 ∂γ2,1 + x3 ∂γ3,1 ), 2 1 X2 = ∂x2 + (x3 ∂γ3,2 − x1 ∂γ2,1 ), 2 1 X3 = ∂x3 − (x1 ∂γ3,1 − x2 ∂γ3,2 ) 2 Γ2,1 = ∂γ2,1 , Γ3,1 = ∂γ3,1 , Γ3,2 = ∂γ3,2 . From (3.9), we derive the explicit expression for the canonical sub-Laplacian of F3,2 ,

3.4 Change of Basis

165

ΔF3,2 = (∂x1 )2 + (∂x2 )2 + (∂x3 )2 1 2 (x2 + x12 )(∂γ2,1 )2 + (x32 + x12 )(∂γ3,1 )2 + (x32 + x22 )(∂γ3,2 )2 + 4 1 1 1 + x2 x3 (∂γ2,1 ∂γ3,1 ) − x1 x3 (∂γ2,1 ∂γ3,2 ) + x1 x2 (∂γ3,1 ∂γ3,2 ) 2 2 2 + (x2 ∂x1 − x1 ∂x2 )∂γ2,1 + (x3 ∂x1 − x1 ∂x3 )∂γ3,1 + (x3 ∂x2 − x2 ∂x3 )∂γ3,2 . (3.10)

3.4 Change of Basis In this section we consider “stratified” changes of basis on a homogeneous Carnot group of step two, according to the definition in Remark 1.4.10, page 61. We use the usual notation, as in the beginning of Section 3.2. Let C, D be two fixed non-singular matrices, with C of dimension m × m and D of dimension n × n. We denote by L the following N × N matrix

C 0 L= . 0 D Finally, we denote again by L the relevant linear change of basis on RN , i.e., the linear map (· denotes row-times-column matrix product)

x C·x N N = . L : R → R , L(x, t) = L · t D·t We define on RN a new composition law ∗ obtained by writing ◦ in the new coordinates defined by (ξ, τ ) = L(x, t). More precisely, we have (ξ, τ ) ∗ (ξ , τ ) := L (L−1 (ξ, τ )) ◦ (L−1 (ξ , τ )) . An explicit computation gives (here D = (dh,k )h,k≤n ) (ξ, τ ) ∗ (ξ , τ ) =

ξi + ξi , i = 1, . . . , m τh + τh + 12 (C −1 )T · ( nk=1 dh,k B (k) ) · C −1 ξ, ξ ,

h = 1, . . . , n

i.e. setting (h) := (C −1 )T · B

n

dh,k B

(k)

· C −1

for every h = 1, . . . , n,

k=1

∗ can be written as follows:

(ξ, τ ) ∗ (ξ , τ ) =

ξi + ξi , i = 1, . . . , m (h) ξ, ξ , h = 1, . . . , n τh + τh + 12 B

.

,

166

3 Carnot Groups of Step Two

We explicitly remark that, if the matrices B (h) ’s are skew-symmetric, the same is (h) ’s. It can easily be proved that H = (RN , ∗, δλ ) is a homotrue for the matrices B geneous Carnot group isomorphic to G = (RN , ◦, δλ ) via L (see also Section 16.3, page 637). It is immediately seen that the canonical basis of H is m

n 1 (h) Bi,j xj (∂/∂τh ), i = 1, . . . , m, (∂/∂ξi ) + 2 h=1

∂/∂τk ,

j =1

k = 1, . . . , n.

i of the i-th vector Xi of the Moreover, it is not difficult to see that the expression X canonical basis of G in the new coordinates (ξ, τ ) on H (i.e. the image via dL of Xi ) coincides at the origin with L · Xi I (0) = (C · ei , 0), where (e1 , . . . , em ) denotes canonical basis of Rm . In particular, given an arbi the m trary sub-Laplacian L = j =1 Yj2 on G, it is always possible to perform a linear change of basis on RN so that L is turned into the canonical sub-Laplacian on a homogeneous Carnot group isomorphic to G.

3.5 The Exponential Map of a Step-two Homogeneous Group We now turn to compute the exponential map of a homogeneous group of step two. As usual, if ξ ∈ Rm and τ ∈ Rn , we write (ξ, τ ) · Z =

m

ξj X j +

j =1

n

τ i Ti ,

i=1

where X1 , . . . , Xm , T1 , . . . , Tn is the Jacobian basis. More explicitly, by (3.8), we have ⎞ ⎛ ξ ⎜ 1 m b(1) ξ x ⎟ ⎜ 2 i,j =1 i,j i j ⎟ ⎟. (ξ, τ ) · Z I = ⎜ .. ⎟ ⎜ . ⎠ ⎝ (n) m 1 b ξ x i j i,j =1 i,j 2 By the definition of exponential map, we have Exp ((ξ, τ ) · Z) = (x(1), t (1)), where (x(s), t (s)) solves (x(s), ˙ t˙(s)) = ((ξ, τ ) · Z)I (x(s), t (s)), i.e. more explicitly,

(x(0), t (0)) = (0, 0),

3.5 The Exponential Map of a Step-two Homogeneous Group

x(s) ˙ = ξ, ˙t (s) = τ +

1 2

167

x(0) = 0,

m

i,j =1 Bi,j ξi

xj (s),

t (0) = 0.

Here, we have set for brevity ⎞ (1) bi,j ⎜ . ⎟ ⎟ := ⎜ ⎝ .. ⎠ . (n) bi,j ⎛

Bi,j

A straightforward computation gives

m 1 Exp ((ξ, τ ) · Z) = ξ, τ + Bi,j ξi ξj , 4

(3.11a)

i,j =1

(1)

(n)

(here we again set Bi,j = (Bi,j , . . . , Bi,j )), whence

m 1 Log (x, t) = x, t − Bi,j xi xj 4

· Z.

(3.11b)

i,j =1

We explicitly remark that, by (3.11a), the exponential map acts like a sort of “identity map” (provided g is equipped with coordinates w.r.t. the Jacobian basis) if and only if the matrices B (i) ’s are skew-symmetric. We now operate a change of coordinates in RN inspired by the exponential map. In other words (see the proof of Theorem 2.2.18, page 131), we equip the algebra g of G by a Lie-group structure isomorphic via Exp to that of (G, ◦), and we identify g with RN via the Jacobian basis. This amounts to equip RN = Rm × Rn with the composition law Z (the reason for this notation will become clearer in further sections) such that Log (Exp ((ξ, τ ) · Z)) ◦ (Exp ((ξ , τ ) · Z)) = (ξ, τ ) Z (ξ , τ ) · Z. We find an explicit expression for Z . It holds Log (Exp ((ξ, τ ) · Z)) ◦ (Exp ((ξ , τ ) · Z))

m m 1 1 Bi,j ξi ξj ◦ ξ , τ + Bi,j ξi ξj = Log ξ, τ + 4 4 i,j =1 i,j =1

m m 1 1 = Log ξ + ξ , τ + τ + Bi,j (ξi ξj + ξi ξj ) + Bi,j ξj ξi 4 2 i,j =1 i,j =1 m m 1 1 = ξ + ξ , τ + τ + Bi,j (ξi ξj + ξi ξj ) + Bi,j ξj ξi 4 2 i,j =1

i,j =1

168

3 Carnot Groups of Step Two

m 1 − Bi,j (ξi + ξi )(ξj + ξj ) · Z 4 i,j =1

m m 1 1 Bi,j ξj ξi − Bi,j ξi ξj ) · Z = ξ + ξ ,τ + τ + 4 4 i,j =1 i,j =1

m 1 Bi,j − Bj,i ξj ξi · Z. = ξ + ξ ,τ + τ + 2 2 i,j =1

As a consequence, the composition law ∗ on RN has the form

m 1 Bi,j − Bj,i ξj ξi , (ξ, τ ) Z (ξ , τ ) = ξ + ξ , τ + τ + 2 2

(3.12)

i,j =1

i.e. it has the form as in (3.6) with the matrices B (k) ’s replaced by the skew-symmetric matrices (k) (k) T (k) = B − (B ) , k = 1, . . . , n, B 2 the skew-symmetric parts of the B (k) ’s. The inverse map on (RN , Z ) is the usual (−x, −t). The homogeneous Lie group (RN , Z , δλ ) is isomorphic to (RN , ◦, δλ ) via the Lie-group isomorphism

m 1 N N Bi,j ξi ξj , Ψ : (R , Z ) → (R , ◦), Ψ (ξ, τ ) := ξ, τ + 4 i,j =1

which “essentially” equals the exponential map (written in the Jacobian coordinates). Since the differential of Ψ is associated to the matrix ⎞ ⎛ Im 0 (k) (k) ⎠, JΨ (ξ, τ ) = ⎝ m Bi,j +Bj,i ξ I j n j =1 2 k≤n,i≤m

(RN ,

we see that the Jacobian basis of Z ) is turned into the Jacobian basis of (RN , ◦) by the change of coordinates Ψ (see, for instance, (2.17b), page 101, or (2.71), page 153). For future references, we summarize what we have just proved in the following proposition. Proposition 3.5.1. Let RN = Rm × Rn be equipped with a homogeneous Lie group structure by the composition law

1 1 (x, t) ◦ (ξ, τ ) = x + ξ, t1 + τ1 + B (1) x, ξ , . . . , tn + τn + B (n) x, ξ 2 2 and the dilation δλ (x, t) = (λx, λ2 t), where B (1) , . . . , B (n) are fixed m × m ma = trices. Then G = (RN , ◦, δλ ) is isomorphic to the homogeneous Lie group G N (R , Z , δλ ), where:

3.6 Prototype Groups of Heisenberg Type

169

1) δλ is the same dilation as above; 2) Z is defined by

1 (1) 1 (n) (x, t) Z (ξ, τ ) = x + ξ, t1 + τ1 + B x, ξ , . . . , tn + τn + B x, ξ 2 2 (i) is the skew-symmetric part of B (i) for every i = 1, . . . , n; the inverse where B is (−x, −t); map on G → G with 3) the Lie-group isomorphism is Ψ : G

1 (1) 1 (n) Ψ (ξ, τ ) = ξ, τ1 + B ξ, ξ , . . . , τn + B ξ, ξ , 4 4 so that Ψ is the identity map iff all the B (i) ’s are skew-symmetric; corresponds (via dΨ ) to the Jacobian basis of G, in 4) the Jacobian basis of G corresponds to that of G; particular the canonical sub-Laplacian of G is the linear map sending the Jacobian basis 5) the exponential map of the group G into the canonical basis of G; of the algebra of G 6) if G is a homogeneous Carnot group, then the same is true for G.

3.6 Prototype Groups of Heisenberg Type Definition 3.6.1 ((Prototype) Heisenberg-type group). Consider the homogeneous Lie group H = (Rm+n , ◦, δλ ) with composition law as in (3.6) (page 158), i.e.

1 (1) 1 (n) (x, t) ◦ (ξ, τ ) = x + ξ, t1 + τ1 + B x, ξ , . . . , tn + τn + B x, ξ , 2 2 where B (1) , . . . , B (n) are fixed m × m matrices, and dilations as in (3.7). Let us also assume that the matrices B (1) , . . . , B (n) have the following properties: 1) B (j ) is an m × m skew-symmetric and orthogonal matrix for every j ≤ n; 2) B (i) B (j ) = −B (j ) B (i) for every i, j ∈ {1, . . . , n} with i = j . If all these conditions are satisfied, H is called a ( prototype) group of Heisenberg-type, in short, a ( prototype) H-type group. A H-type group is a Carnot group, since conditions 1) and 2) imply the linear independence of B (1) , . . . , B (n) . Indeed, if α = (α1 , . . . , αn ) ∈ Rn \ {0}, then 1 αs B (s) |α| n

s=1

170

3 Carnot Groups of Step Two

is orthogonal (hence non-vanishing), as the following computation shows,

T n n 1 1 (s) (s) αs B αs B · |α| |α| s=1 s=1 1 =− 2 αr αs B (r) B (s) |α| r,s≤n 1 2 (r) 2 1 =− 2 αr (B ) − |α| r≤n |α|2

αr αs B (r) B (s)

r,s≤n, r=s

= Im . Here we used the following facts: (B (r) )2 = −Im , since B (r) is skew-symmetric and orthogonal; B (r) B (s) = −B (s) B (r) according to condition 2). The generators of H are the vector fields (see (3.8), page 159) m

n 1 (k) bi,l xl ∂tk , i = 1, . . . , m. (3.13a) Xi = ∂xi + 2 k=1

l=1

Moreover, if we set Tk := ∂/∂tk ,

k = 1, . . . , n,

(3.13b)

then (again from (3.8)) we know that {X1 , . . . , Xm ; T1 , . . . , Tn }

(3.13c)

is the Jacobian basis for H. 2 A direct computation shows that the canonical sub-Laplacian ΔH = m i=1 Xi can be written as follows (see (3.9)) ΔH = Δx +

n 1 (h) B x, B (k) x ∂th tk 4 h,k=1

+

m

B (k) x, ∇x ∂tk +

k=1

n 1 trace(B (k) ) ∂tk . 2 k=1

On the other hand, by conditions 1) and 2), B (h) x, B (h) x = |x|2 , while, for h = k,

B (h) x, B (k) x = 0

since B (h) x, B (k) x = −B (k) B (h) x, x = B (h) B (k) x, x = −B (k) x, B (h) x.

3.6 Prototype Groups of Heisenberg Type

171

We also have trace(B (k) ) = 0, since B (k) is skew-symmetric. Then ΔH takes the very compact form 1 2 |x| Δt + B (k) x, ∇x ∂tk . 4 n

ΔH = Δx +

(3.14)

k=1

By the computations made in Section 3.5 in finding the exponential map of a general homogeneous group of step two, we infer that (see precisely (3.11a), page 167) if {X1 , . . . , Xm ; T1 , . . . , Tn } is the Jacobian basis for H (where the Xi ’s and the Tk ’s are as in (3.13a), (3.13b)) then the exponential map for the ( prototype) H-type group H is Exp : h → H,

Exp (x1 X1 + · · · + xm Xm + t1 T1 + · · · + tn Tn ) = (x, t). (3.15)

Remark 3.6.2. From (3.8) one obtains 1 (Bx)i , ∇t u2 4 m

|∇H u|2 = |∇x u|2 +

i=1

m + (Bx)i , ∇t u ∂xi u,

u ∈ C∞.

i=1

On the other hand, m n (Bx)i , ∇t u2 = B (h) x, B (k) x∂th u ∂tk u i=1

h,k=1 2

= |x| |∇t u|2 and

m n (Bx)i , ∇t u ∂xi u = B (k) x, ∇x u ∂tk u. i=1

k=1

Thus, for every smooth real-valued function u, it holds 1 2 |x| |∇t u|2 + B (k) x, ∇x u ∂tk u. 4 n

|∇H u|2 = |∇x u|2 +

(3.16)

k=1

Remark 3.6.3. The first layer of a H-type group has even dimension m. Indeed, if B is a m × m skew-symmetric orthogonal matrix, we have Im = B · B T = −B 2 , whence 1 = (−1)m (det B)2 . Remark 3.6.4. With the previous notation, if H = (Rm+n , ◦, δλ ) is a H -type group, then z = {(0, t) | t ∈ Rn } is the center of H. Indeed, let (y, t) ∈ H be such that

172

3 Carnot Groups of Step Two

(x, s) ◦ (y, t) = (y, t) ◦ (x, s)

for every (x, s) ∈ H.

This holds iff B (k) x, y = B (k) y, x for any x ∈ Rm and any k ∈ {1, . . . , n}. Then, since (B (k) )T = −B (k) , B (k) y, x = 0 ∀ x ∈ Rm ∀ k ∈ {1, . . . , n}, so that y = 0 because B (k) is orthogonal (hence non-singular). Remark 3.6.5. The classical Heisenberg–Weyl group Hk on R2k+1 is canonically isomorphic to a prototype H-type group. Precisely (see (3.5b), page 157), it is isomorphic to the prototype H-type group (H, ∗) corresponding to the case m = 2k, n = 1 and

0 −Ik (1) . B = Ik 0 The isomorphism ϕ : (R2k+1 , ∗) → (Hk , ◦) is given by ϕ(ξ, η, τ ) = (ξ, η, −4τ ). Moreover, H is in its turn isomorphic to the prototype H-type group with m = 2k, n = 1 and

! 0 −1 0 −1 (1) B = diag ,..., , the block occurring k times. 1 0 1 0 This type of prototype Heisenberg-group is the only (up to isomorphism) H-type group with one-dimensional center (see Chapter 18). Remark 3.6.6. Groups of Heisenberg type with center of dimension n ≥ 2 do exist. For example, the following two matrices ⎛ ⎛ ⎞ ⎞ 0 −1 0 0 0 0 1 0 ⎜1 0 0 0 ⎟ ⎜ 0 0 0 −1 ⎟ ⎟ ⎟ B (2) = ⎜ B (1) = ⎜ ⎝ 0 0 0 −1 ⎠ , ⎝ −1 0 0 0 ⎠ 0 0 1 0 0 1 0 0 satisfy conditions 1)–2) and hence they define in R6 = R4 × R2 a H-type group whose center has dimension 2. The composition law is ⎛

⎞ x1 + ξ1 ⎜ ⎟ x2 + ξ2 ⎜ ⎟ ⎜ ⎟ x3 + ξ3 ⎟. (x, t) ◦ (ξ, τ ) = ⎜ ⎜ ⎟ x4 + ξ4 ⎜ ⎟ 1 ⎝ t1 + τ1 + 2 (−x2 ξ1 + x1 ξ2 − x4 ξ3 + x3 ξ4 ) ⎠ t2 + τ2 + 12 (x3 ξ1 − x4 ξ2 − x1 ξ3 + x2 ξ4 )

3.7 H-groups (in the Sense of Métivier)

The above matrices B (1) and B (2) , together with ⎛ 0 0 0 ⎜ 0 0 1 B (3) = ⎜ ⎝ 0 −1 0 −1 0 0

173

⎞ 1 0⎟ ⎟, 0⎠ 0

define in R7 = R4 × R3 a H-type group whose center has dimension 3. The composition law is ⎞ ⎛ x1 + ξ1 ⎟ ⎜ x2 + ξ2 ⎟ ⎜ ⎟ ⎜ x + ξ 3 3 ⎟ ⎜ ⎟ ⎜ x + ξ 4 4 (x, t) ◦ (ξ, τ ) = ⎜ ⎟. ⎜ t1 + τ1 + 1 (−x2 ξ1 + x1 ξ2 − x4 ξ3 + x3 ξ4 ) ⎟ 2 ⎟ ⎜ ⎟ ⎜ ⎝ t2 + τ2 + 12 (x3 ξ1 − x4 ξ2 − x1 ξ3 + x2 ξ4 ) ⎠ t3 + τ3 + 12 (x4 ξ1 + x3 ξ2 − x2 ξ3 − x1 ξ4 ) Remark 3.6.7. The following result holds (see [Kap80, Corollary 1]). Let m, n be two positive integers. Then there exists a H-type group of dimension m + n whose center has dimension n if and only if it holds n < ρ(m), where ρ is the so-called Hurwitz–Radon function, i.e. ρ : N → N,

ρ(m) := 8p + q,

where m = (odd) · 24p+q , 0 ≤ q ≤ 3.

We explicitly remark that if m is odd, then ρ(m) = 0, whence the first layer of any H-type group has even dimension (as we already proved in Remark 3.6.3). Remark 3.6.8. The groups of Heisenberg-type were introduced by A. Kaplan in [Kap80]. Kaplan’s definition of H-type groups is more abstract than the one given here. In Chapter 18, we shall show that, up to an isomorphism, the two definitions are equivalent.

3.7 H-groups (in the Sense of Métivier) Following G. Métivier [Met80], we give the following definition. Definition 3.7.1 (H-group (in the sense of Métivier)). Let g be a (finite-dimensional real) Lie algebra, and let us denote by z its center. We say that g is of H-type (in the sense of Métivier) if it admits a vector space decomposition [g1 , g1 ] ⊆ g2 , g = g 1 ⊕ g2 g2 ⊆ z with the following additional property: for every η ∈ g∗2 (the dual space of g2 ), the skew-symmetric bilinear form on g1 defined by Bη : g1 × g1 → R, Bη (X, X ) := η([X, X ]) is non-degenerate5 whenever η = 0. 5 We recall that a bilinear map on a finite-dimensional vector space is non-degenerate if any

(or equivalently, if one) of the matrices representing it w.r.t. a fixed basis is non-singular.

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3 Carnot Groups of Step Two

We say that a Lie group is a H-group (in the sense of Métivier), or a HM-group in short, if its Lie algebra is of H-type (in the sense of Métivier). Remark 3.7.2. In this remark, we follow the notation of Definition 3.7.1. First, a (Métivier-)H-type algebra is obviously nilpotent of step two. Moreover, we have [g, g] = [g1 + g2 , g1 + g2 ] ⊆ [g1 , g1 ] [g1 , g1 ] ⊆ [g, g] (since g1 ⊆ g).

(since g2 ⊆ z),

Consequently, it holds [g, g] = [g1 , g1 ].

(3.17a)

g2 = [g, g].

(3.17b)

Finally, we claim that Indeed, from (3.17a) we first derive that [g, g] = [g1 , g1 ] ⊆ g2 (by the very definition of (Métivier-)H-type algebra). We are left to show that g2 ⊆ [g, g]. Suppose to the / [g, g]. This implies, in particular, that contrary that there exists Z ∈ g2 such that Z ∈ Z = {0}. Moreover, since both Z ∈ g2 and [g, g] ⊆ g2 , there certainly exists η ∈ g∗2 such that g2 (Z) = 0 (whence η = 0) and η vanishes identically on [g, g] (here, we are using the fact that Z ∈ / [g, g]). But this implies that, for every X, X ∈ g1 , we have Bη (X, X ) = η([X, X ]) = 0, for

[X, X ] ∈ [g1 , g1 ] = [g, g]

and η|[g,g] ≡ 0.

This is in contradiction with the non-degeneracy of Bη . Collecting together (3.17a) and (3.17b), we see that a (Métivier-)H-type algebra is stratified: indeed we have g = g1 ⊕ g2

with [g1 , g1 ] = g2 and [g1 , g2 ] = {0}.

As a consequence, a HM-group is a Carnot group. Moreover, if g1 is any other complement of g2 = [g, g] in g, it is easy to prove that also the decomposition g = g1 ⊕ g2 satisfies Definition 3.7.1. Collecting the above results, we have proved the following proposition. Proposition 3.7.3 (Characterization. I). A HM-group is a Carnot group G of step two such that if g = g 1 ⊕ g2

([g1 , g1 ] = g2 , [g1 , g2 ] = {0})

is any stratification of the Lie algebra g of G, then the following property holds: For every non-vanishing linear map η from g2 to R, the (skew-symmetric) bilinear form Bη on g1 defined by Equivalently, a bilinear map B on the vector space V is non-degenerate if, for every v ∈ V \ {0}, there exists w ∈ V such that B(v, w) = 0.

3.7 H-groups (in the Sense of Métivier)

Bη (X, X ) := η([X, X ]),

175

X, X ∈ g1 ,

is non-degenerate. When G is expressed in its logarithmic coordinates (making it a homogeneous Carnot group), the above definition is easily re-written as follows. We consider a homogeneous Lie group of step two G = (Rm+n , ◦, δλ ) with the composition law as in (3.6) (page 158), i.e.

1 (1) 1 (n) (x, t) ◦ (ξ, τ ) = x + ξ, t1 + τ1 + B x, ξ , . . . , tn + τn + B x, ξ , 2 2 where B (1) , . . . , B (n) are fixed m × m matrices, and the group of dilations is δλ (x, t) = (λx, λ2 t). For the sake of simplicity, we may also suppose that the matrices B (k) ’s are skew-symmetric (see, for instance, Proposition 3.5.1). We make use of the results and the notation of Section 3.2. Now, if η is a linear map from g2 to R, there exist n scalars η1 , . . . , ηn ∈ R such that η : g2 → R, η(∂ti ) = ηi for all i = 1, . . . , n. In particular, the map Bη , as in Proposition 3.7.3, can be explicitly written (after a simple calculation which we leave to the reader) as follows6 m if X = m v Xi , i=1 vi Xi and X = ni=1 i n (k) then Bη (X, X ) = ηk Bi,j vi vj . − i,j =1

k=1

In other words, the matrix representing the (skew-symmetric) bilinear map Bη w.r.t. the basis X1 , . . . , Xm of g1 is the matrix η1 B (1) + · · · + ηn B (n) . Hence, to ask for Bη to be non-degenerate (for every η = 0) is equivalent to ask that any linear combination of the matrices B (k) ’s is non-singular, unless it is the null matrix (recall that the B (k) ’s are linearly independent). We have thus proved the following proposition. 6 Recall that, by (3.8), page 159, a basis for g is {X , . . . , X } with m 1 1

⎛ ⎞ n m 1 ⎝ (k) ⎠ Xi = (∂/∂xi ) + Bi,l xl (∂/∂tk ), 2 k=1

where we have set

l=1

(k) . B (k) = Bi,l i,l≤m

176

3 Carnot Groups of Step Two

Proposition 3.7.4 (Characterization. II). Let G = (Rm+n , ◦) be a homogeneous Carnot group of step two, with the composition law

1 (1) 1 (n) (x, t) ◦ (ξ, τ ) = x + ξ, t1 + τ1 + B x, ξ , . . . , tn + τn + B x, ξ , 2 2 where B (1) , . . . , B (n) are m×m skew-symmetric linearly independent matrices. Then G is a HM-group if and only if every non-vanishing linear combination of the matrices B (k) ’s is non-singular. In particular, if the above G is a HM-group, then the B (k) ’s are all non-singular m × m matrices, but since the B (k) ’s are also skew-symmetric, this implies that m is necessarily even. Remark 3.7.5 (Any H-type group is a HM-group). Any prototype H-type group (according to Definition 3.6.1, page 169) is a HM-group. Indeed, as it can be seen from the computations on page 169, in a prototype H-type group, for every η = n (k) is |η| times an orthogonal proved that η B (η1 , . . . , ηn ) ∈ Rn , η = 0, we k=1 k n (k) matrix, hence (in particular) k=1 ηk B is non-singular. The converse is not true. For example, consider the group on R5 (the points are denoted by (x, t), x ∈ R4 , t ∈ R) with the composition law

1 (x, t) ◦ (ξ, τ ) = x + ξ, t + τ + Bx, ξ , 2 where

⎛

0 ⎜ −1 B=⎜ ⎝ 0 0

1 0 0 0 0 0 0 −2

⎞ 0 0⎟ ⎟. 2⎠ 0

Then G is obviously a HM-group, for B is a non-singular skew-symmetric matrix. But G is not a prototype H-type group, for B is not orthogonal. But more is true: as we shall prove in Chapter 18 (see Remark 18.2.7 and Corollary 18.2.8, page 695) G is not even isomorphic to any prototype H-type group. Bibliographical Notes. The explicitness of the composition law of a homogeneous Carnot group of step two allows to face a great variety of problems. See, e.g. some recent results concerning with step-two Carnot groups: [Dai00] for mappings with bounded distortion, [FSS03b] for geometric measure theory results, [Mil88] for a microlocal version of some results concerning hypoellipticity, [Ric06] and [SY06] for results on the convex functions, [Tha94] for theorems of Paley–Wiener type. For an introduction to harmonic analysis on the Heisenberg group, see the monograph by S. Thangavelu [Tha98], and for a survey see R. Howe [How80].

3.8 Exercises of Chapter 3

177

3.8 Exercises of Chapter 3 Ex. 1) With reference to Section 3.1, recognize that ΔH1 equals (∂x1 )2 + (∂x2 )2 + 4(x12 + x22 ) (∂x3 )2 + 4 x2 ∂x1 ,x3 − 4 x1 ∂x2 ,x3 . Prove the analogous (general) formula for ΔHn , N

n N 2 2 2 2 ΔHn = (∂xj + ∂yj ) + 4 (xj + yj ) ∂t2 + 4 (yj ∂xj − xj ∂yj ) ∂t . j =1

j =1

j =1

Recognize that the quadratic form related to ΔHn at (x, y, t) is the quadratic form q related to the symmetric matrix ⎛ ⎞ 2y I n ⎠. −2 x A=⎝ 2 2 T T 4 (|x| + |y| ) 2y −2 x For every fixed (x, y, t) ∈ Hn , find all vectors (ξ, η, τ ) ∈ R2n+1 such that (ξ, η, τ ) · A · (ξ, η, τ )T = 0. Compare this to what was shown in (A4), page 65. Ex. 2) Prove (3.9), page 160. Ex. 3) Making use of (3.9), derive an explicit expression for the canonical subLaplacian of Fm,2 generalizing (3.10) (page 165). Ex. 4) Consider the Lie group on R4 (the points are denoted by x = (x1 , x2 , x3 , x4 )) with the composition law given by ⎞ ⎛ x1 + y1 ⎟ ⎜ x2 + y2 ⎟ x◦y =⎜ ⎝ x3 + y3 + x1 y2 − x2 y1 ⎠ . x4 + y3 + x1 y2 − x2 y1 Prove that (R4 , ◦) is a Lie group nilpotent of step two. According to Proposition 3.2.1, this is not a homogeneous Carnot group (for any dilation!) since (we use the usual notation of (3.6), page 158) the relevant matrices

0 1 B (1) = B (2) = −1 0 have the same skew-symmetric parts (whence the latter are not linearly independent). However, according to Ex. 6 in Chapter 2, page 149, since (R4 , ◦) is nilpotent of step two, its algebra g is necessarily stratified! Indeed, setting X1 = ∂x1 − x2 ∂x3 − x2 ∂x4 , X2 = ∂x2 + x1 ∂x3 + x1 ∂x4 , X3 = ∂x3 , X4 = ∂x4 ,

(3.18)

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3 Carnot Groups of Step Two

we have [X1 , X2 ] = 2 ∂x3 + 2 ∂x4 = 2(X3 + X4 ), whence g = span{X1 , X2 , X3 } ⊕ span{X3 + X4 }, and this is a stratification, as in Definition 2.2.3 (page 122). Consequently, this time thanks to Theorem 3.2.2, (R4 , ◦) is isomorphic to a homogeneous Carnot group. Indeed, retracing the arguments in the proof of Theorem 2.2.18 (page 131, see also the notation therein), prove that (R4 , ◦) is isomorphic to (R4 , , δλ ), where δλ (ξ1 , ξ2 , ξ3 , ξ4 ) = (λξ1 , λξ2 , λξ3 , λ2 ξ4 ), and is obtained by the Campbell–Hausdorff multiplication on g equipped with the basis {X1 , X2 , X3 , X4 + X3 }, which, in turn, is given by ⎛ ⎞ ξ 1 + η1 ⎜ ⎟ ξ 2 + η2 ⎟. ξ ◦η =⎜ ⎝ ⎠ ξ 3 + η3 ξ 4 + η4 + ξ1 η2 − ξ 2 η1 A Lie group isomorphism between (R4 , ◦) and (R4 , ) can be calculated noting that (R4 , ) is essentially the Lie algebra of (R4 , ◦): we thus compute the exponential map via the usual system of ODE’s as follows. Since ⎞ ⎛ ξ1 ⎟ ⎜ ξ2 ⎟ ξ1 X1 + ξ2 X2 + ξ3 X3 + ξ4 (X4 + X3 ) I (x) = ⎜ ⎝ ξ3 + ξ4 − ξ1 x 2 + ξ2 x 1 ⎠ ξ4 − ξ1 x 2 + ξ2 x 1 (denote this field by Y I (x)), the solution γ to γ˙ (t) = Y I (γ (t)), γ (0) = 0 is γ (t) = ξ1 t, ξ2 t, (ξ3 + ξ4 )t, ξ4 t , and we have γ (1) = (ξ1 , ξ2 , ξ3 + ξ4 , ξ4 ). Define the map Φ : (R4 , ) → (R4 , ◦),

Φ(ξ ) := (ξ1 , ξ2 , ξ3 + ξ4 , ξ4 )

and prove that it is a Lie group isomorphism, which turns the vector fields in (3.18), respectively, into 1 = ∂ξ1 − ξ2 ∂x4 , X 2 = ∂ξ2 + ξ3 ∂x4 , X X3 = ∂ξ3 , X4 = ∂ξ4 − ∂ξ3 . Ex. 5) Prove in details the last statement in Remark 3.7.2, page 174.

3.8 Exercises of Chapter 3

179

Ex. 6) Consider the following homogeneous Carnot group of step two (which will be called of Kolmogorov type, see Section 4.1.4): this is R5 (the points are, by our usual notation, (x, t) = (x1 , x2 , x3 , t1 , t2 )) with the composition law ⎛ ⎞ x1 + ξ1 ⎜ ⎟ x2 + ξ2 ⎜ ⎟ ⎟. x + ξ (x, t) ◦ (ξ, τ ) = ⎜ 3 3 ⎜ ⎟ ⎝ t1 + τ 1 − x 2 ξ 1 ⎠ t2 + τ 2 − x 3 ξ 1 The relevant matrices are ⎛ ⎞ 0 −2 0 B (1) = ⎝ 0 0 0 ⎠ , 0 0 0

⎛

B (2)

⎞ 0 0 −2 = ⎝0 0 0 ⎠. 0 0 0

Prove that the related canonical sub-Laplacian is 2 ∂x1 − x2 ∂t1 − x3 ∂t2 + (∂x2 )2 + (∂x3 )2 . Now, with a slight change of notation, consider on R1+2n (whose points are denoted by (t, x, y), t ∈ R, x, y ∈ Rn ) the operator 2 L := Δx + ∂ − x, ∇y , n 2 where Δx = i=1 (∂xi ) and ∇y = (∂y1 , . . . , ∂yn ). Find a homogeneous Carnot group structure on R1+2n such that L is the related canonical subLaplacian. Ex. 7) Operate a change of variables on R5 in such a way that the group (R5 , ◦) defined in the previous exercise is turned into a homogeneous Carnot group whose related matrices B (k) ’s are skew-symmetric. (Do the same for the general case of R1+2n .) Prove that the new composition law and the new canonical sub-Laplacian are, respectively, ⎛ ⎞ x1 + ξ1 ⎜ ⎟ x 2 + ξ2 ⎜ ⎟ ⎜ ⎟ x 3 + ξ3 (x, t) ◦ (ξ, τ ) = ⎜ ⎟ ⎝ t1 + τ1 + 1 (x1 ξ2 − x2 ξ1 ) ⎠ 2 t2 + τ2 + 12 (x1 ξ3 − x3 ξ1 ) and

∂x1 −

1 1 x2 ∂t1 − x3 ∂t2 2 2

2

2

2

1 1 + ∂x2 + x1 ∂t1 + ∂x3 + x1 ∂t2 . 2 2

Note that the relevant matrices are

180

3 Carnot Groups of Step Two

(1) B

(2) B

⎛ 0 1 (1) = B − (B (1) )T = ⎝ 1 2 0 ⎛ 0 1 = B (2) − (B (2) )T = ⎝ 0 2 1

⎞ −1 0 0 0⎠, 0 0 ⎞ 0 −1 0 0 ⎠. 0 0

Ex. 8) (The polarized Heisenberg group). Let us denote by (x, y, t) the points of Rn × Rn × R ≡ R2n+1 . Consider the composition law (x, y, t) ◦ (x , y , t ) := x + x , y + y , t + t + x , y (where ·, · denotes the Euclidean inner product) and the dilation δλ (x, y, t) := (λx, λy, λ2 t). pol

Show that Hn = (R2n+1 , ◦, δλ ) is a homogeneous Carnot group and write down its canonical sub-Laplacian. pol Note. Hn is called the polarized Heisenberg group. pol (Matrix representation of H n ). Given x, y ∈ Rn and t ∈ R, define the (n + 2) × (n + 2) matrix (x, y are being considered as column vectors) ⎛ ⎞ 0 yT t m(x, y, t) := In+2 + ⎝ 0 0 x ⎠ , 0 0 0

where In+2 is the identity matrix of order n + 2. Show that Mn := m(x, y, t) | (x, y, t) ∈ R2n+1 is a group with respect to the matrix multiplication. Show also that pol

m : Hn −→ Mn ,

(x, y, t) → m(x, y, t)

is a group isomorphism. pol

(A unitary representation of Hn ). Given x, y ∈ Rn and t ∈ R, define the operators e(x), τ (y), χ(t) from L2 (Rn , C) into itself as follows: e(x)f (ξ ) := eix,ξ f (ξ ), ξ ∈ Rn , τ (y)f (ξ ) := f (ξ + y), ξ ∈ Rn , χ(t)f (ξ ) := ei t f (ξ ), ξ ∈ Rn . (Again, ·, · denotes the Euclidean inner product.) Show that the set of operators

3.8 Exercises of Chapter 3

181

Un := e(x) τ (y) χ(t) | (x, y, t) ∈ R2n+1 is a group with respect to the composition of operators. Show also that pol

π : Hn −→ Un ,

(x, y, t) → e(x) τ (y) χ(t)

is a group isomorphism. Note. e(x) and τ (y) are the unitary operators generated by the position and the momentum operators in quantum mechanics (see [Tha98] for more details).

4 Examples of Carnot Groups

The aim of this chapter is to provide a wide number of explicit examples of Carnot groups of step greater than two, thus enlarging the set of examples given in Chapter 3. Some are already known in literature (for instance, here we collect some results on the filiform groups), some are new. Among the latter, there are what we call the B-groups, the Kolmogorov-type groups (or K-type groups), the Bony-type groups. Moreover, in Section 4.2, we furnish a criterion to recognize when a given operator sum of squares of vector fields in RN is a sub-Laplacian on some homogeneous Carnot group. Precisely, given a set of linearly independent vector fields X1 , . . . , Xm , homogeneous of degree one with respect to a family of dilations {δλ }λ>0 in RN and satisfying a suitable “rank-type” condition, we show how to explicitly define a composition law ◦ on RN making G = (RN , ◦, δλ ) a Carnot group whose generators are X1 , . . . , Xm . Our construction rests on the solvability of relevant systems of ODE’s. In Section 4.3, we illustrate how this method can be applied to produce new examples of Carnot groups. In Chapter 17, Section 17.4, we shall use the results of the present chapter to produce “lifted” groups.

4.1 A Primer of Examples of Carnot Groups 4.1.1 Euclidean Group The additive group (RN , +) is a homogeneous group with respect to the dilations δλ (x) = λ x,

λ > 0.

We call E = (RN , +, δλ ) the Euclidean group. E is a Carnot group of step 1. Its generators are ∂x1 , . . . , ∂xN . Thus, the canonical sub-Laplacian on E is the classical Laplace operator N Δ= ∂x2j . j =1

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4 Examples of Carnot Groups

Moreover, any sub-Laplacian on E has the following form L=

N j =1

Yj2

=

N N j =1

2 bj,i ∂xi

,

i=1

where B = (bi,j )i,j ≤N is a non-singular constant matrix. Hence, L can also be written as N N N L= bj,i bj,k ∂xi ,xk = (B T · B)i,k ∂xi ,xk , i,k=1

j =1

i,k=1

so that L is a second order constant-coefficient strictly elliptic operator. Vice versa, if L = N i,k=1 ai,k ∂xi ,xk , where A = (ai,k )i,k≤N is a positive-definite symmetric matrix, then L is a sub-Laplacian on E. Indeed, this immediately follows 2 is a non-singular symmetric square-root of A), so that by writing N A 2= B (where B L = j =1 Yj , where Yj := N i=1 Bj,i ∂xi . We want to stress that E is the only Carnot group of step 1 (and N generators). 4.1.2 Carnot Groups with Homogeneous Dimension Q ≤ 3 Let G = (RN , ◦, δλ ) be a Carnot group with homogeneous dimension Q≤ 3. With reference to the results and notation in Remark 1.4.8, we recall that Q = rj =1 j Nj , where r and N1 , . . . , Nr are, respectively, the step of G and the dimensions of the layers W (1) , . . . , W (r) of g. Obviously, the group is not the Euclidean group in RN iff W (2) = {0}, i.e. r ≥ 2. In this case, the first layer W (1) must be at least two-dimensional since [W (1) , W (1) ] = W (2) = {0}. This shows that any non-Euclidean Carnot group has homogeneous dimension Q ≥ 4. Thus, if Q ≤ 3, then G is the Euclidean group in RN , i.e. ◦ = + and δλ (x) = λx. The sub-Laplacians on G are the second order elliptic operators with constant coefficients. The canonical sub-Laplacians are d2 /dx12 in R (Q = N = 1), ∂x21 + ∂x22 in R2 (Q = N = 2), and ∂x21 + ∂x22 + ∂x23 in R3 (Q = N = 3). 4.1.3 B-groups Let us consider an N × N matrix B with real entries bi,j (with i, j = 1, . . . , N ). Let us put E(t) := exp(t B), t ∈ R. In R1+N , whose points will be denoted by z = (t, x), t ∈ R, x ∈ RN , let us introduce the following composition law

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185

(t, x) ◦ (t , x ) = (t + t , x + E(t )x). One easily verifies that B = (R1+N , ◦) is a Lie group whose identity is the origin (0, 0), and where the inverse is given by (t, x)−1 = (−t, −E(−t)x). The Jacobian matrix at the origin of the left translation τ(t,x) is the following block matrix 1 0 , Jτ(t,x) (0, 0) = b IN where b stands for the N × 1 column-matrix d E(s)x = BE(s)x s=0 = Bx. ds s=0 Then the Jacobian basis of b, the Lie algebra of B, is given by Y = ∂t + ∇x · Bx,

∂x1 , . . . , ∂xN .

(4.1)

We explicitly remark that, for a general matrix B, the group (B, ◦) may not be nilpotent. Indeed, an easy computation shows that, for any j ∈ {1, . . . , N }, N (B k )i,j ∂xi . [[· · · [∂xj Y ] · · ·]Y, Y ] = k times

i=1

Hence, (B, ◦) is a nilpotent group iff B is a nilpotent matrix. For example, if N = 1 and B = (1), the composition law is

(t, x) ◦ (t , x ) = t + t , x + x exp(t ) , (t, x), (t , x ) ∈ R2 , and the Jacobian basis is ∂t +x∂x , ∂x . Then, R2 equipped with the above composition law is not a stratified group (least of all a homogeneous Carnot group!) since it is not nilpotent. In particular, the second order differential operator on R2 defined by L = ∂x2 + (∂t + x ∂x )2 = (1 + x 2 ) ∂x2 + ∂t2 + 2x ∂x ∂t + x ∂x is a sum of squares of left-invariant vector fields on (R2 , ◦), it satisfies Hörmander’s hypoellipticity condition

rank Lie{∂x , ∂t + x∂x }(t, x) = 2 ∀ (t, x) ∈ R2 , but L is not a sub-Laplacian on any Carnot group. Indeed, L is elliptic at any point (but B is not Euclidean!).

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4 Examples of Carnot Groups

4.1.4 K-type Groups Let us now suppose that the matrix B in special block-form ⎛ 0 0 ⎜ B1 0 ⎜ ⎜ B=⎜ ⎜ 0 B2 ⎜ .. .. ⎝ . . 0 0

the previous example takes the following ··· ··· .. .

0 0 .. .

..

. ···

0 Br

⎞ 0 0⎟ ⎟ .. ⎟ .⎟ ⎟, ⎟ 0⎠ 0

(4.2a)

where Bj is a pj × pj −1 block with rank equal to pj , for every j = 1, 2, . . . , r. Moreover, p0 ≥ p1 ≥ · · · ≥ pr and p0 + p1 + · · · + pr = N . Finally, the 0 blocks in (4.2a) are suitably chosen in such a way that B has dimension N × N. We want to show that the group B related to this matrix is a Carnot group. It will be called a group of Kolmogorov type or, in short, a K-type group. Let us split RN as follows R N = R p0 × R p1 × · · · × R p r and define, for every λ > 0, Dλ x = Dλ (x (0) , x (1) , . . . , x (r) ) = (λx (0) , λ2 x (1) , . . . , λr+1 x (r) ), where x (i) ∈ Rpi for 0 ≤ i ≤ r. We also put δ (t, x) = (λt, D x). λ

(4.2b)

λ

Claim 1. For every λ > 0, the dilation δλ is an automorphism of B. To prove this claim we need the following lemma. Lemma 4.1.1. For every t ∈ R and λ > 0, we have E(λt) Dλ = Dλ E(t)

(4.2c)

where E(t) = exp(tB), B is as in (4.2a) and Dλ is the dilation in (4.2b). Proof. Since B k = 0 for every k ≥ r + 1, one has E(t) =

r

t k B k /k!,

k=0

and (4.2c) holds for every t ∈ R and λ > 0 iff λ k B k Dλ = Dλ B k

∀ k ≥ 0, ∀ λ > 0.

(4.2d)

This identity holds true when k = 0. An easy direct computation shows that it also holds true for k = 1. As a consequence, λ2 B 2 Dλ = λB(λBDλ ) = λB(Dλ B) = (λBDλ )B = (Dλ B)B = Dλ B 2 . Then (4.2d) holds true for k = 2. An iteration of this argument shows (4.2d) for k ≥ 2.

4.1 A Primer of Examples of Carnot Groups

187

From this lemma the Claim 1 easily follows. Indeed, for every z = (t, x), z = ∈ R1+N , we have

(t , x )

(δλ z) ◦ (δλ z ) = = (by Lemma 4.1.1) = = =

(λt, Dλ x) ◦ (λt , Dλ x ) (λt + λt , Dλ x + E(λt )Dλ x) (λ(t + t ), Dλ x + Dλ E(t )x) δλ (t + t , x + E(t )x) δλ (z ◦ z ).

Thus, B = (R1+N , ◦, δλ ) is a homogeneous group whose first layer is R × Rp0 = {(t, x (0) ) | t ∈ R, x (0) ∈ Rp0 }. Moreover, the vector fields in the Jacobian basis related to this first layer are given by (4.2e) Y = ∂t + Bx, ∇x , ∂x1 , . . . , ∂xp0 . Claim 2. We have rank(Lie{Y, ∂x1 , . . . , ∂xp0 }(0, 0)) = 1 + N. Once this claim is proved, it will follow that (R1+N , ◦, δλ ) is a Carnot group of step r + 1 with 1 + p0 generators, which are the vector fields in (4.2e). Thus the related canonical sub-Laplacian is given by ΔB = Y 2 + ΔRp0 ,

where ΔRp0 =

p0

∂x2j .

(4.2f)

j =1

This sub-Laplacian will be said of Kolmogorov type. To prove Claim 2, the following lemma will be useful. Lemma 4.1.2. In Rp × Rq , let us consider the vector field Z = Ay · (∇z )T , where A is a q × p matrix, y ∈ Rp and z ∈ Rq . Suppose rank(A) = q ≤ p. Then span [∂yi , Z] | i = 1, . . . , p = span{∂z1 , . . . , ∂zq }. Proof. Let A = (ai,j )i≤q, j ≤p . Then [∂yi , Z] =

q

aj,i ∂zj ,

i = 1, . . . , p,

j =1

so that, since rank(A) = q,

dim span [∂yi , Z] | i = 1, . . . , p = q. This implies (4.2g).

(4.2g)

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4 Examples of Carnot Groups

We now prove Claim 2. Since B has the form (4.2a), we can write Y = ∂t +

r

Bi x (i−1) · (∇x (i) )T .

i=1

Then, by applying Lemma 4.1.2, we get span [∂xi , Y ] | i = 1, . . . , p0 = span [∂xi , B1 x (0) · (∇x (1) )T ] | i = 1, . . . , p0 = span ∂x (1) | i = 1, . . . , p1 . i

Another application of Lemma 4.1.2 gives span [∂x (1) , Y ] | i = 1, . . . , p1 = span ∂x (2) | i = 1, . . . , p2 . i

i

Iterating this argument, we get Lie{Y, ∂x1 , . . . , ∂xp0 } = Lie{Y, ∂x1 , . . . , ∂xN }. This obviously proves the claim.

Note. The groups of Kolmogorov type were introduced by E. Lanconelli and S. Polidoro [LP94] in studying a class of hypoelliptic ultraparabolic operators including the classical prototype operators of Kolmogorov–Fokker–Planck. The composition law in [LP94] was suggested by the structure of the fundamental solution of the operator ∂x21 + x1 ∂x2 − ∂x3 in R3 given by A.N. Kolmogorov in [Kol34]. Example 4.1.3. With the notation of Section 4.1.4, an example of K-type group is given by the choice p0 = p1 = 1, B1 = (1), whence 0 0 1 0 B= , N = p0 + p1 = 2, exp(s B) = . (1) 0 s 1 Hence, our K-type group B is R3 (whose points are denoted by (t, x1 , x2 )) equipped with the operation (t, x1 , x2 ) ◦ (s, y1 , y2 ) = (t + s, x1 + y1 , x2 + y2 + s x1 ) and the dilation δλ (t, x1 , x2 ) = (λt, λx1 , λ2 x2 ). This is naturally isomorphic to the Heisenberg–Weyl group H1 . Again following the notation of Section 4.1.4, another example of K-type group is given by the choice p0 = p1 = p2 = 1, B1 = B2 = (1), whence ⎛ ⎞ ⎛ ⎞ 1 0 0 0 0 0 B = ⎝ (1) 0 0 ⎠ , N = p0 + p1 + p2 = 3, exp(s B) = ⎝ s 1 0 ⎠ . s2 0 (1) 0 s 1 2

4.1 A Primer of Examples of Carnot Groups

189

Hence, our K-type group B is R4 (whose points are denoted by (t, x1 , x2 , x3 )) equipped with the operation (t, x1 , x2 , x3 ) ◦ (s, y1 , y2 , y3 ) s2 = t + s, y1 + x1 , y2 + x2 + s x1 , y3 + x3 + sx2 + x1 , 2 and the dilation δλ (t, x1 , x2 , x3 ) = (λt, λx1 , λ2 x2 , λ3 x3 ). Alternatively, the same non-trivial block (1) 0

0 (1)

in the latter matrix B can also be realized by the choice 1 0 B1 = , p0 = p1 = 2, 0 1 whence we have N = p0 + p1 = 4 and 0 0 , B = 1 0 0 0 1

where 0 =

0 0 . 0 0

Hence, our K-type group B is R5 (whose points are denoted by (t, x) = (t, x1 , x2 , x3 , x4 )) equipped with the operation (t, x) ◦ (s, y) = (t + s, y1 + x1 , y2 + x2 , y3 + x3 + s x1 , y4 + x4 + s x2 ), and the dilation δλ (t, x1 , x2 , x3 , x4 ) = (λt, λx1 , λx2 , λ2 x3 , λ2 x4 ). Remark 4.1.4. Assume that the matrix B is as in (4.2a). If we define dλ : R1+N → R1+N , dλ (t, x (0) , . . . , x (r) ) = (λ2 t, λx (0) , λ3 x (1) , . . . , λ2r+1 x (r) ), then {dλ }λ>0 is a group of automorphisms of B. For a proof of this statement, we directly refer to [LP94]. This remark shows that (R1+N , ◦, dλ ) is a homogeneous Lie group. It can be also easily proved that the ultraparabolic operator L = Δp0 + Y

(4.3)

is left-invariant (w.r.t. ◦) and homogeneous of degree two with respect to {dλ }λ>0 . Operator (4.3) generalizes the prototypes of the ones introduced by Kolmogorov in [Kol34].

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4 Examples of Carnot Groups

4.1.5 Sum of Carnot Groups Suppose we are given two homogeneous Carnot groups G(1) = (RN , ◦(1) ), G(2) = (RM , ◦(2) ) with dilations (1)

x ∈ G(1) ,

(2)

y ∈ G(2) ,

δλ (x) = (λ x (1) , . . . , λr x (r) ), δλ (y) = (λ y (1) , . . . , λs y (s) ), where x (i) ∈ RNi , i ≤ r,

N1 + · · · + Nr = N

and y (i) ∈ RMi , i ≤ s, M1 + · · · + Ms = M. Let ΔG(1) =

N1 j =1

Xj2

and ΔG(2) =

M1

Yj2

j =1

be the canonical sub-Laplacians on G(1) and G(2) , respectively. We define a homogeneous Carnot group G on RN +M as follows. Suppose r ≤ s. If (x, y) ∈ RN × RM , we consider the following permutation of the coordinates R(x, y) = (x (1) , y (1) , . . . , x (r) , y (r) , y (r+1) , . . . , y (s) ). We then denote the points of G ≡ RN +M by z = R(x, y). We finally define the group law ◦ and the dilation δλ on G as one can expect: for every z = R(x, y), ζ = R(ξ, η) ∈ G, we set z ◦ ζ = R(x ◦(1) ξ, y ◦(2) η),

δλ z = R(δλ(1) x, δλ(2) y).

It is then easily checked that (G, ◦, δλ ) is a homogeneous stratified group of step s and N1 + M1 generators. Moreover, the canonical sub-Laplacian on G is the sum of the sub-Laplacians on G(1) and G(2) : ΔG = ΔG(1) + ΔG(2)

N1 M1 2 = Xj + Yj2 . j =1

j =1

For example, if G(1) is the ordinary Euclidean group on R2 and G(2) is the Heisenberg–Weyl group on R3 , then the “sum” of G(1) and G(2) is the Carnot group on R5 (whose points are denoted z = (x, y) = (x1 , x2 , y1 , y2 , y3 )) with the composition law x1 + x1 , x2 + x2 , , (x, y) ◦ (x , y ) = y1 + y1 , y2 + y2 , y3 + y3 + 2(y2 y1 − y1 y2 ) and dilation δλ (x, y) = (λx1 , λx2 , λy1 , λy2 , λ2 y3 ).

4.2 From a Set of Vector Fields to a Stratified Group

191

4.2 From a Set of Vector Fields to a Stratified Group In the analysis of PDE’s, problem naturally arises: given a linear second the following 2 , where the X ’s are smooth vector fields on RN , does X order operator L = m j j =1 j N there exist a Lie group on R with respect to which L is a sub-Laplacian? And, if the answer is affirmative, is the group law explicitly expressible? The aim of this section is to answer these questions. First of all, we recall some notation. For every k ∈ N, denote W (k) = span XJ | J ∈ {1, . . . , m}k , where, if J = (j1 , . . . , jk ), X(j1 ,...,jk ) = [Xj1 , · · · [Xjk−1 , Xjk ] · · ·]. Assume the vector fields Xj ’s satisfy the following conditions: (H0) X1 , . . . , Xm are linearly independent and δλ -homogeneous of degree one with respect to a suitable family of dilations {δλ }λ>0 of the following type δλ : RN → RN ,

δλ (x) = δλ (x (1) , . . . , x (r) ) := (λx (1) , . . . , λr x (r) ),

where r ≥ 1 is an integer, x (i) ∈ RNi for i = 1, . . . , r, N1 = m and N1 + · · · + Nr = N; (H1) dim(W (k) ) = dim{XI (0) : X ∈ W (k) } for every k = 1, . . . , r; (H2) dim(Lie{X1 , . . . , Xm }I (0)) = N. By the results2 of the previous sections, conditions (H0)–(H1)–(H2) are necessary for m j =1 Xj to be a sub-Laplacian on a suitable homogeneous Carnot group (see Proposition 1.2.13, Remark 1.4.8 and the very definitions of Carnot group and sub-Laplacian). Moreover, the hypotheses (H0)–(H1)–(H2) are independent, as the following examples show: • The vector fields ∂x1 , ∂x2 on R3 satisfy (H0) with respect to the dilation (λx1 , λx2 , λ2 x3 ). Moreover, they satisfy (H1) but not (H2); • The vector fields X1 = ∂x1 + x2 ∂x4 , X2 = ∂x2 , X3 = ∂x3 + x2 ∂x4 + x22 ∂x5 in R5 satisfy (H0) with respect to the dilations (λx1 , λx2 , λx3 , λ2 x4 , λ3 x5 ). Moreover, since [X1 , X2 ] = −∂x4 , [X1 , X3 ] = 0, [X2 , X3 ] = ∂x4 + 2x2 ∂x5 and [X2 , [X2 , X3 ]] = 2∂x5 , the given vector fields satisfy (H2) but not (H1); • the vector fields ∂x1 + x1 ∂x2 , ∂x2 on R2 satisfy (H1) and (H2) but do not satisfy (H0) with respect to any dilation (λσ1 x1 , λσ2 x2 ). (See also Section 4.4.) We are going to show that conditions (H0)–(H1)–(H2) are sufficient for the solvability of our problem. To begin with, we notice that the vector fields in W (k) are δλ -homogeneous of degree k (see Proposition 1.3.10). Moreover, by Proposition 1.3.9 and hypothesis (H1)

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4 Examples of Carnot Groups

W (i) ∩ W (j ) = {0} if i = j . By Remark 1.3.7, we also have W (k) = {0} for every k ≥ r + 1. Then, by Proposition 1.1.7, (4.4) Lie{X1 , . . . , Xm } = W (1) ⊕ · · · ⊕ W (r) . Moreover, by hypotheses (H0) and (H2), dim(W (1) ) = dim(span{X1 , . . . , Xm }) = m and rk=1 dimW (k) = N. The following proposition shows a crucial link between the dimension of the W (k) ’s and the Nk ’s in (H0). Proposition 4.2.1. If X1 , . . . , Xm satisfy dim(W (k) ) = Nk for any k ∈ {1, . . . , r}.

hypotheses

(H0)–(H1)–(H2),

(k)

then

(k)

Proof. Let us set Mk := dim(W (k) ) and fix a basis {Z1 , . . . , ZMk } of W (k) , 1 ≤ k ≤ r. Then (k) (k) {Z1 I (0), . . . , ZMk I (0)} span W (k) I (0), so that, by (H1), it is a basis of W (k) I (0) := {XI (0) : X ∈ W (k) }. By (4.4), the set of vector fields (1) (r) Z1(1) , . . . , ZM , . . . , Z1(r) , . . . , ZM r 1

(4.5)

is a basis of Lie{X1 , . . . , Xm }. Then, by (H2), the column vectors of the matrix

(1) (1) (r) (r) A := Z1 I (0) · · · ZM1 I (0) · · · Z1 I (0) · · · ZMr I (0) span RN . We shall show that they are also linearly independent. By Proposition 1.3.5 (k) and Remark 1.3.7, the vector fields Zj can be written as (k)

Zj =

Ns r

(k,j )

as,i

(s)

(∂/∂xi ),

s=k i=1 (k,j )

where as,i (k,j ) as,i (0)

is a polynomial function δλ -homogeneous of degree s − k. In particular,

= 0 for every k < s ≤ r. As a consequence, the matrix A takes the form

⎛

A(1) ⎜ .. ⎝ . 0

··· .. . ···

⎞ 0 .. ⎟ , . ⎠ (r) A

(k,j ) where A(k) = ak,i (0) 1≤i≤N

The block A(k) has dimension Nk × Mk and rank Mk since

k,

1≤j ≤Mk

.

4.2 From a Set of Vector Fields to a Stratified Group (k)

193

(k)

{Z1 I (0), . . . , ZMk I (0)} is a basis for W (k) I (0) and dim(W (k) I (0)) = dim(W (k) ) = Mk . It follows that Nk ≥ Mk for 1 ≤ k ≤ r. On the other hand, since the column vectors of A span RN , r

r

Mk ≥ N =

k=1

Nk .

k=1

Then Mk = Nk for any k ∈ 1, . . . , r.

Following the previous proof, we infer that the matrix

(1) (1) (r) (r) Z1 I (x) · · · ZN1 I (x) · · · Z1 I (x) · · · ZNr I (x) ,

x ∈ RN ,

takes the following form ⎛

A(1) ⎜ ⎜ ⎜ .. ⎝ .

0 A(2) .. .

··· ··· .. .

0 0 .. .

···

A(r)

⎞ ⎟ ⎟ ⎟, ⎠

where A(1) , . . . , A(r) are square constant non-singular matrices. As a consequence, if the vector fields X1 , . . . , Xm satisfy hypotheses (H0)–(H1)–(H2), then they also satisfy

(H1)∗ dim W (k) I (x) = dim W (k) ∀ k ≤ r, ∀ x ∈ RN .

(H2)∗ dim Lie{X1 , . . . , Xm }I (x) = N ∀ x ∈ RN . Condition (H2)∗ is the well-known Hörmander’s hypoellipticity condition for the 2. X partial differential operator m j =1 j Throughout the remaining part of this section, X1 , . . . , Xm will be a given set of smooth vector fields satisfying hypotheses (H0)–(H1)–(H2). The family {δλ }λ>0 will denote the family of dilations in (H0). We let a = Lie{X1 , . . . , Xm }. (k)

(k)

(1)

(1)

(4.6)

Finally, for every k = 1, . . . , r, Z1 , . . . , ZNk will be a fixed basis for W (k) . We know that (r)

(r)

{Z1 , . . . , ZN } := {Z1 , . . . , ZN1 , . . . , Z1 , . . . , ZNr } is a basis of a. In particular, dim(a) = N. For every ξ = (ξ1 , . . . , ξN ) = (ξ (1) , . . . , ξ (r) ), we set

(4.7)

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4 Examples of Carnot Groups

ξ ·Z =

N

ξj Zj =

j =1

Nk r

(k)

(k)

ξj Zj .

k=1 j =1 (k)

Obviously, a = {ξ · Z : ξ ∈ RN }. Moreover, from the structure of Zj matrix (4.6)) we get r N (k) ξj aj (x1 , . . . , xk−1 ) ∂xk , ξ ·Z = k=1

(see the

j =1

(k)

where aj is a suitable polynomial function independent of xk , . . . , xN . Then, by Remark 1.1.3, the map (x, t) → exp(t ξ · Z)(x) is well defined for every x ∈ RN and t ∈ R. Furthermore,1 Exp : RN −→ RN ,

Exp(ξ ) := exp(ξ · Z)(0)

is a global diffeomorphism with polynomial component functions. Its inverse function, which we shall denote by Log, has polynomial components too. We are now ready to define a composition law on RN , suggested by Corollary 1.3.29. The notation “exp” is introduced in Definition 1.1.2, page 8. Definition 4.2.2. If X1 , . . . , Xm satisfy hypotheses (H0)–(H1)–(H2), we set x, y ∈ RN ,

x ◦ y := exp(Log(y) · Z)(x).

(4.8)

Remark 4.2.3. We shall show that G := (RN , ◦, δλ ) is a Carnot group whose Lie algebra g is a in (4.7). Since the group operation depends only on the Lie algebra itself, it will follow, in particular, that the definition of ◦ is independent of the choice of the basis (k) {Zj | 1 ≤ k ≤ r, 1 ≤ j ≤ Nk } of a. The main task of the proof is to show that ◦ is associative. To this end, we shall use the following result, which is a consequence of the Campbell–Hausdorff–Dynkin formula. Lemma 4.2.4 (Particular case of Campbell–Hausdorff formula). With the hypotheses (H0)–(H1)–(H2) on X1 , . . . , Xm and set a = Lie{X1 , . . . , Xm }, the following result holds: for every X, Y ∈ a, there exists a unique V ∈ a such that

exp(Y ) exp(X)(x) = exp(V )(x) for every x ∈ RN . (4.9) 1 We explicitly note that we are using the notation Exp to denote a map on RN instead of on

an algebra of vector fields.

4.2 From a Set of Vector Fields to a Stratified Group

195

We postpone the proof of this deep result to Part III of the book (see Chapter 15). We would like to stress that V in (4.9) depends only on X and Y , in particular, it is independent of x ∈ RN . Note that no mention of the word “associativity” is made in the above lemma, yet it will be the turning point in proving the associativity of ◦. The statement of Lemma 4.2.4 can be rewritten as follows: for every ξ , η ∈ RN , there exists a unique ζ ∈ RN , which we shall denote by ξ ∗ η, such that

exp(η · Z) exp(ξ · Z)(x) = exp (ξ ∗ η) · Z (x) (4.10) for every x ∈ RN . Then, by Definition 4.2.2, for every x, y ∈ RN , we have

x ◦ y = exp(Log(y) · Z)(x) = exp(Log(y) · Z) exp(Log(x) · Z)(0)

= exp (Log(x) ∗ Log(y)) · Z (0) = Exp Log(x) ∗ Log(y) . Then we have the identity

x ◦ y = Exp Log(x) ∗ Log(y)

∀ x, y ∈ RN ,

which is equivalent to the following one Log(x ◦ y) = Log(x) ∗ Log(y),

∀ x, y ∈ RN .

(4.11)

With this identity at hand, we easily get the proof of the following theorem. Theorem 4.2.5. Let ◦ be the composition law introduced in Definition 4.2.2. Then (RN , ◦) is a Lie group. Proof. 1) I DENTITY ELEMENT. Since Exp(0) = 0 = Log(0), for every x ∈ RN , we have x ◦ 0 = exp(Log(0) · Z)(x) = x, 0 ◦ x = exp(Log(x) · Z)(0) = Exp(Logx) = x. Then 0 is the identity of (RN , ◦). 2) I NVERSE ELEMENT. For any x ∈ RN , we have x ◦ Exp(−Log(x)) = exp(−Log(x) · Z)(x)

= exp(−Log(x) · Z) exp(Log(x) · Z)(0) = 0. An analogous argument shows that Exp(−Log(x)) ◦ x = 0. Then, for every x ∈ RN ,

x −1 = Exp(−Log(x)).

3) A SSOCIATIVITY. For every x, y, z ∈ RN , we have

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4 Examples of Carnot Groups

(x ◦ y) ◦ z = exp(Log(z) · Z)(x ◦ y)

= exp(Log(z) · Z) exp(Log(y) · Z)(x)

(by (4.10)) = exp (Log(y) ∗ Log(z)) · Z (x). On the other hand, by (4.11),

x ◦ (y ◦ z) = exp(Log(y ◦ z) · Z)(x) = exp (Log(y) ∗ Log(z)) · Z (x). This, together with the previous identity, shows the associativity of ◦. To complete the proof of the theorem, we only have to note that the maps (x, y) → x ◦ y = exp(Log(y) · Z)(x),

x → x −1 = Exp(−Log(x))

are smooth.

In order to prove that {δλ }λ>0 is a family of automorphisms of (RN , ◦), we need the following lemma. Lemma 4.2.6. For every x, ξ ∈ RN , we have

δλ exp(ξ · Z)(x) = exp((δλ ξ ) · Z)(δλ (x))

∀ λ > 0.

(4.12)

Proof. The path R t → γ (t) = exp(tξ · Z)(x) is the solution to the Cauchy problem N γ˙ (t) = ξj Zj I (γ (t)), γ (0) = x. j =1

We have N

d (δλ (γ (t))) = δλ (γ˙ (t)) = ξj δλ Zj I (γ (t)) dt j =1

(by Corollary 1.3.6, page 35) =

N

ξj λσj (Zj I )(δλ (γ (t)))

j =1

= ((δλ ξ ) · Z)(δλ (γ (t))). Moreover, δλ (γ (0)) = δλ (x). This shows that μ := δλ (γ ) solves the Cauchy problem μ(t) ˙ = (δλ (ξ ) · Z)I (μ(t)), μ(0) = δλ (x). Thus δλ (γ (t)) = exp(t (δλ ξ ) · Z)(δλ (x)). By replacing t = 1 in this identity, we obtain (4.12).

Theorem 4.2.7. G := (RN , ◦, δλ ) is a homogeneous group.

4.2 From a Set of Vector Fields to a Stratified Group

197

Proof. By Theorem 4.2.5, we only have to prove that δλ is an automorphism of (RN , ◦). By Lemma 4.2.6, for every ξ ∈ RN , we have δλ (Exp(ξ )) = δλ (exp(ξ · Z)(0)) = exp((δλ ξ ) · Z)(0) = Exp(δλ (ξ )), so that, since Log = Exp−1 , Log(δλ (ξ )) = δλ (Log(ξ )). Then, for every x, y ∈ RN , δλ (x ◦ y) = δλ (exp(Log(y) · Z)(x)) (by Lemma 4.2.6) = exp(δλ (Log(y)) · Z)(δλ (x)) = exp(Log(δλ (y)) · Z)(δλ (x)) = (δλ (x)) ◦ (δλ (y)). This completes the proof.

By using the associativity property of the composition law ◦, it is easy to show that the vector fields Z1 , . . . , ZN are invariant with respect to the left translations on (RN , ◦, δλ ). Theorem 4.2.8. The vector fields Z1 , . . . , ZN are left-invariant on G. Proof. If ej = (0, . . . , 1, . . . , 0) (1 being the j -th component), then ej · Z = Zj and x ◦ Exp(t ej · Z) = exp(tZj )(x) for every x ∈ RN and t ∈ R. For any fixed α ∈ RN and u ∈ C ∞ (RN ), let us denote uα (x) := u(α ◦ x). From (1.15) (page 10) and the associativity of ◦, we obtain d uα x ◦ Exp(tej · Z) Zj (u(α ◦ x)) = Zj (uα (x)) = d t t=0

d u (α ◦ x) ◦ Exp(tej · Z) = (Zj u)(α ◦ x), = d t t=0 for every x ∈ RN . This completes the proof.

Corollary 4.2.9. Let g be the Lie algebra of G. Then g = a, where a = Lie{X1 , . . . , Xm }. Proof. By Theorem 4.2.8, we have a ⊆ g. On the other hand, by (4.7) and Proposition 1.2.7, dim(a) = N = dim(g). Thus a = g.

Let us now consider the vector fields Y1 , . . . , YN1 ∈ g such that Yj (0) = ∂xj , 1 ≤ j ≤ N1 . We know that N1 = m. Since Yj is δλ -homogeneous of degree one (see Corollary 1.3.19), by Corollaries 4.2.9 and 1.3.11 (the latter on page 37) and identity (4.4), we have Yj ∈ span{X1 , . . . , Xm }, 1 ≤ j ≤ m. Then, since Y1 , . . . , Ym are linearly independent,

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4 Examples of Carnot Groups

span{Y1 , . . . , Ym } = span{X1 , . . . , Xm }. This identity obviously implies Lie{Y1 , . . . , Ym } = Lie{X1 , . . . , Xm } = a = g. Thus Y1 , . . . , Ym are the of g. Then G = (RN , ◦, δλ ) is a homogeneous mgenerators 2 2 Carnot group, ΔG = j =1 Yj is its canonical sub-Laplacian and L = m j =1 Xj is a sub-Laplacian on G.

We can summarize the results of this section by stating the following theorem. Theorem 4.2.10. Let X1 , . . . , Xm be smooth vector fields in RN satisfying hypotheses (H0)–(H1)–(H2). Let {δλ }λ>0 be the family of dilations defined in (H0). Finally, let ◦ be the composition law on RN introduced in Definition 4.2.2. Then G = (RN , ◦, δλ ) is a homogeneous Carnot group of step r and with m generators whose Lie algebra g is Lie-generated by X1 , . . . , Xm , i.e. g = Lie{X1 , . . . , Xm }. Moreover, the second order partial differential operator L = Laplacian on G.

m

2 j =1 Xj

is a sub-

4.3 Further Examples In this section, we exhibit some non-trivial examples of homogeneous Carnot groups constructed starting from a set of vector fields satisfying hypotheses (H0)–(H1)–(H2) of the previous section. We introduce the notation: given n ∈ N, Bn will denote the following n × n (nilpotent of step n) matrix ⎛ ⎞ 0 0 ··· 0 ⎜1 0 ··· 0 ⎟ ⎜ ⎟ . (4.13) Bn := ⎜ . . . . . . . ... ⎟ ⎝ .. ⎠ 0 ···

1

0

4.3.1 The Vector Fields ∂1 , ∂2 + x1 ∂3 Let us consider in R3 the vector fields X1 = ∂x1 ,

X2 = ∂x2 + x1 ∂x3 .

We denote by x = (x1 , x2 , x3 ) the points of R3 . It is straightforwardly verified that {X1 , X2 } satisfy conditions (H0)–(H1)–(H2) of the previous section with respect to the dilations

4.3 Further Examples

δλ (x1 , x2 , x3 ) = (λx1 , λx2 , λ2 x3 ),

199

λ > 0.

Then, by Theorem 4.2.10, the operator L = ∂x21 + (∂x2 + x1 ∂x3 )2 is a sub-Laplacian (indeed, the canonical one) on a suitable homogeneous Carnot group G = (R3 , ◦, δλ ). We now construct the composition law ◦ by using Definition 4.2.2. With the notation of the previous section, we have W (1) = span{X1 , X2 },

W (2) = span{X3 },

where X3 = [X1 , X2 ] = ∂x3 . For every ξ = (ξ1 , ξ2 , ξ3 ) ∈ R3 , we have ξ ·X =

3

ξj Xj = ξ1 ∂x1 + ξ2 ∂x2 + (ξ2 x1 + ξ3 )∂x3 ,

j =1

so that exp(ξ · X)(x) = γ (1), where γ = (γ1 , γ2 , γ3 ) and ⎧ γ1 (0) = x1 , ⎪ ⎨ γ˙1 (t) = ξ1 , γ˙2 (t) = ξ2 , γ2 (0) = x2 , ⎪ ⎩ γ˙3 (t) = ξ2 γ1 (t) + ξ3 , γ3 (0) = x3 . An easy computation shows that 1 exp(ξ · X)(x) = x1 + ξ1 , x2 + ξ2 , x3 + ξ3 + ξ2 x1 + ξ1 ξ2 . 2 As a consequence, 1 Exp(ξ ) = exp(ξ · X)(0) = ξ1 , ξ2 , ξ3 + ξ1 ξ2 , 2 1 Log(η) = Exp−1 (η) = η1 , η2 , η3 − η1 η2 . 2 Then, by Definition 4.2.2, the composition law determined by X1 , X2 , X3 is given by x ◦ y = exp(Log(y) · X)(x) 1 = exp y1 X1 + y2 X2 + y3 − y1 y2 X3 (x1 , x2 , x3 ) 2 1 1 = x1 + y1 , x2 + y2 , x3 + y 3 − y1 y2 + y2 x1 + y1 y2 , 2 2 i.e. x ◦ y = (x1 + y1 , x2 + y2 , x3 + y3 + x1 y2 ).

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4 Examples of Carnot Groups

4.3.2 Classical and Kohn Laplacians In this section, we re-derive the classical Laplace operator and the Kohn Laplacian (already introduced in Sections 4.1.1 and 3.1, respectively) starting from the relevant vector fields. • The only example of homogeneous Carnot group of step 1 is the usual additive group (RN , +). If δλ denotes the usual dilation on RN δλ (x1 , . . . , xN ) = (λ x1 , . . . , λ xN ), a family of fields satisfying hypotheses (H0)–(H1)–(H2) is necessarily of the form {Xj }j ≤N , where N Xj = ai,j ∂i i=1

2 and A = (ai,j )i,j is a non-singular N ×N matrix so that N j =1 Xj is a strictly-elliptic N constant-coefficient operator. Given ξ, x ∈ R , it holds N ξj X j , exp(ξ · X)(x) = γ (1) set ξ · X := j =1

where γ˙ (r) = A · ξ and γ (0) = x. Hence exp(ξ · X)(x) = x + A · ξ,

Exp(ξ ) = A · ξ,

Log(y) = A−1 · y.

We find out x ◦ y = exp(Log(y) · X)(x) = x + A · Log(y) = x + A · A−1 · y = x + y, the usual additive structure of RN . The canonical sub-Laplacian related to this group is the ordinary Laplace operator N (∂xj )2 . Δ= j =1

• Consider now on R2N +1 (whose points are denoted by z = (x, y, t), x, y ∈ t ∈ R) the 2N vector fields

RN ,

Xj := ∂xj + 2yj ∂t ,

Yj := ∂yj − 2xj ∂t ,

j = 1, . . . , N .

If R2N +1 is equipped with the dilation δλ (z) = (λx, λy, λ2 t), the former 2N vector fields satisfy hypotheses (H0)–(H1)–(H2). We set T := [Xj , Yj ] = −4 ∂t . Given ζ = (ξ, η, τ ), z = (x, y, t) ∈ R2N +1 , one has N exp (ξj Xj + ηj Yj ) + τ T (z) = (μ(1), ν(1), ρ(1)), j =1

4.3 Further Examples

201

where

(μ, ˙ ν, ˙ ρ)(r) ˙ = ξ, η, −4τ + 2 ν(r), ξ − 2 μ(r), η , This gives (set ζ · Z :=

N

j =1 (ξj Xj

(μ, ν, ρ)(0) = (x, y, t).

+ ηj Yj ) + τ T )

exp(ζ · Z)(z) = x + ξ, y + η, t − 4τ + 2 y, ξ − 2 x, η , whence Exp(ζ · Z) = (ξ, η, −4τ ),

Log(z ) = (x , y , −t /4) · Z.

(4.14)

Fixed z, z ∈ R2N +1 , we have

z ◦ z = exp Log(z ) · Z (z) = x + x , y + y , t + t + 2 y, x − 2 x, y . (4.15) We recognize the well-known group (HN , ◦) on R2N +1 . Its canonHeisenberg–Weyl N 2 2 ical sub-Laplacian ΔHN = j =1 (Xj + Yj ) is the Kohn Laplacian on HN . As we know from Chapter 1, ◦ induces a Lie group structure ∗ (of “Campbell–Hausdorfftype”) on the Lie algebra hN of HN in the way re-described hereafter. For (ξ, η, τ ) ∈ RN × RN × R, we agree to set (ξ, η, τ )Z := (ξ, η, τ ) · Z =

N

(ξj Xj + ηj Yj ) + τ T ∈ hN .

j =1

Then the group law ∗ on hN is defined by

(ξ1 , η1 , τ1 )Z ∗ (ξ2 , η2 , τ2 )Z = Log Exp(ξ1 , η1 , τ1 )Z ◦ Exp(ξ2 , η2 , τ2 )Z

= Log (ξ1 , η1 , −4 τ1 ) ◦ (ξ2 , η2 , −4 τ2 )

= Log ξ1 + ξ2 , η1 + η2 , −4τ1 − 4τ2 + 2 η1 , ξ2 − 2 ξ1 , η2 1 1 = ξ1 + ξ2 , η1 + η2 , τ1 + τ2 − η1 , ξ2 + ξ1 , η2 . 2 2 Z If we drop the notation (·)Z and identify hN with R2N +1 via the basis (X1 , . . . , XN , Y1 , . . . , YN , T ), we have somewhat “more intrinsic” group (R2N +1 , ∗) canonically related to HN , where (ξ1 , η1 , τ1 ) ∗ (ξ2 , η2 , τ2 ) 1 1 = ξ1 + ξ2 , η1 + η2 , τ1 + τ2 − η1 , ξ2 + ξ1 , η2 . 2 2

(4.16)

This group is canonically related to HN in the sense that we make precise below: if we consider the change of coordinate system on HN induced by the exponential-type coordinates in (4.14), i.e.

(x, y, t) = Exp (ξ, η, τ )Z = (ξ, η, −4τ ),

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4 Examples of Carnot Groups

then, with respect to this new coordinates, the vector fields Xj and Yj are respectively turned into2 j := ∂ηj + 1 ξj ∂τ . j := ∂ξj − 1 ηj ∂τ , Y X 2 2 j ’s, we turn to construct a related j ’s and Y Starting from these new vector fields X Carnot group (as we did above): after simple computations we obtain the very same group law ∗ as in (4.16). 4.3.3 Bony-type Sub-Laplacians Let us consider in R1+N the operator 2 2 ∂ ∂ 2 ∂ N ∂ L= + t +t + ··· + t , ∂t ∂x1 ∂x2 ∂xN

(t, x1 , . . . , xN ) ∈ R1+N ,

quoted by J.-M. Bony in [Bon69, Rémarque 3.1] as an example of a sum of squares satisfying Hörmander condition but nevertheless with a “very degenerate” characteristic form. L is not a sub-Laplacian on any Carnot group since the vector Clearly, j ∂/∂x vanishes on the hyperplane t = 0. It is however sufficient to field N t j j =1 add a new coordinate in order to lift L to a sub-Laplacian. Indeed, consider on R2+N , whose points are denoted by (t, s, x), t, s ∈ R, x ∈ RN , the following operator L := T 2 + S 2 , where

t2 tN ∂x2 + · · · + ∂x . 2! N! N It is readily verified that the pair T , S satisfies hypothesis (H0) with respect to the family of dilations defined by T := ∂t ,

S := ∂s + t ∂x1 +

δλ (t, s, x) := (λ t, λ s, λ2 x1 , λ3 x2 , . . . , λN +1 xN ). For every k = 1, . . . , N, we then consider the vector field t N −k Xk := [T , [T , · · · [T , S] · · ·]] = ∂xk + t ∂xk+1 + · · · + ∂x .

(N − k)! N k times

With the notation of Section 4.2, we have W (1) = span{T , S} and, for k = 1, . . . , N, W (k+1) = span{Xk }. It is easy to recognize that the hypothesis (H1) is satisfied. Finally, we have 2 Indeed, for every u ∈ C ∞ (HN , R), u = u(x, y, t), we set v := u ◦ Exp, i.e. v =

v(ξ, η, τ ) = u(ξ, η, −4τ ), so that it holds ∂ξj v = (∂xj u) ◦ Exp, ∂ηj v = (∂yj u) ◦ Exp, ∂τ v = −4 (∂t u) ◦ Exp. Consequently, (Xj u) ◦ Exp = ∂ξj v −

1 ηj ∂τ v, 2

(Yj u) ◦ Exp = ∂ηj v +

1 ξj ∂τ v. 2

4.3 Further Examples

203

dim Lie{T , S}I (0) = 2 + N, whence also the hypothesis (H2) holds. As a consequence, L is a sub-Laplacian on a suitable homogeneous Carnot group (G, ◦) on R2+N with step 1 + N and with 2 generators. We now turn to construct the group multiplication ◦ on G by using Definition 4.2.2. Let α, β ∈ R and ξ ∈ RN be fixed. We have j N j j −k ξk Xk = α, β, β t /j + ξk t /(j − k)! αT +βS + k=1

. j =1,...,N

k=1

This yields exp[α, β, ξ ](t, s, x)

N := exp αT + βS + ξk Xk (t, s, x) = (τ, σ, γ )(1), k=1

where (here j runs from 1 to N) τ˙ (r) = α, τ (0) = t, σ˙ (r) = β, σ (0) = s, j γ˙j (r) = β τ j (r)/j ! + k=1 ξk τ j −k (r)/(j − k)!,

γj (0) = xj .

From a direct integration it follows that exp[α, β, ξ ](t, s, x) is given by

j (α + t)j +1 − t j +1 (α + t)j −k+1 − t j −k+1 + α + t, β + s, xj + β ξk . (j + 1)α (j − k + 1)α k=1

We agree to put ((α + t)j +1 − t j +1 )/α = (j + 1)t j when α = 0. We now define the following matrices ⎞ ⎛ 1 0 0 ··· 0 ⎜ α 1 0 ··· 0⎟ ⎟ ⎜ 2! ⎜ 2 .. ⎟ .. α ⎜ α . 1 .⎟ F (α) := ⎜ 3! ⎟, 2! ⎟ ⎜ . . .. .. ⎜ .. .. . . 0⎟ ⎠ ⎝ α N−2 α N−1 α · · · 1 N! (N −1)! 2! ⎛ α ⎞ ⎛ ⎞ α 2! ⎜ α2 ⎟ ⎜ α2 ⎟ ⎜ 3! ⎟ ⎜ 2! ⎟ ⎜ ⎜ 3 ⎟ ⎟ ⎜ α3 ⎟ ⎜ ⎟ V (α) := ⎜ 4! ⎟ , U (α) := ⎜ α3! ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎝ . ⎠ ⎝ . ⎠ αN (N +1)!

and the functions

αN N!

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4 Examples of Carnot Groups

(α, t) := α −1 ((α + t)F (α + t) − t F (t)), F (α, t) := α −1 ((α + t)V (α + t) − t V (t)). V It then holds

(α, t) · ξ + β V (α, t) , exp[α, β, ξ ](t, s, x) = α + t, β + s, x + F

Exp(α, β, ξ ) = α, β, F (α) · ξ + β V (α) ,

Log(τ, σ, y) = τ, σ, F −1 (τ ) · (y − σ V (τ ) . Let (t, s, x) and (τ, σ, y) ∈ R2+N be given. Then we have (t, s, x) ◦ (τ, σ, y)

(τ, t) · F −1 (τ ) · (y − σ V (τ )) + σ V (τ, t) . = τ + t, σ + s, x + F If we now prove that the following identities hold (see also (4.13)) (τ, t) = exp(t BN ) · F (τ ), F

(τ, t) − exp(t BN ) · V (τ ) = U (t), V

(4.17)

then the explicit form of the multiplication ◦ turns out to be (t, s, x) ◦ (τ, σ, y) = (τ + t, σ + s, x + exp(t BN ) · y + σ U (t)) N 1 1 = τ + t, σ + s, x1 + y1 + σ t, . . . , xN + yk t N −k + σ tN . (N − k)! N! k=1

The first identity in (4.17) follows by proving that, for every i, j ∈ {1, . . . , N } with i ≥ j , one has t i−k τ k−j (τ + t)i−j +1 − t i−j +1 = · , (i − j + 1)! τ (i − k)! (k − j + 1)! i

k=j

which readily follows by applying Newton’s binomial formula to the left-hand side. The second identity is equivalent to ti (τ + t)i+1 − t i+1 τ k t i−k = − · , i! (i + 1)! τ (k + 1)! (i − k)! i

i = 1, . . . , N,

k=1

which can be proved analogously. 4.3.4 Kolmogorov-type Sub-Laplacians We now reconsider the examples in Sections 4.1.3 and 4.1.4, and we show how to obtain the composition law of the groups of Kolmogorov type by using the results of Section 4.2. We consider in R1+N (whose points are denoted by (t, x (0) , . . . , x (r) ))

4.3 Further Examples

205

the vector fields introduced in (4.2e). Then, by the results proved in Sections 4.1.3– 4.1.4, they satisfy conditions (H0)–(H1)–(H2) in Section 4.2 with respect to the following family of dilations δλ (t, x (0) , . . . , x (r) ) = (λ t, λ x (0) , . . . , λr+1 x (r) ) (we refer the reader directly to the notation in Sections 4.1.3–4.1.4). For k = 0, . . . , r (k) (k) and j = 1, . . . , pk , we set Zj := ∂/∂ xj . If t, τ ∈ R and x, ξ ∈ RN are fixed, we have pk r+1 (k) (k) exp[τ, ξ ](t, x) := exp τ Y + ξj Zj (t, x) = (μ(1), γ (1)), k=1 j =1

where μ(r) ˙ = τ,

μ(0) = t;

γ˙ (r) = ξ + τ B · γ (r),

γ (0) = x.

This yields exp[τ, ξ ](t, x) = τ + t, exp(τ B) · x + Exp(τ, ξ ) = τ,

1

exp(τ (1 − r)B) · ξ dr ,

0 1

exp(τ (1 − r)B) · ξ dr ,

0

Log(s, y) = s,

1

exp(s(1 − r)B) dr

−1

·y .

0

As a consequence,

(t, x) ◦ (s, y) = t + s, y + exp(sB)x . We explicitly remark that this is the same group multiplication treated in Section 4.1.3. 4.3.5 Sub-Laplacians Arising in Control Theory We here discuss an example of homogeneous Carnot group arising from control theory, and we refer to [Alt99] for a description of the relevance of this example in that context. In RN , we consider the following vector fields X1 := ∂1 + x2 ∂3 + x3 ∂4 + · · · + xN −1 ∂N ,

X2 := ∂2 .

For every k = 3, . . . , N , we have Xk := [Xk−1 , X1 ] = ∂k , whence it is readily verified that X1 and X2 fulfill hypotheses (H0)–(H1)–(H2) with respect to the family of dilations δλ (x1 , x2 , x3 , . . . , xN ) := (λ x1 , λ x2 , λ2 x3 , . . . , λN −1 xN ).

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4 Examples of Carnot Groups

As a consequence, L = X12 + X22 is a sub-Laplacian on a suitable homogeneous Carnot group (G, ◦) on RN with step N − 1 and with 2 generators. In [Alt99] it is given a representation of G by means of matrices of the following form ⎛ ⎞ 1 x2 x3 x4 · · · xN N−2 2 ⎜ ⎟ x x ⎜ 0 1 x1 2!1 · · · (N1−2)! ⎟ ⎜ ⎟ ⎜ .. ⎟ .. ⎜0 0 ⎟ . . 1 x 1 ⎜ ⎟ ≡ (x1 , x2 , . . . , xN ) ∈ G, ⎜ ⎟ 2 . x1 . ⎜0 0 ⎟ . 0 1 ⎜ ⎟ 2! ⎜. . . ⎟ .. ... ... ⎝ .. .. x1 ⎠ 0 0 ··· 0 0 1 whereas the Lie group law is given by the matrix product. We hereafter show how to obtain the composition law following the lines described in Section 4.2. Let ξ ∈ RN be fixed. We have N ξk Xk = (ξ1 , ξ2 , ξ3 + ξ1 x2 , . . . , ξN + ξ1 xN −1 ) = ξ + ξ1 H x, k=1

where H is the following N × N matrix (see also (4.13)) 0 0 . H := 0 BN −2 This gives exp[ξ ](x) := exp( N k=1 ξk Xk )(x) = γ (1), where γ˙ (r) = ξ + ξ1 H γ (r), γ (0) = x, whence r exp(ξ1 (r − t) H )ξ dt. γ (r) = exp(ξ1 r H )x + 0

In particular,

1

exp[ξ ](x) = exp(ξ1 H )x + Exp(ξ ) =

exp(ξ1 (1 − t) H )ξ dt,

0 1

exp(ξ1 (1 − t) H )ξ dt.

0

It is straightforward to recognize that, for every ρ ∈ R, we have 1 0 exp(ρ H ) = . 0 exp(ρ BN −1 ) Given y = (y1 , y ) ∈ RN , the equation y = Exp(ξ ) is equivalent to the following ξ ) ∈ RN ) system (setting ξ = (ξ1 ,

4.3 Further Examples

y 1 = ξ1 ,

y= 0

1

207

ξ dt. exp ξ1 (1 − t)BN −1

As a consequence,

1

Log(y) = y1 , 0

exp(y1 (1 − t)BN −1 ) dt

−1

· y .

For any fixed x, y ∈ RN , this gives x ◦ y = exp(Log(y))(x) = y + exp(y1 H )x = (y1 + x1 , y + exp(y1 BN −1 ) x) N 1 N −j = y1 + x1 , y2 + x2 , y3 + x3 + y1 x2 , . . . , y N + y xj . (N − j )! 1 j =2

4.3.6 Filiform Carnot Groups In this section, we give the definition of filiform Carnot group. To this end, we recall the definition of filiform Lie algebra (see, e.g. [OV94, page 61]). Definition 4.3.1 (Filiform Lie algebra). Let h be a Lie algebra of finite dimension n ≥ 3. For every k ∈ N, we recall that the terms of the lower (or descending) central series for h are h1 := h,

h2 := [h, h],

...,

hk := [h, hk−1 ] = [h[h[· · · [h, h] · · ·]]],

k ∈ N.

k times

Then, the Lie algebra h is called filiform if codimh (hk ) = k

for every k such that 3 ≤ k ≤ n.

In other words, h is filiform if and only if dim(hk ) = n − k

for every k: 3 ≤ k ≤ n (where n = dim(h)).

(4.18)

We explicitly remark that for the lower central series we have h1 ⊇ h2 ⊇ h3 ⊇ · · · ⊇ hk−1 ⊇ hk

∀ k ∈ N.

Remark 4.3.2. If dim(h) = 3, (4.18) says that h is filiform iff dim(h3 ) = 0. Hence, simple arguments show that the only filiform Lie algebras of dimension three are: 1) Lie{X1 , X2 , X3 } with {X1 , X2 , X3 } linearly independent and [X2 , X1 ] = [X3 , X1 ] = [X3 , X2 ] = 0. Any Lie group with such a Lie algebra is isomorphic to the classical (R3 , +).

208

4 Examples of Carnot Groups

2) Lie{X1 , X2 } with {X1 , X2 , [X2 , X1 ]} linearly independent and [X2 , [X2 , X1 ]] = [X1 , [X2 , X1 ]] = 0. Any Lie group with such a Lie algebra is isomorphic to the Heisenberg–Weyl group H1 on R3 . Let us now suppose that n := dim(h) ≥ 4. From (4.18) we have dim(hn−1 ) = 1 and dim(hn ) = 0, whence every filiform algebra is nilpotent, and, precisely, (if n ≥ 4) an n-dimensional filiform algebra is nilpotent of step n−1 (i.e. any commutator of length ≥ n vanishes, and there exists at least one non-vanishing commutator of length = n − 1). A very useful fact is that the converse is also true, as the following proposition states. Proposition 4.3.3 (Characterization). Let h be a Lie algebra of finite dimension n ≥ 3. • If n ≥ 4, then h is filiform if and only if it is nilpotent of step n − 1. • If n = 3, the same is true as in the previous case, except the case of the commutative R3 . Note 4.3.4. Since the linear dimension, the step of nilpotency and being isomorphic to R3 are all invariants of isomorphic Lie algebras, by Proposition 4.3.3, we derive: if g is a Lie algebra isomorphic to a filiform one, then g is filiform too. Proof (of Proposition 4.3.3). The “only if” part follows from Remark 4.3.2. We turn to the “if” part. It can be proved by an inductive argument. For example, we give the proof when n = 4. Suppose that h is nilpotent of step 3. We have to prove that h fulfills (4.18), i.e. dim(h3 ) = 1

and dim(h4 ) = 0.

The second equality follows from the step-3-nilpotence of h. By contradiction, suppose that dim(h3 ) ≥ 2. Since h2 ⊇ h3 and h2 must contain a commutator of length 2 which is not of length 3, then dim(h2 ) ≥ 3. Finally, since there must exist at least two elements in h1 \ h2 , then the inequality dim(h2 ) ≥ 3 implies dim(h1 ) ≥ 5, which contradicts the assumption dim(h1 ) = dim(h) = 4. This completes the proof for n = 4. The proof in the general case follows these ideas and is left to the reader.

Example 4.3.5. Let h be a Lie algebra of finite dimension n satisfying the following commutator identities: if {X1 , . . . , Xn } is a basis (in the sense of vector spaces) forh, we have X3 = [X1 , X2 ],

X4 = [X1 , X3 ],

X5 = [X1 , X4 ],

...,

Xn = [X1 , Xn−1 ],

whereas all other commutators of the Xi ’s vanish identically. In other words, it holds

4.3 Further Examples

⎧ ⎪ ⎨ [X1 , Xi ] = Xi+1 [X1 , Xn ] = 0, ⎪ ⎩ [Xi , Xj ] = 0

209

for every i = 2, . . . , n − 1, for every 1 < i < j ≤ n.

Since h is evidently nilpotent of step n−1, then, by Proposition 4.3.3, h is filiform. In Sections 4.3.3 and 4.3.5 we have furnished two explicit models for a similar algebra. Vice versa, the following remarkable fact holds. Theorem 4.3.6 (Bratzlavsky [Bra74]). Let h be a filiform Lie algebra of finite dimension n. Then there exists a basis {Xi }i for h (in the sense of vector spaces) such that Xi+1 = [X1 , Xi ] ( for every i = 2, . . . , n − 1) and [X1 , Xn ] = 0. Hence any filiform Lie algebra has a basis of the following form: n − 2 times

X1 , X2 , [X1 , X2 ], [X1 , [X1 , X2 ]], [X1 , [X1 , [X1 , X2 ]]], . . . , [X1 , · · · [X1 , X2 ] · · ·] .

=:X3

=:X4

=:X5

=:Xn

In general, nothing is said about the remaining commutators [Xi , Xj ] for 1 < i < j ≤ n. Definition 4.3.7 (Filiform Carnot group). A Carnot group is said filiform if its Lie algebra is a filiform Lie algebra. (Note. If G and H are isomorphic Carnot groups and G is filiform, then H is filiform too. This follows from Note 4.3.4.) By Proposition 4.3.3 and Remark 4.3.2, we have the following result. Proposition 4.3.8. A Carnot group on RN (with N ≥ 3) is filiform if and only if it is nilpotent of step N − 1 (except the trivial case of the usual Euclidean (R3 , +)). Remark 4.3.9. For example, among the Heisenberg–Weyl groups Hn the only filiform one is H1 ; analogously, the only filiform Carnot group of step two (hence nilpotent of step two) has necessarily dimension 3: up to isomorphism, this is H1 . Moreover, the Carnot groups considered in Sections 4.3.3 and 4.3.5 are filiform. Again from Proposition 4.3.3 it follows that a filiform Carnot group G on RN (non-Euclidean) is characterized by a stratification of its algebra g of the following type g = V1 ⊕ · · · ⊕ VN −1

with dim(V1 ) = 2, dim(Vi ) = 1 ∀ i = 2, . . . , N − 1.

In particular, a non-Euclidean filiform Carnot group has necessarily two generators.

210

4 Examples of Carnot Groups

4.4 Fields not Satisfying One of the Hypotheses (H0), (H1), (H2) In this section, we exhibit some examples of vector fields not satisfying one of the hypotheses (H0), (H1) or (H2) on page 191: we formally try to construct a group for each situation and we show that what we obtain is not a Carnot group! 4.4.1 Fields not Satisfying Hypothesis (H0): The Group is not Well Posed We consider on R2 the following two polynomial vector fields X := (1 + x 2 ) ∂x ,

Y := (1 + y 2 ) ∂y .

It is easily seen that they satisfy hypotheses (H1) and (H2) on page 191, but not (H0). Indeed, we have

dim W (1) = 2 = dim span {(1 + x 2 , 0), (0, 1 + y 2 )} ∀ (x, y) ∈ R2 , and all the vector spaces W (k) ’s for k ≥ 2 reduce to {0}, for X and Y commute. However, X and Y are not δλ -homogeneous with respect to any dilation on R2 (for their component functions contain zero degree terms), hence (H0) is not fulfilled. This invalidates our construction of a Carnot group canonically related to X and Y , the construction being heavily dependent on the well-behaved properties of the fields and, in particular, dependent on the delicate Campbell–Hausdorff-type Lemma 4.2.4. It is nonetheless important to remark that a Campbell–Hausdorff formula holds in a more general setting than the one we presented here. For instance, it holds3 for vector fields satisfying the Hörmander condition (i.e. our hypothesis (H2)) and hence in the present situation too. However, we remark that the present case is complicatedby the lack of homogeneity of X and Y , which implies that the exponential series k≥0 X k I /k! may fail to converge everywhere. We formally try to construct a group in the present situation. As done in previous sections, we fix ζ := (ξ, η) ∈ R2 and consider the vector field ζ · Z := ξ X + η Y . We formally let, as usual, Exp(ζ · Z) := (x(1), y(1)), where (x(s), y(s)) solves

(x(s), ˙ y(s)) ˙ = (ζ · Z)I (x(s), y(s)) = ξ (1 + x 2 (s)), η (1 + y 2 (s)) , (x(0), y(0)) = (0, 0). After a simple computation, we get Exp(ζ · Z) = (tan ξ, tan η),

Log(x, y) = (arctan x, arctan y) · Z.

In particular, we explicitly remark that Exp is only defined for 3 The reader is referred to Nagel–Stein–Wainger [NSW85].

4.4 Fields not Satisfying One of the Hypotheses (H0), (H1), (H2)

211

π π π π (ξ, η) ∈ − , + × − ,+ . 2 2 2 2 We now fix (x1 , y1 ), (x2 , y2 ) ∈ R2 , and, again, we formally set (x1 , y1 ) ◦ (x2 , y2 ) := (x(1), y(1)), where (x(s), y(s)) solves

(x(s), ˙ y(s)) ˙ = arctan x2 (1 + x 2 (s)), arctan y2 (1 + y 2 (s)) (x(0), y(0)) = (x1 , y1 ). After a simple computation, we derive (x1 , y1 ) ◦ (x2 , y2 ) = (tan(arctan x1 + arctan x2 ), tan(arctan y1 + arctan y2 )) x1 + x2 y1 + y2 . , = 1 − x1 x2 1 − y1 y2 Where defined, this operation is commutative, associative, (0, 0) is the neutral element and has the inverse (x, y)−1 = (−x, −y). However, the operation is defined only away from the subset of R2 × R2 (x1 , y1 ), (x2 , y2 ) : x1 x2 = 1, or y1 y2 = 1 . Moreover, there does not exist any ε > 0 such that |x| < ε, |y| < ε implies |(x + y)/(1 − xy)| < ε. This makes it impossible to define a group from ◦. Another formal argument is more successful in this case. Indeed, we formally compute the Campbell–Hausdorff-type operation on the “algebra”: we identify (x, y) ∈ R2 with x X + y Y and we set

(ξ1 , η1 ) ∗ (ξ2 , η2 ) ≡ Log Exp(((ξ1 , η1 ) · Z) ◦ (Exp(ξ2 , η2 ) · Z))

= Log (tan ξ1 , tan η1 ) ◦ (tan ξ2 , tan η2 ) tan ξ1 + tan ξ2 tan η1 + tan η2 = Log , 1 − tan ξ1 tan ξ2 1 − tan η1 tan η2

= Log tan(ξ1 + ξ2 ), tan(η1 + η2 ) = (ξ1 + ξ2 , η1 + η2 ) · Z ≡ (ξ1 + ξ2 , η1 + η2 ) ∈ R2 . Thus, the operation on the “algebra” of the formal group related to ◦ is indeed a group law (the usual Euclidean structure on R2 ! reflecting the commutative nature of the algebra generated by the commuting vector fields {X, Y }). Analogously, we consider the change of coordinate system induced by the Log map, i.e. we consider new coordinates defined by (ξ, η) := (arctan x, arctan y).

212

4 Examples of Carnot Groups

With respect to these new coordinates, the vector fields X and Y are respectively turned into4 = ∂η . = ∂ξ , Y X These fields do fulfill hypotheses (H0), (H1) and (H2) of page 191, and the relevant homogeneous Carnot group is obviously the usual additive structure on R2 , exactly as for the operation ∗ above. For another example of polynomial vector fields satisfying hypotheses (H1) and (H2), but not (H0), see Ex. 18 at the end of this chapter. 4.4.2 Fields not Satisfying Hypothesis (H1): The Operation is not Associative We consider on R3 (whose points are denoted by (x, y, t)) the vector fields X := ∂x + y 2 ∂t ,

Y := ∂y .

It holds [X, Y ] = −2y ∂t ,

[X, [X, Y ]] = 0,

[Y, [X, Y ]] = −2∂t ,

and the commutators of length > 3 vanish identically. It is immediately seen that hypothesis (H2) is fulfilled, whereas (H1) is not. Indeed, W (2) = span{−2y ∂t } is one-dimensional, whereas {ZI (0) : Z ∈ W (2) } = {(0, 0, 0)} is zero dimensional. We remark that X and Y are δλ -homogeneous of degree 1 with respect to the (unusual) “dilation”: δλ : R3 −→ R3 ,

δλ (x, y, t) := (λx, λy, λ3 t).

We try formally to consider the relevant “exponential” map: we fix ζ := (ξ, η, τ ) ∈ R3 and consider the vector field ζ · Z := ξ X + η Y + τ [X, Y ]. We formally let, as usual, Exp(ζ · Z) := (x(1), y(1), t (1)), where (x(s), y(s), t (s)) solves

(x(s), ˙ y(s), ˙ t˙(s)) = ξ, η, ξ y 2 (s) − 2τ y(s) (x(0), y(0), t (0)) = (0, 0, 0). 4 Indeed, let v = v(ξ, η) ∈ C ∞ (R2 , R) and set u = u(x, y) := v(arctan x, arctan y). We

also set Log(x, y) := (arctan x, arctan y), so that u = v ◦ Log. We have ∂x u(x, y) =

i.e. ∂x u(x, y) =

1 (∂ξ v) ◦ Log, 1 + x2

1 (∂ξ v) ◦ Log, 1 + tan2 (ξ )

∂y u(x, y) =

1 (∂η v) ◦ Log, 1 + y2

∂y u(x, y) =

1 (∂η v) ◦ Log. 1 + tan2 (η)

This immediately gives Xu(x, y) = (∂ξ v) ◦ Log, Y u(x, y) = (∂η v) ◦ Log.

4.4 Fields not Satisfying One of the Hypotheses (H0), (H1), (H2)

213

After a simple computation, we get

1 2 Exp((ξ, η, τ ) · Z) = ξ, η, ξ η − τ η , 3 and this map is not globally invertible since Exp((0, 0, 0) · Z) = Exp((0, 0, 1) · Z). The singularity of this map reflects the fact that the system of vector fields X, Y, [X, Y ] is pointwise non-everywhere linearly independent! On the other hand, the everywhere well-posedness of the map is a consequence of the δλ -homogeneity of the vector fields X, Y . Since the pointwise dependence of X, Y, [X, Y ] plays a negative rôle, a natural question arises. What if we considered the pointwise everywhere linearly independent vector fields X, Y , [Y, [X, Y ]]? We proceed once again formally: let (ξ, η, τ ), (x, y, t) ∈ R3 be fixed, and let us consider

exp ξ X + ηY + τ [Y, [X, Y ]] (x, y, t) = (γ1 (1), γ2 (1), γ3 (1)), where

(γ˙1 (s), γ˙2 (s), γ˙3 (s)) = (ξ, η, ξ γ22 (s) − 2 τ ), (γ1 (0), γ2 (0), γ3 (0)) = (x, y, t).

A simple computation now gives ξ η2 + ξ y2 + ξ η y − 2 τ + t . (γ1 (1), γ2 (1), γ3 (1)) = x + ξ, y + η, 3 As a consequence, when (x, y, t) = (0, 0, 0), it holds (by the slight abuse of notation Exp(ξ, η, τ ) := Exp(ξ X + η Y + τ [Y, [X, Y ]])) ξ η2 − 2τ . Exp(ξ, η, τ ) = ξ, η, 3 This map is now everywhere invertible, and its inverse function is (by an analogous abuse of notation) x y2 t Log(x, y, t) = x, y, − . 6 2 Finally, given (x1 , y1 , t1 ), (x2 , y2 , t2 ) ∈ R3 , we set (formally following the ideas in Definition 4.2.2) (x1 , y1 , t1 ) ◦ (x2 , y2 , t2 )

= exp x2 X + y2 Y + (x2 y22 /6 − t2 /2)[Y, [X, Y ]] (x1 , y1 , t1 )

= x1 + x2 , y1 + y2 , t1 + t2 + x2 y12 + x2 y2 y1 . This binary operation on R3 is not associative, for

(4.19)

214

4 Examples of Carnot Groups

((0, 1, 0) ◦ (0, 1, 0)) ◦ (1, 0, 0) = (1, 2, 4) = (1, 2, 3) = (0, 1, 0) ◦ ((0, 1, 0) ◦ (1, 0, 0)). Moreover, X and Y are not invariant under left “translations” with respect to ◦ (as we know, the left-invariance is closely linked to the associativity; see the proof of Theorem 4.2.8). We explain more closely the fact that ◦ lacks to be associative by directly studying the Campbell–Hausdorff formula in the present situation. First, we fix x = (x1 , x2 , x3 ), (α1 , α2 , α3 ) and (β1 , β2 , β3 ) in R3 . Then, starting from x, we proceed along the integral path of the vector field A := α1 X + α2 Y + α3 [Y, [X, Y ]], so that, at unit time, we arrive (by definition) in exp(A)(x). Then, starting from exp(A)(x) and following the integral curve of the field B := β1 X + β2 Y + β3 [Y, [X, Y ]], we arrive (at unit time) in exp(B)(exp(A)(x)). Now, the Campbell–Hausdorff formula says that5 we get (at unit time) to this same final point even if we start from x and proceed along the integral path of the vector field C =AB =A+B +

1 1 1 [A, B] + [A, [A, B]] − [B, [A, B]]. 2 12 12

(4.20)

A direct computation shows that it holds 1 C = (α1 + β1 )X + (α2 + β2 )Y + (α1 β2 − α2 β1 )[X, Y ] 2 1 + α3 + β3 + (α2 − β2 )(α1 β2 − α2 β1 ) [Y, [X, Y ]]. 12 As we explained after the statement of Lemma 4.2.4 (page 194), we could certainly construct the desired associative operation on R3 , if C could be expressed as a constant-coefficient linear combination of X, Y, [Y, [X, Y ]]. But here we have C = (α1 + β1 )X + (α2 + β2 )Y 1 1 + α3 + β3 + (α1 β2 − α2 β1 )x2 + (α2 − β2 )(α1 β2 − α2 β1 ) [Y, [X, Y ]], 2 12 which shows that there do not exist three constants γ1 , γ2 , γ3 such that C = γ1 X + γ2 Y + γ3 [Y, [X, Y ]]. This can be seen as the very motivation for the lack of associativity of ◦ as defined in (4.19). 5 We explicitly remark that the Campbell–Hausdorff formula holds in the present case too,

see Chapter 15.

4.5 Exercises of Chapter 4

215

Remark 4.4.1. Nonetheless, we will be able to “lift” the above vector fields X, Y to suitable vector fields (on a larger vector space) satisfying hypotheses (H0)–(H1)– (H2), only exploiting the homogeneity property of X and Y and the fact that they fulfil Hörmander’s condition: this will be properly explained in Chapter 17 (see, in particular, Section 17.4, page 666), thus showing that the hypotheses of the present chapter can be somewhat weakened to produce, in case, “lifted” groups. 4.4.3 Fields not Satisfying Hypothesis (H2): The Group is Undefined We consider on R3 the following two vector fields X := ∂x1 ,

X2 := ∂x2 .

It is easily seen that they satisfy hypothesis (H0) with dilation δλ (x1 , x2 , x3 ) = (λx1 , λx2 , λ2 x3 ) and hypothesis (H1), but they do not satisfy hypothesis (H2) on page 191. Obviously, the exponential map (ξ1 , ξ2 , ξ3 ) → exp(ξ1 X1 + ξ2 X2 ) = (ξ1 , ξ2 , 0) does not even define a bijection from R3 onto R3 . Formally, the related “composition” would be (x1 , x2 , x3 ) ◦ (y1 , y2 , y3 ) = (x1 + y1 , x2 + y2 , 0) which possess no inverse map!

Bibliographical Notes. For some topics on filiform Lie algebras, we followed A.L. Onishchik and E.B. Vinberg [OV94, page 61]. Some of the topics presented in this chapter also appear in [Bon04].

4.5 Exercises of Chapter 4 Ex. 1) Consider on R4 the vector fields 1 x2 ∂3 − 2 1 X2 := ∂2 + x1 ∂3 + 2

X1 := ∂1 −

1 1 x3 ∂4 − x1 x2 ∂4 , 2 12 1 2 x ∂4 . 12 1

216

4 Examples of Carnot Groups

Prove that they fulfill hypotheses (H0)–(H1)–(H2) of page 191 with the dilation δλ (x1 , x2 , x3 , x4 ) := (λx1 , λx2 , λ2 x3 , λ3 x4 ). Set (1)

Z1 := X1 ,

(1)

Z2 := X2 ,

(2)

Z1 := [X1 , X2 ],

(3)

Z1 := [X1 , [X1 , X2 ]].

Following the notation of Section 4.2, verify that exp(ξ · Z)(x) equals ⎛ ⎞ x1 + ξ1 ⎜ ⎟ x 2 + ξ2 ⎜ ⎟, 1 ⎝ ⎠ x3 + ξ3 + 2 (ξ2 x1 − ξ1 x2 ) 1 1 x4 + ξ4 + 2 (ξ3 x1 − ξ1 x3 ) + 12 (x1 − ξ1 )(ξ2 x1 − ξ1 x2 ) whence Exp(ξ ) = ξ and Log(x) = x (i.e. precisely, Exp(ξ · Z) = ξ and Log(x) = x · Z). Finally, prove that the relevant composition law is ⎛ ⎞ x1 + y1 ⎜ ⎟ x2 + y2 ⎜ ⎟. 1 ⎝ ⎠ x3 + y3 + 2 (y2 x1 − y1 x2 ) 1 (x1 − y1 )(y2 x1 − y1 x2 ) x4 + y4 + 12 (y3 x1 − y1 x3 ) + 12 Ex. 2) Write down the canonical sub-Laplacian ΔG of the group obtained in Ex. 1 and observe that ΔG contains the first order differential term 16 x2 ∂4 . Ex. 3) Consider on R4 the vector fields X1 := ∂1 + x2 ∂3 + x22 ∂4 ,

X2 := ∂2 .

Prove that they fulfill hypotheses (H0)–(H1)–(H2) of page 191 with the same dilation as in Ex. 1. Set (1)

Z1 := X1 ,

(1)

Z2 := X2 ,

(2)

Z1 := [X1 , X2 ],

(3)

Z1 := [X2 , [X1 , X2 ]].

Following the notation of Section 4.2, verify that ⎞ ⎛ x 1 + ξ1 ⎟ ⎜ x 2 + ξ2 ⎟. exp(ξ · Z)(x) = ⎜ ⎠ ⎝ x3 − ξ3 + 12 ξ1 ξ2 + ξ1 x2 1 2 2 x4 − 2ξ4 + 3 ξ1 ξ2 − ξ2 ξ3 + ξ1 x2 + ξ1 ξ2 x2 − 2ξ3 x2 Deduce that

and

⎛

⎞ ξ1 ⎜ ⎟ ξ2 ⎟ Exp(ξ · Z) = ⎜ ⎝ ⎠ −ξ3 + 12 ξ1 ξ2 −2ξ4 + 13 ξ1 ξ22 − ξ2 ξ3 ⎛

⎞ x1 ⎜ ⎟ x2 ⎟ · Z. Log(x) = ⎜ ⎝ ⎠ −x3 + 12 x1 x2 1 − 12 x4 − 12 x1 x22 + 12 x2 x3

4.5 Exercises of Chapter 4

217

Finally, prove that the relevant composition law is ⎞ ⎛ x 1 + y1 ⎟ ⎜ x2 + y2 ⎟. x◦y =⎜ ⎠ ⎝ x3 + y3 + y1 x2 x4 + y4 + y1 x22 + 2 x2 y3 Ex. 4) Show that the inverse map for the Lie group in Ex. 3 is given by ⎞ ⎛ −x1 ⎟ ⎜ −x2 ⎟. x −1 = ⎜ ⎠ ⎝ −x3 + x1 x2 −x4 + 2 x2 x3 − x1 x22 Ex. 5) a) Consider the Lie group and the notation in Ex. 3. For every ξ, η ∈ RN , prove that Log(Exp(ξ · Z) ◦ Exp(η · Z)) equals ⎛ ⎞ ξ1 + η1 ⎜ ⎟ ξ 2 + η2 ⎜ ⎟. 1 ⎝ ⎠ ξ3 + η3 + 2 (ξ1 η2 − ξ2 η1 ) 1 1 ξ4 + η4 + 2 (ξ2 η3 − ξ3 η2 ) + 12 (ξ2 − η2 ) (ξ1 η2 − ξ2 η1 ) Denote the above composition law in R4 by ξ ∗ η. Deduce that F = (R4 , ∗) is a Carnot group isomorphic to G = (R4 , ◦). Find the Liegroup isomorphism turning F into G. Considering the natural identification R4 ξ ←→ ξ · Z ∈ g (g being the Lie-algebra of G) define a composition law on g dual to ∗. Compare this operation to the Campbell–Hausdorff composition law in (2.43), motivating your remarks. b) Consider on R4 the change of coordinates modeled on the Log map, i.e. the new coordinates ξ on R4 defined by ⎛ ⎞ x1 ⎜ ⎟ x2 ⎟. ξ = L(x) := ⎜ 1 ⎝ ⎠ −x3 + 2 x1 x2 1 − 12 x4 − 12 x1 x22 + 12 x2 x3 Prove that X1 and X2 are respectively turned6 by L into the following vector fields: 1 := ∂1 − 1 ξ2 ∂3 − 1 ξ22 ∂4 , X 2 12 1 1 1 X2 := ∂2 + ξ1 ∂3 + ξ1 ξ2 − ξ3 ∂4 . 2 12 2 6 I.e. if v = v(ξ ) and u(x) = v(L(x)), then X u(x) = (X i v)(L(x)). i

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4 Examples of Carnot Groups

1 and c) Carry out an exercise similar to Ex. 3 for the above vector fields X 4 X2 on R . Verify that the relevant composition law is the same as ∗ in Ex. 5-(a). Remark that the linear change of coordinates in R4 given by (ξ1 , ξ2 , ξ3 , ξ4 ) → (x1 , x2 , x3 , x4 ) := (ξ2 , ξ1 , −ξ3 , −ξ4 ) 1 and X 2 into X2 and X1 , respectively, of Ex. 1. (Why?) turns X Ex. 6) Let α ∈ R be fixed. Consider on R4 the vector fields 1 1 1 Z1 = ∂1 − x2 ∂3 − x3 + x2 (x1 + α x2 ) ∂4 , 2 2 12 1 1 1 Z2 = ∂2 + x1 ∂3 + − α x3 + x1 (x1 + α x2 ) ∂4 . 2 2 12 Prove that they fulfill hypotheses (H0)–(H1)–(H2) of page 191 with a suitable dilation. Verify that the relevant composition law is the same as ◦ in Ex. 1-(a) of Chapter 1. Ex. 7) For (x1 , y1 ), (x2 , y2 ) ∈ R2 \ {(x, y) ∈ R2 : xy = 1}, set x1 + x2 y1 + y2 . , (x1 , y1 ) ◦ (x2 , y2 ) := 1 − x1 x2 1 − y1 y2 Prove that ◦ is commutative, associative, has (0, 0) as neutral element and has the inverse (x, y)−1 = (−x, −y). However, prove that there does not exist any ε > 0 such that |x| < ε, |y| < ε implies |(x + y)/(1 − xy)| < ε. This makes it impossible to define a group from ◦. Ex. 8) Consider on R4 the vector fields 1 1 x2 ∂3 + x22 ∂4 , 2 6 1 1 X2 := ∂2 + x1 ∂3 + x3 − x1 x2 ∂4 . 2 6

X1 := ∂1 −

Prove that they fulfill hypotheses (H0)–(H1)–(H2) of page 191 with the dilation δλ (x1 , x2 , x3 , x4 ) := (λx1 , λx2 , λ2 x3 , λ3 x4 ). Verify that the relevant composition law is ⎛ ⎞ x 1 + y1 ⎜ ⎟ x2 + y2 ⎟. x◦y =⎜ ⎝ ⎠ x3 + y3 + 12 (x1 y2 − x2 y1 ) 1 x4 + y4 + (x3 y2 − x2 y3 ) + 6 (y2 − x2 ) (x1 y2 − x2 y1 ) Verify that the related Jacobian basis is X1 ,

X2 ,

[X1 , X2 ],

1 − [X2 , [X1 , X2 ]]. 2

4.5 Exercises of Chapter 4

219

Ex. 9) Write the following second order constant-coefficient strictly elliptic operator L on R2 as a sum of squares of vector fields L = 10 (∂x1 )2 + 10 ∂x1 ,x2 + 5 (∂x2 )2 . Is L a sub-Laplacian on a suitable homogeneous Carnot group? Which one? Ex. 10) Following the notation of Section 4.1.4, prove that an example of K-type group is given by the choice p0 = p1 = p2 = · · · = pr = 1, B1 = B2 = · · · = Br = (1), whence ⎞ ⎛ 0 0 ··· 0 0 ⎜1 0 ··· 0 0⎟ ⎟ ⎜ ⎜ .. .. ⎟ .. ⎜ . . .⎟ N = p0 + p1 + · · · + pr = 1 + r, B = ⎜0 1 ⎟, ⎟ ⎜ .. .. . . ⎝. . . 0 0⎠ 0 0 ··· 1 0 so that the relevant K-type group B is R2+r (whose points are denoted by (t, x1 , x2 , x3 , . . . , xr+1 )) equipped with the operation ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ s+t s t ⎟ y1 + x1 ⎜ x1 ⎟ ⎜ y 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ y2 + x2 + s x1 ⎜ x2 ⎟ ⎜ y2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 ⎟ s ⎜ x3 ⎟ ◦ ⎜ y3 ⎟ = ⎜ y + x + s x + x ⎟ 3 3 2 2 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ .. ⎟ ⎜ .. ⎟ ⎜ .. ⎟ ⎝ . ⎠ ⎝ . ⎠ ⎝ . ⎠ 2 r xr+1 yr+1 yr+1 + xr+1 + s xr + s2 xr−1 + · · · + sr! x1 and the dilation δλ (t, x1 , x2 , x3 , . . . , xr+1 ) = (λt, λx1 , λ2 x2 , λ3 x3 , . . . , λr+1 xr+1 ). Ex. 11) Following the notation of Section 4.1.4, find an explicit general expression for the composition law on the K-type group such that r = 1 and B1 = Ip0 . Ex. 12) Consider on R4 the vector fields Z1 = ∂x1 + x1 ∂x4 ,

Z1 = ∂x2 ,

Z1 = ∂x3 + x2 ∂x4 .

Prove that they fulfill hypotheses (H0)–(H1)–(H2) of page 191 with the dilation δλ (x1 , x2 , x3 , x4 ) := (λx1 , λx2 , λx3 , λ2 x4 ). Moreover, [Z1 , Z2 ] = 0 = [Z1 , Z3 ], [Z2 , Z3 ] = ∂x4 . Set Z4 := [Z2 , Z3 ]. Following the notation of Section 4.2, verify that ⎞ ⎛ x 1 + ξ1 ⎟ ⎜ x 2 + ξ2 ⎟, exp(ξ · Z)(x) = ⎜ ⎠ ⎝ x 3 + ξ3 x4 + ξ4 + x1 ξ1 + 12 ξ12 + ξ3 x2 + 12 ξ3 ξ2

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4 Examples of Carnot Groups

whence

⎞ ξ1 ⎟ ⎜ ξ2 ⎟, Exp(ξ · Z) = ⎜ ⎠ ⎝ ξ3 ξ4 + 12 ξ12 + 12 ξ3 ξ2 ⎛ ⎞ y1 ⎜ ⎟ y2 ⎟ · Z. Log(y) = ⎜ ⎝ ⎠ y3 y4 − 12 y12 − 12 y3 y2 ⎛

We explicitly remark that Exp(Z1 ) = (1, 0, 0, 1/2) = (1, 0, 0, 0), which shows that the Jacobian basis of g do not necessarily correspond to the canonical basis of G ≡ RN via the exponential map. (But see also Proposition 2.2.22, page 139.) Finally, prove that the relevant composition law is ⎞ ⎛ x1 + y1 ⎟ ⎜ x2 + y2 ⎟. ⎜ ⎠ ⎝ x3 + y3 x4 + y4 + y3 x2 + y1 x1 Instead, consider the basis {X1 , . . . , X4 } of g corresponding to the canonical basis {e1 , . . . , e4 } of R4 via the exponential map, i.e. Xi := Log(ei ) · Z. Verify that X1 = Z1 −

1 Z2 , 2

X2 = Z2 ,

X3 = Z3 ,

X4 = Z4 .

Now, equip R4 with the following composition law: given ξ, η ∈ R4 , we define ξ ∗ η ∈ R4 as the only vector such that Log(Exp(ξ · X) ◦ Exp(η · X)) = (ξ ∗ η) · X. Obviously, (R4 , ∗) is a Lie group isomorphic to (R4 , ◦). However, check out that ⎞ ⎛ ξ1 + η1 ⎟ ⎜ ξ 2 + η2 ⎟, ξ ∗η =⎜ ⎠ ⎝ ξ 3 + η3 ξ4 + η4 − 12 ξ1 − 12 η1 − 12 ξ3 η2 + 12 ξ2 η3 so that (R4 , ∗) is not a homogeneous Carnot group, for the only “dilations” on R4 which are homomorphisms of (R4 , ∗) have the form δλ (x1 , x2 , x3 , x4 ) = (λα1 x1 , λα2 x2 , λα3 x3 , λα4 x4 ) with α4 = α1 = α2 + α3 .

4.5 Exercises of Chapter 4

221

Ex. 13) According to the definition given in Section 4.1.5, page 190, write the composition law of the sum of the Heisenberg–Weyl groups H1 and H2 . Do the same for H1 and H1 . The groups obtained are prototype H-type groups according to the definition given in Section 3.6, page 169? Why? Ex. 14) Consider the fields on R2 X1 := ∂1 ,

X2 := x1 ∂2 .

Verify that [X1 , X2 ] = ∂2 , whereas all commutators of length > 2 vanish. It holds W (1) I (x) = span{(1, 0), (0, x1 )},

W (2) I (x) = span{(0, 1)}.

Whence dim(W (1) I (x)) = 2 only when x1 = 0, whereas

dim W (2) I (x) = 1 for every x ∈ R2 , whence hypothesis (H1) is not satisfied, but (H2) is. Moreover, X1 and X2 are homogeneous of degree 1 w.r.t. the “dilation” δλ (x1 , x2 ) := (λx1 , λ2 x2 ). Try to construct formally the Exp, Log and ◦ maps as in Section 4.4.2, page 212, and verify that 1 Exp(ξ1 , ξ2 ) = ξ1 , ξ1 ξ2 , 2 2 x2 2 x1 y2 , x ◦ y = x 1 + y1 , x2 + y2 + . Log(x1 , x2 ) = x1 , x1 y1 Observe that these maps are not everywhere defined, ◦ is not associative and X1 , X2 are not invariant w.r.t. ◦. Ex. 15) Let us consider on R2 the following polynomial vector fields X1 := ∂x1 ,

X2 := ∂x2 + x1 ∂x1 .

It is easily seen that they satisfy hypotheses (H1) and (H2) on page 191, but not7 (H0). We explicitly remark that W (k) = span{X1 } for every k ∈ N, k ≥ 2, whence Lie{X1 , X2 } is not a nilpotent algebra. Prove that the same holds for the group formally related to this case, according to the construction of Section 4.2. Following our usual notation, verify that the following facts hold.8 etξ2 − 1 tξ2 γ (t) = x1 e + ξ1 , x2 + tξ2 ξ2 7 Precisely, the only “dilation” map for which X and X are homogeneous is δ (x , x ) = λ 1 2 1 2 (λα x1 , x2 ) (for every α; X1 is δλ -homogeneous of degree α and X2 of degree 0). 8 We agree to denote by (eξ − 1)/ξ the (analytic) function equal to 1 when ξ = 0 and (eξ − 1)/ξ when ξ = 0.

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4 Examples of Carnot Groups

γ˙ = (ξ1 X1 + ξ2 X2 )I (γ ), γ (0) = x, ξ2 e −1 exp(ξ1 X1 + ξ2 X2 )(x1 , x2 ) = ξ1 , ξ2 , ξ2 y2 Log(y1 , y2 ) = y1 y X 1 + y2 X 2 , 2 e −1 x ◦ y = (x1 ey2 + y1 , x2 + y2 ). solves

Moreover, verify that (R2 , ◦) is a Lie group (not nilpotent) and that X1 and X2 Lie-generate the algebra of this group. Ex. 16) Give a complete proof of Proposition 4.3.3. Ex. 17) Let n ∈ N be fixed. Let us consider the Lie algebra kn with a basis {X, Y1 , Y2 , . . . , Yn } with commutator relations [Yi , Yj ] = 0,

1 ≤ i, j ≤ n,

[X, Yj ] = Yj +1 , [X, Yn ] = 0.

1 ≤ j ≤ n − 1,

Then kn is an (n + 1)-dimensional Lie-algebra nilpotent of step n; also, kn is stratified with stratification kn = span{X, Y1 } ⊕ span{Y2 } ⊕ span{Y2 } ⊕ · · · ⊕ span{Yn }. Prove that a model for kn is given by the algebra of vector fields on R1+n (the points are denoted by (x, y) with x ∈ R, y = (y1 , . . . , yn ) ∈ Rn ) spanned by X = ∂x ,

Yj =

n x k−j ∂y , (k − j )! k

j = 1, . . . , n.

k=j

Show that X, Y1 fulfill hypotheses (H0)–(H1)–(H2) of page 191 with the dilation δλ (x, y1 , y2 , . . . , yn ) = (λx, λy1 , λ2 y2 , . . . , λn yn ). Prove that the relevant composition law is (x, y) ◦ (ξ, η) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞

x+ξ y 1 + η1 y 2 + η2 + η 1 x 2 y3 + η3 + η2 x + η1 x2! 2 3 y4 + η4 + η3 x + η2 x2! + η1 x3! .. . 2

n−2

n−1

x x + η1 (n−1)! yn + ηn + ηn−1 x + ηn−2 x2! + · · · + η2 (n−2)!

⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎠

4.5 Exercises of Chapter 4

223

Compare to the Bony-type sub-Laplacians in Section 4.3.3, page 202. Another realization of kn as a matrix algebra can be described as follows. Denote by Kn the set of the (n + 1) × (n + 1) matrices of the form ⎛ ⎞ 0 x 0 ··· 0 yn ⎜ ⎟ ⎜ 0 0 x ... 0 yn−1 ⎟ ⎜ ⎟ ⎜ .. .. .. ⎟ .. ⎜. . 0 . . ⎟ ⎜ ⎟ M(x, y) := ⎜ ⎟. . .. x ⎜ 0 y3 ⎟ ⎜ ⎟ ⎜ 0 x y2 ⎟ ⎜ ⎟ ⎝ 0 y1 ⎠ 0 ··· 0 0 Consider the map ϕ : kn → Kn ,

ϕ xX +

n

yj Yj

= M(x, y).

j =1

Then ϕ is a Lie-algebra isomorphism. Now, consider the set Kn := {exp(M) : M ∈ Kn }, where exp denotes the exponential of a square matrix. It is remarkable to observe that (Kn , ·) (where · denotes the usual product of matrices) is a Lie group with the Lie algebra isomorphic to kn . In particular, Kn is nilpotent of step n and a Carnot group, for kn is stratified. Unfortunately, Kn is not a homogeneous group (but, obviously, it is isomorphic to (kn , ), being the Campbell–Hausdorff multiplication, which is a homogeneous Carnot group if we identify kn to Rn+1 in the obvious way). We sketch the verification that Kn is closed under the operation ·, for this involves the Campbell–Hausdorff formula. We aim to prove that if A, A ∈ Kn , then A·A ∈ Kn . To this end, let M, M ∈ Kn be such that A = exp(M), A = exp(M ). Moreover, since ϕ is an isomorphism between kn and Kn , there exist X, X ∈ kn such that M = ϕ(X), M = ϕ(X ). Then A · A = exp(ϕ(X)) · exp(ϕ(X )) 1 = exp ϕ(X) + ϕ(X ) + [ϕ(X), ϕ(X )] + · · · 2 1 = exp ϕ(X + X + [X, X ] + · · ·) 2

since X X ∈ kn . = exp ϕ(X X ) ∈ Kn , In the second equality, we have used the Campbell–Hausdorff formula for the exponential of matrices, which here involves a finite sum, since in Kn

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4 Examples of Carnot Groups

we have strictly upper triangular matrices. In the third one, we used the fact that ϕ is a Lie-algebra morphism. Incidentally, we have also proved that exp ◦ϕ : (kn , ) → (Kn , ◦) is a Lie-group isomorphism. Ex. 18) With reference to the preceding exercise (we adopt the therein notation), we study the Lie algebra k3 . We have k3 = span{X, Y1 , Y2 , Y3 } with [X, Y1 ] = Y2 , [X, Y2 ] = Y3 , i, j ∈ {1, 2, 3}. Let us also set ⎧ ⎛ 0 x ⎪ ⎪ ⎨ ⎜0 0 K3 = M(x, y) := ⎜ ⎝0 0 ⎪ ⎪ ⎩ 0 0

0 x 0 0

[X, Y3 ] = 0,

[Yi , Yj ] = 0,

⎫ ⎞ y3 ⎪ ⎪ ⎬ y2 ⎟ 4 ⎟ : (x, y) = (x, y1 , y2 , y3 ) ∈ R . y1 ⎠ ⎪ ⎪ ⎭ 0

Verify that K3 (equipped with the usual bracket of matrices) is a Lie algebra and that the map 3 ϕ : k3 → K3 , ϕ xX + yj Yj = M(x, y) j =1

is a Lie-algebra isomorphism. Now, consider the set K3 := {exp(M) : M ∈ K3 }, where exp denotes the exponential of a square matrix. Verify that ⎛ ⎞ 2 1 x x2! y3 + 2!1 xy2 + 3!1 x 2 y1 ⎜0 1 x ⎟ y2 + 2!1 xy1 ⎟. exp(M(x, y)) = ⎜ ⎝0 0 1 ⎠ y1 0 0 0 1 Verify that (K3 , ·) is a Lie group (here, · is the usual product of matrices), by first checking that

exp(M(x, y)) · exp(M(ξ, η)) = exp M (x, y) ◦ (ξ, η) , where

⎞ x+ξ ⎟ ⎜ y1 + η1 ⎟. (x, y) ◦ (ξ, η) = ⎜ 1 ⎠ ⎝ y2 + η2 + 2 (xη1 − ξy1 ) 1 (x − ξ )(xη1 − ξy1 ) y3 + η3 + 12 (xη2 − ξy2 ) + 12 ⎛

Deduce that (K3 , ·) is isomorphic to the Lie group (k3 , ), where is the Campbell–Hausdorff multiplication.

4.5 Exercises of Chapter 4

225

Ex. 19) Analogously to what we did in the preceding two exercises, we study the Lie algebra h1 (related to the Heisenberg group H1 ). We have h1 = span{X, Y, Z} with [X, Y ] = Z, Let us also set

⎧ ⎨

[X, Z] = [Y, Z] = 0. ⎛

0 x H1 = M(x, y, z) := ⎝ 0 0 ⎩ 0 0

⎫ ⎞ z ⎬ y ⎠ : (x, y, z) ∈ R3 . ⎭ 0

Verify that H1 (equipped with the usual bracket of matrices) is a Lie algebra and that the map ϕ : h1 → H1 ,

ϕ(xX + yY + zZ) = M(x, y, z)

is a Lie-algebra isomorphism. Now, consider the set H1 := {exp(M) : M ∈ H1 }, where exp denotes the exponential of a square matrix. Verify that ⎛ ⎞ 1 x z + 12 xy ⎠. exp(M(x, y, z)) = ⎝ 0 1 y 0 0 1 Verify that (H1 , ·) is a Lie group (here, · is the usual product of matrices), by first checking that

exp(M(x, y, z)) · exp(M(ξ, η, ζ )) exp M (x, y, z) ◦ (ξ, η, ζ ) , where

⎛

⎞ x+ξ ⎠. y+η (x, y, z) ◦ (ξ, η, ζ ) = ⎝ z + ζ2 + 12 (xη − ξy)

Deduce that (H1 , ·) is isomorphic to the Lie group (h1 , ), where is the Campbell–Hausdorff multiplication, which, in turn, is isomorphic to the usual Heisenberg–Weyl group H1 on R3 . Ex. 20) We here consider the Lie algebra n4 of the strictly upper triangular matrices of dimension 4. We use the notation ⎫ ⎧ ⎛ ⎞ 0 x 1 x4 x6 ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ⎜ 0 0 x2 x5 ⎟ 6 ⎟ : x = (x1 , . . . , x6 ) ∈ R . n4 = M(x) := ⎜ ⎝ ⎠ 0 0 0 x3 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0 0 0 0 Clearly, n4 is a Lie algebra of dimension 6, nilpotent of step 3. Now, consider the set

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4 Examples of Carnot Groups

N4 := {exp(M) : M ∈ N4 }, where exp denotes the exponential of a square matrix. Verify that exp(M(x)) ⎛ 1 x1 ⎜0 1 =⎜ ⎝0 0 0 0

x4 + 12 x1 x2 x2 1 0

x6 +

1 2!

(x1 x5 + x3 x4 ) + x5 + 12 x2 x3 x3 1

1 3!

x1 x2 x3

⎞ ⎟ ⎟. ⎠

Verify that (N4 , ·) is a Lie group (here, · is the usual product of matrices), by first checking that exp(M(x)) · exp(M(y)) = exp(M(x ◦ y)), where

⎛

x1 + y1 x2 + y2 x3 + y3 x4 + y4 + 12 (x1 y2 − x2 y1 )

⎞

⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟. 1 x◦y =⎜ ⎜ ⎟ x5 + y5 + 2 (x2 y3 − x3 y2 ) ⎜ ⎟ ⎜ x + y + 1 (x y − x y ) + 1 (x y − x y ) ⎟ 6 5 1 3 4 ⎟ ⎜ 6 2 1 5 2 4 3 ⎜ ⎟ 1 (x3 − y3 )(x2 y1 − x1 y2 ) + 12 ⎝ ⎠ 1 + 12 (x1 − y1 )(x2 y3 − x3 y2 ) Deduce that (N4 , ·) is isomorphic to the Lie group (n4 , ), where is the Campbell–Hausdorff multiplication. Verify that the above ◦ defines on R6 a homogeneous Carnot group of step 3 with 3 generators and dilations δλ (x) = (λx1 , λx2 , λx3 , λ2 x4 , λ2 x5 , λ3 x6 ). The first three vector fields of the Jacobian basis are 1 1 1 X1 = ∂x1 − x2 ∂x4 + − x5 + x2 x3 ∂x6 , 2 2 12 1 1 1 X2 = ∂x2 + x1 ∂x4 − x3 ∂x5 − x1 x3 ∂x6 , 2 2 6 1 1 1 x4 + x1 x2 ∂x6 . X3 = ∂x3 + x2 ∂x5 + 2 2 12 The commutator relations are 1 1 x3 ∂x6 , [X2 , X3 ] = ∂x5 + x1 ∂x6 , 2 2 [X1 , [X2 , X3 ]] = ∂x6 = −[X3 , [X1 , X2 ]],

[X1 , X2 ] = ∂x4 −

whereas all other commutators are zero.

5 The Fundamental Solution for a Sub-Laplacian and Applications

In this chapter, we enter the core of the study of the sub-Laplacians L on the homogeneous Carnot groups (and hence on the stratified Lie groups) of homogeneous dimension Q ≥ 3, by showing that they possess a fundamental solution Γ resemblant to the fundamental solution cN |x|2−N of the usual Laplace operator Δ on RN , N ≥ 3. This property is one of the most striking analogies between L and the classical Laplace operator. Indeed, we shall see that it holds Γ = d 2−Q , where Q is the homogeneous dimension of G and d is a symmetric homogeneous norm on G, smooth out of the origin (the relevant definitions will be given in Section 5.1). We shall also call d an L-gauge. To do this, we first fix some results on homogeneous norms and the Carnot– Carathéodory distance. Then, in Section 5.3, we define the fundamental solution Γ , whose existence follows from the hypoellipticity and the homogeneity properties of L. We then collect many of its remarkable properties. As a first application, we provide mean value formulas for L, generalizing to the sub-Laplacian setting the Gauss theorem for classical harmonic functions. These formulas will play a central rôle throughout the book and are proved by only using integration by parts and the coarea theorem (see Theorems 5.5.4 and 5.6.1). From the mean value formulas, we derive Harnack-type inequalities for L and the Brelot convergence property for monotone sequences of L-harmonic functions (see Theorem 5.7.10). Furthermore, as an application of the Harnack theorem, in Section 5.8 we derive several Liouville-type theorems for L. As another application of the properties of the fundamental solution of L, we prove the Sobolev–Stein embedding theorem in the stratified group setting (see Section 5.9). To end with the applications of Γ , we provide three sections devoted to the following topics: some remarks on the analytic-hypoelliptic sub-Laplacians, Lharmonic approximations, and finally an integral representation formula for the fundamental solution by R. Beals, B. Gaveau and P. Greiner [BGG96]. Finally, three appendices close the chapter. The first is devoted to the weak and the strong maximum principles for L. The second one provides an improved ver-

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5 The Fundamental Solution for a Sub-Laplacian and Applications

sion of the pseudo-triangle inequality. In the third appendix, we prove in details the existence of geodesics on Carnot groups. As a direct application of the maximum principles, we give a decomposition theorem for L-harmonic functions, resemblant to the decomposition of a holomorphic function on an annulus of C into the sum of the regular and singular parts from its Laurent expansion. Convention. Throughout this chapter, we fix a stratified group H of step r and m generators. Q denotes the homogeneous dimension of H. Together with H, a stratification V = (V1 , . . . , Vr ) of the algebra of H will be fixed. Moreover, L will be any sub-Laplacian on H related to the given stratification. We recall that any stratification of the algebra of H brings along a homogeneous Carnot group on RN isomorphic to H. Hence, together with the couple (H, V ), we fix G = (RN , ◦, δλ ), a homogeneous Carnot group isomorphic to H, as described in Proposition 2.2.22. We let Ψ : G → H be the Lie-group isomorphism, as in the cited proposition. We still denote by L the sub-Laplacian on G which is Ψ -related to the sub-Laplacian L on H (see (2.68), page 147). Obviously, the “homogeneous version” G of H depends upon the stratification V but not on the sub-Laplacian L. Thus, any definition and result given henceforward for homogeneous Carnot groups has its counterpart (and is actually intended) for any couple (H, V ), where H is an abstract stratified group and V is a stratification for G. Notation. We introduce the notation for the homogeneous Carnot group G = (RN , ◦, δλ ). Its dilations {δλ }λ>0 are denoted by δλ (x) = δλ (x (1) , . . . , x (r) ) = (λx (1) , . . . , λr x (r) ),

x (i) ∈ RNi ,

1 ≤ i ≤ r.

We denote by m := N1 the number of generators of G and assume that the homogeneous dimension Q = N1 + 2 N2 + · · · + r Nr ≥ 3. As we showed in Chapter 1, Section 1.4 (page 56), the sub-Laplacian L on G can be written as follows (see (1.90a)) L=

m

Xj2 = div(A(x)∇ T ),

j =1

where {X1 , . . . , Xm } is a family of vector fields that form a linear basis of the first layer of g, the Lie algebra of G. The matrix A is given by ⎛ ⎞ (X1 I (x))T . ⎠ .. A(x) = X1 I (x) · · · Xm I (x) · ⎝ T (Xm I (x))

5.1 Homogeneous Norms

229

and takes the following block form (see (1.91))

A1,1 A1,2 A= , A2,1 A2,2 where A1,1 is a strictly positive definite constant m × m matrix. The characteristic form of L N1 qL (x, ξ ) := A(x)ξ, ξ = Xj I (x), ξ 2 ,

x, ξ ∈ RN ,

(5.1a)

j =1

is non-negative definite and, for every fixed x ∈ RN , the set {ξ ∈ RN | qL (x, ξ ) = 0} is a linear space of dimension N − m. The vector-valued operator ∇L := (X1 , . . . , Xm )

(5.1b)

is called the L-gradient operator in G. Due to identity (5.1a), we have |∇L u| = 2

m

|Xj u|2 = A∇ T u, ∇ T u ,

u ∈ C 1 (RN , RN ).

(5.1c)

j =1

5.1 Homogeneous Norms Definition 5.1.1. We call homogeneous norm on (the homogeneous Carnot group) G, every continuous1 function d : G → [0, ∞) such that: 1. d(δλ (x)) = λ d(x) for every λ > 0 and x ∈ G; 2. d(x) > 0 iff x = 0. Moreover, we say that d is symmetric if 3. d(x −1 ) = d(x) for every x ∈ G. Example 5.1.2. Define |x|G :=

r

|x

(j )

|

2r! j

1 2r!

,

x = (x (1) , . . . , x (r) ) ∈ G,

(5.2)

j =1

where |x (j ) | denotes the Euclidean norm on RNj . Then | · |G is a homogeneous norm on G smooth out of the origin. It is symmetric if x −1 = −x for any x ∈ G. In general, if G is any Carnot group (with inverse x −1 not necessarily equal to −x) the map x → |Log (x)|G is a symmetric homogeneous norm on G smooth out of the origin. This follows from the facts that Log (δλ (x)) = δλ (Log (x)) and Log (x −1 ) = −Log (x). 1 With respect to the Euclidean topology.

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5 The Fundamental Solution for a Sub-Laplacian and Applications

Example 5.1.3 (Control norm). Let d be the control distance related to a system of generators of G (see Section 5.2). Define d0 (x) := d(x, 0),

x ∈ G.

Then (see Theorem 5.2.8 in Section 5.2) we shall see that d0 is a symmetric homogeneous norm on G. From the next (elementary) proposition it will follow that the homogeneous norms on G are all equivalent. Proposition 5.1.4 (Equivalence of the homogeneous norms). Let d be a homogeneous norm on G. Then there exists a constant c > 0 such that c−1 |x|G ≤ d(x) ≤ c |x|G

∀ x ∈ G,

(5.3)

where | · |G has been defined in (5.2). Proof. Due to the δλ -homogeneity of d and | · |G , inequalities (5.3) hold taking c := max{H, 1/ h}, where H := sup{d(x) : |x|G = 1},

h := inf{d(x) : |x|G = 1}.

We explicitly remark that H < ∞ and h > 0, since the set {x : |x|G = 1} is a compact subset of G not containing the origin and d is a continuous function strictly positive in G \ {0}. Corollary 5.1.5. For every fixed (non-necessarily symmetric) homogeneous norm d on G, there exists a constant c > 0 such that c−1 d(x) ≤ d(x −1 ) ≤ c d(x) ∀ x ∈ G.

(5.4)

Proof. The function x → d(x −1 ) is a homogeneous norm on G. Indeed, recall that δλ is an automorphism of G, whence δλ (x −1 ) = (δλ (x))−1 . Then the assertion follows from Proposition 5.1.4. Any homogeneous norm turns out to be locally Hölder continuous with respect to the Euclidean metric in the following sense. Proposition 5.1.6. Let d be a homogeneous norm on G. Then, for every compact set K ⊂ RN , there exists a constant cK > 0 such that d(y −1 ◦ x) ≤ cK |x − y|1/r where r is the step of G.

∀ x, y ∈ K,

(5.5)

5.1 Homogeneous Norms

231

(See also Proposition 5.15.1 (page 309) in Appendix C for an estimate from below of d(y −1 ◦ x).) Proof. Let K ⊂ RN be a compact set. It is easy to see that there exists a constant c = c(K) > 0 such that |x|G ≤ c |x|1/r for every x ∈ K, where | · |G has been defined in (5.2). We now use Proposition 5.1.4, and we obtain that there exists a constant c = c(K) > 0 such that d(x) ≤ c |x|1/r for every x ∈ K. Hence, (5.5) will follow if we prove that there exists a constant c = c(K) > 0 such that |y −1 ◦ x| ≤ c |x − y| for every x, y ∈ K. If we apply the mean value theorem to the function F (x, y) := y −1 ◦ x, we obtain |y −1 ◦ x| = |F (x, y) − F (x, x)| ≤ max JF (x, t x + (1 − t)y) |x − y| ≤ c |x − y|, t∈[0,1]

and the assertion is proved. Any homogeneous norm satisfies a kind of pseudo-triangle inequality. Proposition 5.1.7 (Pseudo-triangle inequalities. I). Let d be a homogeneous norm on G. Then there exists a constant c > 0 such that: 1) d(x ◦ y) ≤ c(d(x) + d(y)), 2) d(x ◦ y) ≥ 1c d(x) − d(y −1 ), 3) d(x ◦ y) ≥ 1c d(x) − c d(y) for every x, y ∈ G. Proof. Due to the δλ -homogeneity of d, inequality 1) is equivalent to the following one d(x ◦ y) ≤ c if d(x) + d(y) = 1. This inequality holds true taking c := max{d(x ◦ y) : d(x) + d(y) = 1}. Obviously, 1 ≤ c < ∞. From inequality 1) we now obtain d(x) = d((x ◦ y) ◦ y −1 ) ≤ c(d(x ◦ y) + d(y −1 )), whence 2). Now, 3) follows from 2) and Corollary 5.1.5 with a suitable change of the constant c. Given a homogeneous norm d0 on G, the function G × G (x, y) → d(x, y) := d0 (y −1 ◦ x) is a pseudometric on G. Indeed, we have the following proposition. Proposition 5.1.8 (Pseudo-triangle inequalities. II). With the above notation, there exists a positive constant c > 0 such that:

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5 The Fundamental Solution for a Sub-Laplacian and Applications

1) d(x, y) ≤ c d(y, x) for every x, y ∈ G (here c can be taken = 1 iff d0 is also symmetric), 2) d(x, y) ≤ c(d(x, z) + d(z, y)) for every x, y, z ∈ G (the pseudo-triangle inequality for d), 3) d(x, y) = 0 iff x = y. Proof. It immediately follows from Corollary 5.1.5 and Proposition 5.1.7.

5.2 Control Distances or Carnot–Carathéodory Distances We begin with some important definitions. Definition 5.2.1 (X-subunit path). Let X = {X1 , . . . , Xm } be any family of vector fields in RN . A piece-wise regular path γ : [0, T ] → RN is said to be X-subunit if γ˙ (t), ξ 2 ≤

m Xj I (γ (t)), ξ 2

∀ ξ ∈ RN ,

j =1

almost everywhere in [0, T ]. We shall denote by S(X) the set of all X-subunit paths, and we put l(γ ) = T if [0, T ] is the domain of γ ∈ S(X). We explicitly remark that every integral curve of ±Xj (j ∈ {1, . . . , m}) is X-subunit. Convention. We assume RN is X-connected in the following sense (a proof of this fact in the case of stratified vector fields will be given in Theorem 19.1.3 on page 716): For every x, y ∈ RN , there exists γ ∈ S(X),

γ : [0, T ] → RN

such that γ (0) = x and γ (T ) = y.

Then the following definition makes sense. Definition 5.2.2 (X-Carnot–Carathéodory distance). Suppose RN is X-connected. Then, for every x, y ∈ RN , we set

dX (x, y) := inf l(γ ) : γ ∈ S(X), γ (0) = x, γ (T ) = y . (5.6) Under suitable hypotheses, the above inf is actually a minimum. For example, in Appendix C, we show that this occurs if {X1 , . . . , Xm } are generators of the first layer of the stratified algebra of a homogeneous Carnot group (see also [HK00]). Proposition 5.2.3 (dX is a metric). If RN is X-connected, then the function (x, y) → dX (x, y) is a metric on RN , called the X-control distance or the Carnot–Carathéodory distance related to X.

5.2 Control Distances or Carnot–Carathéodory Distances

233

In what follows, when there is no risk of confusion, we shall simply write d instead of dX . Proof. It is quite easy to see that d is non-negative, symmetric and satisfies the triangle inequality. To prove positivity, i.e. d(x, y) = 0 ⇒ (x = y), (5.7) we compare d with the Euclidean metric (which has an interest in its own). Given x ∈ RN and r > 0, define n |Xj I (z)| : |z − x| ≤ r , (5.8a) M(x, r) := sup j =1

where | · | denotes the Euclidean norm. We next show the following inequality M(x, |x − y|) d(x, y) ≥ |x − y|

∀ x, y ∈ RN .

(5.8b)

This will obviously imply (5.7). By contradiction, assume (5.8b) is false for some x and y in RN . Then there exists γ ∈ S(X), γ : [0, T ] → RN , such that γ (0) = x, γ (T ) = y and M(x, |x − y|) T < |x − y|. As a consequence, if we put

t ∗ := sup t ∈ [0, T ] : |γ (s) − x| < |x − y| for 0 ≤ s ≤ t , we have |γ (t ∗ ) − x| =

0

t∗

m γ˙ (s) ds ≤

t∗

|Xj I (γ (s))| ds

j =1 0

≤ M(x, |x − y|) t ∗ ≤ M(x, |x − y|) T < |x − y|. It follows that t ∗ = T and |y − x| = |γ (T ) − x| < |x − y|. This contradiction proves (5.8b). If the vector fields X1 , . . . , Xm are left invariant w.r.t. the translations on a Lie group G = (RN , ◦), then the X = {X1 , . . . , Xm }-control distance has the same property. Indeed, we have the following proposition. Proposition 5.2.4 (Control distance of a left-invariant family). Let G = (RN , ◦) be a Lie group on RN , and let d be the control distance related to a family of left invariant vector fields X = {X1 , . . . , Xm } on G. Then d(x, y) = d(y −1 ◦ x, 0) and

d(x −1 , 0) = d(x, 0)

∀ x, y ∈ G,

(5.9a)

∀ x ∈ G.

(5.9b)

The proof of this proposition will easily follow from the next lemma.

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5 The Fundamental Solution for a Sub-Laplacian and Applications

Lemma 5.2.5. In the hypotheses of Proposition 5.2.4, let γ : [0, T ] → RN be a X-subunit curve. Then α ◦ γ is X-subunit for every α ∈ G. Proof. If we denote by Γ the path α ◦ γ , we have Γ˙ (s) = Jτα (γ (s)) · γ˙ (s) almost everywhere in [0, T ]. Then, for every ξ ∈ RN , T 2 Γ˙ (s), ξ 2 = γ˙ (s), Jτα (γ (s)) ξ ≤ = =

m j =1 m j =1 m

(since γ is X-subunit)

T 2 Xj I (γ (s)), Jτα (γ (s)) ξ 2 Jτα (γ (s)) · Xj I (γ (s)), ξ

(Proposition 1.2.3, page 14)

m 2 2 Xj I (α ◦ γ (s)), ξ = Xj I (Γ (s)), ξ .

j =1

j =1

This proves that Γ is X-subunit. Proof (of Proposition 5.2.4). Let x, y ∈ G, and let γ be a X-subunit path connecting x and y. For every α ∈ G, by the previous lemma, α ◦ γ is a X-subunit path connecting α ◦ x and α ◦ y. Then d(α ◦ x, α ◦ y) ≤ d(x, y). Since x, y, α are arbitrary, this inequality obviously also implies d(x, y) ≤ d(α ◦ x, α ◦ y). Thus, we have proved d(α ◦ x, α ◦ y) = d(x, y) ∀ x, y, α ∈ G.

(5.10)

Choosing in (5.10) α = y −1 , we obtain (5.9a). Putting x = 0 in (5.9a), we obtain d(y −1 , 0) = d(0, y) = d(y, 0), which is (5.9b). This completes the proof. The control distance related to homogeneous vector fields is homogeneous. More precisely, the following assertion holds. Proposition 5.2.6 (Control distance of a homogeneous family). Let d be the control distance related to a family X = {X1 , . . . , Xm } of smooth vector fields in RN . Assume the Xj ’s are λ -homogeneous of degree one with respect to the “dilations” λ : R N → R N ,

λ (x1 , . . . , xN ) = (λσ1 x1 , . . . , λσN xN ),

where σ1 , . . . , σN are positive real numbers. Then d(λ (x), λ (y)) = λ d(x, y) ∀ x, y ∈ RN ∀ λ > 0. For the proof of this proposition we need the following lemma.

(5.11)

5.2 Control Distances or Carnot–Carathéodory Distances

235

Lemma 5.2.7. In the hypotheses of Proposition 5.2.6, let γ : [0, T ] → RN be a X-subunit curve. Then, for every λ > 0, the curve Γ : [0, λ T ] → RN ,

Γ (s) := λ (γ (s/λ))

is a X-subunit path. Proof. For every ξ ∈ RN , we have (note that λ is a linear and symmetric map for it is represented by a diagonal matrix) 2 2 Γ˙ (s), ξ 2 = λ−2 λ (γ˙ (s/λ)), ξ = λ−2 γ˙ (s/λ), λ (ξ ) (since γ is X-subunit) ≤ λ−2

m 2 Xj I (γ (s/λ)), λ (ξ ) j =1

= (Corollary 1.3.6, page 35) =

m j =1 m

−1 2 λ λ (Xj I (γ (s/λ))), ξ 2 Xj I (λ (γ (s/λ))), ξ .

j =1

Since λ (γ (s/λ)) = Γ (s), this proves the lemma. Proof (of Proposition 5.2.6). Let x, y ∈ RN , and let γ : [0, T ] → RN be a Xsubunit curve connecting x and y. By the previous lemma, Γ (s) = λ (γ (s/λ)) (0 ≤ s ≤ λ t) is a X-subunit curve, so that, since Γ connects λ (x) and λ (y), d(λ (x), λ (y)) ≤ l(Γ ) = λ T = λ l(γ ). Then, being γ an arbitrary X-subunit curve connecting x and y, d(λ (x), λ (y)) ≤ λ d(x, y) ∀ x, y ∈ RN ∀ λ > 0.

(5.12)

x ) and 1/λ ( y ), respecThis inequality obviously implies (replace x and y with 1/λ ( tively, and then λ with 1/ λ; then remove “∼”) d(x, y) ≤

1 d(λ (x), λ (y)) λ

∀ x, y ∈ RN ∀ λ > 0.

Then (5.12) holds with the equality sign and the proposition is proved. Theorem 5.2.8 (Control distance of a homogeneous Carnot group). Let G be a homogeneous Carnot group on RN , and let d be the control distance related to any family of generators for G. Then G x → d0 (x) := d(x, 0) is a symmetric homogeneous norm on G.

(5.13)

236

5 The Fundamental Solution for a Sub-Laplacian and Applications

Proof. By means of Propositions 5.2.3, 5.2.4 and 5.2.6, we are only left to prove that d0 is continuous. For the proof of this fact, we refer to Theorem 19.1.3 on page 716. Remark 5.2.9. The homogeneous norm (5.13), in general, is not smooth. Corollary 5.2.10. Let G be a Carnot group. Denote by d the control distance related to a family of generators for G. Then, for every compact subset K of G, there exists a positive constant C(K) such that d(x, y) ≤ C(K) |x − y|1/r

∀ x, y ∈ K,

where r denotes the step of G. (See also Proposition 5.15.1 (page 309) in Appendix C for an estimate from below of d(x, y).) Proof. It follows from Propositions 5.1.6, 5.2.4 and Theorem 5.2.8.

5.3 The Fundamental Solution Throughout the sequel, we shall make use of some maximum principles for subLaplacians, which (for the reader’s convenience) we postpone to Appendix A of the present chapter (see Section 5.13). For our purposes, it is convenient to give the definition of fundamental solution of a sub-Laplacian L on a homogeneous Carnot group as follows. Definition 5.3.1 (Fundamental solution). Let G be a homogeneous Carnot group on RN . Let L be a sub-Laplacian on G. A function Γ : RN \ {0} → R is a fundamental solution for L if: (i) Γ ∈ C ∞ (RN \ {0}); (ii) Γ ∈ L1loc (RN ) and Γ (x) −→ 0 when x tends to infinity; (iii) LΓ = −Dirac0 , being Dirac0 the Dirac measure supported at {0}. More explicitly (recall that L∗ = L, being L∗ the formal adjoint of L), Γ Lϕ dx = −ϕ(0) ∀ ϕ ∈ C0∞ (RN ). (5.14) RN

Theorem 5.3.2 (Existence of the fundamental solution). Let L be a sub-Laplacian on a homogeneous Carnot group G (whose homogeneous dimension Q is > 2). Then there exists a fundamental solution Γ for L. (Note. Such a fundamental solution is indeed unique, as it will be proved in Proposition 5.3.10.)

5.3 The Fundamental Solution

237

Proof. The existence of such a fundamental solution may be proved by means of very general arguments from the theory of distributions, based on the hypoellipticity of L and of its formal adjoint L∗ (= L), jointly with the well-behaved homogeneity properties of L. Indeed, from the hypoellipticity of L (see property (A0), page 63) we infer the existence of a “local” fundamental solution satisfying LΓ = −Dirac0 on a neighborhood of the origin (see F. Trèves [Tre67, Theorems 52.1, 52.2]). Moreover, by using the homogeneity properties of L, a “local-to-global” argument can be performed. It is out of our scopes here to give the details. The complete proof is due to G.B. Folland and can be found in [Fol75, Theorem 2.1] (see also L. Gallardo [Gal82] for some further properties of Γ obtained via probabilistic techniques). An alternative proof can be found in [BLU02, Theorem 3.9]. From the integral identity (5.14) and condition (i) in the above Definition 5.3.1 we immediately get the L-harmonicity of Γ out of the origin. Indeed, if we replace in (5.14) a test function ϕ with support in RN \ {0}, by the smoothness of Γ out of the origin, we can integrate by parts obtaining (LΓ ) ϕ dx = 0 ∀ ϕ ∈ C0∞ (RN \ {0}). RN

This obviously implies LΓ = 0 in RN \ {0}.

(5.15)

A simple change of variable and the left-invariance of L w.r.t. the translations on G give the following theorem. Theorem 5.3.3 (Γ left-inverts L). Let L be a sub-Laplacian on a homogeneous Carnot group G. If Γ is a fundamental solution for L, then Γ (y −1 ◦ x) Lϕ(x) dx = −ϕ(y) ∀ ϕ ∈ C0∞ (RN ) (5.16) RN

and every y ∈ RN . Proof. The change of variable z = y −1 ◦ x gives −1 Γ (y ◦ x) Lϕ(x) dx = Γ (z) (Lϕ)(y ◦ z) dz. RN

RN

(5.17)

On the other hand, since L is left-invariant on G, then (Lϕ)(y ◦ z) = L(ϕ(y ◦ z)). Replacing this identity in (5.17) and using (5.14) with ϕ(·) replaced by ϕ(y ◦ ·), one gets the thesis.

238

5 The Fundamental Solution for a Sub-Laplacian and Applications

Remark 5.3.4. The integral identity (5.16) means that L(Γ (y −1 ◦ ·)) = −Diracy in the weak sense of distributions. Here Diracy denotes the Dirac measure supported at {y}. Due to identity (5.16), we can say that Γ is a left inverse of L. We next prove that Γ is a right inverse too. Theorem 5.3.5 (Γ right-inverts L). Let L be a sub-Laplacian on a homogeneous Carnot group G. If Γ is a fundamental solution for L, then, for every ϕ ∈ C0∞ (RN ), the function RN y → u(y) :=

RN

Γ (y −1 ◦ x) ϕ(x) dx

(5.18a)

is smooth and satisfies the equation Lu = −ϕ.

(5.18b)

The proof of this theorem requires some prerequisites. First of all, we note that a change of variable in the integral at the right-hand side of (5.18a) gives Γ (z) ϕ(y ◦ z) dz. u(y) = RN

Then we can differentiate under the integral sign and get the smoothness of u. Moreover, if supp(ϕ) ⊆ {x : d(x) ≤ R} (here d denotes a fixed homogeneous norm on G), then |ϕ(z)| dz |u(y)| ≤ sup{|Γ (z)| : d(y ◦ z) ≤ R} RN =: C(y) |ϕ(z)| dz. (5.19) RN

On the other hand, by Corollary 5.1.5 and Proposition 5.1.7, d(z) ≥

1 d(y) − cd(y ◦ z), c

for a suitable positive constant c independent of x, y, z. As a consequence, 1 C(y) ≤ sup |Γ (z)| : d(z) ≥ d(y) − cR , c so that, since Γ (z) vanishes as z goes to infinity, inequality (5.19) implies lim u(y) = 0.

y→∞

(5.20)

We then show a crucial property of the (ε, G)-mollifiers. The relevant definition is the following one.

5.3 The Fundamental Solution

239

Definition 5.3.6 (Mollifiers). Let G = (RN , ◦, δλ ) be a homogeneous Carnot group on RN . Let O be a fixed non-empty open neighborhood of the origin 0. Let also be given a function J ∈ C0∞ (RN , R), J ≥ 0, such that supp(J ) ⊂ O and J = 1. RN

For any ε > 0, we set Jε (x) := ε −Q J (δ1/ε (x)). Let u ∈ L1loc (RN ). We set, for every x ∈ RN , uε (x) := (u ∗G Jε )(x) := u(y) Jε (x ◦ y −1 ) dy RN = u(z−1 ◦ x) Jε (z) dz.

(5.21)

δε (O)

We call uε a mollifier of u (or (ε, G)-mollifier) related to the kernel J . Note that this mollifier depends only on G = (RN , ◦, δλ ) and J . For the use of mollifiers in a context of subelliptic PDE’s, see also [CDG97]. Example 5.3.7. Let G = (RN , ◦, δλ ) be a homogeneous Carnot group on RN . Let be a fixed homogeneous symmetric norm on G. We set a notation which will be used throughout the book. For every x ∈ G and every r > 0, we set

B (x, r) := y ∈ G : (x −1 ◦ y) < r . We say that B (x, r) is the -ball with center x and radius r. Also, fixed a point x ∈ G and a set A ⊂ G, we let

-dist(x, A) := inf (x −1 ◦ a) : a ∈ A . We call -dist(x, A) the -distance of x from A. The notation dist (x, A) will also be available. Let now a function J ∈ C0∞ (RN ), J ≥ 0 be given such that J = 1. supp(J ) ⊂ B (0, 1) and RN

For any ε > 0, we set Jε (x) := ε −Q J (δ1/ε (x)). Let u ∈ L1loc (Ω), Ω ⊆ RN open. For the open set Ωε := {x ∈ Ω : -dist(x, ∂Ω) > ε}, we define

uε (x) := (u ∗G Jε )(x) := = B (0,ε)

for every x ∈ Ωε .

B (x, ε)

u(y) Jε (x ◦ y −1 ) dy

u(y −1 ◦ x) Jε (y) dy

(5.22)

240

5 The Fundamental Solution for a Sub-Laplacian and Applications

We call uε a mollifier of u (or (ε, G)-mollifier) related to the homogeneous norm . Note that this mollifier depends only on G = (RN , ◦, δλ ), J and . Remark 5.3.8. Let the notation in Definition 5.3.6 be fixed. Let u ∈ L1loc (RN ). Then the following fact holds: ()

uε → u as ε → 0 in L1loc (RN ).

Indeed, we have (perform the change of variable y = δ1/ε (z)) u(z−1 ◦ x) ε −Q J (δ1/ε (z)) dz = u(δε (y −1 ) ◦ x)J (y) dy. uε (x) = δε (O)

O

J = 1), −1 dx |uε (x) − u(x)| dx = u(δ (y ) ◦ x) J (y) dy − u(x) ε RN O −1 dx = u(δ (y ) ◦ x) − u(x) J (y) dy ε RN O u(δε (y −1 ) ◦ x) − u(x) dx J (y) dy. ≤

As a consequence (recall that RN

O

RN

O

Given σ > 0, by well-known results, there exists ε = ε(σ, G, O) > 0 such that if 0 < ε < ε, then the integral in braces is ≤ σ for every fixed y ∈ O. This proves that |uε (x) − u(x)| dx ≤ σ J (y) dy = σ ∀ 0 < ε < ε, RN

O

i.e. () holds. Proposition 5.3.9 (L-harmonicity of the mollifier). Let G = (RN , ◦, δλ ) be a homogeneous Carnot group on RN . Let L be a sub-Laplacian on G. Let u ∈ L1loc (RN ) be a weak solution to Lu = 0 in RN , i.e. u Lϕ dy = 0 ∀ ϕ ∈ C0∞ (RN ). (5.23) RN

Then, if uε denotes the mollification on G w.r.t. any kernel J (as in Definition 5.3.6), we have (5.24) Luε = 0 in G for every ε > 0. Proof. First, note that, being supp(Jε ) ⊂ δε (O), u(y −1 ◦ x) Jε (y) dy. uε (x) = RN

For every test function ϕ, we thus have (Fubini–Tonelli’s theorem certainly applies)

5.3 The Fundamental Solution

RN

uε (x) Lϕ(x) dx =

RN

=

RN

=

RN

=

RN

241

Lϕ(x) u(y −1 ◦ x) Jε (y) dy dx N R

Jε (y) (Lϕ)(x)u(y −1 ◦ x) dx dy N R

Jε (y) (Lϕ)(y ◦ z) u(z) dz dy N R

Jε (y) L z → ϕ(y ◦ z) u(z) dz dy. RN

The inner integral in the far right-hand side is equal to zero by the hypothesis (5.23). Then uε (x) Lϕ(x) dx = 0 ∀ ϕ ∈ C0∞ (RN ), RN

so that the claimed (5.24) follows from the hypoellipticity of L and the fact that L∗ = L. With Proposition 5.3.9 at hand, it is easy to prove the uniqueness of the fundamental solution. Proposition 5.3.10 (Uniqueness of the fundamental solution). Let L be a subLaplacian on a homogeneous Carnot group G. The fundamental solution of L (whose existence is granted by Theorem 5.3.2) is unique. Proof. Let Γ and Γ be fundamental solutions for L. Then the function u = Γ − Γ has the following properties: u ∈ L1loc (RN ), u(x) → 0 as x → ∞ and u Lϕ dy = 0 ∀ ϕ ∈ C0∞ (RN ). RN

As a consequence, by Proposition 5.3.9, Luε = 0 in RN for every ε > 0. Thus, since uε (x) → 0 as x → ∞ (argue as in (5.20)), the maximum principle in Section 5.13 implies uε = 0 in RN . On the other hand, uε → u as ε → 0 in L1loc (RN ) (see Remark 5.3.8). Then u = 0 almost everywhere in RN , so that Γ = Γ in RN \ {0}. We now prove that Γ is “G-symmetric” with respect to the origin. More precisely, the following assertion holds. Proposition 5.3.11 (Symmetry of Γ ). Let L be a sub-Laplacian on a homogeneous Carnot group G. Let Γ be the fundamental solution of L. Then Γ (x −1 ) = Γ (x)

∀ x ∈ G \ {0}.

Proof. Given ϕ ∈ C0∞ (RN ), define Γ (y −1 ◦ x) Lϕ(y) dy, u(x) := − RN

x ∈ G.

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5 The Fundamental Solution for a Sub-Laplacian and Applications

The function u is smooth and vanishes at infinity (see (5.20)). Moreover, for every ψ ∈ C0∞ (RN ), Lu(x) ψ(x) dx = u(x) Lψ(x) dx RN RN

−1 Lϕ(y) Γ (y ◦ x) Lψ(x) dx dy =− N RN R = Lϕ(x) ψ(x) dx (by Theorem 5.3.3). RN

This proves that L(u − ϕ) = 0 in G. Thus, since u − ϕ vanishes at infinity, by the maximum principle, u = ϕ in RN . In particular, ϕ(0) = u(0) = − Γ (y −1 ◦ x)Lϕ(y) dy ∀ ϕ ∈ C0∞ (RN ), RN

so that x → Γ (x −1 ) is a fundamental solution of L (see Definition 5.3.1). The uniqueness of the fundamental solution (Proposition 5.3.10) implies Γ (x −1 ) = Γ (x) for every x ∈ G \ {0}. Finally, we are in the position to prove Theorem 5.3.5. Proof (of Theorem 5.3.5). Let u be the function defined in (5.18a). Then u ∈ C ∞ (RN ) and, for any test function ψ ∈ C0∞ (RN ), one has (Lu)(y) ψ(y) dy = u(y) Lψ(y) dy RN RN

Lψ(y) Γ (y −1 ◦ x) ϕ(x) dx dy = N RN R

ϕ(x) Γ (y −1 ◦ x)Lψ(y) dy dx = N RN R

ϕ(x) Γ (x −1 ◦ y) Lψ(y) dy dx. = RN

RN

Here we used Proposition 5.3.11 ensuring that Γ (y −1 ◦ x) = Γ ((x −1 ◦ y)−1 ) = Γ (x −1 ◦ y). Now, by identity (5.16), the inner integral at the far right-hand side is equal to −ψ(x). Then (Lu)(y) ψ(y) dy = − ϕ(x) ψ(x) dx ∀ ψ ∈ C0∞ (RN ). RN

RN

This gives identity (5.18b).

From the uniqueness of Γ we easily obtain its δλ -homogeneity. Proposition 5.3.12 (δλ -homogeneity of Γ ). Let L be a sub-Laplacian on a homogeneous Carnot group G. Let Γ be the fundamental solution of L. Then Γ is δλ -homogeneous of degree 2 − Q, i.e. Γ (δλ (x)) = λ2−Q Γ (x)

∀ x ∈ G \ {0} ∀ λ > 0.

5.3 The Fundamental Solution

243

Proof. For any fixed λ > 0, define Γ (x) := λQ−2 Γ (δλ (x))

∀ x ∈ G \ {0}.

It is quite obvious that Γ ∈ C ∞ (RN \ {0}) ∩ L1loc (RN ) and Γ (x) → 0 as x → ∞. Moreover, for every test function ϕ ∈ C0∞ (RN ), Γ (x) Lϕ(x) dx = λQ−2 Γ (δλ (x)) Lϕ(x) dx RN

RN

(by using the change of variable y = δλ (x)) Γ (y) (Lϕ)(δ1/λ (y)) dy = λ−2 RN

(since L is δλ -homogeneous of degree 2) Γ (y)L ϕ(δ1/λ (y)) dy = −ϕ(δ1/λ (0)) = −ϕ(0). = RN

This proves that Γ is a fundamental solution of L. Then, by Proposition 5.3.10, Γ = Γ and the assertion is proved. From the strong maximum principle in Theorem 5.13.8 we obtain another important property of Γ . Proposition 5.3.13 (Positivity of Γ ). Let L be a sub-Laplacian on a homogeneous Carnot group G. Let Γ be the fundamental solution of L. Then Γ (x) > 0 ∀ x ∈ G \ {0}. Proof. Let ϕ ∈ C0∞ (RN ), ϕ ≥ 0. Define Γ (y −1 ◦ x)ϕ(x) dx, u(y) := RN

y ∈ G.

The function u is smooth, vanishes at infinity and satisfies the equation Lu = −ϕ (see (5.20) and Theorem 5.3.5). Then Lu ≤ 0 in G and lim u(y) = 0. y→∞

By the maximum principle in Corollary 5.13.6, it follows that u ≥ 0 in G. Hence Γ (y −1 ◦ x) ϕ(x) dx ≥ 0 ∀ ϕ ∈ C0∞ (RN ), ϕ ≥ 0. RN

Thus, Γ ≥ 0, so that, since it is L-harmonic in G\{0}, the strong maximum principle in Theorem 5.13.8 implies Γ ≡ 0 or Γ (x) > 0 for any x = 0. The first case would contradict identity (5.14). Then Γ > 0 at any point of G \ {0}. This ends the proof.

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5 The Fundamental Solution for a Sub-Laplacian and Applications

Corollary 5.3.14 (Pole of Γ ). Let L be a sub-Laplacian on a homogeneous Carnot group G. The fundamental solution Γ of L has a pole at 0, i.e. lim Γ (x) = ∞.

x→0

(5.25)

Proof. Since Γ is smooth and strictly positive out the origin, we have h := min{Γ (x) : d(x) = 1} > 0. Here d denotes any fixed homogeneous norm on G. Then, by Proposition 5.3.12, Γ (x) = d(x)2−Q Γ (δ1/d(x) (x)) ≥ h d 2−Q (x). From this inequality (5.25) immediately follows. 5.3.1 The Fundamental Solution in the Abstract Setting The aim of this brief section is to show how to derive a “fundamental solution” for a sub-Laplacian L on an abstract stratified group H, starting from the fundamental on a homogeneous-Carnot-group copy G solution of the related sub-Laplacian L of H. Many other alternative “more intrinsic” definitions may be certainly provided, for example, by making use of the integration on an abstract Lie group. We considered more “in the spirit” of our exposition to pass through the homogeneous group G. Besides, this also makes unnecessary to furnish the (lengthy) theory of integration on manifolds. m 2 Let H be a stratified group with an algebra h. Let L = j =1 Xj be a subLaplacian on H, and let V = (V1 , . . . , Vr ) be the stratification of h related to L, according to Definition 2.2.25, page 144. Let also E be a basis for h adapted to the stratification V . By Proposition 2.2.22 on page 139 (whose notation we presently follow), there exists a homogeneous Carnot group G = (RN , E ) which is isomorphic to H. Let Ψ : G → H be the isomorphism as in Proposition 2.2.22-(1). With the therein notation, we have Ψ = Exp ◦ πE−1 , where, for every ξ ∈ RN , πE−1 (ξ ) is the vector field in h having ξ as N-tuple of the coordinates w.r.t. the basis E. m be the vector fields in g (the algebra of G) which are Ψ -related 1 , . . . , X Let X to X1 , . . . , Xm , respectively, i.e. j ) = Xj dΨ (X

for every j = 1, . . . , m.

We set := L

m j )2 , (X j =1

5.3 The Fundamental Solution

245

is a sub-Laplacian on the homogeneous Carnot group G) we let Γ and (since L denote its (unique) fundamental solution. A possible definition of a fundamental solution for L is Γ := Γ ◦ Ψ −1 .

(5.26)

Our task here is to show how Γ depends on the (arbitrary) choice of the above basis E. We show that, roughly speaking “up to a multiplicative factor”, Γ in (5.26) is intrinsic (see below for the precise statement). To this aim, let E1 and E2 be two bases of h adapted to the stratification V . With the above notation, let (for i = 1, 2) Gi = (RN , Ei ) and gi denotes the algebra of Gi . Moreover, we set Ψi := Exp ◦ πE−1 , i

i = 1, 2.

(5.27)

Again for i = 1, 2, we also let (i) m (i) , . . . , X X 1

be the vector fields in gi which are Ψi -related to X1 , . . . , Xm , respectively, i.e. (i) ) = Xj dΨi (X j We set i := L

for every j = 1, . . . , m.

m (i) )2 , (X j

(5.28)

i = 1, 2.

j =1

i . We claim that Finally, Γi denotes the fundamental solution of L . Γ2 = c1,2 · Γ1 ◦ (Ψ1−1 ◦ Ψ2 ), where c1,2 := det πE1 ◦ πE−1 2 This will give

Γ2 ◦ Ψ2−1 = c1,2 · Γ1 ◦ Ψ1−1 ,

(5.29) (5.30)

which proves the “almost” intrinsic nature of the definition (5.26) of Γ . We explicitly remark that πE1 ◦πE−1 is a linear automorphism of RN , so that its determinant is well2 posed. (Note. We explicitly remark that the uniqueness of the fundamental solution of L up to a multiplicative factor is a matter of fact. Indeed, even in the homogeneous Carnot setting, the fundamental solution depends on the measure within the integral in (5.14) of Definition 5.3.1. In order to have a unique fundamental solution, the choice of the Lebesgue measure on RN was quite natural (though arbitrary). Instead, in the context of an abstract Lie group H, the Haar measure is unique only up to a multiplicative factor, and there is no effective way to prefer a Haar measure instead of another. These remarks show that the constant in (5.30) is perfectly justified.) We now turn to the claimed (5.29). By invoking Proposition 5.3.10, set Γ1,2 := c1,2 · Γ1 ◦ (Ψ1−1 ◦ Ψ2 ),

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5 The Fundamental Solution for a Sub-Laplacian and Applications

then (5.29) will follow if we show that Γ1,2 satisfies (i), (ii) and (iii) of Defini2 . tion 5.3.1 w.r.t. the sub-Laplacian L To begin with, observe that (thanks to (5.27)) −1 −1 = πE1 ◦ πE−1 Ψ1−1 ◦ Ψ2 = Exp ◦ πE−1 ◦ Exp ◦ π , (5.31) E 1 2 2 and this last map is a linear isomorphism of RN . As a consequence, since Γ1 ∈ C ∞ (RN \ {0}), we immediately infer that the same holds for Γ1,2 . This proves (i). Moreover, since Γ1 ∈ L1loc (RN ) and Γ1 (x) −→ 0 when x tends to infinity, the same holds for Γ1,2 , again thanks to (5.31). This proves (ii). Finally, we prove (iii). First, note that from (5.28) one gets (1) ) = Xj = dΨ2 (X (2) ) dΨ1 (X j j

for every j = 1, . . . , m,

i.e. for every j = 1, . . . , m, (1) = d(Ψ −1 ◦ Ψ2 )(X (2) ). X j j 1 Set Ψ1,2 := Ψ1−1 ◦ Ψ2 . This gives 1 = dΨ1,2 (L 2 ), L i.e. it holds 1 f ) ◦ Ψ1,2 = L 2 (f ◦ Ψ1,2 ) (L

∀ f ∈ C ∞ (RN , R).

(5.32)

Let now ϕ ∈ C0∞ (RN ). Then we have 2 ϕ) = c1,2 · 2 ϕ)(x) dx Γ1,2 (L Γ1 (Ψ1,2 (x))(L RN

RN

, see (5.31)) (by the linear change of variable y = Ψ1,2 (x) = πE1 ◦ πE−1 2 −1 dy 2 ϕ) Ψ (y) Γ1 (y)(L (see (5.32)) = c1,2 · 1,2 c1,2 RN 1 ϕ ◦ Ψ −1 (y) dy Γ1 (y)L = 1,2 RN

1 and ϕ ◦ Ψ −1 ∈ C ∞ (RN )) (Γ1 is the fundamental solution of L 1,2 0 −1 (0) = −ϕ(0). = − ϕ ◦ Ψ1,2 This proves (iii), and the proof is complete.

5.4 L-gauges and L-radial Functions On every homogeneous Carnot group, there exist distinguished smooth symmetric homogeneous norms playing a fundamental rôle for the sub-Laplacians. We call these norms gauges, according to the following definition.

5.4 L-gauges and L-radial Functions

247

Definition 5.4.1 (L-gauge). Let L be a sub-Laplacian on a homogeneous Carnot group G. We call L-gauge on G a homogeneous symmetric norm d smooth out of the origin and satisfying (5.33) L(d 2−Q ) = 0 in G \ {0}. An L-radial function on G is a function u : G \ {0} → R such that u(x) = f (d(x))

∀ x ∈ G \ {0}

for a suitable f : (0, ∞) → R and a given L-gauge d on G. The L-gauges are deeply related to the fundamental solution of L. Proposition 5.4.2. Let L be a sub-Laplacian on a homogeneous Carnot group G. Let Γ be the fundamental solution of L. Then (Γ (x))1/(2−Q) if x ∈ G \ {0}, d(x) := 0 if x = 0 is an L-gauge on G. Proof. The assertion follows from condition (i) in Definition 5.3.1, (5.15), Propositions 5.3.11, 5.3.12, 5.3.13 and Corollary 5.3.14. In the next section, we shall show the reverse part of Proposition 5.4.2 (see Theorem 5.5.6): if d is an L-gauge on G, then there exists a positive constant βd such that Γ = βd d 2−Q is the fundamental solution of L. As a consequence, by Proposition 5.3.10, the L-gauge is unique up to a multiplicative constant. In Section 9.8, we shall also prove the following fact: if d is a homogeneous norm on G, smooth out of the origin and such that L(d α ) = 0 in G \ {0} for a suitable α ∈ R, α = 0, then α = 2 − Q and d is an L-gauge on G (see Corollary 9.8.8, page 461, for the precise statement). The sub-Laplacian of an L-radial function takes a noteworthy form. Proposition 5.4.3. Let L be a sub-Laplacian on a homogeneous Carnot group G. Let f (d) be a smooth L-radial function on G \ {0}. Then

Q−1 L(f (d)) = |∇L d|2 f (d) + f (d) , (5.34) d 2 where, if L = m j =1 Xj , we set ∇L = (X1 , . . . , Xm ).

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5 The Fundamental Solution for a Sub-Laplacian and Applications

Proof. An easy computation gives (see also Ex. 6, Chapter 1) L(f (d)) =

m

Xj2 (f (d))

m m 2 = f (d) (Xj d) + f (d) Xj2 (d),

j =1

so that

j =1

j =1

L(f (d)) = f (d) |∇L d|2 + f (d) L(d).

(5.35)

(Note that (5.35) holds with the only assumption that f and d are smooth functions on some open subsets of G and R, respectively, and f ◦ d is defined.) Applying this formula to the function f (s) = s 2−Q and keeping in mind (5.33), we obtain 0 = (1 − Q) d −Q |∇L d|2 + d 1−Q L(d), hence

Q−1 . d Identity (5.34) follows by replacing this last identity in (5.35). L(d) = |∇L d|2

If is any (sufficiently smooth) homogeneous norm on G, the integration of a -radial function over a “-radially-symmetric” domain reduces to an integration of a single variable function. Proposition 5.4.4. Let L be a sub-Laplacian on a homogeneous Carnot group G. Let be any homogeneous norm on G smooth on G \ {0}. Let f () be a function defined on the -ball B (0, r) of radius r centered at the origin, B (0, r) := {x ∈ G : (x) < r}. Then, if f () ∈ L1 (B (0, r)), it holds f ((x)) dx = Q ω

r

s Q−1 f (s) ds,

(5.36)

0

B (0,r)

where ω denotes the Lebesgue measure of B (0, 1), ω := |B (0, 1)|. Proof. The coarea formula gives f ((x)) dx = B (0,r)

0

r

f (s)

{=s}

1 N −1 ds. dH |∇ |

(5.37)

On the other hand, by using the δλ -homogeneity of , we have

r 1 dH N −1 ds = |B (0, r)| = ω r Q 0 {=s} |∇| for every r > 0. Differentiating this last identity with respect to r, we obtain 1 dH N −1 = Qω r Q−1 . (5.38) {=r} |∇| By using this identity in (5.37), we immediately get (5.36).

5.4 L-gauges and L-radial Functions

249

Corollary 5.4.5. Let L be a sub-Laplacian on a homogeneous Carnot group G. Let be any homogeneous norm on G. The function α is locally integrable in RN if and only if α > −Q. Proof. If is also smooth on G \ {0}, by Proposition 5.4.4, r α (x) dx = Qω s α+Q−1 ds, 0

B (0,r)

and the assertion trivially follows. If is only continuous, we argue as follows. If α ≥ 0, the assertion is trivial. If α < 0, we have N α (x) dx = α (x) dx B (0,r)

k+1 k k=0 {r/2 ≤
≤ (r/2)α

N 1 dx 2kα {r/2k+1 ≤(x)
(by x = δr/2k (y))

k=0

= (r/2)α

N 1 r Q dy 2kα 2k {1/2≤(y)<1} k=0

= c r Q (r/2)α

N

2−k(α+Q) ,

k=0

hence, if α > −Q, then α is integrable on B (0, r). The reverse assertion follows by the same computation as above, by taking the obvious bound from below. A couple of examples of explicit L-gauges are in order. See also Example 5.10.3 and Chapter 18. Example 5.4.6 (Δ-gauge). The classical Laplace operator in RN , N ≥ 3, Δ :=

N

∂x2j

j =1

is the canonical (sub-)Laplacian on the Euclidean group E := (RN , +, δλ ) with δλ x = λ x. The homogeneous dimension of E is N and a Δ-gauge function is the Euclidean norm 1/2 N 2 xj , x = (x1 , . . . , xN ). x → |x| := j =1

Indeed, | · | is smooth and strictly positive out of the origin, δλ -homogeneous of degree 1 and, as it is well known, Δ(|x|2−N ) = 0 ∀ x = 0.

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5 The Fundamental Solution for a Sub-Laplacian and Applications

On every group of Heisenberg type, noteworthy explicit L-gauges are known. They were discovered by A. Kaplan [Kap80]. Example 5.4.7 (ΔH -gauges). Let H = (Rm+n , ◦, δλ ) be a (prototype) group of Heisenberg type (see Section 3.6, page 169). Denoting by (x, t) the points of H, x ∈ Rm , t ∈ Rn , we know that the canonical sub-Laplacian ΔH on H takes the form (see (3.14) on page 171) 1 B (k) x, ∇x ∂tk . ΔH = Δx + |x|2 Δt + 4 n

(5.39a)

k=1

Here, the B (k) ’s are n skew-symmetric m × m orthogonal matrices satisfying the relation B (i) B (j ) = −B (j ) B (i) for every i, j ∈ {1, . . . , n} with i = j . The dilations {δλ }λ>0 are given by δλ (x, t) = (λ x, λ2 t). Then Q = m + 2n

(5.39b)

is the homogeneous dimension of H. We want to show that 1/4 d(x, t) := |x|4 + 16 |t|2

(5.39c)

is a ΔH -gauge. Here |x| and |t| denote respectively the Euclidean norm of x ∈ Rm and t ∈ Rn . First of all, we remark that d is strictly positive and smooth out of the origin, it is δλ -homogeneous of degree one and symmetric, since (x, t)−1 = (−x, −t). Now, we aim to compute ΔH (d 2−Q ). To this end, it is convenient to fix the following notation: v(r, s) := r 4 + 16 s 2 , r = |x|, s = |t|;

α = (2 − Q)/4.

Then G(x, t) := (d(x, t))2−Q = (r 4 + 16 s 2 )α = v α (r, s). Since B (k) is skew-symmetric, we have B (k) x, ∇x G(x, t) = 4α v α−1 r 2 B (k) x, x = 0 for every x ∈ Rm , t ∈ Rn and 1 ≤ k ≤ n. Then, by using (5.39a), r2 ΔH G = Δx G + Δt G 4

r2 α−1 Δx v + Δt v = αv 4

r2 α−2 2 2 |∇x v| + |∇t v| . + α(α − 1) v 4

(5.39d)

5.5 Gauge Functions and Surface Mean Value Theorem

251

Now, thanks to the radial symmetry of v with respect to x and t, we easily obtain Δx v = (8 + 4m)r 2 ,

r2 Δt v = 8nr 2 4

(5.39e)

and

r2 |∇t v|2 = (16 rs)2 . 4 Replacing (5.39e) and (5.39f) in (5.39d), we get |∇x v|2 = 16r 6 ,

(5.39f)

ΔH G = 4α v α−1 r 2 (2 + m + 2n) + 16α(α − 1) v α−2 r 2 (r 4 + 16 s 2 ) = 4α v α−1 r 2 (2 + Q + 4(α − 1)) = 4α v α−1 r 2 (Q − 2 + 4α) = 0 (being α = (2 − Q)/4). Then ΔH (d 2−Q ) = 0 in H \ {0}, and d in (5.39c) is a ΔH -gauge. Remark 5.4.8 (Cylindrically-symmetric functions on H). In the notation of the above Example 5.4.7, we shall call cylindrically-symmetric any function u on H such that u(x, t) = v(|x|, t) for some function v = v(r, t), r ∈ R, r > 0, t ∈ Rn . Assume v is smooth. Then B (k) x, ∇x u(x, t) = B (k) x, x/|x| ∂r v(|x|, t) = 0, since B (k) x, x = 0 for every x ∈ Rm . As a consequence, keeping in mind (3.14) and (3.16), we obtain the following form of ΔH and |∇H | for cylindrically-symmetric functions u(x, t) = v(|x|, t): m−1 1 vr + r 2 Δt , r 4 1 |∇H u|2 = vr2 + r 2 |∇t v|2 . 4 ΔH u = vrr +

(5.39g)

Here, r = |x| and vr = δr v, vrr = δrr v.

5.5 Gauge Functions and Surface Mean Value Theorem Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an Lgauge. The aim of this section is to prove a mean value theorem on the boundary of the d-balls for the L-harmonic functions. When L is the classical Laplace operator, our result will give back the Gauss theorem for classical harmonic functions. We shall obtain the mean value theorem as a byproduct of a representation formula for general

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5 The Fundamental Solution for a Sub-Laplacian and Applications

C 2 functions. These representation formulas will play a major rôle throughout the book. For any x ∈ G and r > 0, we recall that we have already defined the d-ball of center x and radius r as follows: Bd (x, r) := {y ∈ G : d(x −1 ◦ y) < r}.

(5.40)

Then Bd (x, r) = x ◦ Bd (0, r). By using the translation invariance of the Lebesgue measure and the δλ -homogeneity of d, one also easily recognizes that |Bd (x, r)| = r Q |Bd (0, 1)| =: ωd r Q .

(5.41)

We explicitly remark that ∂Bd (x, r) := {y ∈ G : d(x −1 ◦ y) = r} is a smooth manifold of dimension N − 1. Indeed, by Sard’s lemma, this holds true for almost every r > 0. The assertion then follows for every r > 0, since ∂Bd (x, r) is diffeomorphic to ∂Bd (x, 1) via the dilation δr . (Note that, so far, d may be any homogeneous norm smooth out of the origin.) Definition 5.5.1 (The kernels of the mean value formulas). Let L be a subLaplacian on the homogeneous Carnot group G, and let d be an L-gauge. We set, for x ∈ G \ {0}, ΨL (x) := |∇L d|2 (x). Moreover, for every x, y ∈ G with x = y, we define the functions ΨL (x, y) := ΨL (x −1 ◦ y)

and KL (x, y) :=

|∇L d|2 (x −1 ◦ y) . |∇(d(x −1 ◦ ·))|(y)

(5.42)

Remark 5.5.2. We explicitly remark that ΨL is δλ -homogeneous of degree zero, a fact which will be used repeatedly. We would like to recall that ΨL appears in the “radial” form of L, see (5.34). We also explicitly remark that, while ΨL is translation-invariant (i.e. ΨL (α ◦ x, α ◦ y) = ΨL (x, y)), the function KL does not necessarily share the same property. Moreover, when L = Δ is the classical Laplace operator, then ΨL = KL = 1. For a general sub-Laplacian L, we shall prove the following fact (see Proposition 9.8.9, page 462): The function ΨL is constant if and only if G is the Euclidean group.

5.5 Gauge Functions and Surface Mean Value Theorem

253

For instance, consider the following example. Example 5.5.3. Let H = (Rm+n , ◦, δλ ) be a group of Heisenberg type. Denoting by (x, t) the points of H, x ∈ Rm , t ∈ Rn , we proved in Example 5.4.7 that the “Folland function” d(x, t) := (|x|4 + 16 |t|2 )1/4 is a ΔH -gauge. Using (5.39g), we obtain ΨH (x, t) = |∇H d(x, t)|2

6 2 2 4 r r rs r s2 = + 16 3 = + 16 4 , d d d d4 d where r = |x| and s = |t|. Then ΨH (x, t) =

|x|2 |x|4 + 16 |t|2

,

(x, t) = (0, 0).

Let Ω ⊆ G be an open set and u, v ∈ C 2 (Ω). By using the divergence form of the sub-Laplacian L (see (1.90a) on page 64), L = div(A(x) · ∇ T ), we easily get v Lu − u Lv = div(v A · ∇ T u) − div(u A · ∇ T v).

(5.43a)

Let us now assume that Ω is bounded with boundary ∂Ω of class C 1 and exterior normal ν = ν(y) at any point y ∈ ∂Ω. Then, if u, v are of class C 2 in a neighborhood of Ω, integrating (5.43a) on Ω and using the divergence theorem, we obtain the Green identity2 v A · ∇ T u, ν − u A · ∇ T v, ν dH N −1 . (5.43b) (v Lu − u Lv) dH N = Ω

∂Ω

Hereafter, dH N (respectively, dH N −1 ) stands for the N -dimensional (respectively, (N − 1)-dimensional) Hausdorff measure in RN . If we choose v ≡ 1 in (5.43b), we 2 Another way to write the Green identity is the following one: if L = m X 2 , we have j =1 j

Ω

(v Lu − u Lv) dH N =

∂Ω

⎛ ⎝v

m j =1

Xj u Xj I, ν − u

m

⎞ Xj v Xj I, ν ⎠dH N −1 .

j =1

This follows from (5.43b), recalling that (see (1.90b), page 64) A is the N × N symmetric matrix A(x) = σ (x) σ (x)T , where σ (x) is the N × m matrix whose columns are X1 I (x), . . . , Xm I (x).

254

5 The Fundamental Solution for a Sub-Laplacian and Applications

obtain3

Lu dH

N

=

Ω

A · ∇ T u, ν dH N −1 ,

(5.43c)

∀ u ∈ C0∞ (Ω).

(5.43d)

∂Ω

so that

Lu dH N = 0 Ω

Let us now consider an arbitrary open set O ⊆ RN such that Bd (x, r) ⊂ O for a suitable r > 0. For 0 < ε < r, we define the “d-ring” Dε,r := Bd (x, r) \ Bd (x, ε) = {y ∈ RN : ε < d(x −1 ◦ y) < r}. Given u ∈ C 2 (O), we apply the Green identity (5.43b) to the functions u and v := d 2−Q (x −1 ◦ ·) on the open set Dε,r . Since v is L-harmonic in G \ {0} (note that here we apply, for the first time and with crucial consequences, the fact d is an L-gauge), we obtain v Lu = Sr (u) − Sε (u) + Tε (u) − Tr (u), (5.43e) Dε,r

where we have used the following notation v A · ∇ T u, ν dH N −1 , Sρ (u) := ∂Bd (x,ρ) u A · ∇ T v, ν dH N −1 . Tρ (u) := ∂Bd (x,ρ)

Since v is constant on ∂Bd (x, ρ), keeping in mind (5.43c), we have 2−Q T N −1 2−Q A · ∇ u, ν dH =ρ Lu dH N , (5.43f) Sρ (u) = ρ ∂Bd (x,ρ)

Bd (x,ρ)

so that, by means of (5.41), Sε (u) = ε 2 O(|Lu|) −→ 0,

as ε → 0.

(5.43g)

To evaluate Tρ (u), we first remark that on ∂Bd (x, ρ) one has ν=

∇(d(x −1 ◦ ·)) , |∇(d(x −1 ◦ ·))|

and A · ∇ T (d(x −1 ◦ ·)), ∇ T (d(x −1 ◦ ·)) |∇(d(x −1 ◦ ·))| −1 ΨL (x ◦ ·) . (see (5.1c) and (5.42)) = (2 − Q) ρ 1−Q |∇(d(x −1 ◦ ·))|

A · ∇ T v, ν = (2 − Q) d 1−Q (x −1 ◦ ·)

3 Or, equivalently (see the previous note),

Ω

Lu dH N =

m

∂Ω j =1

Xj u Xj I, ν dH N −1 .

5.5 Gauge Functions and Surface Mean Value Theorem

Therefore, keeping in mind the second definition in (5.42), 1−Q Tρ (u) = (2 − Q) ρ u(y) KL (x, y) dH N −1 (y),

255

(5.43h)

∂Bd (x,ρ)

so that Tε (u) = (u(x) + o(1)) Tε (1),

as ε → 0.

(5.43i)

To compute Tε (1), we observe that (5.43e) with u = 1 gives Tε (1) = Tr (1) for 0 < ε < r < ∞. Hence, for every ε > 0,

Tε (1) = T1 (1) = (2 − Q) ∂Bd (x,1)

KL (x, ·) dH N −1 .

(5.43j)

(5.43k)

Finally, since v = d 2−Q (x, ·) ∈ L1 (Bd (x, r)) (see Corollary 5.4.5) lim v Lu dH N = v Lu dH N . ε→0 Dε,r

D0,r

Therefore, as ε → 0, identity (5.43e) together with (5.43f)-(5.43k) give v Lu dH N = r 2−Q Lu dH N + T1 (1) u(x) Bd (x,r) Bd (x,r) − (2 − Q) r 1−Q u KL (x, ·) dH N −1 .

(5.43l)

∂Bd (x,r)

We now observe that

T1 (1) = (2 − Q) ∂Bd (x,1)

does not depend on x, i.e.

(Q − 2) ∂Bd (x,1)

KL (x, ·) dH N −1

KL (x, ·) dH N −1

= (Q − 2) ∂Bd (0,1)

Indeed, from (5.43j) we have 1 1 = T1 (1) Tr (1) r Q−1 dr Q 0 1 = (2 − Q)

KL (0, ·) dH N −1 =: (βd )−1 .

(5.43m)

KL (x, y) dH N −1 (y) dr (by the coarea formula) = (2 − Q) ΨL (x, y) dH N −1 (y) Bd (x,1) = (2 − Q) ΨL dH N . 0

∂Bd (x,r)

Bd (0,1)

256

5 The Fundamental Solution for a Sub-Laplacian and Applications

Here we used the very definition (5.42) of the functions ΨL and KL and the left invariance of ΨL . Incidentally, we have also proved that −1 (βd ) = Q(Q − 2) ΨL dH N . (5.43n) Bd (0,1)

From identity (5.43l), moving terms around, we obtain (Q − 2) βd u(x) = u KL (x, ·) dH N −1 r Q−1 ∂Bd (x,r) − βd (d 2−Q (x −1 ◦ ·) − r 2−Q ) Lu dH N .

(5.44)

Bd (x,r)

We have thus proved the following fundamental result. Theorem 5.5.4 (Surface mean value theorem). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge on G. Let O be an open subset of G, and let u ∈ C 2 (O, R). Then, for every x ∈ O and r > 0 such that Bd (x, r) ⊂ O, we have u(x) = Mr (u)(x) − Nr (Lu)(x), where

(5.45)

(Q − 2) βd Mr (u)(x) = KL (x, z) u(z) dH N −1 (z), r Q−1 ∂Bd (x,r) Nr (w)(x) = βd (d 2−Q (x −1 ◦ z) − r 2−Q ) w(z) dH N (z),

(5.46)

Bd (x,r)

and βd and KL are defined, respectively, in (5.43m) and (5.42). In particular, if Lu = 0, i.e. u is L-harmonic in O, we have u(x) = Mr (u)(x).

(5.47) RN ,

Remark 5.5.5. When L = Δ is the classical Laplace operator in N ≥ 3, the kernel KL is constant (KL ≡ 1). Then, in this case, 1 N −1 Mr (u)(x) = u(y) dH (y) =: − − u(y) dH N −1 y σN r N −1 |x−y|=r |x−y|=r (see (5.43m) and (5.46)) and (5.47) gives back the Gauss theorem for classical harmonic functions. From Theorem 5.5.4, we straightforwardly also obtain the following result (see also Corollary 9.8.8 on page 461 for yet another improvement). Theorem 5.5.6 (“Uniqueness” of the L-gauges. I). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let d be an L-gauge on G, and let βd be the positive constant defined in (5.43m). Then Γ = βd d 2−Q is the fundamental solution of L.

(5.48)

5.6 Superposition of Average Operators

257

Proof. Let ϕ ∈ C0∞ (Ω) and choose r > 0 such that supp(ϕ) ⊂ Bd (0, r). Then, by the mean value formula (5.45) (being u ≡ 0 on ∂Bd (0, r)), ϕ(0) = −βd (d 2−Q (z) − r 2−Q ) Lϕ(z) dH N (z). Bd (0,r)

On the other hand, by identity (5.43d), r 2−Q Lϕ dH N = 0. Bd (0,r)

Then, if Γ is the function defined in (5.48), Γ ∈ L1loc (RN ) thanks to Corollary 5.4.5 and −ϕ(0) = Γ (z) Lϕ(z) dH N (z) RN

C0∞ (Ω).

Moreover, Γ is smooth in G \ {0} and Γ (z) → 0 as z → ∞, for every ϕ ∈ since Q − 2 > 0. Thus, by Definition 5.3.1, Γ is the fundamental solution of L. From Theorem 5.5.4 we obtain the following asymptotic formula for L. Theorem 5.5.7 (Asymptotic surface formula for L). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. Let Ω ⊆ G be open, and let u ∈ C 2 (Ω, R). Then, for every x ∈ Ω, we have lim

r→0+

Mr (u)(x) − u(x) = ad Lu(x), r2

(5.49a)

where ad := βd

(d 2−Q (y) − 1) dy.

(5.49b)

Bd (0,1)

Proof. The change of variable z = x ◦ δr (y) in the integral defining Nr gives Nr (1)(x) = ad r 2

for every x ∈ G.

As a consequence, if u ∈ C 2 (Ω, R), from (5.45) we obtain Mr (u)(x) − u(x) = Nr (Lu)(x) = Nr (Lu − Lu(x))(x) + Nr (1)(x) Lu(x) = ad r 2 (Lu(x) + o(1)), as r → 0+ . This proves (5.49a).

5.6 Superposition of Average Operators. Solid Mean Value Theorems. Koebe-type Theorems As in the previous section, L and d will respectively denote a sub-Laplacian and an L-gauge on the homogeneous Carnot group G. We shall denote by Bd (x, r) the

258

5 The Fundamental Solution for a Sub-Laplacian and Applications

d-ball with center x ∈ G and radius r ≥ 0 and by Q the homogeneous dimension of G. We shall also assume, as usual, Q ≥ 3. Given an open set O ⊆ RN and a real number r > 0, we let Or := {x ∈ O | d-dist(x, ∂O) > r}, where

d-dist(x, ∂O) := inf d(y −1 ◦ x). y∈∂O

Since the d-balls are connected, one easily recognizes that Bd (x, ρ) ⊆ O for every x ∈ O and 0 < ρ < d-dist(x, ∂O). It follows that Bd (x, ρ) ⊆ O

∀ x ∈ Or and 0 < ρ ≤ r.

1 Let ϕ : R → R be an L -function vanishing out of the interval ]0, 1[ and such that R ϕ = 1. For r > 0, define

1 t ϕ , r r

ϕr (t) :=

t ∈ R.

Let us now consider a function u ∈ C 2 (O). Then, if x ∈ Or , from (5.45) we obtain u(x) = Mρ (u)(x) − Nρ (Lu)(x)

for 0 < ρ ≤ r.

We now multiply both sides of this identity times ϕr (ρ) and integrate with respect to ρ. We thus get u(x) = Φr (u)(x) − Φr∗ (Lu)(x),

where

∞

Φr (u)(x) :=

x ∈ Or ,

(5.50a)

ϕr (ρ) Mρ (u)(x) dρ

(5.50b)

ϕr (ρ) Nρ (w)(x) dρ.

(5.50c)

0

and Φr∗ (w)(x) :=

∞

0

By using the coarea formula, the average operator Φr can be written as follows (we agree to let u = 0 out of O) Φr (u)(x) = u(z) φr (x −1 ◦ z) dz (5.50d) RN

with

φr (z) := r −Q φ δ1/r (z)

and φ(z) := (Q − 2)βd ΨL (z) We note that φ vanishes out of Bd (0, 1) and

ϕ(d(z)) . d(z)Q−1

(5.50e)

5.6 Superposition of Average Operators

RN

∞

φ(z) dz = (Q − 2)βd 0

∞

=

ϕ(ρ) ρ Q−1

d(z)=ρ

259

|∇L d(z)|2 dH N −1 (y) dρ |∇d(z)|

ϕ(ρ) dρ = 1.

0

Here we used (5.42), (5.43h), (5.43k) and (5.43m). If the function ϕ is smooth and its support is contained in ]0, 1[, then φ ∈ C0∞ (RN ), supp(φ) ⊆ Bd (0, 1) and x → Φr (u)(x) is a smooth map in Or , whenever u is just an L1loc (O)-function. We would like to explicitly remark that in the integral (5.50d) the function z → φr (x −1 ◦ z) vanishes out of Bd (x, r). As a consequence, if x ∈ Or , that integral is performed on a compact set contained in O, since Bd (x, r) ⊆ O. If we choose Q t Q−1 if 0 < t < 1, ϕ(t) = 0 otherwise, then

Φr (u) = Mr (u) and Φr∗ (w) = Nr (w),

where Mr (u)(x) := and nd Nr (w)(x) := Q r

Bd (x,r)

r

ρ 0

md rQ

Q−1

ΨL (x −1 ◦ y) u(y) dy,

(5.50f)

2−Q −1 2−Q d w(y) dy dρ, (x ◦ y) − ρ

Bd (x,ρ)

(5.50g)

being md := Q(Q − 2)βd and nd := Q βd .

(5.50h)

Hence, from (5.50a), we obtain the following theorem. Theorem 5.6.1 (Solid mean value theorem). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge on G. Let O be an open subset of G, and let u ∈ C 2 (O, R). Then, for every x ∈ O and r > 0 such that Bd (x, r) ⊂ O, we have u(x) = Mr (u)(x) − Nr (Lu)(x), where

(5.51)

260

5 The Fundamental Solution for a Sub-Laplacian and Applications

md Mr (u)(x) = Q ΨL (x −1 ◦ y) u(y) dH N (y), r Bd (x,r)

2−Q −1 nd r Q−1 d ρ (x ◦ y) − ρ 2−Q w(y) dy dρ, Nr (w)(x) = Q r Bd (x,ρ) 0 and md , nd and ΨL are defined in (5.50h) (see also (5.43m)) and (5.42). In particular, if Lu = 0, i.e. u is L-harmonic in O, we have u(x) = Mr (u)(x).

(5.52)

Remark 5.6.2. When L = Δ is the classical Laplace operator in RN , N ≥ 3, one has ΨL ≡ 1. Then, in this case, 1 Mr (u)(x) = u(y) dy =: −− u(y) dH N (y) ωN r N |x−y| 0 such that Bd (x, r) ⊂ O, (ii) u(x) = Mr (u)(x) for every x ∈ O and r > 0 such that Bd (x, r) ⊂ O. Then u ∈ C ∞ (O) and

Lu = 0 in O.

Proof. Assume condition (i) is satisfied. Then u(x) = Mρ (u)(x) for 0 < ρ ≤ r. (5.53) Let ϕ ∈ C0∞ (]0, 1[, R) be such that R ϕ = 1. Multiply both sides of (5.53) times ϕr (ρ) = ϕ(ρ/r)/r. An integration with respect to ρ gives u(x) = Φr (u)(x)

∀ x ∈ Or ,

where Φr (u) is the integral operator (5.50b). From (5.50b) we get u(x) = u(z) φr (x −1 ◦ z) dz, O

where φr (z) = r −Q φ(δ1/r (z)) and φ is the smooth function defined by (5.50e). It follows that u ∈ C ∞ (O). Condition (i) and identity (5.45) now give Nr (Lu)(x) = 0

5.6 Superposition of Average Operators

261

for every x ∈ O and r > 0 such that Bd (x, r) ⊂ O. Since the kernel appearing in the integral operator Nr is strictly positive, this implies Lu = 0 in O. Thus, the theorem is proved if condition (i) is fulfilled. Let us now assume (ii). Since ρ → Mρ (u)(x) is continuous on ]0, r] and r Q Mr (u)(x) = Q ρ Q−1 Mρ (u)(x) dρ, r 0 from (ii) we get Qr Q−1 u(x) =

d Q d (r u(x)) = Q dr dr

r

ρ Q−1 Mρ (u)(x) dρ

0

= Qr Q−1 Mr (u)(x). Hence u(x) = Mr (u)(x) for every x ∈ O and r > 0 such that Bd (x, r) ⊂ O. Then u satisfies condition (i), so that u ∈ C ∞ (O) and Lu = 0 in O. Remark 5.6.4 (Another Koebe-type result). In the hypotheses of Theorem 5.6.3, if u : O → R is continuous and satisfies u(x) = Φr (u)(x)

∀ r > 0 ∀ x ∈ Or ,

where Φr is the integral operator in (5.50d) related to a smooth function ϕ ∈ C0∞ (]0, 1[, R) (see also (5.50b)), then u ∈ C ∞ (O). As a consequence, by identity (5.50a), Φr∗ (Lu) = 0 in Or for every r > 0. It follows that Lu = 0 in O.

From Theorem 5.6.1 we obtain another asymptotic formula for the sub-Laplacians. Theorem 5.6.5 (Asymptotic solid formula for L). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. Let Ω ⊆ G be open, and let u ∈ C 2 (Ω, R). Then, for every x ∈ Ω, we have lim

r→0+

Mr (u)(x) − u(x) = ad Lu(x), r2

where ad = Q ad /(Q + 2) (and ad is as in (5.49b)). Proof. From (5.50g) we obtain (by recalling the definition (5.49b) of ad )

2−Q Q βd r Q+1 Nr (1)(x) = Q ρ (y) − 1 dy dρ = r 2 ad . d r 0 Bd (0,1) Then, by Theorem 5.6.1 (arguing as in the proof of Theorem 5.5.7), we get Mr (u)(x) − u(x) = ad r 2 (Lu(x) + o(1)), and (5.54) follows.

as r → 0+ ,

(5.54)

262

5 The Fundamental Solution for a Sub-Laplacian and Applications

5.7 Harnack Inequalities for Sub-Laplacians In this section, we shall prove some type of Harnack inequalities for sub-Laplacians. Our main tool will be the solid average operator Mr in (5.50f). Let d be an L-gauge on G, let cd be the positive constant of the pseudo-triangle inequality (see Proposition 5.1.8) d(x −1 ◦ y) ≤ cd d(x −1 ◦ z) + d(z−1 ◦ y) ∀ x, y, z ∈ G, (5.55) and let ΨL = |∇L d|2 be the kernel appearing in the average operator Mr in (5.50f). The following lemma will play a fundamental rôle in this section: it will allow us to compare the average on different d-balls of a given non-negative function. Lemma 5.7.1. Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. For every r > 0, there exists a point z0 = z0 (r) ∈ G satisfying the following conditions: (i) d(z0 ) = λ r, (ii) ΨL (x −1 ◦ y) ≥ μ and ΨL (y −1 ◦ x) ≥ μ for every x ∈ Bd (z0 , r) and every y ∈ Bd (0, 2 cd r). Here λ ≥ 1 and μ > 0 are real constants independent of r (depending only on G, d and L). Proof. We split the proof into three steps. (I) The set {y ∈ G \ {0} | ΨL (y) = 0} has empty interior. Indeed, suppose by contradiction ΨL (y) = 0 for every y in a neighborhood U of a suitable y0 = 0. Then, since ΨL = |∇L d|2 , this gives ∇L ΨL ≡ 0 on U , so that, by Proposition 1.5.6 (page 69), d is constant in an open set containing y0 . As a consequence, the function r → d(δr (y0 )) = r d(y0 ) is constant near r = 1. This implies d(y0 ) = 0, which is a contradiction with the assumption y0 = 0. (II) From step (I) and the homogeneity of ΨL we infer the existence of a point y0 ∈ G, d(y0 ) = 1, such that ΨL (y0 ) > 0 and ΨL (y0−1 ) > 0. Then, by the continuity of ΨL out of the origin, there exist two positive constants σ and μ (with σ ≤ 1) such that ΨL (ξ ◦ y0 ◦ η) ≥ μ,

ΨL (ξ ◦ y0−1 ◦ η) ≥ μ

(5.56)

for every ξ , η ∈ G satisfying the inequalities d(ξ ), d(η) ≤ 2 cd σ . (III) For any fixed r > 0, we let z0 := δλ r (y0 ) with λ = 1/σ . Let x ∈ Bd (z0 , r) and y ∈ Bd (0, 2 cd r). From the second inequality in (5.56) we obtain ΨL (x −1 ◦ y) = ΨL (x −1 ◦ z0 ) ◦ z0−1 ◦ y (by the homogeneity of ΨL ) = ΨL δ1/(λ r) (x −1 ◦ z0 ) ◦ y0−1 ◦ δ1/(λ r) (y) ≥ μ,

5.7 Harnack Inequalities

263

since d(δ1/(λ r) (x −1 ◦ z0 )) ≤ σ and d(δ1/(λ r) (y)) ≤ 2 cd σ . Analogously, from the first inequality in (5.56) we obtain ΨL (y −1 ◦ x) = ΨL y −1 ◦ z0 ◦ (z0−1 ◦ x) = ΨL δ1/(λ r) (y −1 ) ◦ y0 ◦ δ1/(λ r) (z0−1 ◦ x) ≥ μ, since d(δ1/(λ r) (z0−1 ◦ x)) ≤ σ and d(δ1/(λ r) (y −1 )) ≤ 2 cd σ . To state the next theorem, it is convenient to introduce the following constants θ0 := cd (1 + cd (1 + λ)),

θ := cd (λ + θ0 ).

(5.57)

Theorem 5.7.2 (Non-homogeneous Harnack inequality). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. Let Ω ⊆ G be open. Finally, let x0 ∈ Ω and r > 0 be such that Bd (x0 , θ r) ⊂ Ω (here θ is as in (5.57), see also Lemma 5.7.1 and (5.55)). Then, for any p ∈ (Q/2, ∞], we have (5.58) sup u ≤ c inf u + r 2−Q/p Lu Lp (B (x , θ r)) d 0 B (x , r) Bd (x0 ,r)

d

0

for every non-negative smooth function u : Ω → R. Here c is a positive constant depending only on G, d, L and p and not depending on u, r, x0 and Ω. Proof. Since L is left invariant, we may assume x0 = 0. We split the proof in several steps. Mr will be the average operator in (5.50f) and z0 = z0 (r) the point of G given by Lemma 5.7.1. (I) There exists an absolute constant4 c > 0 such that Mr (u)(x) ≤ c Mθ0 r (u)(z0 )

∀ x ∈ Bd (0, r).

(5.59)

Indeed, for every x ∈ Bd (0, r), we have Bd (x, r) ⊆ Bd (0, 2 cd r) ⊆ Bd (0, θ0 r) ⊂ Ω, whence (being u non-negative) Mr (u)(x) ≤

sup ΨL

G\{0}

m d

rQ

u(z) dz

Bd (x,r)

(by the first inequality in Lemma 5.7.1-(ii)) c1 ΨL (z0−1 ◦ z) u(z) dz, ≤ Q r Bd (x,r) where c1 = md /μ supG\{0} ΨL . We remark that c1 < ∞, since ΨL is smooth out of the origin and δλ -homogeneous of degree zero. Then, since we also have 4 Hereafter, we call absolute constant any positive real constant independent of u, r, x 0

and Ω.

264

5 The Fundamental Solution for a Sub-Laplacian and Applications

Bd (x, r) ⊆ Bd (z0 , θ0 r) ⊆ Bd (0, θ r) ⊂ Ω, we get (again by the non-negativity of u) c1 c1 Mr (u)(x) ≤ Q ΨL (z0−1 ◦ z) u(z) dz = Mθ0 r (u)(z0 ). r md Bd (z0 ,θ0 r) This proves (5.59) with c = c1 /md . (II) There exists an absolute constant c > 0 such that Mr (u)(z0 ) ≤ c Mθ0 r (u)(y)

∀ y ∈ Bd (0, r).

(5.60)

This inequality can be proved just by proceeding as in the previous step, by using the second inequality in Lemma 5.7.1-(ii) and the inclusions Bd (z0 , r) ⊆ Bd (y, θ0 r) ⊆ Bd (0, θ r) ⊂ Ω. (III) Let finally x, y ∈ Bd (0, r). Then, by repeatedly using the solid mean value theorem 5.6.1, we have u(x) = Mr (u)(x) − Nr (Lu)(x) ≤ = =

(by (5.59))

cMθ0 r (u)(z0 ) − Nr (Lu)(x) c u(z0 ) + Nθ0 r (Lu)(z0 ) − Nr (Lu)(x) c Mr (u)(z0 ) − Nr (Lu)(z0 ) + Nθ0 r (Lu)(z0 ) − Nr (Lu)(x).

On the other hand, from (5.60), Mr (u)(z0 ) ≤ c Mθ0 r (u)(y) = c u(y) − Nθ0 r (Lu)(y) . By using this last estimate in the previous one, we infer that u(x) is bounded from above by a suitable absolute constant c times u(y) + |Nθ0 r (Lu)(y)| + |Nr (Lu)(z0 )| + |Nθ0 r (Lu)(z0 )| + |Nr (Lu)(x)| for all x, y ∈ Bd (0, r). An elementary computation based on Hölder’s inequality shows that |Nr (w)(x)| ≤ cp r 2−Q/p w

Lp (Bd (x, r))

(5.61)

for Q/2 < p ≤ ∞, p = p/(p − 1) and

1/p cp := nd (d(z)2−Q − 1) dz . d(z)<1

Thus, keeping in mind the inclusions Bd (y, θ0 r), Bd (z0 , r), Bd (z0 , θ0 r), Bd (x, r) ⊆ Bd (0, θ r), from the upper bound of u(x) and from (5.61) we obtain (5.58) when x0 = 0. This ends the proof.

5.7 Harnack Inequalities

265

Theorem 5.7.2 contains the following “homogeneous” Harnack inequality. Corollary 5.7.3 (Homogeneous invariant Harnack inequality). Let L be a subLaplacian on the homogeneous Carnot group G, and let d be an L-gauge. Let Ω be an open subset of G, and let u : Ω → R be a non-negative smooth solution to Lu = 0. Then (5.62) sup u ≤ c inf u Bd (x0 ,r)

Bd (x0 ,r)

for every x0 ∈ Ω and r > 0 such that Bd (x0 , θ r) ⊂ Ω. The constant c depends only on G, L and d and does not depend on u, r, x0 and Ω. By using a covering argument, from the non-homogeneous Harnack inequality of Theorem 5.7.2 one obtains the following theorem. Theorem 5.7.4 (Non-homogeneous, non-invariant Harnack inequality). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. Let Ω be an open subset of G, and let K and K0 be compact and connected subsets of Ω such that K ⊂ int(K0 ). Then, for every p ∈ (Q/2, ∞], there exists a positive constant c = c(K, K0 , Ω, L, d, p, Q) such that sup u ≤ c inf u+ Lu Lp (K ) (5.63) 0 K K

for every u ∈ C ∞ (Ω, R), u ≥ 0. Proof. Let {Dj | j = 1, . . . , q} be a finite family of d-balls Dj = Bd (xj , rj ) such that (here θ is as in (5.57)): q (i) K ⊂ j =1 Dj , (ii) θ Dj := Bd (xj , θ rj ) ⊂ K0 for any j = 1, . . . , q, (iii) Dj ∩ Dj +1 = ∅ for every j ∈ {1, . . . , q − 1}. We explicitly remark that such a covering exists, since K is compact, connected and contained in the interior of K0 . By Theorem 5.7.2, we have sup u ≤ c inf u+ Lu Lp (θ Dj ) , j = 1, . . . , q, Dj

Dj

for every u ∈ C ∞ (Ω, R), u ≥ 0. The constant c is independent of u. Then, inequality (5.63) will follow by a repeated application of the following elementary lemma. Lemma 5.7.5. Let A1 and A2 be arbitrary sets such that A1 ∩ A2 = ∅. Suppose u : A1 ∩ A2 → R is a non-negative function satisfying (5.64) sup u ≤ c inf u + Li , i = 1, 2, Ai

Ai

266

5 The Fundamental Solution for a Sub-Laplacian and Applications

for suitable constants c≥1 and Li ≥ 0, i = 1, 2. Then sup u ≤ c2 inf u + L1 + L2 . A1 ∪A2

A1 ∪A2

Proof. We have to show that u(x) ≤ c{u(y) + L1 + L2 }

(5.65)

for every x, y ∈ A1 ∪ A2 . Now, if x, y ∈ A1 or x, y ∈ A2 , then inequality (5.65) directly follows from (5.64). Suppose x ∈ A1 and y ∈ A2 and choose a point z ∈ A1 ∩ A2 . By hypothesis (5.64), u(x) ≤ c {u(z) + L1 }

and u(z) ≤ c {u(y) + L2 }.

Hence u(x) ≤ c {c (u(y) + L2 ) + L1 }, and (5.65) follows. From Theorem 5.7.4 one obtains the following improvement of Theorem 5.7.2. Corollary 5.7.6 (Non-homogeneous invariant Harnack inequality). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. Let Ω ⊆ G be open, and let r, R and R0 be real constants such that 0 < r < R < R0 . Assume r R , ≤ ρ and Bd (x0 , R0 ) ⊂ Ω R R0 for suitable ρ < 1 and x0 ∈ Ω. Then, for every p ∈ (Q/2, ∞], there exists a constant c > 0 such that sup u ≤ c inf u + R 2−Q/p Lu Lp (B (x , R)) d 0 B (x , r) Bd (x0 ,r)

d

(5.66)

0

for every u ∈ C ∞ (Ω, R), u ≥ 0. The constant c depends only on G, L, d, p and ρ and does not depend on u, r, R, R0 , Ω and x0 . Proof. Since L is left invariant, we may assume x0 = 0. Let us put uR (x) := u(δR (x)),

x ∈ δ1/R (Ω).

Then, by applying Theorem 5.7.4 to the function uR , the compact sets K = Bd (0, ρ) and K0 = Bd (0, 1) and the open set Bd (0, 1/ρ) (which is contained in Bd (0, R0 /R) ⊆ δ1/R (Ω)), we obtain sup uR ≤ c inf uR + LuR Lp (Bd (0,1)) Bd (0,ρ)

Bd (0,ρ)

=c

inf uR + R 2−Q/p Lu Lp (Bd (0,R)) ,

Bd (0,ρ)

5.7 Harnack Inequalities

267

with c > 0 depending only on the parameters in the assertion of the corollary. Note that we have also used the δλ -homogeneity of L (of degree 2). From this inequality (5.66) follows, since sup uR = Bd (0,ρ)

sup Bd (0,R ρ)

u ≥ sup u and Bd (0,r)

inf uR ≤ inf u.

Bd (0,ρ)

Bd (0,r)

This ends the proof. Theorem 5.7.4 and the δλ -homogeneity of L easily imply the following Harnack inequality on rings. Corollary 5.7.7 (Harnack inequality on rings). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. Let u be a smooth nonnegative function on the ring

AR,c := x ∈ G | cR < d(x) < R/c , where R > 0 and 0 < c < 1. Let 0 < a < b < c. Then, for every p ∈ (Q/2, ∞], there exists a constant c > 0 such that sup u ≤ c inf u + R 2−Q/p Lu Lp (A ) . (5.67) R,b AR,a AR,a The constant c depends only on G, L, p, d, a, b and c and does not depend on u and R. Proof. The change of variable x → δR (x) reduces (5.67) to the analogous inequality with R = 1. This last one follows from Theorem 5.7.4. Remark 5.7.8 (The Harnack inequality in the abstract setting). According the convention in the incipit of the chapter, given an abstract stratified group H, via the isomorphism Ψ between H and a homogeneous Carnot group G, all the Harnack inequalities of the present section do possess a counterpart in H. For example, it suffices to consider the results in Remark 2.2.28 in order to derive the following result from Theorem 5.7.4. Theorem 5.7.9 (A Harnack inequality in the abstract setting). Let H be an abstract stratified group, and let L be a sub-Laplacian on H. Let Ω be an open subset of H, and let K be a compact and connected subset of Ω. Then there exists a positive constant c = c(H, L, Ω, K) such that sup u ≤ c inf u, K

K

for every non-negative function u ∈ C ∞ (Ω, R) satisfying Lu = 0 in Ω. We close this section by giving the following “monotone convergence” theorem.

268

5 The Fundamental Solution for a Sub-Laplacian and Applications

Theorem 5.7.10 (The Brelot convergence property). Let H be an abstract stratified group, and let L be a sub-Laplacian on H. Let Ω ⊆ H be open and connected. Let {un }n∈N be a sequence of L-harmonic functions in Ω, i.e. un ∈ C ∞ (Ω, R) and Lun = 0 in Ω

for every n ∈ N.

Assume the sequence {un }n∈N is monotone increasing and sup{un (x0 )} < ∞

(5.68)

n∈N

at some point x0 ∈ Ω. Then there exists an L-harmonic function u in Ω such that {un }n∈N is uniformly convergent on every compact subset of Ω to u. Proof. By the results in Remark 2.2.28, it suffices5 to consider the case when L is a sub-Laplacian on a homogeneous Carnot group G. Let K be a compact subset of Ω. Since Ω is connected, there exists a compact and connected set K ∗ such that K ⊆ K ∗ ⊂ Ω and x0 ∈ K0 . Then, by Theorem 5.7.4, sup(un − um ) ≤ sup(un − um ) ≤ cinfK ∗ (un − um ) K

K∗

≤ c(un (x0 ) − um (x0 ))

for every n ≥ m ≥ 1.

The constant c is independent of n and m. Then, by condition (5.68), {un }n is uniformly convergent on K. Since K is an arbitrary compact subset of Ω, {un }n∈N is locally uniformly convergent to a continuous function u : Ω → R. On the other hand, by the solid mean value Theorem 5.6.1, for every x ∈ Ω and r > 0 such that Bd (x, r) ⊂ Ω, we have un (x) = Mr (un )(x)

∀ n ∈ N.

Letting n tend to infinity (by the uniform convergence un → u), we get u(x) = Mr (u)(x)

∀ x ∈ Ω, r > 0 : Bd (x, r) ⊂ Ω,

and now the Koebe-type Theorem 5.6.3 implies u ∈ C ∞ (Ω, R) and Lu = 0 in Ω.

5 Indeed, following the notation in Remark 2.2.28 and in the assertion of the above theorem,

the following facts hold: the (abstract) sub-Laplacian L is Ψ -related to the sub-Laplacian on G; set := Ψ −1 (Ω), is open and connected L un := un ◦ Ψ , Ω x0 := Ψ −1 (x0 ), then Ω in G (recall that Ψ is a homeomorphism), the sequence { un }n∈N is monotone increasing un = 0 on Ω. Finally, if K is a compact subset of H, then bounded in on Ω, x0 and L := Ψ −1 (K) is a compact subset of G, and K un − u|. sup |un − u| = sup | K

K

5.8 Liouville-type Theorems

269

The above Brelot convergence property implies the following strong minimum principle (see also Section 5.13 for a more exhaustive investigation of maximum– minimum principles). Corollary 5.7.11 (Strong minimum principle). Let L be a sub-Laplacian on an abstract stratified group H. A non-negative solution to Lu = 0 on an open connected set Ω ⊆ H vanishes identically iff it vanishes at a point. Proof. Apply the result of Theorem 5.7.10 to the sequence {n · u | n ∈ N}.

5.8 Liouville-type Theorems The classical Liouville theorem for entire harmonic functions also holds in the subLaplacian setting. Indeed, the Harnack inequality of Corollary 5.7.3 implies the following theorem: Theorem 5.8.1 (Liouville theorem for sub-Laplacians). Let H be an abstract stratified group. Let L be a sub-Laplacian on H. Let u ∈ C ∞ (H, R) be a function satisfying u ≥ 0 and Lu = 0 in H. Then u is constant. Proof. By the results in Remark 2.2.28, it suffices6 to consider the case when L is a sub-Laplacian on a homogeneous Carnot group G. Define m := inf u and v := u − m. G

Then v ≥ 0 and Lv = 0 in G. From Harnack inequality (5.62) we obtain sup v ≤ c Bd (0,r)

inf v,

Bd (0,r)

c independent of r.

From this inequality, letting r tend to infinity, we obtain 0 ≤ sup v ≤ c inf v = 0, G

which implies v ≡ 0 and u ≡ m.

G

Theorem 5.8.1 can be viewed as a particular case of the following stronger version of the Liouville property for L. 6 Indeed, the (abstract) sub-Laplacian L is Ψ -related to the sub-Laplacian L on G and, set

u = 0 on G. Once we have proved that u := u ◦ Ψ , we have u ≥ 0 and L u is constant on G, the same follows for u = u ◦ Ψ −1 .

270

5 The Fundamental Solution for a Sub-Laplacian and Applications

Theorem 5.8.2 (Liouville-type theorem-polynomial lower bound). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let u be an entire solution to Lu = 0, i.e. a function u ∈ C ∞ (G, R) such that Lu = 0 in G. Assume there exists a polynomial function p on G such that u≥p

in G.

Then7 u is a polynomial function and degG u ≤ degG p. The inequality degG u ≤ degG p can be strict: take, for example, u ≡ 0, p = −x12 , so that degG u = 0 < 2 = degG p. Remark 5.8.3. All the results in this section involving polynomial functions are obviously related to the fixed coordinates on a homogeneous Carnot group. Abstract counterparts of these results are available in the obvious way. A polynomial function on the abstract stratified group H is a function P : H → R such that p := P ◦ Exp is a polynomial function on the vector space h (the Lie algebra of H), i.e. p is a polynomial function when expressed in coordinates w.r.t. any (or equivalently, w.r.t. at least one) basis for h. All the details are left to the reader. Theorem 5.8.2 is an easy consequence of the following one. Theorem 5.8.4 (Liouville-type theorem-polynomial sub-Laplacian). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let u be a smooth function on G satisfying u ≥ 0 and Lu = w in G, where w is a polynomial function. Then u is a polynomial function and degG u ≤ 2 + degG w. More precisely,

degG u =

degG w 2 + degG w

if degG u = 0, if degG u ≥ 2.

(5.69)

The case degG u = 1 cannot occur, since a polynomial of G-degree 1 cannot be a non-negative function. The proof of this theorem will follow from a representation formula that we shall deduce from identity (5.50a) in Section 5.6. We first show how Theorem 5.8.2 can be obtained from Theorem 5.8.4. Proof (of Theorem 5.8.2). We first recall that L is a differential operator, δλ -homogeneous of degree two, and its coefficients are polynomial functions. Then, if u ≥ p and Lu = 0, we have u − p ≥ 0 and L(u − p) = w, 7 See Definition 1.3.3 on page 33 for the definition of the G-degree deg (p) of a polynomial G

p and the G-length |α|G of the multi-index α.

5.8 Liouville-type Theorems

271

where w := −Lp is a polynomial whose G-degree does not exceed max{0, degG p − 2}. From Theorem 5.8.4 it follows that u − p is a polynomial and that degG (u − p) ≤ 2 + degG (w) ≤ 2 + max{0, degG p − 2} = max{2, degG p}. Hence, u = (u − p) + p is a polynomial function and its G-degree does not exceed max{2, degG p}. If degG p ≥ 2, this gives the assertion of Theorem 5.8.2. It remains to consider the cases degG p = 0 and degG p = 1 (indeed, in these cases the above argument only proves that degG u ≤ 2, which is weaker than degG u ≤ degG p). In both cases, Lp ≡ 0, so that the hypotheses of Theorem 5.8.2 rewrites as u − p ≥ 0,

L(u − p) = 0

in G.

Hence, by Liouville Theorem 5.8.1, u − p is constant, whence u = (u − p) + p = p + constant, so that u is a polynomial function of the same G-degree as p. We next prove a representation formula having its own interest and useful for the proof of Theorem 5.8.4. Proposition 5.8.5. Let L be a sub-Laplacian on the homogeneous Carnot group (G, ∗), and let d be an L-gauge on G. Let u ∈ C ∞ (G, R) be such that Lu = w

in G,

(5.70a)

(α)

(5.70b)

where w is a polynomial function. Then u(x) = Φr (u)(x) −

CQ w (α) (x) r 2+|α|G

|α|G ≤m

for any x ∈ G and r > 0. Φr denotes the integral operator (5.50b) related to a smooth function ϕ ∈ C0∞ ((0, 1), R). Moreover, m is the G-degree of w, w (α) (x) := Dyα |y=0 w(x ∗ y) (α)

and, for any α with |α|G ≤ m, CQ is a positive constant depending only on α, Q and d. In particular, w (α) is a polynomial function with G-degree not exceeding m − |α|G . Proof. Identity (5.50a) in Section 5.6 and hypothesis (5.70a) give u(x) = Φr (u)(x) − Φr∗ (w)(x),

272

5 The Fundamental Solution for a Sub-Laplacian and Applications

where Φr∗ (w)(x) =

∞

ϕr (ρ) Nρ (w)(x) dρ, and (d 2−Q (x −1 ∗ y) − ρ 2−Q ) w(y) dH N (y) Nρ (w)(x) = βd Bd (x,ρ) (d 2−Q (y) − ρ 2−Q ) w(x ∗ y) dH N (y). = βd 0

Bd (0,ρ)

We now claim that w(x ∗ y) =

w (α) (x) yα , α!

(5.71)

|α|G ≤m

where w (α) = Dyα |y=0 w(x ∗ y) is a polynomial function of G-degree ≤ m − |α|G . Taking this claim for granted for a moment, we have Φr∗ (w)(x)

∞ w (α) (x) ρ dρ βd = ϕ (d 2−Q (y) − ρ 2−Q ) y α dH N (y) α! r r 0 Bd (x,ρ) |α|G ≤m

(by using the change of variables y = δρ (z) and ρ = r σ ) (α) = CQ w (α) (x) r 2+|α|G , |α|G ≤m

where (α) CQ

βd = α!

∞

0

ϕ(σ )

(d

2−Q

(z) − 1) z dH (z) dσ. α

N

Bd (0,1)

Then, we are left with the proof of (5.71). Since w is a polynomial function and (x, y) → x ∗ y has polynomial components too, one has w(x ∗ y) = cα,β x α y β |α|G +|β|G ≤n

for a suitable positive integer n and real constants cα,β . We have to prove only that n ≤ m. Now, since degG w ≤ m, w(z) = cγ zγ , cγ ∈ R for any γ . |γ |G ≤m

Then, since δλ is an automorphism of the group G, cα,β λ|α|G +|β|G x α y β = w(δλ (x) ∗ δλ (y)) |α|G +|β|G ≤n

= w(δλ (x ∗ y)) =

cγ λ|γ |G (x ∗ y)γ

|γ |G ≤m

for every x, y ∈ G and λ > 0. As a consequence,

5.8 Liouville-type Theorems

273

cα,β x α y β = 0 ∀ x, y ∈ G,

m<|α|G +|β|G ≤n

so that

w(x ∗ y) =

cα,β x α y β ,

|α|G +|β|G ≤m

and the claim follows. This completes the proof. We are now in the position to prove Theorem 5.8.4. Proof (of Theorem 5.8.4). In the Harnack inequality (5.58) of Theorem 5.7.2, take x0 = 0, r = 2 |x| and p = ∞. We obtain

u(x) ≤ c u(0) + |x|m+2 , (5.72) where c is a positive absolute constant. Let us recall the notation used in Corollary 1.5.5 (page 68). If L is the sum of squares of the vector fields X1 , . . . , XN1 and if β = (i1 , . . . , ik ) is a multi-index with components in {1, . . . , N1 }, we set X β := Xi1 ◦ · · · ◦ Xik and |β| = k. Let us now use the representation formula of Proposition 5.8.5. Since w (α) in (5.70b) has G-degree ≤ m − |α|, for any non-negative multi-index β with |β| > m, we have X β u(x) = X β Φr (u)(x) ∀ x ∈ G. Then, since the Xj ’s are left-invariant on (G, ∗) and δλ -homogeneous of degree one, we have (see also (5.50d) and (5.50e)) X β u(x) = u(y) X β φr (x −1 ∗ y) dy G −|β| (y −1 ∗ x) dy, =r u(y) X β φ r RN

(z) := where φ

φ(z−1 )

and

β = r −Q X β φ ∗ δ1/r . X φ r

Hence, X β u(x) = r −|β|

(z) dz. u(x ∗ δr (z−1 )) X β φ

(5.73)

Bd (0,1)

Using inequality (5.72) in (5.73), we obtain |X β u(x)| ≤ c r −|β| (1 + |x| + r)m+2 for every x ∈ G and r > 0. The constant c depends on u(0), but it is independent of x and r. Letting r tend to infinity, we obtain X β u(x) = 0

∀x ∈ G

and for every β with |β| > m + 2. By the cited Corollary 1.5.5, this implies that u is a polynomial function of G-degree ≤ m + 2.

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5 The Fundamental Solution for a Sub-Laplacian and Applications

Using the same argument as above, we can prove the following improvement of Theorem 5.8.4. Theorem 5.8.6 (Liouville: polynomial sub-Laplacian and bound). Let L be a subLaplacian on the homogeneous Carnot group G. Let u : G → R be a smooth function satisfying u ≥ p and Lu = w in G, where p and w are polynomial functions. Then u is a polynomial and degG u ≤ max{degG p, 2 + degG w}. Proof. Set v = u − p. We have v ≥ 0,

Lv = w − Lp

in G.

Since L has polynomial coefficients, we can apply Theorem 5.8.4 to derive degG (u − p) ≤ 2 + degG (w − Lp) ≤ 2 + max{degG w, degG Lp} max{degG w, degG p − 2} if degG p ≥ 2, ≤ 2+ max{degG w, 0} if degG p ≤ 1 max{2 + degG w, degG p} if degG p ≥ 2, = 2 + degG w if degG p ≤ 1 = max{degG p, 2 + degG w}. Summing up all the above statements, we obtain the following Liouville theorem “of polynomial type”. Theorem 5.8.7 (Liouville-type theorem). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let u : G → R be smooth and such that u≥p

and Lu = w

in G,

where p and w are polynomial functions. Then u is a polynomial, and if w ≡ 0, degG p degG u ≤ max{degG p, 2 + degG w} otherwise. 5.8.1 Asymptotic Liouville-type Theorems We close this section by giving some more “asymptotic” Liouville-type theorems, easy consequences of Theorem 5.8.2. Theorem 5.8.8 (Asymptotic Liouville. I). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let u be an entire solution to Lu = 0. Assume there exists a real number m ≥ 0 such that u(x) = O(m (x)),

as x → ∞.

(5.74)

5.8 Liouville-type Theorems

275

Then u is a polynomial function and degG u ≤ [m].

(5.75)

Here is (any fixed) homogeneous norm on G and [m] denotes the integer part of m, i.e. [m] ∈ Z and [m] ≤ m < [m] + 1. Proof. By condition (5.74) and Proposition 5.1.4, we get u(x) ≥ p(x) where

p(x) = −c 1 +

r

∀ x ∈ G, [m]+1 |x

|

(j ) 2 r!/j

,

j =1

and c is a suitable positive constant. Then, by Theorem 5.8.2, u is a polynomial function. Let n := degG u. Assume, by contradiction, n ≥ [m] + 1. Writing u(x) =

n

uk (x),

k=0

where uk is δλ -homogeneous of degree k, from condition (5.74) we get n k=0

u(x) (x)k−n uk δ1/(x) (x) = −→ 0, (x)n

as (x) → ∞.

Hence, un (y) = 0 for every y ∈ G such that (y) = 1, which implies degG u ≤ n−1, a contradiction. Then n ≤ [m] and the proof is complete. Theorem 5.8.8 together with Theorem 5.6.1 give the following corollary. Corollary 5.8.9 (Asymptotic Liouville. II). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. Let u be an entire solution to Lu = 0. Assume there exists x0 ∈ G and a real number m ≥ 0 such that |u(y)| dy = O(r m ), as r → ∞. (5.76) − Bd (x0 ,r)

Then u is a polynomial function and degG u ≤ [m]. In (5.76) we have set

1 − := . |D| D D

276

5 The Fundamental Solution for a Sub-Laplacian and Applications

Proof. Let x ∈ G \ {0} and put r = max{d(x), d(x0 )}. By the pseudo-triangle inequality (5.55), Bd (x, r) ⊆ Bd (x0 , 2 cd r). Then, from the solid mean value Theorem 5.6.1, we obtain |u(x)| = |Mr (u)(x)| ≤ Mr (|u|)(x) md |u(y)| dy, ≤ Q sup ΨL r Bd (x0 ,2cd r) G\{0} |u(x)| ≤ c −

so that

|u(y)| dy,

Bd (x0 ,2cd r)

being c > 0 independent of x and r. This inequality and (5.76) give u(x) = O(r m ) = O(d(x)m ),

as x → ∞.

Then, by Theorem 5.8.8, u is a polynomial function with the G-degree ≤ [m].

Remark 5.8.10. By using Proposition 5.1.4, one easily recognizes that condition (5.76) in the previous corollary is equivalent to the following one |u(y)| dy = O(r m+Q ), as r → ∞, (5.77) Br

where Br := {y ∈ G : (y) < r} is the ball of radius r centered at the origin, with respect to a homogeneous norm on G. Corollary 5.8.11 (Asymptotic Liouville. III). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let u be an entire solution to Lu = 0, and 1 ≤ p ≤ ∞. Assume (with the notation of Remark 5.8.10) u Lp (Br ) = O(r m+Q/p ),

as r → ∞.

(5.78)

Then u is a polynomial function of the G-degree not exceeding [m]. Proof. The Hölder inequality gives |u(y)| dy ≤ c r Q(1−1/p) u Lp (Br ) . Br

Then (5.78) implies (5.77), and the assertion follows from Remark 5.8.10 and Corollary 5.8.9.

5.9 Some Results on G -fractional Integrals and the Sobolev–Stein Embedding Inequality Let L be a sub-Laplacian on the homogeneous Carnot group G of homogeneous dimension Q. Let 0 < α < Q. Given a function f : G → R, we formally define

5.9 Sobolev–Stein Embedding Inequality

Iα (f )(x) :=

G

277

f (y) dy, (d(x, y))Q−α

where d(x, y) stands for d(y −1 ◦ x) and z → d(z) is an L-gauge function on G. By analogy with the Euclidean setting, we shall call Iα the G-fractional integral of order α. The following theorem, when G is the usual Euclidean group (RN , +), gives back a celebrated theorem by Hardy, Littlewood and Sobolev. Theorem 5.9.1 (Hardy–Littlewood–Sobolev for sub-Laplacians). Let L be a subLaplacian on the homogeneous Carnot group G, and let d be an L-gauge. Suppose 1 < α < Q and 1 < p < Q α . Let q > p be defined by 1/q = 1/p − α/Q. Then there exists a positive constant C = C(α, p, G, L, d) such that Iα (f )q ≤ C f p

for every f ∈ Lp (G).

Here, we use the notation · r to denote the Lr norm in G ≡ RN with respect to the Lebesgue measure. In the classical Euclidean setting, several proofs of this theorem are known. The simplest proof seems to be due to L.I. Hedberg [Hed72] and it makes use of the maximal function theorem by Hardy–Littlewood–Wiener. This last theorem holds in general metric spaces equipped with a doubling measure.8 In particular, it holds in our context. Fixed a sub-Laplacian L and an L-gauge d, we define the L-maximal function ML (f ) of a function f ∈ Lp (G, C), 1 < p < ∞, as follows ML (f )(x) := sup − |f (y)| dy, x ∈ G, r>0

Bd (x,r)

−

1 = . |D| D D The function x → ML (f )(x) is lower semicontinuous. Indeed, if ML (f )(x) > α, there exists r > 0 such that −Bd (x,r) |f (y)|dy > α. Then, since |f (y)| dy x → − where we used the notation

Bd (x,r)

is a continuous function (see Ex. 5 at the end of this chapter), there exists δ > 0 such that 8 A Radon measure μ on a quasi-metric space (X, d) is doubling if there exists a positive

constant Cd such that

0 < μ(B(x, 2r)) ≤ Cd μ(B(x, r))

for every d-ball with center at x and radius r.

278

5 The Fundamental Solution for a Sub-Laplacian and Applications

ML (f )(z) ≥ −

if d(x −1 ◦ z) < δ.

|f (y)| dy > α

Bd (z,r)

The L-maximal function theorem is the following one. L-Maximal Function Theorem. Let 1 < p < ∞. With the above notation, there exists a positive constant C = C(p, G, L, d) such that ML (f )p ≤ C f p

for every f ∈ Lp (G, C).

A proof of this theorem in general doubling metric spaces (which is out of our scopes here) can be found in the monograph [Ste81, Chapter 2], by E.M. Stein. Starting from this result, we now prove Theorem 5.9.1 by using the idea in L.I. Hedberg’s paper [Hed72]. Proof (of Theorem 5.9.1). For every fixed t > 0 and x ∈ G, we have

+ |f (y)| (d(x, y))α−Q dy Iα (f )(x) = =:

d(x,y)≤t d(x,y)≥t (t) (t) Iα (f )(x) + Eα (f )(x).

The Hölder inequality gives (t) Eα (f )(x) ≤ f p

1−1/p (α−Q)p/(p−1)

(d(x, y))

d(x,y)≤t

= (see (5.36)) C f p

∞

dy

1−1/p

τ Q−1+(α−Q)p/(p−1) dτ

t

= C f p t α−Q/p . On the other hand, one has Iα(t) (f )(x)

=

∞ −k−1
≤C

≤C

t/2

k=0 ∞

t/2k

|f (y)| (d(x, y))α−Q dy

{d(x,y)≤t2−k }

|f (y)| dy

α−Q −k Q 2 t ML (f )(x)

k=0

= C t α ML (f )(x)

∞

2−k α .

k=0

Then, summing up the above estimates, we derive Iα (f )(x) ≤ C t α−Q/p f p + t α ML (f )(x) , for every t > 0. We now choose

x ∈ G,

5.9 Sobolev–Stein Embedding Inequality

t=

f p ML (f )(x)

so that

(p α)/Q

Iα (f )(x) ≤ C f p

Hence, being q (1 − pα/Q) = p, we have q Iα (f )q

≤

(pqα)/Q Cf p

279

p/Q ,

(ML (f )(x))1−(p α)/Q . G

(ML (f )(x))p dx

(by the L-maximal function theorem) (p q α)/Q

≤ Cf p

p

q

f p = Cf p ,

since (p q α)/Q + p = q. The theorem is proved. From this theorem and the representation formula (5.16), one easily obtains an inequality extending the classical Sobolev embedding theorem to the homogeneous Carnot groups. Theorem 5.9.2 (Sobolev–Stein embedding). Let L be a sub-Laplacian on the homogeneous Carnot group G of homogeneous dimension Q. Suppose 1 < p < Q. Then there exists a positive constant C = C(p, G, L) such that for every u ∈ C0∞ (RN , R), uq ≤ C ∇L up where 1/q = 1/p − 1/Q

i.e. q =

Qp . Q−p

Proof. Let u ∈ C0∞ (RN , R). Using the representation formula (5.16), we have u(x) = − Γ (x −1 ◦ y) Lu(y) dy. Keeping in mind that L = right-hand side, we obtain

G

m

2 j =1 Xj

u(x) =

RN

and Xj∗ = −Xj , by integrating by parts at the

(∇L Γ )(x −1 ◦ y) ∇L u(y) dy.

(5.79)

On the other hand, out of the origin, we have ∇L Γ = βd ∇L (d 2−Q ) = (2 − Q) βd d 1−Q ∇L d, so that, since ∇L d is smooth in G \ {0} and δλ -homogeneous of degree zero, |∇L Γ | ≤ C d 1−Q , for a suitable constant C > 0 depending only on L. Using this inequality in (5.79), we get

280

5 The Fundamental Solution for a Sub-Laplacian and Applications

|u(x)| ≤ C

G

|∇L u(y)| d(x, y)1−Q dy = C I1 (|∇L u|)(x).

Then, by the Hardy–Littlewood–Sobolev Theorem 5.9.1, uq ≤ C I1 (|∇L u|)q ≤ C ∇L up , where 1/q = 1/p − 1/Q This ends the proof.

⇔q=

Qp . Q−p

5.10 Some Remarks on the Analytic Hypoellipticity of Sub-Laplacians In this section, we collect some results on the hypoellipticity (especially in the analytic sense) of sub-Laplacians. It is far from our scopes here to give proofs of the results of this section, the interested reader will be properly referred to the existing literature. First of all, we recall the relevant definition. Definition 5.10.1 ((Analytic) hypoellipticity). We say that a differential operator L defined on an open set Ω ⊆ RN is hypoelliptic (respectively, analytic hypoelliptic) in Ω if, for every open set Ω ⊆ Ω and every f ∈ C ∞ (Ω , R) (respectively, f real analytic in Ω ), any solution u to the equation Lu = f on Ω (in the weak sense of distributions) is of class C ∞ (Ω , R) (respectively, is real analytic on Ω ). In the sequel, we shall write u ∈ C ω (Ω) to mean that u is real analytic on Ω. Moreover, we may also use the notation C ∞ -hypoelliptic and C ω -hypoelliptic to mean, respectively, hypoelliptic and analytic hypoelliptic. In the very special case of a homogeneous differential operator L with constant coefficients in RN , the problem of hypoellipticity is completely solved by the following result (see, e.g. [Hor69]). Let L be a homogeneous differential operator with constant coefficients in RN . Then the following statements are equivalent: 1) L is C ∞ -hypoelliptic in RN , 2) L is C ω -hypoelliptic in RN , 3) L is elliptic in RN . Moreover, if L has constant coefficients (but it is not necessarily homogeneous), then the equivalence of (2) and (3) still holds true. As a consequence of the above result, all sub-Laplacians on the Euclidean group E = (RN , +) (see Section 4.1.1) are C ∞ and C ω -hypoelliptic.

5.10 Analytic Hypoellipticity of Sub-Laplacians

281

We next focus our attention to the C ∞ -hypoellipticity. Let {X1 , . . . , Xm } be C ∞ vector fields on RN . We recall the well-known rank (or bracket) condition (also referred to as Hörmander’s hypoellipticity condition): dim Lie{X1 , . . . , Xm }I (x) = N ∀ x ∈ RN , i.e. for every x ∈ RN , there exists a set of N iterated brackets of the Xi ’s which are linearly independent at x. The following celebrated result holds (see Hörmander [Hor67]). the rank condition is suffiIf {X1 , . . . , Xm } are C ∞ -vector fields on RN , then n 2 cient for the C ∞ -hypoellipticity of the operator L = j =1 Xj . Moreover, if the coefficients of the Xj ’s are analytic, then the rank condition is also necessary for the C ∞ -hypoellipticity of L. (See also Derridj [Der71], Helffer–Nourrigat [HN79], Kohn [Koh73], Ole˘ınik– Radkeviˇc [OR73], Rothschild–Stein [RS76].) See also Bony [Bon69,Bon70] for a partial converse of the rank condition, namely If the sum of squares L of smooth vector fields is C ∞ -hypoelliptic, then the rank condition holds on an open set, dense in RN . (See also Fedi˘ı [Fed70], Kusuoka–Stroock [KuSt85], Bell–Mohammed [BM95] for examples of sums of squares which are C ∞ -hypoelliptic but do not satisfy the rank condition everywhere.) The problem of C ∞ -hypoellipticity is thus completely solved for any subLaplacian L on any homogeneous Carnot group, since L is a sum of squares of polynomial vector fields satisfying the rank condition. It is also interesting to remark that, for any homogeneous left-invariant differential operator L on a stratified Lie group (hence in particular for our sub-Laplacians), the C ∞ -hypoellipticity of L is equivalent to a Liouville-type property for L, namely the property stating that the only bounded functions u on G such that Lu = 0 are the constant functions. (See [Rot83]; see also [HN79,Gel83].) We finally turn our attention to the C ω -hypoellipticity. The problem of analytic hypoellipticity is more involved and only partial results are known. To begin with, we consider the rank condition, which played a central rôle for C ∞ -hypoellipticity. Unfortunately, if L is a sum of squares of analytic vector fields, then the rankcondition is not sufficient for analytic hypoellipticity. (See, for instance, Trèves [Tre78], Tartakoff [Tar80], Grigis–Sjöstrand [GS85]; see also explicit counterexamples in [BG72,Hel82,PR80,HH91,Chr91].) In the sequel of the section, we collect some of these results (in particular, several explicit negative ones) for our subLaplacians on Carnot groups. The first one is encouraging: The canonical sub-Laplacian on the Heisenberg–Weyl group Hn is analytic hypoelliptic. It is not difficult to prove this result making use of the real analyticity (out of the origin) of the fundamental solution Γ for the canonical sub-Laplacian on Hn , for instance (Q = 2n + 2 denotes the homogeneous dimension of Hn ) Γ (x, t) = cQ

(|x|4

1 . + |t|2 )(Q−2)/4

(5.80)

282

5 The Fundamental Solution for a Sub-Laplacian and Applications

Since (see Kaplan [Kap80]; see also Example 5.4.7, page 250, and Chapter 18) the fundamental solution Γ for the canonical sub-Laplacian on every H-type group has exactly the same form as in (5.80), then the canonical sub-Laplacian on any H-type group is analytic-hypoelliptic. This is only a partial result of what is true on (a class of operators containing the) Heisenberg-type groups, as we shall see below; but, unfortunately (as we shall also see in a moment), it must be soon realized that C ω -hypoellipticity rarely occurs within the non-Euclidean setting of Carnot groups. To this end, we cite a first “negative” result (see Helffer [Hel82]): If G is a Carnot group of step two, and L is a sub-Laplacian on G, then a necessary condition for L to be C ω -hypoelliptic is that G is a HM-group. (See Definition 3.7.3, page 174, for the definition of HM-group.) Hence, at least within the setting of step-two Carnot groups, in order to find a C ω -hypoelliptic sub-Laplacian, we must restrict our attention to the sub-class of HM-groups. Fortunately, a complete answer on the C ω -hypoellipticity for subLaplacians is available on HM-groups. This is given by the following result (see Métivier [Met81]). If G is a HM-group and L ∈ Um (g) (i.e. L is a homogeneous operator of degree m in the relevant enveloping algebra), then L is C ω -hypoelliptic if and only if L is C ∞ -hypoelliptic. Consequently, since any sub-Laplacian L on G belongs to U2 (g) and L is C ∞ hypoelliptic by the rank condition, then L is also C ω -hypoelliptic. This result covers quite large classes of remarkable cases: for example, since the Heisenberg–Weyl groups, the Iwasawa-groups, and (more generally) the H-type groups all belong to the HM-group class, we now know that all their sub-Laplacians are C ω -hypoelliptic. On the converse, it is easy to exhibit a step-two group where no sub-Laplacian is real analytic: take any Carnot group where the first layer has odd dimension (for, in that case, it cannot be a HM-group; see Example 5.10.2 below). The problem finally rest on the investigation of Carnot groups of step strictly greater than two. Unfortunately, a complete answer to the analytic-hypoellipticity in that case is still an open problem. Quoting Rothschild [Rot84], it is reasonable to conjecture that if G is not a HM-group, then there is no L ∈ Um (g) (hence, no subLaplacian) which is analytic-hypoelliptic. For example, from a result by M. Christ [Chr93, Theorem 1.5] we infer that if G is a filiform Carnot group of dimension ≥ 4, then no sub-Laplacian on G is analytic-hypoelliptic. In Examples 5.10.4, 5.10.5, 5.10.6 below, we exhibit some other explicit negative results. To begin, we give two examples: the first (respectively, the second) is an example of a sub-Laplacian on a homogeneous Carnot group of step two which is not (respectively, which is) analytic-hypoelliptic; in the latter case, we explicitly write the (analytic) fundamental solution. Example 5.10.2. Let R4 (the points are denoted by (x, y, z, t) with x, y, z, t ∈ R) be equipped with the dilation δλ (x, y, z, t) = (λx, λy, λz, λ2 t) and the composition law

5.10 Analytic Hypoellipticity of Sub-Laplacians

⎛ ⎞ ⎛ ⎞ ⎛ ξ x ⎜y ⎟ ⎜η⎟ ⎜ ⎝ ⎠◦⎝ ⎠=⎝ ζ z t τ t

283

⎞ x+ξ y+η ⎟ ⎠. z+ζ +τ +xη

Then G = (R4 , ◦, δλ ) is a homogeneous Carnot group of step two, and ΔG = (∂x )2 + (∂y + x ∂t )2 + (∂z )2 is its canonical sub-Laplacian. It is easily seen that G is isomorphic to the sum (in the sense of Section 4.1.5) of the Heisenberg-Weyl group H1 on R3 and the usual Euclidean group (R, +) (note that the canonical sub-Laplacians on both these groups are analytic hypoelliptic!). It can be proved that ΔG is not analytic hypoelliptic! Indeed, by the cited result of Helffer [Hel82], this follows from the fact that G is not a HM-group, since the first layer of the stratification has odd dimension. This example is a particular case of the family of not analytic hypoelliptic subLaplacians (see Rothschild [Rot84]) L=

n (∂xj )2 + (∂yj + xj ∂t )2 + (∂z )2 ,

(5.81)

j =1

which, in turn, are inspired by a famous counterexample by Baouendi–Goulaouic [BG72]. Indeed, in [BG72] it is proved that the operator L :=

n (∂xj )2 + (xj ∂t )2 + (∂z )2 j =1

on Rn+2 (the points are denoted by (x, z, t), x = (x1 , . . . , xn ) ∈ Rn , z ∈ R, t ∈ R) is not C ω -hypoelliptic on Rn+2 . Now, we notice that the operator L in (5.81) is a “lifted” version of L, so that it is easy to prove that L is not C ω -hypoelliptic on R2n+2 if L is not C ω -hypoelliptic on Rn+2 . Indeed, suppose to the contrary that L is C ω -hypoelliptic on R2n+2 . Then take a function f = f (x, z, t) real analytic on an open set Ω ⊆ Rn+2 and a solution u = u(x, z, t) to Lu = f on Ω. If we set f(x, y, z, t) := f (x, z, t) and u(x, y, z, t) := u(x, z, t), then we notice that L u(x, y, z, t) = Lu(x, z, t) = f (x, z, t) = f(x, y, z, t)

(5.82)

:= {(x, y, z, t) : (x, z, t) ∈ Ω, y ∈ R}. Since f is clearly on the open set Ω then (5.82) and the supposed C ω -hypoellipticity of L imply real analytic on Ω, u∈ This obviously means that u is real analytic on Ω. Thus we have shown that C ω (Ω). L is C ω -hypoelliptic on Rn+2 , contrarily to what is proved in [BG72]. Example 5.10.3. This example is taken from Balogh–Tyson [BT02]. Let us consider the group G on R5 (the points are denoted by (x1 , x2 , x3 , x4 , t) ∈ G, (x1 , x2 , x3 , x4 ) ∈ R4 corresponds to the first layer of the stratification, t ∈ R to the second

284

5 The Fundamental Solution for a Sub-Laplacian and Applications

one) with dilation δλ (x1 , x2 , x3 , x4 , t) = (λx1 , λx2 , λx3 , λx4 , λ2 t) and the composition law ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ξ1 x 1 + ξ1 x1 x 2 + ξ2 ⎜ x2 ⎟ ⎜ ξ2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ x 3 + ξ3 ⎜ x3 ⎟ ◦ ⎜ ξ3 ⎟ = ⎜ ⎟. ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ x4 ξ4 x 4 + ξ4 1 t τ t + τ + 2 (x2 ξ1 − x1 ξ2 + 2x4 ξ3 − 2x3 ξ4 ) Following our conventional notation, G is the homogeneous Carnot group of step two (with m = 4 generators and n = 1) with relevant matrix ⎛ ⎞ 0 1 0 0 ⎜ −1 0 0 0 ⎟ B (1) = ⎝ ⎠. 0 0 0 2 0 0 −2 0 Then G is obviously a HM-group, for B is a non-singular skew-symmetric matrix. In particular, by the cited result of Métivier [Met81], the canonical sub-Laplacian L is C ω -hypoelliptic. We can see this directly, for the relevant fundamental solution Γ has been explicitly written by Balogh–Tyson in [BT02] (making use of a remarkable formula by Beals–Gaveau–Greiner, see [BGG96]; we describe this formula closely in Section 5.12, page 291): it is apparent that Γ is analytic out of the origin! Indeed, it holds Γ (x1 , x2 , x3 , x4 , t) = c d 2−Q (x1 , x2 , x3 , x4 , t), where c is a suitable positive constant, Q = 6 is the homogeneous dimension of G, and d is the homogeneous norm defined by d(x1 , x2 , x3 , x4 , t)

2

1/8 1 2 1 2 x1 + x2 + x32 + x42 + t 2 = 2 2 " 3/8

2 1 2 1 2 1 2 1 2 x + x + x32 + x42 + t 2 · x1 + x2 + 2 2 2 1 2 2 " −1/8

2 1 2 1 2 1 2 1 2 2 2 2 2 2 x + x + x3 + x4 + t · x1 + x2 + x3 + x4 + . (5.83) 2 2 2 1 2 2 This formula gives the remarkable example of an explicit fundamental solution of a group which is not a H-type group! We then turn our attention to groups of step greater than two. Following the idea in the argument at the end of Example 5.10.2, we can give infinite examples of groups of arbitrarily high step with a sub-Laplacian which is not C ω -hypoelliptic: it suffices to take the sum (in the sense of Section 4.1.5) of the group G of Example 5.10.2 with another Carnot group. Let us now quote some other examples taken or inspired by the existing literature.

5.10 Analytic Hypoellipticity of Sub-Laplacians

285

Example 5.10.4. This example is taken from a paper by Christ [Chr95] (see also [Hel82,PR80]). Consider the Carnot group on R4 with the composition 9 low ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ξ1 x 1 + ξ1 x1 x 2 + ξ2 ⎟ ⎜ x2 ⎟ ⎜ ξ2 ⎟ ⎜ ⎠. ⎝ ⎠◦⎝ ⎠=⎝ x3 ξ3 x 3 + ξ3 − ξ2 x 1 x4 ξ4 x4 + ξ4 + 2ξ3 x1 − ξ2 x12 It is easily seen that G is a filiform homogeneous Carnot group of step three and two generators, with dilations δλ (x1 , x2 , x3 , x4 ) = (λx1 , λx2 , λ2 x3 , λ3 x4 ), and such that the first two vector fields of the relevant Jacobian basis are X1 = ∂x1 ,

X2 = ∂x2 − x1 ∂x3 − x12 ∂x4 .

Then, it can be proved that the canonical sub-Laplacian on G 2 ΔG = X12 + X22 = (∂x1 )2 + ∂x2 − x1 ∂x3 − x12 ∂x4 is not C ω -hypoelliptic. (Compare the group in this example to the Bony-type subLaplacian with N = 2 in Section 4.3.3, page 202.) Example 5.10.5. It is known (see [Chr91,Chr93,HH91,PR80]) that in R3 (with coordinates (t, s, x)) the operator L = (∂t )2 + (∂s − t m ∂x )2

(5.84)

is not C ω -hypoelliptic for any m ∈ N, m ≥ 2. In this example, fixed m as above, we give a suitable sub-Laplacian L “lifting” L: as a consequence (arguing as at the end of Example 5.10.2), L cannot be C ω -hypoelliptic, since L does not possess this property. Take N ∈ N, N ≥ m ≥ 2, and consider the following Bony-type sub-Laplacian (see Section 4.3.3, page 202): we equip R2+N (whose points are denoted by (t, s, x), t, s ∈ R, x ∈ RN ) by the composition law (t, s, x1 , x2 , x3 , . . . , xN ) ◦ (τ, σ, ξ1 , ξ2 , ξ3 , . . . , ξN ) ⎛ t +τ ⎜ s+σ ⎜ ⎜ x1 + ξ1 + σ t ⎜ 2 ⎜ x2 + ξ2 + ξ1 t + σ t2! =⎜ 2 3 ⎜ x3 + ξ3 + ξ2 t + ξ1 t2! + σ t3! ⎜ ⎜ .. ⎜ . ⎝ xN + ξN + ξN −1 t + ξN −2 9 Compare to Ex. 3, Chapter 4, page 216.

t2 2!

+ · · · + ξ1

t N−1 (N −1)!

⎞

+σ

tN N!

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

286

5 The Fundamental Solution for a Sub-Laplacian and Applications

and the group of dilations defined by δλ (t, s, x1 , x2 , x3 , . . . , xN ) = (λt, λs, λ2 x1 , λ3 x2 , λ4 x3 , . . . , λN +1 xN ). Then, G = (R2+N , ◦, δλ ) is a filiform homogeneous Carnot group of step N + 1 (note that, since N ≥ m ≥ 2, then the step N + 1 is ≥ 3) and two generators, and

2 t2 tN ΔG = (∂t )2 + ∂s + t ∂x1 + ∂x2 + · · · + ∂xN 2! N! is its canonical sub-Laplacian. We now prove that ΔG is not C ω -hypoelliptic (starting from the fact that L in (5.84) is not). Indeed, since L is not C ω -hypoelliptic, there exists an open set Ω ⊆ R3 and a function u = u(t, s, x) on Ω such that Lu ∈ C ω (Ω) but u ∈ / C ω (Ω). Let us now consider the function (recall that m ≤ N ) u = u(t, s, x1 , . . . , xN ) := u(t, s, −m!xm ) defined on the open subset of R2+N

:= (t, s, x1 , . . . , xN ) | (t, s, −m! xm ) ∈ Ω . Ω As a consequence, it holds u(t, s, x1 , . . . , xN ) = (Lu)(t, s, −m! xm ) on Ω, ΔG (for Lu ∈ C ω (Ω)) and (for u ∈ whence ΔG u ∈ C ω (Ω) u∈ / C ω (Ω) / C ω (Ω)). This proves that ΔG is not analytic-hypoelliptic. Example 5.10.6. Let m, k ∈ N be such that 0 ≤ m ≤ k. Consider the operator on R3 (whose points are denoted by (x1 , x2 , x3 )) L = (∂x1 )2 + (x1m ∂x2 )2 + (x1k ∂x3 )2 .

(5.85)

O.A. Ole˘ınik and E.V. Radkeviˇc [OR72] (see also [Him98]) proved that L is C ω hypoelliptic if and only if m = k. Our aim in this example is to “lift” (in a suitable sense) the vector fields on R3 appearing in (5.85) ∂x1 ,

x1m ∂x2 ,

x1k ∂x3

(5.86)

to three vector fields (in a larger space, namely R3+m+k ) generating a homogeneous Carnot group. It will easily follow (arguing as in the last paragraph of Example 5.10.5) that the relevant sub-Laplacian is not C ω -hypoelliptic if L is not. When m = k = 0, then L = ΔR3 is the ordinary Laplace operator on R3 ; when m = 0, k = 1, we obtain a non-analytic-hypoelliptic operator already considered in Example 5.10.2 (see also Baouendi–Goulaouic [BG72]). Let us now suppose that k > m ≥ 1. We equip R3+m+k with the following coordinates (the semicolon will denote different layers in a suitable homogeneous Carnot group structure)

5.11 Harmonic Approximation

287

P = (x1 , y1 , z1 ; y2 , z2 ; y3 , z3 ; . . . , ym , zm ; x2 , zm+1 ; zm+2 ; zm+3 ; . . . ; zk ; x3 ). We define a group of dilations by setting (recall that k ≥ m + 1) δλ (P ) = λx1 , λy1 , λz1 ; λ2 y2 , λ2 z2 ; λ3 y3 , λ3 z3 ; . . . ; λm ym , λm zm ; λm+1 x2 , λm+1 zm+1 ; λm+2 zm+2 ; λm+3 zm+3 ; . . . ; λk zk ; λk+1 x3 . Let us also consider on R3+m+k the following vector fields X = ∂x1 , Y = ∂y1 + x1 ∂y2 + x12 ∂y3 + · · · + x1m−1 ∂ym + x1m ∂x2 , Z = ∂z1 + x1 ∂z2 + x12 ∂z3 + · · · + x1k−1 ∂zk + x1k ∂x3 . It is then not difficult to see that X, Y, Z are δλ -homogeneous of degree one and they fulfill hypotheses (H0)–(H1)–(H2) of page 191. Hence, by the results of Section 4.2 (page 191), we can define a suitable homogeneous Carnot group structure on R3+m+k such that ΔG = X 2 + Y 2 + Z 2 is the relevant canonical sub-Laplacian. Now, since ΔG (u(x1 , x2 , x3 )) = (Lu)(x1 , x2 , x3 ) for any smooth function u on R3+m+k depending only on x1 , x2 , x3 , we can argue as in the last paragraph of Example 5.10.5 to infer that ΔG is not C ω -hypoelliptic (since L is not).

5.11 Harmonic Approximation Let L be a sub-Laplacian on the homogeneous Carnot group G. In this section, we give some conditions ensuring that a L-harmonic function defined in a neighborhood of a compact set K contained in an open set Ω can be uniformly approximated on K by a sequence of L-harmonic functions in Ω. In some of the results of this section, we assume that L is analytic-hypoelliptic (see Section 5.10). Γ = d 2−Q , will denote its fundamental solution. To begin with, we prove the following lemma. Lemma 5.11.1. Let L be a sub-Laplacian on the homogeneous Carnot group G = (RN , ◦, δλ ), and let Γ = d 2−Q be its fundamental solution. Suppose also that L is analytic-hypoelliptic.

288

5 The Fundamental Solution for a Sub-Laplacian and Applications

Let K ⊆ G be compact. Then there exists R = R(K, G, L) > 0 such that, for every z ∈ G with d(z) > R, it holds 1 α D Cα (z) · y α , Cα (z) = Γ (z−1 ◦ y) , Γ (z−1 ◦ y) = α! y=0 N α∈(N∪{0})

the series converging uniformly on the d-disc {y ∈ G : d(y) < d(K)}, where d(K) := supz∈K d(z). Proof. Let ζ ∈ G be such that d(ζ ) = 1. Since η → Γ (ζ −1 ◦ η) is analytic close to η = 0 and ∂Bd (0, 1) = {ζ ∈ G : d(ζ ) = 1} is compact, there exists ρ > 0, independent of ζ , such that Γ (ζ −1 ◦ η) = aα (ζ ) · ηα , uniformly on Bd (0, ρ), α∈(N∪{0})N

1 α aα (ζ ) = D Γ (ζ −1 ◦ η) . α! η=0

where

Let us now choose R > 0 such that R > d(K)/ρ. If z ∈ G, d(z) > R, then Γ (z−1 ◦ y) = d 2−Q (z) Γ δd −1 (z) z−1 ◦ δd −1 (z) (y) α aα δd −1 (z) z−1 · δd −1 (z) (y) = d 2−Q (z)

=:

α∈(N∪{0})N

Cα (z) · y α .

α∈(N∪{0})N

The series converges uniformly for y ∈ G such that d(y) d δd −1 (z) (y) = < ρ. d(z) In particular, this holds for y ∈ Bd (0, R ρ) ⊇ Bd (0, d(K)).

The next lemma does not require the analytic-hypoellipticity of L. Lemma 5.11.2. Let L be a sub-Laplacian on the homogeneous Carnot group G = (RN , ◦, δλ ), and let Γ = d 2−Q be its fundamental solution. Let u be an L-harmonic function on the ball Bd (0, r). Suppose u(x) =

∞

uk (x),

x ∈ Bd (0, r),

k=1

where uk is a continuous δλ -homogeneous function in the whole G of δλ -degree mk , k ∈ N. If the series is uniformly convergent on the compact subsets of Bd (0, r) and mk = mh for every k = h, then every uk is L-harmonic in G.

5.11 Harmonic Approximation

289

Proof. Let ϕ ∈ C0∞ (RN , R) with supp ϕ ⊆ Bd (0, 1). Since u is L-harmonic in Bd (0, r), we have 0= u(x) L ϕ(δλ−1 (x)) dx ∀ λ < r. Bd (0,r)

The homogeneity of L and the change of variable y = δλ−1 (x) give 0 = λQ−2 u(δλ (y)) Lϕ(y) dy Bd (0,r/λ)

= λQ−2 =λ =λ

∞

uk (δλ (y)) Lϕ(y) dy

k=1 Bd (0,r/λ) ∞ Q−2 mk

uk (y) Lϕ(y) dy

λ

Q−2

k=1 ∞

Bd (0,r/λ)

λ

uk (y) Lϕ(y) dy.

mk

k=1

Bd (0,1)

Note that, in the last equality, we were able to replace Bd (0, r/λ) by Bd (0, 1), since λ < r and ϕ is supported in a compact set in Bd (0, 1). Then uk (y) Lϕ(y) dy = 0 ∀ ϕ ∈ C0∞ (Bd (0, 1)) ∀ k ∈ N. Bd (0,1)

This means that Luk = 0 in Bd (0, 1) in the weak sense of distributions. Since L is hypoelliptic, this implies the L-harmonicity of uk in Bd (0, 1), and so in G, due to the δλ -homogeneity of uk , k ∈ N. We are now ready to prove the announced approximation theorem. Theorem 5.11.3 (An approximation theorem). Let L be a sub-Laplacian on the homogeneous Carnot group G. Suppose that L is analytic-hypoelliptic. Let Ω ⊆ G be an open set such that ∂Ω = ∂Ω. Let K ⊆ Ω be compact and satisfy the following condition: if ω is a bounded connected component of Ω \ K, then ∂ω ⊆ ∂Ω.

(5.87)

Then, for every function h which is L-harmonic in a neighborhood of K, there exists a sequence (hn )n∈N of L-harmonic functions in Ω such that lim hn = h,

n→∞

uniformly on K.

Proof. By general results from functional analysis, it is well known that it suffices to prove the following statement.10 10 This is also known as “Caccioppoli’s completeness method”.

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5 The Fundamental Solution for a Sub-Laplacian and Applications

Let μ be a signed Radon measure supported on K and satisfying h dμ = 0

(5.88)

K

for every L-harmonic function h in Ω. Then (5.88) holds for every function h which is L-harmonic just in a neighborhood of K. The crucial part of the proof is the following assertion. Claim. If μ satisfies the previous hypotheses, then u(x) := Γ (y −1 ◦ x) dμ(y) = Γ (x −1 ◦ y) dμ(y) (5.89) K

K

is identically zero in Ω \ K. We first show how to complete the proof of the theorem by using this claim. Let h be an L-harmonic function in an open set Ω0 ⊇ K, Ω0 ⊆ Ω. Choose a function φ ∈ C0∞ (Ω0 ) such that φ = 1 in an open set Ω1 ⊇ K, Ω 1 ⊆ Ω0 . Then φ h ∈ C0∞ (Ω0 ) and φ h = h in Ω1 . By the representation formula (5.16), we have Γ (x −1 ◦ y)L(φ h)(x) dx (φ h)(y) = − Ω0 =− Γ (x −1 ◦ y) L(φ h)(x) dx, y ∈ Ω0 . Ω0 \Ω 1

It follows that h(y) dμ(y) = (φ h)(y) dμ(y) K K

−1 Γ (x ◦ y) L(φ h)(x) dx dμ(y) =: (). =− K

Ω0 \Ω 1

Thus, by interchanging the integrals and keeping in mind (5.89), we infer L(φ h)(x)u(x) dx = 0, () = − Ω0 \Ω 1

since u = 0 in Ω \ K ⊇ Ω0 \ Ω 1 . Thus, we are left with the proof of the Claim. Let ω be a connected component of Ω \ K. We have to prove that u ≡ 0 in ω. We first suppose that ω is bounded. Then ∅ = ∂ω ⊆ ∂Ω. In particular, this implies that ∂Ω = ∅, hence Ω = G for ∂Ω = ∂Ω. For every x0 ∈ G \ Ω, the function y → Γ (x0−1 ◦ y) is L-harmonic in Ω. Therefore, by the assumption (5.88), Γ (y −1 ◦ x0 ) dμ(y) = 0. u(x0 ) = K

This proves that u ≡ 0 in G \ Ω. As a consequence,

5.12 An Integral Representation Formula for Γ

D α u(x) = 0

for every multi-index α

291

(5.90)

and for every x ∈ G \ Ω. Since u ∈ C ∞ (G \ K), it follows that (5.90) also holds at any point x ∈ ∂Ω = ∂Ω. In particular, since ∂ω ⊆ ∂Ω, (5.90) holds at some point x ∈ ∂ω. Then ω ≡ 0 in a neighborhood of x, since u is L-harmonic, hence real analytic, close to x. The connectedness of ω and again the analyticity of u imply that u ≡ 0 in ω. Let us now assume that ω is unbounded. By Lemma 5.11.1, for every x ∈ ω with d(x) sufficiently large, we have Cα (x) · y α , uniformly in Bd (0, d(K)). Γ (x −1 ◦ y) = α∈(N∪{0})N

Then

u(x) =

Γ (x

−1

◦ y) dμ(y) =

K

∞

um (x, y) dμ(y),

m=0 K

where

um (x, y) =

Cα (x) · y α .

|α|G =k

The function um is δλ -homogeneous of degree m and the series ∞

um (x, ·)

m=0

is uniformly convergent on Bd (0, d(K)) to the L-harmonic function y → Γ (x −1 ◦ y). Then, by Lemma 5.11.2, um (x, ·) is L-harmonic in G, so that, by the assumption (5.88), um (x, y) dμ(y) = 0 ∀ m ≥ 0. K

Thus we have proved that u(x) = 0 for every x ∈ ω, with d(x) sufficiently large. Since ω is connected and u is analytic in ω, this implies u ≡ 0 in ω and completes the proof of the theorem.

5.12 An Integral Representation Formula for the Fundamental Solution on Step-two Carnot Groups The aim of this section is to state a remarkable result in the paper [BGG96] by R. Beals, B. Gaveau and P. Greiner. This result provides a somewhat explicit integral representation formula of the fundamental solution of the canonical sub-Laplacian on a general Carnot group of step two. We shall see in Section 16.3 (page 637 in Part III) that, given a Carnot group G1 and an arbitrary sub-Laplacian L on G1 , there exists a Carnot group G2 isomorphic

292

5 The Fundamental Solution for a Sub-Laplacian and Applications

to G1 such that L corresponds (via the related isomorphism in the relevant Lie algebras) to the canonical sub-Laplacian ΔG2 on G2 . Moreover, if G2 (whence G1 ) has step two, we saw in Proposition 3.5.1 (page 168) that we can perform another Liegroup isomorphism sending G2 into the homogeneous Carnot group G3 such that the composition law on G3 is given by (we follow our usual notation)

1 (1) 1 (n) (x, t) ◦ (ξ, τ ) = x + ξ, t1 + τ1 + B x, ξ , . . . , tn + τn + B x, ξ (5.91) 2 2 and the matrices B (i) ’s are skew-symmetric. This last isomorphism also sends the canonical sub-Laplacian of G2 into the canonical sub-Laplacian of G3 . Now, the cited result in [BGG96] furnishes an integral formula for the canonical subLaplacian on a homogeneous Carnot group of step two whose composition law ◦ has precisely the above form and the matrices B (i) ’s are skew-symmetric. Our argument above shows that (and how) we can obtain a representation formula for the fundamental solution of any (not necessarily canonical) sub-Laplacian on any homogeneous Carnot group of step two. We now state the remarkable result in [BGG96]. We explicitly remark that in [BGG96] the formalism of complex Hamiltonian mechanics is followed: we slightly change the therein notation. Theorem 5.12.1 (Beals–Gaveau–Greiner, [BGG96]). Let G = Rm+n (whose points are denoted by (x, t), x ∈ Rm , t ∈ Rn ) be equipped with a homogeneous Carnot group structure by the dilation δλ (x, t) = (λx, λ2 t) and the composition law matrices of in (5.91), where the B (k) ’s are n skew-symmetric linearly independent order m × m. Consider the canonical sub-Laplacian ΔG = ni=1 Xi2 , where m n 1 (k) Xi = ∂/∂xi + bi,l xl ∂/∂tk , i = 1, . . . , m 2 k=1

l=1

(k) (here bi,l denotes the entry of position (i, l) of B (k) ). Then, for every (x, t) with x = 0, the fundamental solution Γ of ΔG is given by √ det(V(B(τ ))) Γ (x, t) = cQ dτ, (5.92) 1 Rn ( 2 W(B(τ )) · x, x − ι t, τ )Q/2−1

where ι is the imaginary unit of C and cQ is the dimensional constant cQ =

( Q 2 − 1) . 2 (2 π)Q/2

(5.93)

Here we used the following notation: Q = m + 2 n is the homogeneous dimension of G, in (5.93) is Euler’s Gamma function, τ = (τ1 , . . . , τn ) ∈ Rn , B(τ ) =

1 (τ1 B (1) + · · · + τn B (n) ), 2

5.13 Appendix A. Maximum Principles

293

V and W are the real-analytic functions prolonging z/ sin(z) and z/ tan(z), respectively, at z = 0, i.e. V(z) =

∞ (−1)j z2j , (2j + 1)!

W(z) =

j =0

∞ (−1)j 22j B2j 2j z (2j )! j =0

(here the B2j ’s are the Bernoulli numbers). Moreover, for every t ∈ Rn \{0}, we have Γ (0, t) = lim Γ (x, t). 0=x→0

We remark that a general integral formula for Γ is provided in [BGG96] comprising the case x = 0 too, by shifting the contour Rn into the complex domain Cn (see [BGG96, Theorem 3, page 315]). The δλ -homogeneity of Γ in (5.92) (of degree 2 − Q) should be noted. As we saw in Example 5.10.3 (page 283), formula (5.92) can, in some cases, give explicit fundamental solutions. This can be done by using the fact that, if λ1 (τ ), . . . , λm (τ ) and v1 (τ ), . . . , vm (τ ) denote the eigenvalues and corresponding eigenvectors of the matrix B(τ ) (over the complex field), normalized in such a way that |vj (τ )| = 1 for j = 1, . . . , m, we have m # det V(B(τ )) =

λj (τ ) , sin(λj (τ ))

j =1 m

W(B(τ )) · x, x =

j =1

2 λj (τ ) x, vj (τ ) . tan(λj (τ ))

(In the last formula, the inner product is, of course, that of Cm .)

5.13 Appendix A. Maximum Principles In this section, we shall prove some weak and strong maximum principles for L, an arbitrary sub-Laplacian on a homogeneous Carnot group G. To begin with, we prove some elementary lemmas. Lemma 5.13.1. Let Ω ⊂ RN be a bounded open set and let u : Ω → R be an arbitrary function. Then there exists a point x0 ∈ Ω such that lim sup u(x) = sup u. x→x0

(5.94)

Ω

Proof. We argue by contradiction and assume that (5.94) is false. Then, for every x ∈ Ω, there exists an open neighborhood Vx of x such that sup u < sup u. Ω∩Vx

Ω

(5.95)

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5 The Fundamental Solution for a Sub-Laplacian and Applications

The family {Vx : x ∈ Ω} is an open covering of Ω, so that, since Ω is compact, we have p $ Ω⊆ Vxj , p ∈ N, j =1

for suitable x1 , . . . , xp ∈ Ω. Then sup u = max Ω

sup u : j = 1, . . . , p .

(5.96)

Ω∩Vxj

On the other hand, by (5.95), the right-hand side of (5.96) is strictly less than supΩ u. This contradiction proves the lemma. Lemma 5.13.2. Let A and B be N ×N symmetric matrices with constant real entries. Assume A ≥ 0 and B ≤ 0. Then trace(A · B) ≤ 0. Proof. Let R := A1/2 be a symmetric square root of A. Then trace(A·B) = trace(R· R · B) = trace(R · B · R) = trace(R T · B · R) ≤ 0, since B ≤ 0. Lemma 5.13.3. Let L be a sub-Laplacian on the homogeneous Carnot group G. Let Ω ⊆ G be an arbitrary open set, and let u : Ω → R be a C 2 real function. Assume that u has a local maximum at x0 ∈ Ω. Then Lu(x0 ) ≤ 0.

(5.97)

Proof. We know that L = div(A · ∇ T ), where A is a N × N symmetric matrix with polynomial entries and A(x) ≥ 0 at any point x ∈ RN . Then L = trace(A · D 2 u) + b, ∇u ,

(5.98)

where D 2 u = (∂xi xj )i,j ≤N is the Hessian matrix of u and b is the vector-valued function whose j -th component is given by bj =

N

∂xi ai,j .

(5.99)

i=1

Since u has a local maximum at x0 , we have ∇u(x0 ) = 0 and D 2 u(x0 ) ≤ 0. Then, by Lemma 5.13.2, Lu(x0 ) = trace(A(x0 ) · D 2 u(x0 )) ≤ 0. This ends the proof.

We are now able to give a simple proof of the following weak maximum principle.

5.13 Appendix A. Maximum Principles

295

Theorem 5.13.4 (Weak maximum principle). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let Ω be a bounded open subset of G. Let u : Ω → R be a C 2 function such that Lu ≥ 0 in Ω, (5.100) lim supx→y u(x) ≤ 0 for every y ∈ ∂Ω. Then u ≤ 0 in Ω. Proof. We know that the matrix A in (5.98) has the following block form (see also (1.91), page 64)

A1,1 A1,2 A= , A2,1 A2,2 where A1,1 = (ai,j )i,j ≤m is a constant m × m symmetric matrix strictly positive definite. Then a1,1 > 0. Let b1 be given by (5.99) with j = 1. Define λ := 2 sup x∈Ω

b1 (x) , a1,1

M :=

sup

exp(λ x1 ),

(x1 ,...,xN )∈Ω

and h(x) = h(x1 , . . . , xN ) := M − exp(λ x1 ). A trivial computation shows that h(x) ≥ 0 and

Lh(x) < 0 for every x ∈ Ω.

(5.101)

For an arbitrary ε > 0, let us now consider the function uε := u − ε h. Due to inequalities (5.101) and condition (5.100), we have Luε > 0 in Ω

and lim sup uε (x) ≤ 0 for every y ∈ ∂Ω.

(5.102)

x→y

By Lemma 5.13.1, there exists a point x0 ∈ Ω such that lim sup uε (x) = sup uε . x→x0

(5.103)

Ω

We want to show that x0 ∈ ∂Ω. Arguing by contradiction, we assume x0 ∈ Ω. Then, by the continuity of u in Ω, uε (x0 ) = lim sup uε (x), x→x0

so that, by (5.103), uε (x0 ) = maxΩ uε . As a consequence, by Lemma 5.13.3, Luε (x0 ) ≤ 0. This contradicts the first inequality in (5.102). Thus x0 ∈ ∂Ω. Then, by (5.103) and the second condition in (5.102), sup uε = lim sup uε (x) ≤ 0. Ω

x→x0

Hence, u − ε h = uε ≤ 0 in Ω for every ε > 0. Letting ε tend to zero, we obtain u ≤ 0 in Ω. The theorem is thus completely proved.

296

5 The Fundamental Solution for a Sub-Laplacian and Applications

Note 5.13.5. The previous proof can be applied also to continuous functions satisfying the inequality Lu ≥ 0 in the asymptotic sense of Exercises 8 and 9 at the end of the chapter (see also Ex. 10). Corollary 5.13.6. Let L be a sub-Laplacian on the homogeneous Carnot group G. Let Ω be an unbounded open subset of G. Let u : Ω → R be a C 2 function such that ⎧ in Ω, ⎪ ⎨ Lu ≥ 0 lim supx→y u(x) ≤ 0 for every y ∈ ∂Ω, (5.104) ⎪ ⎩ lim sup |x|→∞ u(x) ≤ 0. Then u ≤ 0 in Ω. Proof. Let ε > 0 be arbitrary but fixed. The third condition in (5.104) implies the existence of a real positive constant R such that u(x) − ε < 0 in Ω \ ΩR ,

(5.105)

where ΩR := {x ∈ Ω : |x| < R}. It follows that L(u − ε) = Lu ≥ 0 in ΩR , lim supx→y u(x) ≤ 0 for every y ∈ ∂ΩR . Then, by Theorem 5.13.4, u − ε ≤ 0 in ΩR . This inequality, together with (5.105), gives u ≤ ε in Ω for every ε > 0. Hence u ≤ 0 in Ω. A particular case of Corollary 5.13.6 is the following one. Corollary 5.13.7. If L is as in Corollary 5.13.6, the only entire L-harmonic function vanishing at infinity is the null function. Proof. Let u : G → R be an entire L-harmonic function vanishing at infinity, i.e. u ∈ C ∞ (G, R) satisfies Lu = 0 in G, lim|x|→∞ u(x) = 0. Then, by applying Corollary 5.13.6 both to u and −u, we get u ≡ 0.

The rest of this section is devoted to the proof of the following strong maximum principle. Theorem 5.13.8 (Strong maximum principle). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let Ω be a connected open subset of G. Let u : Ω → R be a C 2 function such that u ≤ 0 and Lu ≥ 0 in Ω.

(5.106)

Suppose there exists a point x0 ∈ Ω such that u(x0 ) = 0. Then u(x) = 0 for every x ∈ Ω.

5.13 Appendix A. Maximum Principles

297

The proof of this theorem requires several preliminary results. In what follows, we shall denote by | · | the standard Euclidean norm and by D(z, r) the ball D(z, r) := {x ∈ RN : |x − z| < r}. Definition 5.13.9. Let F be a relatively closed subset of Ω. We say that a vector ν ∈ RN \ {0} is orthogonal to F at a point y ∈ Ω ∩ ∂F if D(y + ν, |ν|) ⊆ (Ω \ F ) ∪ {y}.

(5.107)

If this inclusion holds, we shall write ν⊥F at y. We also put

F ∗ := y ∈ Ω ∩ ∂F | there exists ν: ν⊥F at y . With the above notation, we explicitly remark that F ∗ = ∅ if F ⊂ Ω, F = Ω. Indeed, since Ω is connected, Ω ∩ ∂F is not empty. Take a point z ∈ Ω ∩ ∂F , a ball D(z, R) ⊆ Ω and a point x0 ∈ D(z, R/2). Let y ∈ Ω ∩ ∂F be such that r := |x0 − y| = dist(x0 , ∂F ). Then y ∈ F ∗ and ν := 2r (x0 − y)⊥F at y. The following Hopf-type lemma will be crucial for the proof of the strong maximum principle in Theorem 5.13.8. Lemma 5.13.10 (A Hopf-type lemma for sub-Laplacians). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let Ω ⊆ G be open, and let u : Ω → R be a C 2 function satisfying the inequalities in (5.106). Let F := {x ∈ Ω : u(x) = 0}.

(5.108)

Assume ∅ = F = Ω. Then, for every y ∈ F ∗ and ν⊥F at y, we have qL (y, ν) = 0,

(5.109)

where qL (x, ξ ) := A(x) · ξ, ξ is the characteristic form of L defined in (5.1a). Proof. Let y ∈ F ∗ and ν⊥F at y. Then D(y + ν, |ν|) ⊆ (Ω \ F ) ∪ {y}. Since F ∗ ⊆ F ∩ Ω, y is a maximum point for u (see (5.108)). Then ∇u(y) = 0. We now argue by contradiction assuming that (5.109) is false. Hence qL (y, ν) > 0. Let us now consider the function h(x) := exp(−λ |x − z|2 ) − exp(−λ r 2 ),

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5 The Fundamental Solution for a Sub-Laplacian and Applications

where z = y + ν and r = |ν|. The positive constant λ will be fixed later on. A direct and easy computation shows that Lh(y) = 4λ2 exp(−λ r 2 ) · qL (y, ν) + O(1/λ) , as λ → ∞. Then, we can choose and fix λ > 0 in such a way that Lh > 0 in a suitable neighborhood V of y. Obviously, we may assume V ⊂ Ω. Let us now consider the bounded open set U := V ∩ D(z, r). Note that ∂U = Γ1 ∪ Γ2 , where Γ1 = V ∩ ∂D(z, r) and Γ2 = D(z, r) ∩ ∂V . Since Γ2 is a compact subset of Ω \F and u < 0 in Ω \F , there exists ε > 0 such that u + ε h < 0 in Γ2 . On the other hand, being h = 0 on ∂D(z, r) and u ≤ 0 in Ω, we have u+ε h ≤ 0 on Γ1 . Then, since L(u+ε h) ≥ ε Lh ≥ 0 in U , from the maximum principle of Theorem 5.13.4, we obtain u + ε h ≤ 0 in U . Since u(y) = h(y) = 0, this implies h(y + t ν) − h(y) u(y + t ν) − u(y) ≤ −ε t t

for 0 < t < 1.

(5.110)

Letting t tend to zero in this inequality, we get ∇u(y), ν ≤ −ε ∇h(y), ν = −2ε exp(−λ r 2 ) r 2 . This contradicts the condition ∇u(y) = 0 and completes the proof.

Corollary 5.13.11. Let the hypotheses and notation of the previous lemma hold. Let 2 also L = m j =1 Xj . Then we have Xj I (y), ν = 0

∀ y ∈ F ∗ ∀ ν⊥F at y

and for every j = 1, . . . , m. Proof. It follows from the previous lemma, by just noticing that qL (x, ξ ) =

m Xj I (y), ξ 2 . j =1

Another crucial definition is the following one. Definition 5.13.12 ((Positively) invariant set w.r.t. a vector field). Let X ∈ T (RN ) be a smooth vector field in RN , and let F be a relatively closed subset of Ω. We say that F is positively X-invariant if, for any integral curve γ of X, γ : [0, T ] → Ω such that γ (0) ∈ F , we have γ (t) ∈ F for every t ∈ [0, T ]. We say that F is X-invariant if it is positively X-invariant with respect to both X and −X.

5.13 Appendix A. Maximum Principles

299

It is easy to verify that the condition XI (y), ν ≤ 0

∀ y ∈ F ∗ ∀ ν⊥F at y

(5.111)

is necessary for the positive X-invariance of F . Indeed, let y ∈ F ∗ , ν⊥F at y and γ : [0, T ] → Ω be an integral curve of X such that γ (0) = y. Let F be positively X-invariant. Since D(y + ν, |ν|) ⊆ (Ω \ F ) ∪ {y}, we have |γ (t) − (y + ν)|2 ≥ |ν|2 and |γ (0) − (y + ν)|2 = |ν|2 for every t ∈ [0, T ]. This means that the real function t → |γ (t) − (y + ν)|2 has a minimum at t = 0. As a consequence, d 0≤ |γ (t) − (y + ν)|2 = γ˙ (0), γ (0) − (y + ν) = XI (y), −ν . d t t=0 Hence (5.111) holds. We will show that this condition is also sufficient for F to be positively X-invariant. To prove this statement, we need the following elementary lemma. Lemma 5.13.13. Let g : [0, T ] → R be a continuous function such that g(t + h) − g(t) ≤M h

lim sup h→0−

∀ t ∈ (0, T ],

(5.112)

for a suitable M ∈ R. Then g(t) ≤ g(0) + M t

∀ t ∈ [0, T ].

Proof. Let ε > 0 be fixed. Condition (5.112) implies that the real function t → g(t) − g(0) − (M + ε)t has a maximum at t = 0. Indeed, suppose to the contrary that there exist ε0 > 0 and t0 ∈ (0, T ] such that g(t) − g(0) − (M + ε) t ≤ g(t0 ) − g(0) − (M + ε0 ) t0

∀ t ∈ [0, T ].

In particular, for t = t0 + h and h < 0 small enough, this gives g(t0 + h) − g(t0 ) ≥ (M + ε0 ), h which contradicts the hypothesis. Then, g(t) − g(0) − (M + ε) t ≤ 0 for every t ∈ [0, T ]. Letting ε tend to zero, we obtain the assertion. Proposition 5.13.14 (Nagumo–Bony). Let X ∈ T (RN ) be a smooth vector field in RN , and let F be a relatively closed subset of Ω. Then F is positively X-invariant if and only if (5.113) XI (y), ν ≤ 0 ∀ y ∈ F ∗ ∀ ν⊥F at y.

300

5 The Fundamental Solution for a Sub-Laplacian and Applications

Proof. We only need to show the “if” part. Let γ : [0, T ] → Ω be an integral curve of X such that x0 := γ (0) ∈ F . Define δ(t) := dist(γ (t), F ),

0 ≤ t ≤ T.

We have to prove that δ(t) = 0 for every t ∈ [0, T ]. Let V be a bounded neighborhood of x0 containing γ ([0, T ]), and let L :=

|XI (x) − XI (z)| |x − z| x,z∈V , x=z sup

(5.114)

be the Lipschitz constant of X on V . We may suppose L T < 1/2 and V = D(x0 , r) with D(x0 , 2r) ⊆ Ω. We claim that L(t) := lim sup h→0−

δ(t + h) − δ(t) ≤ L δ(t) h

∀ t ∈ (0, T ].

(5.115)

If δ(t) = 0, inequality (5.115) is trivial, since h < 0 and δ(t + h) ≥ 0. Suppose δ(t) > 0 and choose a sequence hn ↑ 0 such that L(t) = lim

n→∞

δ(t + hn ) − δ(t) . hn

Let us now denote x := γ (t) and xn := γ (t + hn ). Since γ ([0, T ]) ⊂ D(x0 , r) and D(x0 , 2r) ⊆ Ω, for every n ∈ N there exists a point zn ∈ F ∩ Ω such that |xn − zn | = dist(xn , F ) = δ(t + hn ). Obviously, we may suppose that zn → z ∈ F ∩ D(x0 , r), so that, since xn → x, |x − z| = lim |xn − z| = lim dist(xn , F ) n→∞

n→∞

= dist(x, F ) = δ(t). Moreover, ν :=

1 (x − z)⊥F at z. 2

Then δ(t + hn ) − δ(t) = |xn − zn | − |x − z| ≥ |xn − zn | − |x − zn | xn − x, zn − x . ≥ |xn − x| ≥ − |x − zn | Hence

) L(t) ≤ lim

n→∞

* ) * x − zn xn − x x−z = γ˙ (t), , |x − zn | hn |x − z|

2 XI (x), ν |x − z| 2 = XI (z) − XI (x), ν + XI (z), ν . |x − z| =

(5.116)

5.13 Appendix A. Maximum Principles

301

From (5.116) and (5.113), together with (5.114), we finally get L(t) ≤ L |x − z| = L δ(t). This completes the proof of (5.115). This inequality, combined with Lemma 5.13.13, gives δ(t) ≤ δ(t) − δ(0) ≤ L T sup δ, [0,T ]

so that sup[0,T ] δ ≤ 1/2 · sup[0,T ] δ. Hence δ ≡ 0, and the proof is complete. Corollary 5.13.15. The closed set F is X-invariant if and only if XI (y), ν = 0,

∀y ∈ F∗

∀ ν⊥F at y.

Proof. It straightforwardly follows from Proposition 5.13.14 and Definition 5.13.12. This corollary, together with Corollary 5.13.11, immediately gives the following result. m 2 Corollary 5.13.16. Let L = j =1 Xj be a sub-Laplacian on the homogeneous Carnot group G. Let u : Ω → R be a C 2 function satisfying the inequalities (5.106). Let F := {x ∈ Ω : u(x) = 0}. Assume ∅ = F = Ω. Then F is invariant with respect to X1 , . . . , Xm . Proposition 5.13.17. Let F be a relatively closed subset of the open set Ω ⊆ RN . Assume ∅ = F = Ω. Then a := {X ∈ T (RN ) : F is X-invariant} is a Lie algebra of vector fields. Proof. Let X, Y ∈ a, and let λ, μ be real constants. By Corollary 5.13.15, we have XI (y), ν = Y I (y), ν = 0 for every y ∈ F ∗ and for every ν⊥F at y. Then λ XI (y) + μ Y I (y), ν = 0 ∀ y ∈ F ∗ ∀ ν⊥F at y. Hence a is a linear space. The next lemma will complete the proof of the proposition. Lemma 5.13.18. In the notation of Proposition 5.13.17, if X, Y [X, Y ] ∈ a.

∈ a, then

Proof. Let y ∈ F ∗ and ν⊥F at y. For every t > 0 define √ √ √ √ Γ (t) := exp(− tY ) ◦ exp(− tX) ◦ exp( tY ) ◦ exp(− tX) (y).

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5 The Fundamental Solution for a Sub-Laplacian and Applications

Here ◦ denotes the composition of maps, whereas “exp” denotes the exponential of a vector field as introduced in Definition 1.1.2, page 8. Let T > 0 be such that Γ (t) ∈ Ω for 0 ≤ t ≤ T . By using the Taylor expansion (1.7) of exp (on page 7) with n = 2, a direct computation gives Γ (t) = y + t JY I (y) · XI (y) − JXI (y) · Y I (y) + o(t) = y + t[X, Y ]I (y) + o(t), as t ↓ 0. Then lim t↓0

Γ (t) − y = [X, Y ]I (y). t

(5.117)

On the other hand, since F is X and Y invariant, Γ (t) ∈ F for every t ∈ [0, T ]. As a consequence, since D(y + ν, |ν|) ⊆ (Ω \ F ) ∪ {y} and Γ (0) = y, |Γ (t) − (y + ν)|2 ≥ |ν|2 = |Γ (0) − (y + ν)|2 . Then, by using also (5.117), we have d 0≥ |Γ (t) − (y + ν)|2 = 2[X, Y ]I (y), ν . d t t=0 Hence

[X, Y ]I (y), ν ≤ 0 ∀ y ∈ F ∗ ∀ ν⊥F at y.

By swapping X with Y , we also get [Y, X]I (y), ν ≤ 0, so that [X, Y ]I (y), ν = 0 ∀ y ∈ F ∗ ∀ ν⊥F at y. Then, by Corollary 5.13.15, [X, Y ] ∈ a.

Finally, we are in the position to give the proof of Theorem 5.13.8. Proof (of Theorem 5.13.8). Let us define F := {x ∈ Ω : u(x) = 0 }. By hypothesis, x0 ∈ F . Then F is a non-empty relatively closed subset of Ω. We have to prove that F = Ω. By contradiction, assume F = Ω. Then, since Ω is connected, F ∗ = ∅. Let y ∈ F ∗ , and let ν⊥F at y. By Proposition 5.13.17, ZI (y), ν = 0 ∀ Z ∈ g. Since dim(g) = N, this obviously implies ν = 0. On the other hand, by the very definition of a vector orthogonal to F , we have ν ∈ RN \ {0}. This contradiction completes the proof.

5.13 Appendix A. Maximum Principles

303

5.13.1 A Decomposition Theorem for L-harmonic Functions In this section, we give a decomposition theorem for L-harmonic functions, resemblant to the decomposition of a holomorphic function on an annulus of C into the sum of the regular and singular parts from its Laurent expansion (for the classical case of the Laplace operator, see also S. Axler, P. Bourdon, W. Ramey [ABR92]). Our main tool is the maximum principle from the previous section (precisely, we use Corollary 5.13.7, page 296). In the sequel, we assume Q ≥ 3. Moreover, in the proof of Theorem 5.13.20, we adopt the following notation: G = (RN , ◦, δλ ) is a homogeneous Carnot group, L is a sub-Laplacian on G, Γ = d 2−Q is the fundamental solution for L (see Proposition 5.4.2). If d is the above L-gauge, A is any subset of G and λ > 0, we agree to set Aλ := {x ∈ G | d-dist(x, A) < λ}, where

d-dist(x, A) := inf{d(x −1 ◦ a) | a ∈ A}.

Moreover, we use the following simple lemma. Lemma 5.13.19. Let K be a compact subset of G, and let f be bounded on K. Then the function F : G → R,

Γ (y −1 ◦ x) f (y) dy

F (x) := K

is L-harmonic on G \ K and vanishes at infinity. Moreover, ifμ is a Radon measure on RN with compact support K, the same is true for G(x) := RN Γ (y −1 ◦x) dμ(y). Proof. It easily follows by differentiation under the integral sign (recall also that Γ is locally integrable and vanishes at infinity). We are now ready to state and prove the following assertion. Theorem 5.13.20 (The decomposition theorem). Let the hypotheses in the incipit of this section hold. Let K ⊂ Ω ⊆ G, with K compact and Ω open. If u is Lharmonic in Ω \ K, then u has a decomposition of the form u = r + s, where r is L-harmonic in Ω and s is L-harmonic in G \ K. Furthermore, it can be assumed that s vanishes at infinity, and in this case the above decomposition is unique. Proof. Suppose the theorem holds true whenever Ω is bounded. We show that it holds true even for an unbounded Ω. Indeed, let u ∈ H(Ω \ K), where K is compact and Ω is an (unbounded) open set. Let R > 0 be such that K ⊂ Bd (0, R). Set := Ω ∩ Bd (0, R). Then K ⊂ Ω and, by our assumption, u can be decomposed as Ω s ∈ H(G \ K) and s → 0 at infinity. We consider the u = r + s, where r ∈ H(Ω), function r := u−s on Ω. Then r is L-harmonic in Ω \K and extends L-harmonically

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5 The Fundamental Solution for a Sub-Laplacian and Applications

(which is an open neighborhood of K). This across K, since r coincides with r on Ω ends the proof, since u = r + s, with r ∈ H(Ω), s ∈ H(G \ K) and s → 0 at infinity. We can then suppose that Ω is bounded. We fix the following notation (see also Fig. 5.1): B(λ) := closure of (∂Ω)λ , C(λ) := closure of Kλ , A(λ) := Ω \ (B(λ) ∪ C(λ)).

Fig. 5.1. Figure for the proof of Theorem 5.13.20

Since K is compact and ∂Ω is closed, we can choose λ > 0 small enough so that B(λ) ∩ C(λ) = ∅. Since Ω is bounded, we can choose a cut-off function ψλ ∈ C0∞ (RN ) such that supp(ψλ ) ⊂ Ω \ K,

ψλ ≡ 1 on A(λ).

We consider the function u ψλ , and we agree to consider this function trivially prolonged on RN to be zero. Hence this trivial prolongation belongs to C0∞ (RN ). By (5.16) in Theorem 5.3.3 (page 237) (being ψλ ≡ 1 on A(λ)), Γ (x −1 ◦ y) L(u ψλ )(y) dy u(x) = (u ψλ )(x) = − RN

+ + + =− A(λ) B(λ) C(λ) RN \(A(λ)∪B(λ)∪C(λ))

Γ (y −1 ◦ x) L(u ψλ )(y) dy ∀ x ∈ A(λ). (5.118) + =− B(λ)

C(λ)

In the last equality we used the fact that ψλ ≡ 1 on A(λ) jointly with Lu = 0 on Ω, and the fact that ψλ is supported in Ω. We now set Γ (y −1 ◦ x) L(u ψλ )(y) dy for x ∈ Ω \ B(λ), rλ (x) := − B(λ) sλ (x) := − Γ (y −1 ◦ x) L(u ψλ )(y) dy for x ∈ G \ C(λ). C(λ)

5.13 Appendix A. Maximum Principles

305

Hence, (5.118) gives the decomposition u(x) = rλ (x) + sλ (x)

∀ x ∈ A(λ).

(5.119)

From Lemma 5.13.19, we infer that rλ and sλ are L-harmonic on the respective sets of definition and that sλ vanishes at infinity. Let now 0 < μ < λ. Then obviously A(λ) ⊂ A(μ). From the decomposition in (5.119), we get an analogous decomposition u(x) = rμ (x) + sμ (x)

∀ x ∈ A(μ).

(5.120)

We claim that the decompositions (5.119) and (5.120) are compatible, i.e. rλ (x) = rμ (x) sλ (x) = sμ (x)

∀ x ∈ Ω \ B(λ), ∀ x ∈ G \ C(λ).

(5.121)

To prove the claim, we first remark that from (5.119) and (5.120) we obtain rλ (x) − rμ (x) = sμ (x) − sλ (x) Now, let us consider the following function sμ (x) − sλ (x) S : G → R, S(x) := rλ (x) − rμ (x)

∀ x ∈ A(λ).

(5.122)

for every x ∈ G \ C(λ), for every x ∈ Ω \ B(λ).

We claim that S has the following properties: i) S is well-posed: indeed, thanks to (5.122), sμ − sλ coincides with rλ − rμ on the set {G \ C(λ)} ∩ {Ω \ B(λ)} = A(λ); ii) S vanishes at infinity: indeed, this is true for sμ and sλ ; iii) S is L-harmonic on G: indeed, this is true for sμ − sλ on the open set G \ C(λ), and this is true for rλ − rμ on the open set Ω \ B(λ) (recall that C(μ) ⊂ C(λ) and B(μ) ⊂ B(λ)). Now, by the maximum principle (precisely, see Corollary 5.13.7, page 296) we infer that S ≡ 0, which is equivalent to the claimed (5.121). Now, let us fix a decreasing sequence of positive λn ’s such that λn → 0 as n → ∞. We set r(x) := rλn (x) s(x) := sλn (x)

∀ x ∈ Ω (where n ∈ N is such that x ∈ Ω \ B(λn )), ∀ x ∈ G \ K (where n ∈ N is such that x ∈ G \ C(λn )).

Thanks to (5.121), the definition of r(x) and s(x) are unambiguous.11 Now, the decomposition u(x) = r(x) + s(x) ∀x ∈Ω \K 11 We are also using the trivial fact that Ω \B(λ ) ↑ Ω and RN \C(λ ) ↓ RN \K as n → ∞. n n

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5 The Fundamental Solution for a Sub-Laplacian and Applications

follows from the analogous decomposition (5.119) (and the fact that A(λ) ↑ Ω \ K as n → ∞). Moreover, it is easily seen that r ∈ H(Ω), s ∈ H(G \ K) and s vanishes at infinity. This gives the desired decomposition of u as in the assertion. Finally, the uniqueness of the decomposition in the assertion (under the assumption that s vanishes at infinity) is another easy consequence of the same maximum principle quoted above. Indeed, suppose we are given two decompositions r2 + s2 = u = r1 + s1

on Ω \ K,

where ri ∈ H(Ω),

si ∈ H(G \ K),

lim si (x) = 0,

x→∞

i = 1, 2.

Then setting S : G → R,

S(x) :=

s1 (x) − s2 (x) r2 (x) − r1 (x)

for every x ∈ G \ K, for every x ∈ Ω,

we see that S ∈ H(G), S vanishes at infinity, and we end the proof following the same arguments as in the previous paragraph.

5.14 Appendix B. The Improved Pseudo-triangle Inequality Let G = (RN , δλ , ◦) be a homogeneous Carnot group, and let d be a symmetric homogeneous norm on G, smooth out of the origin.12 For example, d could be an L-gauge on G for some sub-Laplacian L on G. We know from Proposition 5.1.7 (page 231) that (even without assumptions on smoothness or symmetry of d) d satisfies the pseudo-triangle inequality d(a ◦ b) ≤ c (d(a) + d(b))

for every a, b ∈ G.

Here c≥1 is a constant depending on d and G. The aim of this brief appendix is to prove the following improvement of the pseudo-triangle inequality. For the following result, see also [DFGL05, (2.6)]. Proposition 5.14.1 (The improved pseudo-triangle inequality). Let d be a symmetric homogeneous norm on the homogeneous Carnot group G. Furthermore, suppose d is smooth out of the origin. Then there exists a constant β ≥ 1 depending only on d and G such that d(y ◦ x) ≤ β d(y) + d(x)

for every x, y ∈ G.

(5.123)

12 In other words (see the definition at the beginning of Section 5.1, page 229), the present d

has the following properties: d ∈ C(G, [0, ∞)) ∩ C ∞ (G \ {0}); d(δλ (x)) = λ d(x) for every λ > 0 and x ∈ G; d(x) = 0 iff x = 0; d(x −1 ) = d(x) for every x ∈ G.

5.14 Appendix B. The Improved Pseudo-triangle Inequality

307

Proof. Since (5.123) holds when y = 0, we can suppose y = 0. Since d is δλ homogeneous of degree one, (5.123) is equivalent to d(δ1/d(y) (y) ◦ δ1/d(y) (x)) ≤ β + d(δ1/d(y) (x)).

(5.124a)

By using the symmetry of d and setting ξ −1 = δ1/d(y) (y),

η−1 = δ1/d(y) (x),

(5.124a) is equivalent to (note that d(δ1/d(y) (y)) = 1) d(η ◦ ξ ) − d(η) ≤ β

for every ξ, η ∈ G: d(ξ ) = 1.

(5.124b)

By the usual13 pseudo-triangle inequality for d, (5.124b) holds when η ∈ Bd (0, M), i.e. d(η) ≤ M (where M = M(d, G) % 1 will be chosen in the sequel). Indeed, if η ∈ Bd (0, M), we have d(η ◦ ξ ) − d(η) ≤ c (d(η) + d(ξ )) − d(η) = (c − 1)d(η) + c≤(c − 1)M + c =: β. We can hence suppose η ∈ / Bd (0, M). Roughly speaking, we will show that we can drop η from (5.124b), by an argument of left-translation along curves which are supported away from zero, when M is large enough. Then, (5.124b) will follow from the classical mean value theorem. We now make this precise. Set Z := Log (ξ ) ∈ g, where Log is the logarithmic map related to G and g is the Lie algebra of G. Consider the integral curve γ of Z starting from η, i.e. with our usual notation γ (t) = exp(tZ)(η) = η ◦ exp(tZ)(0) = η ◦ Exp (tZ). Here we used Corollary 1.2.24 (page 24) and the definition of exponential map (see page 24). Obviously, we have γ (0) = η and γ (1) = η ◦ Exp (Z) = η ◦ Exp (Log (ξ )) = η ◦ ξ . If we show that we can choose M % 1 such that d(γ (t)) ≥ 1

for every t ∈ [0, 1]

(5.124c)

(recall that γ depends on η, besides ξ ), then [0, 1] t → u(t) := d(γ (t)) is smooth (for d is smooth out of the origin by hypothesis) so that the classical Lagrange mean value theorem applies and gives ˙ = sup (∇d)(γ (t)), γ˙ (t) d(η ◦ ξ ) − d(η) = u(1) − u(0) ≤ sup |u(t)| t∈[0,1] t∈[0,1] = sup (∇d)(γ (t)), (ZI )(γ (t)) t∈[0,1] = sup (Zd)(γ (t)). (5.124d) t∈[0,1]

Let X1 , . . . , XN be the Jacobian basis for the Lie algebra g of G. Hence, 13 See Proposition 5.1.7-1, page 231.

308

5 The Fundamental Solution for a Sub-Laplacian and Applications N

Z = Log (ξ ) =

(5.124e)

pj (ξ ) Zj ,

j =1

where the pj ’s are polynomials, so that there exists a constant C such that sup |pj (ξ )| ≤ C1

for all j = 1, . . . , N ,

(5.124f)

d(ξ )=1

since {ξ ∈ G : d(ξ ) = 1} is a compact set (see, e.g. Proposition 5.1.4, page 230). Moreover, Zj is δλ -homogeneous of degree σj ≥ 1 (see Corollary 1.3.19, page 42), so that Zj d is δλ -homogeneous of degree 1 − σj ≤ 0. Consequently, it is bounded on G \ Bd (0, 1), say sup |(Zj d)(ζ )| ≤ C2

for all j = 1, . . . , N.

(5.124g)

d(ζ )≥1

We now use again the claimed (5.124c) and, by collecting together (5.124f) to (5.124g), we derive that (5.124d) yields d(η ◦ ξ ) − d(η) ≤ sup (Zd)(ζ ) d(ζ )≥1

N = sup pj (ξ ) (Zj d)(ζ ) ≤ N C1 C2 . d(ζ )≥1 j =1

This proves (5.124b). We are then left to prove (5.124c). From the pseudo-triangle inequality for d (see Proposition 5.1.7-2, page 231) we have d(γ (t)) = d(η ◦ Exp (tZ)) ≥ ≥

1 d(η) − d(Exp (tZ)) c

1 1 M− sup d(Exp (t Log (ξ ))) =: M − m(d, G), c c t∈[0,1], d(ξ )=1

whence (5.124c) follows by choosing M = c(1 + m(d, G)). The finiteness of m(d, G) follows from N d(Exp (t Log (ξ ))) = d Exp tpj (ξ ) Zj j =1

≤ sup d Exp |ζj |≤C1

N

ζj Zj

< ∞,

j =1

uniformly for t ∈ [0, 1], d(ξ ) = 1. Here we used (5.124f) and the fact that N q(ζ ) := Exp ζj Zj j =1

has polynomial coefficient functions (see (1.75a), page 50). This completes the proof.

5.15 Appendix C. Existence of Geodesics

309

Note that, β being the constant in Proposition 5.14.1, if we replace x, y in (5.123) by, respectively, y −1 ◦ z and z−1 ◦ x, we get d(y −1 ◦ x) ≤ β d(y −1 ◦ z) + d(z−1 ◦ x)

for every x, y, z ∈ G.

(5.125a)

Moreover, by interchanging y and z in the above inequality (and using the symmetry of d) one gets |d(y −1 ◦ x) − d(z−1 ◦ x)| ≤ β d(y −1 ◦ z)

for every x, y, z ∈ G.

(5.125b)

5.15 Appendix C. Existence of Geodesics Let G = (RN , δλ , ◦) be a homogeneous Carnot group. Let g = W (1) ⊕ W (2) ⊕ · · · ⊕ W (r) be a stratification of the Lie algebra of G, as in Remark 1.4.8 (page 59). Let X = {X1 , . . . , Xm } be any basis of W (1) . We consider the related Carnot–Carathéodory distance dX . The aim of this section is to prove that the “inf” defining dX in (5.6) on page 232 is actually a minimum. In other words, fixed any x, y ∈ G, we show the existence of a X-subunit curve γ : [0, T ] → G connecting x and y (i.e. γ (0) = x, γ (T ) = y) such that T = dX (x, y). We shall call any such curve a X-geodesic (for x and y). The existence of geodesics can be proved in many general cases (see [Bus55] and [HK00]). Our argument here will make crucial use of the δλ -homogeneity and left-invariant properties of the system X. The resulting proof will be quite simple (for a more general proof, see the note after Theorem 5.15.5). First, we recall that, by Propositions 5.2.4, 5.2.6 and Theorem 5.2.8, dX (x, y) = d0 (y −1 ◦ x)

for every x, y ∈ G,

(5.126)

where d0 (z) := dX (z, 0),

z ∈ G,

(5.127)

and d0 is a homogeneous (symmetric) norm on G. As usual, we denote the dilation of G by δλ (x) = δλ (x1 , . . . , xN ) = (λσ1 x1 , . . . , λσN xN ),

λ > 0, x ∈ G,

where 1 = σ1 ≤ · · · ≤ σN = r are consecutive integers and r is the step of nilpotency of G. We are ready to prove the following result. Proposition 5.15.1. Let G be a homogeneous Carnot group, and let d be any homogeneous norm on G. For every compact set K ⊂ G, there exists a constant cK > 0 such that (cK )−1 |x − y| ≤ d(y −1 ◦ x) ≤ cK |x − y|1/r

∀ x, y ∈ K,

where r is the step of G and | · | is the Euclidean norm on G ≡ RN .

(5.128)

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5 The Fundamental Solution for a Sub-Laplacian and Applications

In particular, (5.128) holds true if d(y −1 ◦ x) is replaced by dX (x, y), where dX is the control distance related to any basis of generators (of the first layer of the stratification of the Lie algebra) of G. (Note. More generally, the estimates in (5.128) also hold when d = dX (with a suitable r), where X is a system of smooth vector fields satisfying the Hörmander condition, see [Lan83,NSW85,VSC92,Gro96]. The first equality in (5.128) holds for a general Carnot–Carathéodory distance d = dX , see [HK00] and Ex. 25 at the end of the chapter.) Proof. Once (5.128) has been proved, the last assertion of the proposition follows from (5.126), by taking d = d0 (d0 as in (5.127)). We then turn to prove (5.128). If | · | denotes the absolute value on R, set : G → [0, ∞),

(x) =

N

|xj |1/σj .

(5.129)

j =1

Obviously, is a homogeneous (symmetric) norm on G. By the equivalence of all homogeneous norms on G (see Proposition 5.1.4), there exists a constant c = c(, d, G) ≥ 1 such that c−1 (x) ≤ d(x) ≤ c (x)

∀ x ∈ G.

(5.130)

Thus (5.128) will follow if we show that, given a compact set K ⊂ G, (cK )−1 |x − y| ≤ (y −1 ◦ x) ≤ cK |x − y|1/r

∀ x, y ∈ K,

(5.131)

for a suitable constant cK > 0. Now, we recall the result in Corollary 1.3.18 (page 41): for every j ∈ {1, . . . , N } and every x, y ∈ G, we have (j ) (y −1 ◦ x)j = xj − yj + Pk (x, y) (xk − yk ), k: σk <σj (j )

where Pk (x, y) is a polynomial function. Thus, the very definition of gives 1/σj N (j ) −1 (y ◦ x) = Pk (x, y) (xk − yk ) . (5.132) xj − yj + j =1

k: σk <σj

(j )

Since any Pk is a continuous function, we immediately get from (5.132) the estimate from above: for every x, y ∈ K

1/σj N −1 |xj − yj | + c |xk − yk | (y ◦ x) ≤ j =1 N

≤c

j =1

k: σk <σj

k: σk ≤σj

1/σj N |xk − yk | ≤ c

j =1 k: σk ≤σj

|xk − yk |1/σj

5.15 Appendix C. Existence of Geodesics

≤ c

N

|xj − yj |1/σj ≤ c

j =1

≤ c N

N

1/σN

N

311

|xj − yj |1/σN

j =1

≤ c N 1+1/σN |x − y|1/σN .

|xj − yj |

j =1

This gives the estimate from above in (5.131), since σN = r. (Yet another proof of the estimate from above can be obtained by the Lagrange mean value theorem.14 ) The estimate from below is just a little more delicate. We fix the notation: for every j = 2, . . . , N, let nj be the cardinality of the set {k : σk < σj }. Let α be a vector in Rn with n = n2 + · · · + nN , and let us denote α in the following way (j ) (j ) α = α (2) , . . . , α (N ) with α (j ) = α1 , . . . , αnj , j = 2, . . . , N. With the notation in (5.132), if K is a compact subset of G, there exists a constant (j ) M ≥ 1 such that (recall that the Pk ’s are polynomial functions) (j )

sup |Pk (x, y)| ≤ M

∀ j = 2, . . . , N ∀ k : σk < σj .

x,y∈K

As a consequence, (5.132) gives (here M = n M) (y

−1

1/σj N (j ) ◦ x) = Pk (x, y) (xk − yk ) x j − y j + j =1

k: σk <σj

14 Indeed, fix any y ∈ G and set

fy : RN → R,

fy (x) := (y −1 ◦ x)j .

Given any x, x0 ∈ G, by the Lagrange mean value theorem, we have |fy (x) − fy (x0 )| ≤

()

sup

|ξ −x0 |≤|x−x0 |

|∇fy (ξ )| |x − x0 |.

Now, take x0 = y and observe that fy (x0 ) = fy (y) = 0. Moreover, if x, y belong to a compact set K, and if RK % 1 is such that K ⊆ B(0, RK ), then sup

|ξ −y|≤|x−y|

|∇fy (ξ )| ≤

sup |ξ |≤RK +diam(K)

|∇fy (ξ )| =: Mj < ∞,

for fy (ξ ) is a polynomial function in ξ and y. Thus, () gives (set M = maxj ≤N Mj ) |(y −1 ◦ x)j | ≤ M |x − y|

∀ x, y ∈ K ∀ j ≤ N.

Then, for every x, y ∈ K, we have (y −1 ◦ x) =

N j =1

N |(y −1 ◦ x)j |1/σj ≤ c |x − y|1/σj ≤ c |x − y|1/r . j =1

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5 The Fundamental Solution for a Sub-Laplacian and Applications

≥ inf

|α|≤M

1/σj N (j ) x j − y j + αk (xk − yk ) =: (). j =1

k: σk <σj

Being x, y ∈ K, we obviously have (whenever α ≤ M) (j ) x j − y j + α (x − y ) k k ≤ c(K, M) < ∞, k k: σk <σj

hence (notice that 1/σj ≤ 1) N (j ) c(K, M)1/r xj − yj + αk (xk − yk ) () ≥ inf c(K, M) |α|≤M k: σk <σj j =1

= c (K) inf fα (x − y) =: ( ), |α|≤M

where fα : R → R, N

N (j ) fα (z) = αk zk . zj + j =1

k: σk <σj

We explicitly remark that fα is homogeneous of degree 1 with respect to the Euclidean dilations on RN . As a consequence (if x = y, otherwise there is nothing to prove),

x−y ≥ |x − y| inf fα (ξ ). fα (x − y) = |x − y| fα |ξ |=1 |x − y| This gives ( ) ≥ |x − y| c (K)

inf

|α|≤M, |ξ |=1

fα (ξ ) ≥ |x − y| c (K).

Indeed, since fα (ξ ) is a continuous function in ξ and α, and (for every α) fα (·) is positive outside the origin (as a simple inductive argument shows), there exist α0 and ξ0 (with |α0 | ≤ M and |ξ0 | = 1) such that inf

|α|≤M, |ξ |=1

fα (ξ ) = fα0 (ξ0 ) > 0.

This gives the lower estimate in (5.131) and completes the proof. From Proposition 5.15.1 we obtain useful corollaries, as we show below. In the sequel, we write dE to denote the usual Euclidean metric on G ≡ RN . Corollary 5.15.2. Let G be a homogeneous Carnot group. Let dX be the control distance related to any basis X of generators (of the first layer of the stratification of the Lie algebra) of G.

5.15 Appendix C. Existence of Geodesics

313

Then A ⊆ G is bounded in the metric space (G, dX ) if and only if it is bounded in the Euclidean metric space (G, dE ). More precisely, there exists a constant c≥1 such that (for every R > 0) BdX (0, R) ⊆ BdE (0, R1 (R))

and BdE (0, R) ⊆ BdX (0, R2 (R)),

(5.133)

where R1 (R) = cmax{R, R r },

R2 (R) = cmax{R, R 1/r }.

(Note. The fact that a bounded set in (RN , dE ) is also bounded in (RN , dX ) holds for a dX related to a general system of vector fields satisfying the Hörmander condition, see the note after Proposition 5.15.1. The reverse assertion may be false even in the Hörmander case, see Ex. 27.) Proof. By (5.130), we know that there exists a constant c = c(, dX , G) ≥ 1 such that (5.134) c−1 (x) ≤ dX (x, 0) ≤ c (x) ∀ x ∈ G, where is as in (5.129). Now, it is obvious that a set A ⊆ G is bounded in (G, dE ) iff there exists R > 0 such that A ⊆ B (0, R). This remark jointly with (5.134) proves the first assertion of the corollary. Precisely (see the definition of ), it holds (for every R > 0) BdE (0, R) ⊆ B 0, N max{R, R 1/r } , B (0, R) ⊆ BdE 0, N max{R, R r } . These inclusions, together with (5.134), prove (5.133). Obviously, (5.133) also gives the inclusions BdX (0, R3 (R)) ⊆ BdE (0, R) where

and BdE (0, R4 (R)) ⊆ BdX (0, R),

R3 (R) = c−1 min{R, R 1/r },

(5.135)

R1 (R) = c−r min{R, R r }.

Corollary 5.15.3. Let the hypothesis of Corollary 5.15.2 hold. Then the map id : (G, dX ) → (G, dE ), id(x) = x is a homeomorphism. As a consequence, the topologies of the metric spaces (G, dX ), (G, dE ) coincide. (Note. The continuity of id : (G, dX ) → (G, dE ) holds for a general Carnot– Carathéodory distance dX , see Ex. 25 at the end of the chapter. For a general dX , the reverse continuity may be false, see Ex. 26 (see also [BR96]). Instead, this reverse continuity holds for all smooth vector fields satisfying the Hörmander condition, see the note after Proposition 5.15.1 and Ex. 25.)

314

5 The Fundamental Solution for a Sub-Laplacian and Applications

Proof. Let {xn }n∈N be a sequence in G. Suppose xn → x in (G, dE ). Then there exists a compact set K ⊂ RN such that x, xn ∈ K for every n ∈ N. Hence, by the second inequality in (5.128), we derive dX (x, xn ) ≤ cK |x − xn | ∀ n ∈ N, so that, letting n → ∞, xn → x in (G, dX ) too. Vice versa, suppose xn → x in (G, dX ). Then the sequence {xn }n is bounded in (G, dX ). Hence, by Corollary 5.15.2, it is also bounded in (G, dE ). Consequently, there exists a compact subset K of RN such that x, xn ∈ K for every n ∈ N. Hence, by the first inequality in (5.128), we derive |x − xn | ≤ cK dX (x, xn ) ∀ n ∈ N, so that, letting n → ∞, xn → x in (G, dE ) too. We have thus proved that dX

dE

xn −→ x ⇒ id(xn ) −→ x,

dE

dX

xn −→ x ⇒ id−1 (xn ) −→ x.

As a consequence, by the well-known characterization of continuity (in a metric space) as sequential-continuity, we infer that id : (G, dX ) → (G, dE ) is a homeomorphism. This implies that id and id−1 are open maps, i.e. (G, dX ) and (G, dE ) have the same open sets. By collecting together the above lemmas, we obtain the following result. Proposition 5.15.4. Let G be a homogeneous Carnot group. Let dX be the control distance related to any basis X of generators (of the first layer of the stratification of the Lie algebra) of G. Then, a subset of G is, respectively, open, closed, bounded or compact in the metric space (G, dX ) if and only if the same holds in the Euclidean metric space G ≡ RN . In particular, the compact subsets of (G, dX ) are precisely the closed and bounded subsets of RN or, equivalently, the closed and bounded subsets of (G, dX ). Proof. Let A ⊆ G. Then, by Corollary 5.15.3, A is, respectively, open, closed or compact in the metric space (G, dX ) if and only if the same holds in (G, dE ). By Corollary 5.15.2, A is bounded in (G, dX ) if and only if the same holds in (G, dE ). This also gives the last assertion of the proposition and completes the proof. We are in a position to prove the following result. Theorem 5.15.5 (Existence of X-geodesics). Let G be a homogeneous Carnot group. Let dX be the control distance related to any basis X of generators (of the first layer of the stratification of the Lie algebra) of G. Then, for every x, y ∈ G, there exists a X-geodesic connecting x and y, i.e. there exists a X-subunit path γ : [0, T ] → G such that γ (0) = x, γ (T ) = y and T = dX (x, y).

5.15 Appendix C. Existence of Geodesics

315

(Note. Since we proved that the compact subsets of (G, dX ) are precisely the closed and bounded subsets of (G, dX ) (see Proposition 5.15.4) this result also follows from the general results in [Bus55, page 25].) Proof. Fix x, y ∈ G. By definition of dX , there exists a sequence {γn }n∈N of Xsubunit paths γn : [0, Tn ] → G,

γn (0) = x, γn (Tn ) = y,

Tn ' dX (x, y).

By definition of X-subunit path, γn is an absolutely continuous curve such that, for every n ∈ N, there exists En ⊆ [0, Tn ] such that Fn := [0, Tn ] \ En has vanishing Lebesgue measure and γ˙n (t), ξ 2 ≤

m Xj I (γn (t)), ξ 2

∀ ξ ∈ RN ∀ t ∈ En .

(5.136)

j =1

Since Tn ≤ Tn+1 for all n, it is not restrictive to suppose that every γn is defined on [0, T1 ], by prolonging γn to be y on (Tn , T1 ]. We still denote this prolongation by γn . Observe that the prolongation is still a X-subunit path (connecting x to y) because, for t ∈ (Tn , T1 ], γn is constant, whence the far left-hand side in (5.136) is 0. The resulting prolongation is also trivially absolutely continuous. We claim that the family of functions F = {γn }n on [0, T1 ] is uniformly bounded and equicontinuous. • F is equicontinuous: Suppose we have proved that F is uniformly bounded, i.e. (5.137) ∃M >0 : sup |γn (t)| ≤ M ∀ n ∈ N. t∈[0,T1 ]

Then the equicontinuity of F follows. Indeed, for every t, t ∈ [0, T1 ], it holds t t |γn (t) − γn (t )| = γ˙n (s) ds ≤ |γ˙n (s)| ds t t t γ˙n (s), ξ ds (see the note15 ) = sup t ξ ∈RN : |ξ |=1

(by (5.136)) ≤

sup

t ξ ∈RN : |ξ |=1

(Cauchy–Schwartz) ≤

t

t

sup

t ξ ∈RN : |ξ |=1

m Xj I (γn (s)), ξ 2

ds

j =1 m

1/2 |Xj I (γn (s))| |ξ |

j =1

15 Here, we use a well-known fact: if v ∈ RN , then

|v| =

1/2

sup ξ ∈RN : |ξ |=1

|v, ξ |.

2

2

ds

316

5 The Fundamental Solution for a Sub-Laplacian and Applications

= (by (5.137))

t

t

(X1 I, . . . , Xm I )(γn (s)) ds

≤ |t − t | sup (X1 I, . . . , Xm I )(η) ≤ M |t − t |. |η|≤M

In the last inequality, we used the smoothness of the Xj ’s. Note that this proves more than the equicontinuity, namely, F is a uniform Lipschitz-continuous family. • F is uniformly bounded: We are then left to prove (5.137). By the very definition of dX , for every t ∈ [0, T1 ], we have dX (x, γn (t)) ≤ t ≤ T1 . As a consequence, by the triangle-inequality for the distance dX , we get dX (0, γn (t)) ≤ dX (0, x) + dX (x, γn (t)) ≤ dX (0, x) + T1 < ∞. Hence, the set {γn (t) : n ∈ N, t ∈ [0, T1 ]} is bounded in (G, dX ), so that, due to Corollary 5.15.2, the set is bounded in the Euclidean metric. This is precisely (5.137). We are then entitled to apply the Arzelà–Ascoli theorem, ensuring that there exists a subsequence of {γn }n∈N which converges uniformly on [0, T1 ], say to γ : [0, T1 ] → G. For the sake of brevity, we still denote this uniformly convergent subsequence by {γn }n∈N . Set T := dX (x, y). We aim to prove that the curve γ ∗ : [0, T ] → G,

γ ∗ := γ |[0,T ]

is X-subunit and γ connects x to y, which will end the proof since γ is defined on [0, T ]. First, γn (0) = x for every n ∈ N yields γ (0) = x, by pointwise convergence. Second, since γn converges uniformly to γ on [0, T1 ], and {Tn }n is a sequence in [0, T1 ] converging to T , we infer γn (Tn ) → γ (T ), so that

γ ∗ (T ) = γ (T ) = lim γn (Tn ) = y, n→∞

because γn (Tn ) = y for every n ∈ N. This proves that γ ∗ connects x to y. Finally, we prove that γ is X-subunit. To begin with, we remark that γ is absolutely continuous. Indeed, in proving the equicontinuity of F, we showed that there exists a constant M such that |γn (t) − γn (t )| ≤ M |t − t |

for every n ∈ N and every t, t ∈ [0, T1 ].

Letting n → ∞, this shows that γ is Lipschitz continuous, whence it is absolutely continuous. Obviously, γ is X-subunit iff, for every fixed ξ ∈ RN , the functions

5.15 Appendix C. Existence of Geodesics

m f (t) := Xj I (γ (t)), ξ 2

317

1/2

±

± γ˙ (t), ξ

j =1

are non-negative a.e. on [0, T1 ]. To this end, it suffices to prove that (notice that f± ∈ L1 (0, T1 )) T1

0

f ± (t) ϕ(t) dt ≥ 0 ∀ ϕ ∈ C0∞ (0, T1 ), ϕ ≥ 0.

(5.138)

First, being γn X-subunit, it holds

m

T1

0≤ 0

Xj I (γn (t)), ξ 2

± γ˙n (t), ξ ϕ(t) dt

j =1

m 1/2 2 Xj I (γn ), ξ ϕ±

T1

=

1/2

0

T1

γ˙n , ξ ϕ =: An ± Bn

0

j =1

for every n ∈ N and every ϕ ∈ C0∞ (0, T1 ), ϕ ≥ 0 (exploit (5.136)), whence 0 ≤ An ± Bn

for every n ∈ N.

(5.139)

Now, by dominated convergence (indeed, recall that Xj is smooth and {γn }n∈N is uniformly bounded), the limit of An is easily obtained: An = 0

T1

m

1/2 Xj I (γn ), ξ

2

n→∞

T1

ϕ −→

0

j =1

m

1/2 Xj I (γ ), ξ

2

ϕ. (5.140)

j =1

Furthermore, the limit of Bn can be obtained by recalling that absolutely continuous functions support integration by parts: Bn =

T1

γ˙n , ξ ϕ =

0 n→∞

−→ −

N i=1

N i=1

T1

ξi 0

γi ϕ˙ =

T1

ξi

(γ˙n )i ϕ = −

0 N i=1

N i=1

T1

ξi

T1

γ˙i ϕ =

0

ξi

T1

(γn )i ϕ˙

0

γ˙ , ξ ϕ.

(5.141)

0

(Passing to the limit, we used another dominated convergence argument, recalling that γn → γ uniformly.) As a consequence, letting n → ∞ in (5.139), from (5.140) and (5.141) we infer 1/2 T1 T1 m 2 Xj I (γ ), ξ ϕ± γ˙ , ξ ϕ, 0≤ 0

j =1

0

which is exactly (5.138) (by the very definition of f ± ). This completes the proof.

318

5 The Fundamental Solution for a Sub-Laplacian and Applications

As a consequence of Theorem 5.15.5, we have the following segment property for the Carnot–Carathéodory distance on a homogeneous Carnot group. (See also B. Franchi and E. Lanconelli [FL83] for the segment property in a sub-elliptic context.) Corollary 5.15.6 (The segment property). Let G be a homogeneous Carnot group. Let dX be the control distance related to any basis X of generators (of the first layer of the stratification of the Lie algebra) of G. Then the metric space (G, dX ) has the segment property, i.e. for every x, y ∈ G, there exists a continuous curve γ : [0, T ] → G with γ (0) = x, γ (T ) = y, and such that dX (x, y) = dX (x, γ (t)) + dX (γ (t), y) for every t ∈ [0, T ]. For instance, γ can be any X-geodesic connecting x and y, whose existence is granted by Theorem 5.15.5. Proof. Let x, y ∈ G be fixed. Let γ be a X-geodesic connecting x and y: the existence of such a X-geodesic is granted by Theorem 5.15.5. We claim that dX (x, y) = dX (x, γ (t)) + dX (γ (t), y)

for every t ∈ [0, T ].

Fix t ∈ [0, T ]. First, by the very definition of dX (being γ a X-subunit curve), we have () dX (x, γ (t)) ≤ t. (Actually, the equality will hold.) We next prove that ()

dX (γ (t), y) ≤ T − t.

(Actually, the equality will hold.) Indeed, consider the curve γ : [0, T − t] → G,

γ (s) := γ (s + t).

Obviously, γ is X-subunit and γ (0) = γ (t), γ (T − t) = γ (t) = y. Hence, again by the definition of dX , one has dX (γ (t), y) = dX ( γ (0), γ (T − t)) ≤ T − t, i.e. () holds. Consequently, by () and () together with the triangle-inequality, we get T = dX (x, y) ≤ dX (x, γ (t)) + dX (γ (t), y) ≤ t + (T − t) = T . Hence, these inequalities are in fact equalities, and the proof is complete. (Incidentally, this also proves that the equality holds in () and ().) Bibliographical Notes. Some of the topics presented in this chapter also appear in [BL01].

5.16 Exercises of Chapter 5

319

For other equivalent definitions of the Carnot–Carathéodory distance, see, e.g. [JSC87,NSW85]. See [HK00] and the references therein for applications of Carnot– Carathéodory distances in PDE’s problems. In particular, see the collection of papers [BR96] for an introduction to the geometry of C-C spaces. For explicit estimates of the Carnot–Carathéodory distance for “diagonal” vector fields, see [FL82]; for the segment property, see [FL83]. See also [GN98]. The gauge functions on the Heisenberg group and on H-type groups were discovered by G.B. Folland [Fol75] and A. Kaplan [Kap80], respectively. For mean value formulas for the Hörmander sum of squares, see [Lan90,CGL93]; see [Gav77] for mean value formulas for the Heisenberg group; for a survey on mean value formulas in the classical setting of Laplace’s operator, see [NV94] (see also the list of references therein for related results in a non-Riemannian setting). The bibliography on Harnack-type inequalities for sub-elliptic operators is extremely vast: see, e.g. [FL83] for the first result on these topics. See also [BM95, Bon69,FGW94,FL82,GL03,LM97a,LK00,SaC90,SCS91,Varo87]. For Liouville-type theorems for homogeneous operators, see [Gel83,KoSt85, LM97a,Luo97]. For results on the maximum principles and propagation, see [Ama79, Bon69,Hil70,Hop52,PW67,Red71,Spe81,Tai88].

5.16 Exercises of Chapter 5 Ex. 1) Give a detailed proof of Lemma 5.13.19, page 303. Ex. 2) The following operator L :=

n (∂xj )2 + (∂yj + xj ∂t )2 + (∂z )2 j =1

in R2n+2 (the points are denoted by (x, y, z, t) with x = (x1 , . . . , xn ) ∈ Rn , y = (y1 , . . . , yn ) ∈ Rn , z ∈ R, t ∈ R) is not analytic-hypoelliptic (see [Rot84]). Find an explicit homogeneous Carnot group G such that L is the canonical sub-Laplacian of G. Ex. 3) Construct explicitly the Carnot group referred to in Example 5.10.6, page 286. (Hint: Model the needed composition law on the composition law in Example 5.10.5, page 285.) Ex. 4) Consider the Kolmogorov-type group in Ex. 7 of Chapter 3, page 179. By means of formula (5.92), verify that the fundamental solution of its canonical sub-Laplacian is given by (when x = 0)

320

5 The Fundamental Solution for a Sub-Laplacian and Applications

Γ (x, t) =c

R2

×

dτ1 dτ2

|τ | sinh |τ |

|τ | 2 2 2 {(τ2 x2 − τ1 x3 )2 + 2 tanh |τ | (|τ | x1 + (τ1 x2 + τ2 x3 ) )}

52

{. . . the above braces . . .}2 + (t1 τ1 + t2 τ2 )2

where c is the dimensional constant c=

3 . 27 π 3

Ex. 5) Prove that if f ∈ L1loc (RN ), then x → −

|f (y)| dy

Bd (x,r)

is a continuous function. Ex. 6) Provide a detailed proof of (5.61) at page 264. Ex. 7) Let u ∈ C 2 (Bd (0, R)) ∩ H Bd (0, R) \ Bd (0, r) , 0 < r < R. Prove that there exist constants C0 , C1 such that Mρ (u)(0) = C0 +

C1 ρ Q−2

for every ρ ∈ (r, R).

Ex. 8) (The surface asymptotic sub-Laplacian). Let Ω ⊆ G be open, and let u ∈ C(Ω, R). Given x ∈ Ω, we say that the surface asymptotic subLaplacian of u at x is non-negative, written ALu(x) ≥ 0, if lim inf r→0+

Mr (u)(x) − u(x) ≥ 0. r2

If AL(−u)(x) ≥ 0, we write ALu(x) ≤ 0. Prove the following statements: (i) If ALu ≥ 0 and ALv ≥ 0, then AL(λ u + μ v) ≥ 0 for every λ, μ ≥ 0. (ii) If ALu ≥ 0 and λ ≤ 0, then AL(λ u) ≤ 0. (iii) If u ∈ C 2 (Ω, R) and Lu ≥ 0, then ALu(x) ≥ 0 for every x ∈ Ω. (iv) If x0 ∈ Ω is a local maximum point for u, then ALu(x0 ) ≤ 0. Ex. 9) (The solid asymptotic sub-Laplacian). Let Ω ⊆ G be open, and let u ∈ C(Ω, R). Given x ∈ Ω, we say that the solid asymptotic sub-Laplacian of u at x is non-negative, written ALu(x) ≥ 0, if lim inf r→0+

Mr (u)(x) − u(x) ≥ 0. r2

If AL(−u)(x) ≥ 0, we write ALu(x) ≤ 0. Prove the statements of the previous exercise with AL replaced by AL.

5.16 Exercises of Chapter 5

321

Ex. 10) (Maximum principle for AL and AL). Let Ω ⊆ G be open and bounded. Let u ∈ C(Ω, R) be such that ALu ≥ 0 in Ω, ALu ≥ 0 in Ω, or lim sup u(y) ≤ 0 ∀ x ∈ ∂Ω. lim sup u(y) ≤ 0 ∀ x ∈ ∂Ω, y→x

y→x

Prove that u ≤ 0 in Ω. (Hint: Proceed as in the proof of the weak maximum principle, Theorem 5.13.4.) Ex. 11) Provide the detailed proof of the result in Remark 5.6.4. Ex. 12) Let L be a sub-Laplacian on a homogeneous Carnot group G on RN . Suppose that L has no first order differential terms, i.e. L(xk ) ≡ 0 for every k = 1, . . . , N . Let ψ ∈ C 2 (G, G) be such that L(u ◦ ψ) = (L u) ◦ ψ

∀ u ∈ C 2 (G).

(5.142)

Following the usual notation for the “stratification” of the variables of G, we set ψ = (ψ (1) , . . . , ψ (r) ) (here ψ (i) ∈ C 2 (G, RNi ), N1 + · · · + Nr = N and Ni is the dimension of the i-th layer in the stratification of the algebra of G). Prove that, for every 1 ≤ j ≤ N1 , it holds (1) 0 = L ψj (5.143) (1) 2 1 = ∇L ψ . j

(Hint: Use (1.112) in Ex. 12, Chapter 1, page 83.) (1) (1) (1) (1) Let now uj := (ψj )2 . Then uj ≥ 0, and from (5.143) we have Luj = (1)

2. Using Liouville Theorem 5.8.4, derive that uj (1) uj

is a polynomial of G-

= p1 + p2 where p1 and p2 are degree ≤ 2 and therefore polynomials of ordinary degrees 2 and 1, respectively. Deduce that p2 ≡ 0. + (1)

(x (1) )

(x (2) ),

(1)

Since |ψj | =

(1)

uj , using, for example, Theorem 5.8.8, derive that ψj

is a polynomial of G-degree ≤ 1, depending at most on x (1) . Now suppose by induction that ψ (1) , . . . , ψ (n) have polynomial component p functions of G-degree respectively 1, . . . , n at most. Write L = k=1 Xk2 (k) with Xk = N i=1 σi ∂i . Prove that (using again (1.112)) if i = n + 1 and 1 ≤ j ≤ Ni+1 , it holds (n+1) 0 = L ψj (5.144) p (k) 2 (n+1) 2 σ (ψ) = ∇L ψ . (n+1)

Let uj

k=1

n+1, j

(n+1) 2 ) ;

then uj

:= (ψj

(n+1)

L uj

j

(n+1)

=2

≥ 0 and, from (5.144), derive

p (k) 2 σn+1, j (ψ). k=1

322

5 The Fundamental Solution for a Sub-Laplacian and Applications (k)

On the other hand, σn+1, j is a polynomial of G-degree n thus depending only on x (1) , . . . , x (n) . Consequently, by induction hypothesis, (k) 2 (k) 2 σn+1, j (ψ) = σn+1, j (ψ (1) , . . . , ψ (n) ) (n+1)

has G-degree at most 2 n. Therefore (why?), uj

+

is a polynomial of G-

(n+1) (n+1) | = uj ≤ C(1 + |x|)n+1 . degree at most 2 + 2n, consequently |ψj (n+1) This last estimate, together with Lψj = 0 and Theorem 5.8.2, proves (n+1) is a polynomial of G-degree at most n + 1. We have proved the that ψj

following assertion. Proposition 5.16.1. Let ψ be a C 2 (G, G) map such that L ◦ ψ = ψ ◦ L, where L is a sub-Laplacian without the first order differential terms. Following the usual notation of the coordinates on G, we let ψ = (ψ (1) , . . . , ψ (r) ), where ψ (j ) ∈ C 2 (G, RNj ) for every j = 1, . . . , r. Then each component of ψ (j ) is a polynomial function of G-degree less or equal to j . Ex. 13) (Maps commuting with Laplace operator Δ). Consider the Laplace oper 2 on RN . Let ψ ∈ C 2 (RN , RN ) be such that ator Δ = N ∂ i=1 i,i Δ(u ◦ ψ) = (Δ u) ◦ ψ

∀ u ∈ C 2 (RN ).

(5.145)

With the aid of Ex. 12, Chapter 1, page 83, prove that (5.145) holds if and only if ⎧ i = 1, . . . , N, ⎪ ⎨ Δψi = 0, ⎪ ⎩

i = 1, . . . , N, |∇ψi |2 = 1, ∇ψi , ∇ψj = 0, 1 ≤ i, j ≤ N,

i = j.

Derive that there exists an N ×N orthogonal matrix M and a vector C ∈ RN such that ψ(x) = Mx + C. This means that the set of the maps commuting with the Laplace operator coincides with the group of the isometries of RN . Ex. 14) (Maps commuting with ΔH 1 ). Let us consider the Kohn Laplacian ΔH1 on the Heisenberg-Weyl group H1 ≡ R3 . We characterize the maps ψ ∈ C 2 (R3 , R3 ) such that ΔH1 (u ◦ ψ) = (ΔH1 u) ◦ ψ

∀ u ∈ C 2 (H1 ).

(5.146)

Set ψ(x, y, t) = (ξ(x, y, t), η(x, y, t), τ (x, y, t)). With the aid of Ex. 12, Chapter 1, page 83, prove that (5.146) is equivalent to

5.16 Exercises of Chapter 5

⎧ ⎪ ⎪ ⎪ (i) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (ii) ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ (iii) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (iv) ⎪ ⎪ ⎪ ⎪ (v) ⎪ ⎪ ⎪ ⎩ (vi)

323

1 = |∇H1 ξ |2 , 0 = ΔH1 ξ, 1 = |∇H1 η|2 , 0 = ΔH1 η, 4 (ξ 2 + η2 ) = |∇H1 τ |2 , 0 = ΔH1 τ,

2 η = ∇H1 ξ, ∇H1 τ , − 2 ξ = ∇H1 η, ∇H1 τ , 0 = ∇H1 ξ, ∇H1 η .

Using Liouville Theorem 5.8.4, derive that ξ and η are polynomials of H1 degree ≤ 1, τ is a polynomial of H1 -degree ≤ 2. Hence, there exist constants such that ξ(x, y, t) = c0 + c1 x + c2 y, η(x, y, t) = d0 + d1 x + d2 y, τ (x, y, t) = e0 + e1 x + e2 y + e3 t. From the first equation in (i) and the first equation in (ii) and from (vi) it follows that there exists θ ∈ [0, 2 π[ such that (c1 , c2 ) = (cos θ, sin θ ),

(d1 , d2 ) = ±(− sin θ, cos θ ).

From (iv) and (v) one gets

2d0 c1 − 2d1 c0 ± 2 y e 1 + 2 y e3 , = e2 − 2 x e3 2d0 c2 − 2d2 c0 ∓ 2 x i.e. e3 = ±1, Recalling that on

H1

e1 = 2d0 c1 − 2d1 c0 ,

e2 = 2d0 c2 − 2d2 c0 .

it holds

(α, β, γ ) ◦ (x, y, t) = (x + α, y + β, t + γ + 2xβ − 2yα), we have proved that the only maps commuting with ΔH1 have the form ⎛ ⎞ ⎛ x ⎞ c0 M · ψ+ (x, y, t) = ⎝ d0 ⎠ ◦ ⎝ + y ⎠ , e0 t ⎛ ⎞ ⎛ x ⎞ c0 M · ψ− (x, y, t) = ⎝ d0 ⎠ ◦ ⎝ − y ⎠ , e0 −t where M+ and M− are (respectively) isometries with the determinant equal to +1 (respectively, −1) and (c0 , d0 , e0 ) is a fixed point in H1 .

324

5 The Fundamental Solution for a Sub-Laplacian and Applications

Ex. 15) (Translation-formula in surface integrals over a d-sphere). Let f : G → R be non-negative (or with suitable summability properties). Let d be a smooth, homogeneous and symmetric norm on G, and let S(x0 , r) be the d-sphere with center x0 and radius r > 0. Finally, let H N −1 denote the Hausdorff (N − 1)-dimensional measure on RN . Prove that f (x) dH N −1 (x) = f (x0 ◦ y) Kd (x0 , y) dH N −1 (y), S(x0 ,r)

S(0,r)

where Kd (x0 , y) :=

|∇(d(x0−1 ◦ ·))|(x0 ◦ y) . |∇d(y)|

|∇d(y) · Jτ

(x0 ◦ y)|

Prove also that Kd (x0 , y) =

x0−1

|∇d(y)|

∇d(y) = · Jτ −1 (x0 ◦ y). x0 |∇d(y)|

(Hint: Consider the identity (why does it hold?) f (x) dH N (x) = f (x0 ◦ y) dH N (y) S(x0 ,r)

S(0,r)

rewritten (by the coarea formula) as r f (x) 0

S(x0 ,λ)

=

r

0

dH N −1 (x) |∇d(x0−1 ◦ ·)|(x)

dr

dH N −1 (y) dr f (x0 ◦ y) |∇d(y)| S(0,λ)

and differentiate w.r.t. r.) Ex. 16) In the sequel, d is a symmetric homogeneous norm on G. If x ∈ G and A ⊆ G is any set, we call the d-distance of x to A, the following real nonnegative number distd (x, A) := inf d(x −1 ◦ a). a∈A

Prove the following result. Lemma 5.16.2. Let A ⊂ G be any set. For every x ∈ G, there exists a x ∈ A (the closure of A) such that distd (x, A) = d(x −1 ◦ a x ). (Hint: By definition, there exists {aj }j in A such that d(x −1 ◦ aj ) → distd (x, A). Obviously, {aj }j is bounded, otherwise d(x −1 ◦ aj ) ≥ 1c d(aj )− d(x) → ∞. Then, extract a subsequence ajn → a x ∈ A, as n → ∞, and use the continuity of d.) Then prove the following result.

5.16 Exercises of Chapter 5

325

Proposition 5.16.3. Let A ⊆ G be any set. The following assertions hold: (a) For every x ∈ G, we have distd (x, A) = distd (x, A); (b) The d-distance from A, i.e. the function distd (·, A) : G → [0, ∞),

x → distd (x, A),

is a continuous function. Hint: (a). Let α ∈ A be such that distd (x, A) = d(x −1 ◦ α). Let aj ∈ A be such that aj → α as j → ∞. Then consider the following inequalities: distd (x, A) ≥ distd (x, A) = d(x −1 ◦ α) = lim d(x −1 ◦ aj ) ≥ distd (x, A). j →∞

The last inequality follows from d(x −1 ◦ aj ) ≥ distd (x, A) for every j ∈ A, since aj ∈ A, and by the very definition of distd (x, A). (b). Provide the details of the following arguments: Let x0 ∈ G be fixed. It suffices to show that from every sequence {xj }j ∈N in G converging to x0 we can extract a subsequence {xjn }n∈N such that lim distd (xjn , A) = distd (x0 , A).

n→∞

(5.147)

It is not restrictive to suppose that A is closed. For every j ∈ N, there exists aj ∈ A such that distd (xj , A) = d(xj−1 ◦ aj ). (5.148) It is easy to see that {aj }j ∈N is bounded. Hence, we can extract a converging subsequence from {aj }j ∈N , say ajn → a ∈ A as n → ∞. From (5.148) we infer lim distd (xjn , A) = lim d(xj−1 ◦ ajn ) = d(x0−1 ◦ a). n

n→∞

n→∞

(5.149)

Thus, (5.147) will follow if we show that d(x0−1 ◦ a) = distd (x0 , A). Suppose to the contrary that we have d(x0−1 ◦ a) = distd (x0 , A). This may occur iff (i) : d(x0−1 ◦ a) < distd (x0 , A)

or (ii) : d(x0−1 ◦ a) > distd (x0 , A).

Case (i) is impossible by the very definition of distd (x0 , A) (since a ∈ A). Case (ii) is absurd too, as we show below. Let a0 ∈ A be such that distd (x0 , A) = d(a0−1 ◦ x0 ). Then, by (5.149), we derive d(x0−1 ◦ a) = lim distd (xjn , A) ≤ lim d(a0−1 ◦ xjn ) = This contradicts (ii).

n→∞ d(a0−1

n→∞

◦ x0 ) = distd (x0 , A).

326

5 The Fundamental Solution for a Sub-Laplacian and Applications

Ex. 17) Prove the following improvement of Proposition 5.16.3 above, when the extra hypothesis d is smooth holds. Proposition 5.16.4. Let d be a smooth homogeneous norm as in Proposition 5.14.1, and let A ⊆ G. Prove that the function distd (x, A) is Lipschitz continuous with respect to d, i.e. |distd (ξ, A) − distd (η, A)| ≤ β d(ξ, η)

∀ ξ, η ∈ G,

where β is the same constant as in Proposition 5.14.1. (Hint: Let a ∈ A, and write (5.125a) with x = a, y = ξ , z = η, d(ξ −1 ◦ a) ≤ d(η−1 ◦ a) + β d(ξ −1 ◦ η). Then take the infimum over a ∈ A, and derive distd (ξ, A) ≤ distd (η, A) + β d(ξ, η). Then interchange ξ and η.) Ex. 18) (T : Another mean integral operator). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let Ω ⊆ G be a bounded open set. For every x ∈ Ω, let us put rx :=

1 distd (x, ∂Ω), 2

where d is an L-gauge function. For every u ∈ L1loc (Ω), define T (u)(x) := Mrx (u)(x),

(5.150)

where Mr is the average operator defined in (5.50f). Prove the following statements: (i) T (u) is continuous in Ω, (ii) u ≤ v ⇒ T (u) ≤ T (v), (iii) u ∈ L∞ (Ω) ⇒ supΩ |T (u)| ≤ ess supΩ |u|. Ex. 19) (The strong maximum principle related to T ). Let T be as in the previous exercise, and let u be a continuous function in Ω such that T (u) ≤ u. If Ω is connected and u attains its minimum in Ω, then u = constant in Ω. Ex. 20) (The weak maximum principle related to T ). Let T be as in the previous exercises, and let u be a continuous function in Ω such that T (u) ≤ u in Ω, lim infx→y u(x) ≥ 0 ∀ y ∈ ∂Ω. Prove that u ≥ 0 in Ω. Ex. 21) Let u be an L-harmonic function in an open set Ω ⊆ G, Ω = G. Assume u ∈ Lp (Ω), 1 ≤ p ≤ ∞. Then c uLp (Ω) , |u(x)| ≤ distd (x, ∂Ω) where c is independent of u and Ω, and d is an L-gauge function. (Hint: Formula (5.52) may prove useful.)

5.16 Exercises of Chapter 5

327

Ex. 22) (Another Koebe-type result). Let Ω ⊆ G be open, and let u ∈ C(Ω, R). Suppose that one of the following conditions is satisfied: (i) Mρ (u)(x) = Mr (u)(x) for every ρ, r > 0 such that 0 < ρ ≤ r and Bd (x, r) ⊂ Ω, (ii) Mρ (u)(x) = Mr (u)(x) for every ρ, r > 0 such that 0 < ρ ≤ r and Bd (x, r) ⊂ Ω. Show that u ∈ C ∞ (Ω, R) and Lu = 0 in Ω. Ex. 23) Prove the following Harnack inequality on d-spheres. Theorem 5.16.5. There exists a constant C > 1 such that sup h ≤ C ∂Bd (0,r)

inf

∂Bd (0,r)

h

(5.151)

for every 0 < r ≤ 1/2 and every L-harmonic non-negative function h on Bd (0, 1) \ {0}. Ex. 24) Let d be any homogeneous norm on G, smooth on G \ {0}. Prove that d(x)=r

dH N −1 (x) = cd r Q−1 |∇d(x)|

for every r > 0,

where cd = Q H N (Bd (0, 1)). (Hint: By the coarea formula and the δλ homogeneity of d, we have

r dH N −1 (x) dρ = dH N = r Q H N (Bd (0, 1)). {d(x)=ρ} |∇d(x)| {d(x)
j =1

Suppose γ : [0, T ] → RN is X-subunit, γ (0) = x and T < R/M(x, R). Then γ ([0, T ]) ⊆ BE (x, R). (Hint: Suppose to the contrary that γ ([0, T ]) BE (x, R). Hence, there exists a least t ∈ (0, T ], such that y := γ (t) ∈ / BE (x, R). From the minimality of t and the continuity of γ , we certainly have y ∈ ∂BE (x, R), i.e.

328

5 The Fundamental Solution for a Sub-Laplacian and Applications

|y − x| = R. Notice also that dX (x, y) ≤ t ≤ T . Hence, (5.8b) on page 233 gives R = |x − y| ≤ M(x, |x − y|) dX (x, y) ≤ M(x, R)T < R. This is absurd.) Deduce the following result. Corollary 5.16.7. Let X = {X1 , . . . , Xm } be a system of locally Lipschitzcontinuous vector fields on RN . Then, for every bounded set D ⊂ RN , there exist positive numbers R0 * 1, R1 % 1 (depending only on D and X) such that BdX (x, R) ⊆ BE (0, R1 ) for every x ∈ D and every R ∈ [0, R0 ]. In particular, for every x0 ∈ RN , every Carnot–Carathéodory ball BdX (x0 , R) with small radius (dependently on x0 ) is bounded in the Euclidean metric. (Hint: With the notation of Lemma 5.16.6, let M := sup{M(x, 1) : x ∈ D}. Obviously, M < ∞, for the Xj ’s are continuous. Set R0 := (4 M)−1 . Let x ∈ D and R ≤ R0 . (Note that M ≥ M(x, 1), since x ∈ D.) Let also y ∈ BdX (x, R), i.e. dX (x, y) < R. Then there exists a X-subunit curve γ : [0, T ] → RN such that γ (0) = x, γ (T ) = y and T < dX (x, y) + (4M)−1 < 2 R0 = (2M)−1 < 1/M(x, 1). Then, we can apply Lemma 5.16.6 with R = 1 and derive that γ ([0, T ]) ⊆ BE (x, 1). In particular, this gives |y − x| = |γ (T ) − x| < 1, i.e. y ∈ BE (x, 1). Due to the arbitrariness of y ∈ BdX (x, R), this proves $ BdX (x, R) ⊆ BE (x, 1) ⊆ D1 := BE (x, 1) RN . x∈D

This ends the proof, by finding a suitable R1 % 1 such that D1 ⊆ BE (0, R1 ).) Proposition 5.16.8. Let X = {X1 , . . . , Xm } be a system of locally Lipschitzcontinuous vector fields on RN . Let K be a compact subset of RN . Then there exists a constant c = c(K, X) > 0 such that dX (x, y) ≥ c |x − y|

for every x, y ∈ K.

Proof. If dX (x, y) = ∞, there is nothing to prove. Hence, we can suppose dX (x, y) < ∞. Let γ : [0, T ] → RN be any X-subunit curve such that γ (0) = x, γ (T ) = y. Let R := |x − y|. Set

5.16 Exercises of Chapter 5

M := sup

n

329

|Xj I (z)| : z ∈ BE (x, 1 + diam(K)), x ∈ K .

j =1

Note that M = M(K, X) < ∞, since K is compact and the Xj ’s are continuous. Obviously, we do not have γ ([0, T ]) ⊆ BE (x, R), because |γ (T ) − x| = |y − x| = R is not < R. Hence, by Lemma 5.16.6, T ≥ R/M(x, R) ≥ R/M =: c R = c|x − y|. (Indeed, M(x, R) ≤ M.) Passing to the infimum over the above γ ’s, we get dX (x, y) ≥ c |x − y|. Derive the following assertion from Corollary 5.16.7 and Proposition 5.16.8. Proposition 5.16.9. Let X = {X1 , . . . , Xm } be locally Lipschitz-continuous vector fields on RN . Suppose RN is X-connected. Let dX be the relevant Carnot–Carathéodory distance. Then the map id : (G, dX ) → (G, dE ) is continuous. (Hint: Dealing with metric spaces, we can prove sequential continuity. Let xn → x0 in (G, dX ). We aim to prove that xn → x0 in (G, dE ). By Corollary 5.16.7, there exists ε = ε(x0 ) > 0 such that BdX (x0 , ε) is bounded in (G, dE ). By definition of limit, there exists n = n(ε) ∈ N such that xn ∈ BdX (x0 , ε) whenever n ≥ n. Hence the set {xn : n ≥ n} ∪ {x0 } is contained in a compact subset of RN . We can thus apply Proposition 5.16.8 to derive that, for a suitable c > 0, c |xn − x0 | ≤ dX (xn , x0 )

for all n ≥ n.

Letting n → ∞, we get xn → x0 in (G, dE ).) Now, consider a system X of Hörmander vector fields in RN and the relevant dX . Recall that, by the Carathéodory–Chow–Rashevsky theorem (see Chapter 19 for suitable references), RN is X-connected. Moreover, by known results (see, e.g. [NSW85]) the inequalities in Proposition 5.15.1 hold (for a suitable r > 0) when d(y −1 ◦ x) is replaced by dX (x, y). In particular, given a compact subset K of RN , there exist r, c > 0 (depending on K, X) such that dX (x, y) ≤ c |x − y|1/r

∀ x, y ∈ K.

Arguing as in the first paragraph of the proof of Corollary 5.15.3, derive the following assertion: Given a system X of Hörmander vector fields in RN , the map id : (G, dE ) → (G, dX ) is continuous (indeed, it is a homeomorphism, see Proposition 5.16.9).

330

5 The Fundamental Solution for a Sub-Laplacian and Applications

Ex. 26) Consider the vector fields in R2 defined by X1 = ∂x1 ,

X2 = max{0, x1 } ∂x2 .

Consider the relevant dX , where X = {X1 , X2 }. Is R2 X-connected? Prove that the identity map id : (R2 , dE ) → (R2 , dX ) is not continuous. (Hint: It may be useful to notice that the integral curves of X1 are the lines parallel to the x1 axis; the integral curve of X2 through (x0 , y0 ) is (x0 , y0 ) if x0 ≤ 0, t → (x0 , y0 + x0 t) if x0 > 0, i.e. the single point (x0 , y0 ) if x0 ≤ 0 or the lines parallel to the x2 axis if x0 > 0.) Ex. 27) Consider the vector fields in R2 defined by X1 = ∂x1 ,

X2 = (1 + x22 ) ∂x2 .

Verify that the system X = {X1 , X2 } satisfies the Hörmander condition. Consider the relevant dX . Is R2 X-connected? Prove that there exists a set Ω ⊂ R2 which is bounded w.r.t. dX but unbounded in the Euclidean metric. Hint: Observe that the integral curve of X2 (hence, a X-subunit path!) starting from (x0 , y0 ) is t → x0 , tan(t + arctan(y0 )) . Hence, for example,

1 π dX (0, 0), 0, tan − ≤ π/2 − 1/n < π/2, 2 n but

π 1 dE (0, 0), 0, tan − −→ ∞ 2 n

as n → ∞.

Ex. 28) Prove the following linear algebra result. Lemma 5.16.10. Let v1 , . . . , vm be vectors in RN . Suppose w ∈ RN is such that m () w, x 2 ≤ vj , x 2 ∀ x ∈ RN . j =1

Then there exist scalars α1 , . . . , αm such that w=

m j =1

αj vj

and

m j =1

αj2 ≤ 1.

5.16 Exercises of Chapter 5

331

Proof. Let us first prove that w ∈ span{v1 , . . . , vm } =: V . From the decomposition RN = V ⊕ V ⊥ it follows w = a + b,

b ∈ V ⊥.

a ∈ V,

If we choose x := b in (), we get m

b4 = a + b, b 2 ≤

vj , b 2 = 0,

j =1

whence b = 0, i.e. w = a ∈ V . Up to a permutation of the vj ’s, it is not restrictive to suppose that (v1 , . . . , vq ) is a basis of V . Then we have vj =

q

βj,k vk

∀ j = 1, . . . , m.

k=1

The m × q matrix B whose (j, k)-th entry is βj,k has the block form Iq B= , , B where Iq is the identity matrix of order q. Let γ1 , . . . , γq ∈ R be such that q w = k=1 γk vk . We characterize all the scalars α1 , . . . , αm ∈ R such that w= m j =1 αj vj . From the identity m q q γk v k = w = αj βj,k vk k=1 j =1

k=1

and the linear independence of v1 , . . . , vq we infer γk =

(•)

m

αj βj,k

∀ k = 1, . . . , q.

j =1

Let us suppose that, besides (•), there also exists a solution x¯ ∈ RN to the m-equation system vj , x = αj , (SL) j = 1, . . . , m. This will give m

2 αj2

j =1

=

m

2 αj vj , x¯

j =1

= w, x¯ 2 ≤ whence

m

2 j =1 αj

=

- m

m vj , x¯ 2 =

. 2 αj vj , x¯

j =1 m

j =1

≤ 1, and the proof is complete.

j =1

αj2 ,

332

5 The Fundamental Solution for a Sub-Laplacian and Applications

We remark that, in order to (SL) to be solvable, it is necessary and sufficient that the rank of the coefficient-matrix of (SL) (i.e. q) equals the rank of the complete-matrix of (SL): thanks to the dependence of vq+1 , . . . , vm w.r.t. v1 , . . . , vq , this is equivalent to αj =

(••)

q

βj,k αk

∀ j = q + 1, . . . , m.

k=1

Hence, the proof is complete if we show that there exists (α1 , . . . , αm ) ∈ Rm satisfying (•) and (••), i.e. a solution to the m-equation and m-indeterminate system α1 , . . . , αm

(SL)’

⎧ m ⎪ ⎪ ⎪ αj βj,k = γk , ⎪ ⎨

k = 1, . . . , q,

⎪ ⎪ ⎪ ⎩

j = q + 1, . . . , m.

j =1 q ⎪

βj,k αk − αj = 0,

k=1

The coefficient-matrix of (SL)’ is tB , Iq . , −Im−q B We show that this matrix is invertible, whence (SL)’ is solvable. In general, if A is a real q × (m − q) matrix, the block matrix

t A A A 0 Iq 2 satisfies P = Im + · P := t A −Im−q 0 0 tA

0 A

.

Observe that P 2 is the sum of a positive-definite matrix plus a positive semi2 definite one. In particular, √ P is positive-definite, hence it is not singular. This gives | det P | = det P 2 > 0, so that P is not singular too. This ends the proof. By means of Lemma 5.16.10, derive the following equivalent characterization of X-subunit curve. Proposition 5.16.11. Let X = {X1 , . . . , Xm } be a system of locally Lipschitz-continuous vector fields on an open set Ω ⊆ RN . Let γ : [0, T ] → Ω be an absolutely continuous curve. Then γ is Xsubunit if and only if there exist measurable functions cj : [0, T ] → R, j = 1, . . . , m, such that, almost everywhere on [0, T ], γ˙ (t) =

m j =1

cj (t) Xj I (γ (t)),

and

m 2 cj (t) ≤ 1. j =1

(5.152)

5.16 Exercises of Chapter 5

333

Proof. Suppose γ : [0, T ] → Ω, satisfies (5.152). The Cauchy-Schwartz inequality in Rm immediately yields m m m 2 2 2 2 γ˙ (t), ξ ≤ Xj (γ (t)), ξ Xj (γ (t)), ξ , (cj (t)) · ≤ j =1

j =1

j =1

i.e. γ is a X-subunit curve. Vice versa, let γ be X-subunit. If we apply Lemma 5.16.10 with the choice vj := Xj (γ (t)),

j = 1, . . . , m,

w := γ˙ (t),

then (5.152) is satisfied for suitable scalar functions cj ’s defined a.e. on [0, T ]. It is easy to prove16 the measurability of the cj ’s. Ex. 29) Let u be an L-harmonic function in an open set Ω ⊆ G. Let Bd (x0 , r) ⊂ Ω. Then, for every multi-index α, there exists a constant Cα > 0 (independent of u, x0 and r) such that α X u(x0 ) ≤ Cα r −|α|G sup |u|. Bd (x0 ,r)

(Hint: Since u is L-harmonic, the representation formula (5.50a) (see also (5.50d)) gives −1 u(x0 ) = u(z) φr (x ◦ z) dz = u(z) φ,r (x −1 ◦ z) dz, RN

RN

where φ,r (ζ ) = φr (ζ −1 ). Now, derive the integral.)

16 Indeed, notice that γ˙ is measurable (since γ is a absolutely continuous), t → X (γ (t)) is j

continuous and the operations which provide the components of a vector w.r.t. a system of vectors are continuous. We leave the details to the reader.

Part II

Elements of Potential Theory for Sub-Laplacians

6 Abstract Harmonic Spaces

In this chapter, we present some topics from the theory of abstract harmonic spaces. Although this theory is extremely vast, we only focus on the results which are crucial for the scopes of the next chapters. In this abstract setting, we shall only deal with the Perron–Wiener–Brelot method for the Dirichlet problem, the study of harmonic minorants and majorants and balayage theory. Our aim here is to furnish the background material for an exhaustive potential theory for the sub-Laplacians L, which will be developed through the rest of Part II. The lack of explicit Poisson integral formulas for L forced us to follow the abstract approach to this theory. In the subsequent chapters, we shall use this basic results, together with the particular structure of the fundamental solutions, to develop the deepest part of the potential theory for L. The following scheme describes our approach to the abstract potential theory for sub-Laplacians. Notation and definitions will be explained in due course.

L-Harmonic Spaces (Capacity, Polarity, etc.) S∗ -Harmonic Spaces (Bouligand’s Theorem) S-Harmonic Spaces (Wiener Resolutivity Theorem) Harmonic Spaces (PWB theory)

338

6 Abstract Harmonic Spaces

Convention. Throughout this chapter (E, T ) will denote a topological Hausdorff space, locally connected and locally compact. We also assume that the topology T has a countable basis.

6.1 Preliminaries Let A ⊆ E and u : A → [−∞, ∞]. If x is a point of A, we define lim inf u(y) := sup inf u , y→x

V ∈Ux

lim sup u(y) := inf y→x

V ∈Ux

V ∩A

sup u ,

V ∩A

where Ux denotes the family of the neighborhoods of x. (Note the slight difference w.r.t. the usual definition of lim sup and lim inf in that, for example, in the lim inf definition, the inner infV ∩A is usually replaced by inf(V \{x})∩A . Our choice will be soon motivated.) The function u is called lower semicontinuous (l.s.c., in short) at x ∈ A if u(x) = lim infy→x u(y). If u(x) = lim supy→x u(y), then u will be called upper semicontinuous (u.s.c., in short) at x. If u is l.s.c. (u.s.c.) at any point of A, then u will be said l.s.c. (u.s.c.) on A. We now list some elementary properties of semicontinuous functions, whose proofs will be left as exercises. (P1) Suppose that f, g are l.s.c. (u.s.c.) on A. Then f +g, λf with λ ≥ 0, max{f, g}, min{f, g} are l.s.c. (u.s.c.) on A. Moreover, λf is u.s.c. if f is l.s.c. and λ < 0. (P2) f : A → [−∞, ∞] is l.s.c. on A if and only if {f > t} := {x ∈ A : f (x) > t} is open for every t ∈ R. Similarly, f is u.s.c. on A if and only if {f < t} := {x ∈ A : f (x) < t} is open for every t ∈ R. (P3) Let (uα )α∈A be a family of l.s.c. (u.s.c.) functions on A. Then supα∈A uα (infα∈A uα , respectively) is l.s.c. (u.s.c., respectively). (P4) If A is compact and u : A → ]−∞, ∞] (u : A → [−∞, ∞[, respectively) is l.s.c. (u.s.c., respectively), then u attains its minimum (maximum, respectively) on A. (P5) If A is compact and u : A → ]−∞, ∞] (u : A → [−∞, ∞[, respectively) is l.s.c. (u.s.c., respectively), then there exists a sequence (fj ) of real continuous functions on A such that fj ≤ fj +1 and fj → u pointwise on A (fj ≥ fj +1 and fj → u pointwise on A, respectively).

6.1 Preliminaries

339

If u : A → [−∞, ∞], one defines the lower regularization u of u as follows u : A → [−∞, ∞],

u(x) = lim inf u(y). y→x

Similarly, one defines the upper regularization uˇ of u as follows uˇ : A → [−∞, ∞],

u(x) ˇ = lim sup u(y). y→x

The functions u and uˇ are l.s.c. and u.s.c., respectively, and u ≤ u ≤ u. ˇ We now state and prove some propositions on the envelopes of families of functions which will play a crucial rôle in our subsequent exposition. Our proofs are simple adaptation to an abstract setting of Lemmas 3.7.1 and 3.7.4 in [AG01]. Proposition 6.1.1 (Envelopes I). Let (uα )α∈A be a family of l.s.c. functions from A (⊆ E) to ]−∞, ∞]. Then there exists a countable set B ⊆ A such that sup uα = sup uα .

α∈B

α∈A

Proof. Let u := supα∈A uα . Then u is l.s.c. (see (P3)) and {uα > t}. {u > t} = α∈A

Since (E, T ) is locally compact and endowed with a countable basis of open sets, there exists a countable set At ⊆ A such that {u > t} = {uα > t}. (6.1) α∈At

Define B := t∈Q At and v := supα∈B uα . Let us prove that v = u. Assume by contradiction v = u. Since v ≤ u, this implies the existence of a point y ∈ A such that v(y) < u(y). Let τ ∈ Q satisfy v(y) ≤ τ < u(y). By using (6.1), we can find ατ ∈ Aτ such that τ < uατ (y). Then v(y) = sup uα (y) ≥ uατ (y) > τ ≥ v(y). α∈B

This contradiction completes the proof. Proposition 6.1.2 (Envelopes II: Choquet). Let (uα )α∈A be a family of functions from A (⊆ E) to [−∞, ∞], and let u := infα∈A uα . Then there exists a countable set B ⊆ A such that u = v , where v = inf uα . α∈B

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6 Abstract Harmonic Spaces

Proof. By replacing uα with arctan uα , we may assume that u is bounded from below. Let (Bn )n∈N be a sequence of open sets such that {Bn : n ∈ N} forms a basis of the topology T and such that, for every m ∈ N, Nm := {n ∈ N : Bn = Bm } is infinite. For every n ∈ N, we now choose xn ∈ Bn and αn ∈ A such that u(xn ) < inf u + Bn ∩A

1 n

and uαn (xn ) < u(xn ) +

1 . n

Let us define B := {αn : n ∈ N} and v := infα∈B uα . Then, for every m ∈ N and n ∈ Nm , we have inf v = inf v ≤ uαn (xn ) < u(xn ) +

Bm ∩A

Bn ∩A

1 2 2 < inf u + = inf u + , n Bn ∩A n Bm ∩A n

hence, being Nm infinite, inf v ≤ inf u for every m ∈ N.

Bm ∩A

Bm ∩A

As a consequence, for every x ∈ A, inf v ≤ v (x) = sup {Bm : x∈Bm } Bm ∩A

sup

inf u = u(x),

{Bm : x∈Bm } Bm ∩A

so that v ≤ u. Since the reverse inequality is obvious, we are done.

6.2 Sheafs of Functions. Harmonic Sheafs We begin with the main definition of this section. Definition 6.2.1 (Sheaf of functions. Harmonic sheaf). Suppose we are given, for every open set V ∈ T , a family F(V ) of extended real-valued functions u : V → [−∞, ∞]. We say that the map F : V → F(V ) is a sheaf of functions on E if the following properties hold: if V1 , V2 ∈ T with V1 ⊆ V2 and u ∈ F(V2 ), then u|V1 ∈ F(V1 ), if (Vα )α∈A ⊆ T and u : α∈A Vα → [−∞, ∞] is such that u|Vα ∈ F(Vα ) for every α ∈ A, then u ∈ F( α∈A Vα ).

(6.2a) (6.2b)

A sheaf of functions F on E is called harmonic if F(V ) is a linear subspace of C(V , R), the vector space of the real continuous functions defined on V . When F is a harmonic sheaf on E and V is an open set in T , a function u ∈ F(V ) will be called F-harmonic.

6.2 Sheafs of Functions. Harmonic Sheafs

341

6.2.1 Regular Open Sets. Harmonic Measures. Hyperharmonic Functions Definition 6.2.2 (H-regular set). Let H be a harmonic sheaf on E. We say that an open set V ∈ T is H-regular if the following conditions are satisfied: (R1) V is compact and ∂V = ∅; (R2) for every continuous function f : ∂V → R, there exists a unique H-harmonic function in V , denoted by HfV , such that lim HfV (x) = f (y)

x→y

for every y ∈ ∂V ;

(R3) if f ≥ 0, then HfV ≥ 0. When V is H-regular, from the linearity of F(V ) and the uniqueness assumption in condition (R2) it follows that HfV+g = HfV + HgV ,

V Hλf = λHfV

for every f, g ∈ C(∂V , R) and for every λ ∈ R. Then, also keeping in mind (R3), for every H-regular open set V and for every x ∈ V , the map C(∂V , R) f → HfV (x) ∈ R is linear and positive. Hence, the following definition is well posed. Definition 6.2.3 (H-harmonic measure. I). Let H be a harmonic sheaf on E. Let V ∈ T be an H-regular set. Then there exists a Radon measure μVx on C(V , R) such that HfV (x) = f (y) dμVx (y) ∀ f ∈ C(∂V , R). ∂V

The measure

μVx

is called the H-harmonic measure related to V and x.

We provide a definition which will be used throughout the sequel. Definition 6.2.4 (H-hyperharmonic function). Let H be a harmonic sheaf on (E, T ). Let Ω ∈ T . A function u : Ω → ]−∞, ∞] is called H-hyperharmonic in Ω if: (i) u is lower semi-continuous; (ii) for every H-regular open set V ⊂ V ⊆ Ω, one has u(x) ≥ u(y) dμVx (y) ∀ x ∈ V. ∂V

We shall denote by H∗ (Ω) the set of the H-hyperharmonic functions in Ω. Since

∂V

u dμVx

= sup ∂V

ϕ dμVx

condition (ii) can be rewritten as follows

| ϕ ∈ C(∂V , R), ϕ ≤ u ,

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6 Abstract Harmonic Spaces

(ii)’ for every H-regular open set V ⊆ V ⊆ Ω and for every ϕ ∈ C(∂V , R) such that ϕ ≤ u|∂V , one has HϕV ≤ u|V . A function v : Ω → [−∞, ∞[ will be called H-hypoharmonic if −v ∈ H∗ (Ω). We denote by H∗ (Ω) := −H∗ (Ω) the family of the H-hypoharmonic functions in Ω. Exercise 6.2.5. Let Ω ⊆ E be open, and let F ⊆ H∗ (Ω). Suppose that u0 := inf F ∗ is

l.s.c. inV Ω. Then u0 ∈V H (Ω). (Hint: if V ⊆ V ⊆ Ω is H-regular, then u(x) ≥ ∂V u dμx ≥ ∂V u0 dμx for every x ∈ V and u ∈ F.)

We will see later that H∗ (Ω) is a sheaf of functions. Now, we state a proposition whose easy proof will be left as an exercise. Proposition 6.2.6. Let Ω ⊂ E be open, and let u, v ∈ H∗ (Ω). Then: (i) u + v ∈ H∗ (Ω), (ii) λ u ∈ H∗ (Ω) for every λ ≥ 0, (iii) min{u, v} ∈ H∗ (Ω). An analogous proposition also holds, obviously, for H∗ (Ω). 6.2.2 Directed Families of Functions In this section we shall show some results on directed families of functions, extending well-known theorems related to monotone sequences of functions. The results we are going to show will be applied in the subsequent section to families of harmonic and hyperharmonic functions. Let F be a family of functions from A ⊆ E to [−∞, ∞]. Definition 6.2.7 (Up (down) directed family). We say that F is up directed, and we shall write F ↑, if, for every u, v ∈ F, there exists w ∈ F such that u≤w

and v ≤ w.

Analogously, if, for every u, v ∈ F, there exists w ∈ F such that u ≥ w, v ≥ w, we say that F is down directed, and we shall write F ↓.

6.2 Sheafs of Functions. Harmonic Sheafs

343

Theorem 6.2.8 (Dini–Cartan). Let A ⊆ E be compact, and let F be a down directed family of upper semicontinuous functions. Assume inf F := inf u = 0. u∈F

Then F is uniformly convergent to zero, that is, for every ε > 0, there exists u ∈ F such that u(x) < ε for every x ∈ A. Proof. Let ε > 0. Since inf F = 0, for every x ∈ A, there exists ux ∈ F such that ux (x) < ε. The upper semicontinuity of ux implies the existence of a neighborhood Vx of x such that ux (y) < ε for every y ∈ Vx . Since A is compact, we can find x 1 , . . . , xp ∈ A

such that A ⊆

p

Vxj .

j =1

On the other hand, since F is down directed, there exists u ∈ F such that u ≤ uxj for every j ∈ {1, . . . , p}. As a consequence, given any point y ∈ A, there exists j ∈ {1, . . . , p} such that y ∈ Vxj , so that u(y) ≤ uxj (y) < ε, and the assertion follows. Corollary 6.2.9. Let F be an up directed (respectively, down directed) family of l.s.c. (respectively, u.s.c.) functions on a compact set A ⊆ E. Assume that v := sup F

(respectively, v := inf F)

is continuous. Then F is uniformly convergent to u, that is, for every ε > 0, there exists u ∈ F such that v − u < ε (respectively, u − v < ε). Proof. Apply the previous theorem to the family Fv := {v − u : u ∈ F} (respectively Fv := {u − v : u ∈ F}). Now, we prove one of the main results of this section. Theorem 6.2.10 (Levi–Cartan). Let A ⊆ E be compact, and let μ be a Radon measure in A. Suppose we are given an up directed family F of l.s.c. functions. Then sup u dμ. u dμ = (6.3) sup u∈F

A

A

u∈F

Analogously, if F is a down directed family of u.s.c. functions, then inf u dμ. u dμ = inf u∈F

A

A

u∈F

Proof. We only prove the first part of the theorem. The second one will follow by applying (6.3) to the family −F := {−u : u ∈ F}. The function

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6 Abstract Harmonic Spaces

v := sup u u∈F

is l.s.c. since the functions in F are l.s.c. Therefore, v dμ = sup φ dμ : φ ∈ C(A, R), φ ≤ v . A

(6.4)

A

Since u ≤ v for any u ∈ F, the inequality u dμ ≤ v dμ sup u∈F

A

(6.5)

A

readily follows. To prove the reverse one, let us pick φ ∈ C(A, R) such that φ ≤ v and consider Fφ := {min{u, φ} : u ∈ F}. This family is up directed and sup Fφ = min{v, φ} = φ. Then, we can apply Corollary 6.2.9: for every ε > 0, there exists u0 ∈ F such that φ − min{u0 , φ} < ε. As a consequence, φ dμ ≤ εμ(A) + min{u0 , φ} dμ ≤ εμ(A) + u0 dμ A A A ≤ εμ(A) + sup u dμ. u∈F

A

Noticing that μ(A) < ∞, since A is compact, and letting ε tend to zero, we get φ dμ ≤ sup u dμ. A

u∈F

A

Thanks to (6.4) this inequality implies v dμ ≤ sup u dμ. A

u∈F

A

This equality, together with (6.5), completes the proof. We close the section by proving the following theorem, an easy consequence of Proposition 6.1.1. Theorem 6.2.11 (Cornea). Let F be an up directed family of continuous functions in an open set Ω ∈ T . Suppose F has the following property: for every increasing sequence (fn )n in F, one has supn∈N fn ∈ C(Ω, R). Then the function v := sup u u∈F

is continuous in Ω.

(6.6)

6.3 Harmonic Spaces

345

Proof. Proposition 6.1.1 implies the existence of a sequence (gn ) in F such that v = supn gn . Since F is up directed, we can construct a sequence (fn ) in F such that f1 = g1 , f2 ≥ max{g2 , f1 }, . . . , fn+1 ≥ max{gn+1 , fn }, . . . . Then F fn ↑ v. Assumption (6.6) implies that v is continuous.

6.3 Harmonic Spaces Definition 6.3.1 (Harmonic space). Let H be a harmonic sheaf on E. We say that (E, H) is a harmonic space if the following axioms are satisfied: (A1) (Positivity). For every relatively compact open set Ω ⊆ E, there exist h0 ∈ H∗ (Ω) and k0 ∈ H∗ (Ω) satisfying1 : inf h0 , inf k0 > 0 Ω

Ω

and h0 (x) < ∞

∀ x ∈ Ω.

(A2) (Convergence). If {un }n∈N is a monotone increasing sequence of H-harmonic functions on an open set Ω ∈ T such that x ∈ Ω : sup un (x) < ∞ n∈N

is dense in Ω, then u := lim un n→∞

is H-harmonic in Ω. (A3) (Regularity). The family Tr of the H-regular open sets is a basis for the topology T . (A4) (Separation). For every x, y ∈ E with x = y, there exist u, v ∈ H∗ (E) such that u(x) v(y) = u(y) v(x). The following remark is often useful in applying the abstract harmonic theory to the linear partial differential equations without zero order terms. Remark 6.3.2. If the constant functions are H-harmonic in E, then the positivity axiom is satisfied. Moreover, the separation assumption (A4) can be restated as follows. (A4)’ For every x, y ∈ E with x = y, there exists u ∈ H∗ (E) such that u(x) = u(y). Indeed, (A4)’ imply (A4) by taking in the latter v ≡ 1. 1 An H-hyperharmonic function taking real values in a dense subset of its domain will

be called H-superharmonic, see Section 6.5. Then we can say that h0 , −k0 are Hsuperharmonic.

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6 Abstract Harmonic Spaces

If Ω ∈ T , V ⊆ V ⊆ Ω is H-regular and u ∈ H(Ω), it readily follows from the property (R2) of the regular sets that V u|V = Hu| . ∂V

Hence u(x) =

∂V

∀ x ∈ V.

u dμVx

An important consequence of the regularity axiom is that this property characterizes the H-harmonic functions. Proposition 6.3.3 (Characterization of H-harmonicity). Let (E, H) be a harmonic space. Let Ω ∈ T and u ∈ C(Ω, R) be such that u dμVx for every x ∈ V (6.7) u(x) = ∂V

and for every H-regular open set V ⊆ V ⊆ Ω. Then u ∈ H(Ω). Proof. Let us define V := {V ∈ Tr | V ⊆ Ω}. We claim that Ω=

V.

(6.8)

V ∈V

Taking this claim for granted for a moment, from (6.7) we get V u|V = Hu| ∂V

∀ V ∈ V.

Then u is H-harmonic in V for every V ∈ V. Since H is a sheaf, (6.8) now implies that u ∈ H(Ω). So we are left with the proof of (6.8). For every x ∈ Ω, there exists a regular open set W contained in Ω and containing x (by (A3)). Since ∂W is compact, there exists V ∈ Tr such that V ⊆ W,

x ∈ V,

V ∩ ∂W = ∅

(we are again using (A3), together with the Hausdorff separation property of the topology T ). Consequently, x ∈ V ⊆ V ⊆ W ⊆ Ω. This shows (6.8) and completes the proof of the proposition. Exercise 6.3.4. For every open set Ω ⊆ E, one has H∗ (Ω) ∩ H∗ (Ω) = H(Ω). Corollary 6.3.5. Let (E, H) be a harmonic space. Let {un }n∈N be a sequence in H(Ω), Ω ∈ T , such that un → u,

as n → ∞,

uniformly on every compact subset of Ω. Then u ∈ H(Ω).

6.3 Harmonic Spaces

347

Proof. First of all, u is continuous in Ω, since it is the limit of a sequence of continuous functions, locally uniformly convergent. Moreover, for every regular open set V ⊆ V ⊆ Ω and for every x ∈ V , we have u(x) = lim un (x) = lim un dμVx (un ∈ H(Ω)) n→∞ n→∞ ∂V lim un dμVx (un is uniformly convergent on V ) = ∂V n→∞ u dμVx . = ∂V

Then, by Proposition 6.3.3, u ∈ H(Ω).

6.3.1 Directed Families of Harmonic and Hyperharmonic Functions The results proved in the previous section are the ad hoc tools to show the following two theorems, which are consequences of the convergence axiom (A2) in Section 6.3. Theorem 6.3.6 (Up directed families in H(Ω)). Let (E, H) be a harmonic space, and let Ω be an open subset of E. Suppose we are given a family F ⊆ H(Ω) such that: (i) F is up directed, (ii) v := supu∈F u is finite in a dense subset of Ω. Then v is H-harmonic in Ω. Proof. First, we prove that v is continuous in Ω by using Cornea’s Theorem 6.2.11. Let (fn ) be a monotone increasing sequence in F. Obviously, f := sup fn ≤ sup u, u∈F

n∈N

so that, by hypothesis (ii), f < ∞ in a dense subset of Ω. The convergence axiom (A2) implies that f ∈ H(Ω), hence it is continuous. This shows that F satisfies the assumptions of Theorem 6.2.11, so that v ∈ C(Ω, R). Let us now take an Hregular open set V ⊆ V ⊆ Ω and a point x ∈ Ω. Since ∂V is compact and μVx is a Radon measure in ∂V , by Levi–Cartan’s Theorem 6.2.10 we obtain sup u dμVx = u dμVx = v dμVx . v(x) = sup u(x) = sup u∈F

u∈F

∂V

Then, by Proposition 6.3.3, u ∈ H(Ω).

∂V

u∈F

∂V

Corollary 6.3.7. In the hypotheses of the previous theorem, let F ⊆ H(Ω) be such that F ↓ and inf F > −∞ in a dense subset of Ω. Then inf F ∈ H(Ω). Proof. Apply Theorem 6.3.6 to the family −F := {−u : u ∈ F}.

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6 Abstract Harmonic Spaces

Theorem 6.3.8 (Up directed families in H∗ (Ω)). Let (E, H) be a harmonic space, and let Ω ⊆ E be open. If F ⊆ H∗ (Ω) and F ↑, then v := sup u ∈ H∗ (Ω). u∈F

Proof. The function v is l.s.c. Moreover, for every H-regular open set V ⊆ V ⊆ Ω and for every x ∈ V , by using Theorem 6.2.10, we obtain v(x) = sup u(x) ≥ sup sup u dμVx , u dμVx = so that v(x) ≥

u∈F

∂V

u∈F

∂V

∂V

u∈F

v dμVx . This proves the theorem.

Exercise 6.3.9. Let V ⊆ E be an H-regular open set, and let f : ∂V → R be l.s.c. and such that f dμVx < ∞ ∀ x ∈ V.

∂V

Then x → ∂V f is H-harmonic in V .

(Hint: ∂V f dμVx = sup{HϕV (x) : ϕ ∈ C(∂V , R), ϕ ≤ f }.) dμVx

Exercise 6.3.10. Let V ⊆ E be an H-regular open set, and let f : ∂V → R

be μVx -summable for every x ∈ V . Then x → ∂V f dμVx is H-harmonic in V . (Hint: Use Ex. 6.3.9 and Corollary 6.3.7.)

6.4 B-hyperharmonic Functions. Minimum Principle Definition 6.4.1 (B-hyperharmonic function). Let (E, H) be a harmonic space and let Ω be an open subset of E. Assume that, for every x ∈ Ω, we are given a basis B(x) of H-regular neighborhoods of x with closure contained in Ω. A l.s.c. function u : Ω → ]−∞, ∞] will be called B-hyperharmonic in Ω if

u(x) ≥ ∂V

u dμVx

for every x ∈ Ω and every V ∈ B(x). We shall denote by B-H∗ (Ω) the family of the B-hyperharmonic functions in Ω. Obviously, H∗ (Ω) ⊆ B-H∗ (Ω). We will see in Corollary 6.4.9 that H∗ (Ω) = B-H∗ (Ω). An application of the Levi–Cartan Theorem 6.2.10 gives the following lemma.

6.4 B-hyperharmonic Functions. Minimum Principle

349

Lemma 6.4.2. Let K ⊆ Ω be compact, and let μ be a measure supported inside K. Suppose we are given a family K of compact subset of K such that: (i) K ↓, that is, for every A, B ∈ K, there exists C ∈ K contained in A ∩ B; (ii) supp(μ) ⊆ A for every A ∈ K. Then supp(μ) ⊆

A.

A∈K

Proof. Let us put B := A∈K A. We have to show that μ(Ω \ A) = 0. Obviously, since μ(Ω \ K) = 0 and B ⊆ K, this is equivalent to prove that μ(B) = μ(K). Observing that F := {χA : A ∈ K} is a down directed family of u.s.c functions, we may apply the Levi–Cartan Theorem 6.2.10 to obtain inf χA dμ = inf μ(B) = χB dμ = χA dμ = inf μ(A) = μ(K). K

K

A∈K

A∈K K

A∈K

The last equality follows from the assumptions supp(μ) ⊆ A ⊆ K for every A ∈ K. This ends the proof. Definition 6.4.3 (B-invariant set). Let A ⊆ Ω be relatively closed in Ω, that is A = Ω ∩ A0 with A0 closed in E. We say that A is B-invariant if for every y ∈ A and every V ∈ B(y), one has supp(μVy ) ⊆ A. We now show several lemmas needed for proving a minimum principle for Bhyperharmonic functions, which will be the main achievement of this section. The first one is a trivial corollary of Lemma 6.4.2. Lemma 6.4.4. Let K ⊆ Ω be compact, and let K be a family of closed B-invariant subset of K. Assume K ↓. Then

A B= A∈K

is B-invariant. The second lemma shows that the zero set of a non-negative B-hyperharmonic function is a B-invariant set. Lemma 6.4.5. Let u ∈ B-H∗ (Ω), u ≥ 0. Then u−1 (0) := {x ∈ Ω : u(x) = 0} is B-invariant. Proof. First of all, since u is l.s.c. and u−1 (0) = {x ∈ Ω : u(x) ≤ 0},

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6 Abstract Harmonic Spaces

the set u−1 (0) is relatively closed in Ω. Moreover, if y ∈ u−1 (0) and V ∈ B(y), we have 0 = u(y) ≥ u dμVy (for u ∈ H∗ (Ω)) ≥0

V

(recall that u ≥ 0).

Then u = 0 μVy -almost everywhere on ∂V . That is, since supp(μVy ) ⊆ ∂V ⊆ Ω, we have 0 = μVy (∂V \ u−1 (0)) = μVy (Ω \ u−1 (0)). This means that supp(μVy ) ⊆ u−1 (0), and the lemma is proved. The argument used in this proof can be used for proving the following assertion. Lemma 6.4.6. Let A ⊆ Ω be B-invariant, and let u ∈ B-H∗ (Ω) such that u ≥ 0 in A. Then A0 := {x ∈ A : u(x) = 0} is B-invariant. The next lemma shows that every non-empty compact B-invariant set contains a non-empty minimal B-invariant set. Indeed, we have the following result. Lemma 6.4.7 (Minimal B-invariant set). Let K ⊆ Ω be a non-empty compact B-invariant set. Then there exists A0 ⊆ K such that: (i) A0 is non-empty and B-invariant, (ii) A0 is minimal, that is, A ⊆ A0 , A = ∅, A B-invariant ⇒ A = A0 . Proof. Define E := {A ⊆ K : A closed, A = ∅, A B-invariant}. We have to prove that (E, ⊆) has a minimal element. We will use Zorn’s lemma stating the existence of such an element if every linearly ordered family of elements of E has a minorant in E. Then, let E0 ⊆ E be linearly ordered with respect to the inclusion ⊆. Define

A. B := A∈E0

B is compact and non-empty, since every finite intersection of elements of E0 is nonempty, for (E0 , ⊆) is linearly ordered. Moreover, by Lemma 6.4.4, B is invariant. Then B ∈ E0 and, obviously, B ⊆ A for every A ∈ E0 . Thus E0 has a minorant in E. The lemma is proved. Finally, we are ready to prove our minimum principle.

6.4 B-hyperharmonic Functions. Minimum Principle

351

Theorem 6.4.8 (Minimum principle for B-H∗ functions). Let Ω be a relatively compact open set in a harmonic space (E, H), and let u ∈ B-H∗ (Ω). Assume lim inf u(x) ≥ 0

∀ y ∈ ∂Ω.

x→y

(6.9)

Then u(x) ≥ 0 for every x ∈ Ω. Proof. Argue by contradiction and assume the existence of a point x ∈ Ω such that u(x) < 0. Let h0 ∈ H∗ (Ω) be such that infΩ h0 > 0 and h0 < ∞ at any point of Ω (see the positivity axiom (A1)). Let us prove the following claim: there exists a strictly positive real constant α such that u + αh0 ≥ 0 in Ω and K := {x ∈ Ω : u(x) + αh0 (x) = 0} is compact. Consider the function h :=

1 max{0, −u}. h0

Condition (6.9), together with the upper semicontinuity of h, the compactness of Ω, and the condition u(x) < 0, implies the existence of maxΩ h =: α, with 0 < α < ∞. Moreover: (i) u + αh0 ≥ u + h h0 = u + max{0, −u} ≥ 0 in Ω. (ii) K := {x ∈ Ω : u(x) + αh0 (x) = 0} is non-empty, since the points of K are the maximum points of h. (iii) K = K. Indeed, first notice that K is relatively closed in Ω being the null set of a non-negative l.s.c. function. Then K = Ω ∩ K. If we prove that K ⊆ Ω, we are done. For this it is enough to show that if y ∈ ∂Ω, then y ∈ / K. Now, condition (6.9) implies lim inf(u(x) + αh0 ) ≥ α inf h0 > 0, x→y

Ω

so that u + αh0 > 0 in a suitable neighborhood of y. This, obviously, implies that y ∈ / K. (iv) K is compact, since K = K ⊆ Ω and Ω is compact. By Lemma 6.4.5, K is B-invariant and, by Lemma 6.4.6, there exists A0 ⊆ K, compact, non-empty, B-invariant and minimal. We claim that (6.10) A0 = {x0 } for a suitable x0 ∈ Ω. Assuming this claim for true for a moment, we show how to reach a contradiction, thus proving the theorem. Let V ∈ B(x0 ). Since A0 is Binvariant, it has to be supp(μVx0 ) ⊆ A0 = {x0 }. On the other hand, again by the positivity axiom, we can find k0 ∈ H∗ (Ω) such that k0 (x0 ) > 0. It follows that 0 < k0 (x0 ) ≤ k0 dμΩ x0 , ∂V

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6 Abstract Harmonic Spaces

so that μVx0 (∂V ) > 0. This is impossible, since supp(μVx0 ) ⊆ {x0 } and x0 ∈ / ∂V . Then, we are left with the proof of (6.10). By contradiction, assume it is false. Then, since H∗ (E) separates different points of E, there exists w ∈ H∗ (E) such that w ≡ ∞. Fix such a hyperharmonic function and choose a real β > 0 such that βu + w < 0 somewhere in A0 (recall that if u + αh0 ≡ 0 in A0 and α > 0, then u < 0 in A0 ). Letting βu + w , −γ := min A0 h0 we have γ > 0, βu + w + γ h0 ≥ 0 in A0 and A1 := {x ∈ A0 : (βu + w + γ h0 )(x) = 0} = ∅. By Lemma 6.4.6, A1 is B-invariant. Then, since A1 ⊆ A0 and A0 is minimal, it has to be A1 = A0 , so that βu + w + γ h0 ≡ 0 in A0 . On the other hand, being A0 ⊆ K, u + αh0 ≡ 0 in A0 . It follows that w = (βα − γ )h0 := λh0 in A0 . Summing up: for every w ∈ H∗ (E), w ≡ ∞ in A0 , there exists λ ∈ R such that w = λh0 . This implies that H∗ (E) cannot separate different points of A0 , in contradiction with the separation axiom (A4). Then A0 has to be a singleton. This proves (6.10) and completes the proof of the theorem. We now show some consequences of the previous theorem. First of all, we show that the B-hyperharmonicity is equivalent to the hyperharmonicity. Corollary 6.4.9. Let Ω be an arbitrary open set in E, and let u ∈ B-H∗ (Ω) for a suitable B. Then u ∈ H∗ (Ω). Proof. Let V be an H-regular set with the closure contained in Ω. We have to prove that u(x) ≥ u dμVx ∀ x ∈ V. (6.11) ∂V

For every z ∈ V , define BV (z) = {W ∈ B(z) : W ⊆ V }. Let us now choose φ ∈ C(∂V , R), φ ≤ u|∂V . The function u − HφV is BV hyperharmonic in V . Moreover, lim inf (u − HφV )(x) ≥ u(y) − φ(y) ≥ 0

V x→y

Then, by Theorem 6.4.8, u − HφV ≥ 0 in V , that is, u(x) ≥ φ dμVx ∀x ∈V

∀ y ∈ ∂V .

(6.12)

∂V

and for every φ ∈ C(∂V , R), φ ≤ u|∂V . Taking the supremum at the right-hand side with respect to these functions φ’s, we obtain (6.11).

6.5 Subharmonic and Superharmonic Functions. Perron Families

353

Corollary 6.4.10. Let (E, H) be a harmonic space, and let Ω ⊆ E be open. The map Ω → H∗ (Ω) is a sheaf of functions. Proof. Property (6.2a) (page 340) is trivial. Let us prove (6.2b). Let (Ωα )α∈A be a family of open subsets of E, and let Ωα , u : Ω → [−∞, ∞], Ω := α∈A

be a function such that u|Ωα ∈ H∗ (Ωα ) for every α ∈ A. Then u is l.s.c and u > −∞ at any point. For every x ∈ Ω, define B(x) := {V ∈ Tr : x ∈ V ⊆ V ⊆ Ωα for a suitable α ∈ A}. One readily verifies that u ∈ B-H∗ (Ω). Then, by the previous corollary, we infer that u ∈ H∗ (Ω). Exercise 6.4.11. Let (E, H) be a harmonic space. Let Ω ⊆ E be open, and let x0 ∈ Ω. If a function u : Ω → ]−∞, ∞] satisfies: (i) u ∈ H∗ (Ω \ {x0 }), (ii) lim supx→x0 u(x) = ∞, then u ∈ H∗ (Ω).

6.5 Subharmonic and Superharmonic Functions. Perron Families Definition 6.5.1 (H-super- and H-sub-harmonic function). Let (E, H) be a harmonic space, and let Ω ⊆ E be open. A function u ∈ H∗ (Ω) will be said Hsuperharmonic if, for every H-regular open set V ⊆ V ⊆ Ω, the function V x → u dμVx ∂V

is H-harmonic in V . The set of the H-superharmonic functions in Ω will be denoted by S(Ω). A function v : Ω → [−∞, ∞[ will be said H-subharmonic in Ω if −v ∈ S(Ω). We shall denote by S(Ω) := −S(Ω) the set of the H-subharmonic functions in Ω. From Theorem 6.3.6 we readily obtain the following characterization of the Hsuperharmonic functions.

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6 Abstract Harmonic Spaces

Theorem 6.5.2 (Characterization of S(Ω)). Let u ∈ H∗ (Ω). Then u ∈ S(Ω) if and only if D := {x ∈ Ω : u(x) < ∞} is dense in Ω, that is, D ⊇ Ω. Proof. By contradiction, suppose D is not dense in Ω. Then there exists an Hregular open set V such that V ⊆ {x ∈ Ω : u(x) = ∞}. (6.13)

Invoking the positivity axiom, we may assume ∂V dμVx > 0 for every x ∈ V . Indeed, by this axiom, one can suppose the existence of a strictly positive H∗ function k0 in an open set containing V . This implies k0 dμVx ≥ k0 (x) > 0, ∂V

hence μVx (∂V ) > 0 for every x ∈ Ω. Then, from (6.13) we get u dμVx = ∞ ∀ x ∈ V, ∂V

showing that x → ∂V u dμVx is not H-harmonic in V . Vice versa, suppose D ⊇ Ω, and let V ⊆ V ⊆ Ω be an H-regular set. Consider F := {HφV : φ ∈ C(∂V , R), φ ≤ u|∂V }. F is an up directed family of H-harmonic functions in V such that sup F ≤ u < ∞ in D, since u ∈ H∗ (Ω). Theorem 6.3.6 implies that v := sup F is H-harmonic in V . This completes the proof, since u dμVx , x ∈ V . v(x) = ∂V

A useful criterion for subharmonicity is given by the next corollary. Corollary 6.5.3. Let (E, H) be a harmonic space and let Ω ⊆ E be open. Let u ∈ H∗ (Ω). Assume there exists v ∈ S(Ω) such that u ≤ v. Then u ∈ S(Ω). Proof. Notice that {x ∈ Ω : u(x) < ∞} ⊇ {x ∈ Ω : v(x) < ∞} and apply Theorem 6.5.2. We also have the following result.

6.5 Subharmonic and Superharmonic Functions. Perron Families

355

Proposition 6.5.4. Let u, v ∈ S(Ω). Then: (i) λu ∈ S(Ω) for every λ ≥ 0, (ii) u + v ∈ S(Ω), (iii) min{u, v} ∈ S(Ω). Proof. Exercise.

Let us now introduce the following crucial definition. Definition 6.5.5 (Perron-regularization). Let (E, H) be a harmonic space, and let Ω ⊆ E be open. Given u ∈ H∗ (Ω) and an H-regular set V ⊆ V ⊆ Ω, define u(x), x∈ / V, uV : Ω → ]−∞, ∞], uV (x) :=

V x ∈ V. ∂V u dμx , The function uV will be called the Perron-regularization of u related to V . The main properties of the Perron-regularization are given in by the following theorem. Theorem 6.5.6 (Properties of the Perron-regularization). Suppose that u ∈ H∗ (Ω) and V ⊆ V ⊆ Ω is an open H-regular set, then: (i) uV ≤ u, (ii) uV ∈ H∗ (Ω), (iii) uV ≤ vV if u, v ∈ H∗ (Ω) and u ≤ v. Moreover, if u ∈ S(Ω), then (iv) uV ∈ S(Ω) and u|V ∈ H(V ). Proof. (i) It simply follows from the very definition of hyperharmonic function. (ii) We split the proof of this property in two steps. (ii-a) uV is lower semicontinuous. For this, we have to prove that lim inf uV (x) ≥ uV (y) x→y

for every y ∈ Ω.

The statement holds true if y ∈ V , since, in this case, uV = u in a neighborhood of y. On the other hand, for y ∈ V , we have uV (y) = u dμVy = sup{HϕV | ϕ ∈ C(∂V , R), ϕ ≤ u|∂V }. (6.14) ∂V

Then uV is H-hyperharmonic in V (see Theorem 6.3.8), hence lower semicontinuous at any point y ∈ V . Let us now suppose y ∈ ∂V . In this case uV (y) = u(y), so that

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6 Abstract Harmonic Spaces

lim inf uV (x) = lim inf u(x) ≥ u(y) = uV (y).

V x→y

V x→y

Then we only have to prove that lim inf uV (x) ≥ u(y).

V x→y

(6.15)

Let ϕ ∈ C(∂V , R), ϕ ≤ u|∂V . Then uV ≥ HϕV in V , so that lim inf uV (x) ≥ lim inf HϕV (x) = ϕ(y).

V x→y

V x→y

Since u|∂V = sup{ϕ ∈ C(∂V , R) : ϕ ≤ u|∂V }, (6.15) follows from the previous inequality. (ii-b) uV ∈ B-H∗ (Ω) with B defined as follows ⎧ ⎪ if x ∈ V , ⎨ {B : B H-regular open set, x ∈ B ⊆ B ⊆ V } B(x) = {B : B H-regular open set, x ∈ B ⊆ B ⊆ Ω \ V } if x ∈ / V, ⎪ ⎩ {B : B H-regular open set, x ∈ B ⊆ B ⊆ Ω} if x ∈ ∂V . Since uV is H-hyperharmonic in Ω \ V and in V (see (6.14)), we only have to prove that uV (y) ≥ uV dμB ∀y ∈ ∂B, ∀ B ∈ B(y). y ∂B

We have uV (y) = u(y) (since y ∈ ∂V ) ≥ u dμB (u ∈ H∗ (Ω)) y ∂B uV dμB (u ≥ uV ). ≥ y ∂B

B-H∗ (Ω).

Thus, by Corollary 6.4.9, u ∈ H∗ (Ω). This shows that uV ∈ (iii) This is quite obvious. (iv) Since uV ∈ H∗ (Ω) and uV ≤ u, if u ∈ S(Ω) then uV ∈ S(Ω) (see Corollary 6.5.3). The harmonicity of uV in V follows from the very definition of subharmonic function. We are now ready to give another crucial definition. Definition 6.5.7 (Perron family). Let (E, H) be a harmonic space, and let Ω ⊆ E be open. A family of functions F ⊆ H∗ (Ω) will be said a Perron family if the following conditions are satisfied: (i) F ↓, that is, F is down directed, (ii) F has an H-subharmonic minorant, i.e. there exists v0 ∈ S(Ω) such that v0 ≤ u

for every u ∈ F,

6.5 Subharmonic and Superharmonic Functions. Perron Families

357

(iii) uV ∈ F for every u ∈ F and for every H-regular open set V ⊆ V ⊆ Ω, (iv) F contains at least one H-superharmonic function, i.e. there exists u0 ∈ F such that u0 ∈ S(Ω). The following important result holds. Theorem 6.5.8 (The fundamental theorem on Perron families). Let (E, H) be a harmonic space, and let F be a Perron family on an open set Ω ⊆ E. Then u := inf F ∈ H(Ω). Proof. Let V ⊆ V ⊆ Ω be an arbitrary fixed H-regular set. It is enough to prove that u is H-harmonic in V . Define FV = {uV |V : u ∈ F, u ∈ S(Ω)} and prove the following list of claims: (C1) FV = ∅. This follows from assumption (iv) in Definition 6.5.7. (C2) FV ⊆ H(V ). This is granted by Theorem 6.5.6-(iv). (C3) FV ↓. Indeed, let u, v ∈ F ∩ S(Ω). Since F is down directed, there exists w ∈ F such that w ≤ min{u, v}. The function w is H-superharmonic, for it is bounded from above by min{u, v} which is H-superharmonic (see Corollary 6.5.3). It follows that wV ∈ FV and wV ≤ min{uV , vV }, proving that FV ↓. (C4) FV has an H-subharmonic minorant, so that inf FV > −∞ in a dense subset of Ω. This directly follows from assumptions (ii) and (iii) in Definition 6.5.7. (C5) u|V = inf FV . Indeed, the inequality u|V ≤ inf FV is trivial for FV ⊆ {u|V : u ∈ F}. On the other hand, let u0 ∈ F ∩ S(Ω) be the function given by assumption (iv) in Definition 6.5.7. For every u ∈ F, take v ∈ F satisfying v ≤ min{u, u0 } (F ↓). Since u0 ∈ S(Ω), also v ∈ S(Ω). It follows that v|V ∈ FV and that inf FV ≤ vV ≤ v|V ≤ u|V . Therefore, inf FV ≤ u|V

∀u ∈ F,

and the inequality inf FV ≤ u|V follows. Summing up: (C1)–(C4) and Corollary 6.3.7 imply inf FV ∈ H(V ). Hence, from (C5) we get u|V ∈ H(V ). The proof is complete.

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6 Abstract Harmonic Spaces

6.6 Harmonic Majorants and Minorants From the fundamental theorem on Perron families, one easily obtains conditions for the existence of least harmonic majorants of subharmonic functions and of greatest harmonic minorants of superharmonic functions. Theorem 6.6.1 (The least H-superharmonic majorant). Let (E, H) be a harmonic space, and let Ω ⊆ E be open. Let v ∈ S(Ω) be such that v ≤ u0 for a suitable u0 ∈ S(Ω). Then v has a least H-superharmonic majorant h on Ω, and h ∈ H(Ω). Proof. Consider the family F := {u ∈ S(Ω) : v ≤ u ≤ u0 }. If u ∈ F and V is an H-regular open set with V ⊆ Ω, we have v ≤ vV ≤ uV ≤ u ≤ u0 . Hence uV ∈ F. It follows that F is a Perron family, with the H-subharmonic minorant v and the H-superharmonic majorant u0 . Then h := inf F ∈ H(Ω), by Theorem 6.5.8, and h ≥ v in Ω. Hence h is the least element of F and h is H-harmonic in Ω. Corollary 6.6.2. In a harmonic space (E, H), every H-superharmonic function with an H-subharmonic minorant in an open set Ω has a greatest H-subharmonic minorant which is H-harmonic in Ω. The following proposition holds. Proposition 6.6.3. Let Ω ⊆ E be open, and let v1 , v2 ∈ S(Ω). Assume v1 , v2 have an H-superharmonic majorant. Then v1 + v2 has a least H-superharmonic majorant given by h1 + h2 , where hi is the least H-(super)harmonic majorant of vi . Proof. Let h be the least H-(super)harmonic majorant of v1 + v2 . Obviously, h ≤ h1 + h2 . On the other hand, being h ≥ v1 + v2 , the function h − v1 is an Hsuperharmonic majorant of v2 . Hence h − v1 ≥ h2 . It follows that h − h2 ≥ v1 and h − h2 is H-harmonic in Ω. Then h − h2 ≥ h1 , that is h ≥ h1 + h2 . This completes the proof.

6.7 The Perron–Wiener–Brelot Operator

359

6.7 The Perron–Wiener–Brelot Operator Throughout this section, Ω will denote an open set, with compact closure and nonempty boundary, in a harmonic space (E, H). We shall construct the Perron–Wiener– Brelot operator f → HfΩ from a suitable linear set of real extended functions f : ∂Ω → [−∞, ∞] to the linear space H(Ω) of the H-harmonic functions in Ω. When the H-Dirichlet problem u ∈ H(Ω), (H-D) limx→y u(x) = f (y) ∀ y ∈ ∂Ω, has a solution u, it will turn out that u = HfΩ . For this reason, sometimes we shall call HfΩ the generalized solution to (H-D) in the sense of Perron–Wiener–Brelot. Our construction, in the case of classic harmonic functions, i.e. of the solutions to the “usual” Laplace equation Δu = 0, goes back to O. Perron, N. Wiener and M. Brelot. Definition 6.7.1 (Upper and lower functions). Let (E, H) be a harmonic space, and let Ω ⊆ E be open and such that Ω is compact and ∂Ω = ∅. Given a function f : ∂Ω → [−∞, ∞], we set2 Ω U f := u ∈ H∗ (Ω) : lim inf u ≥ f, inf u > −∞ ∂Ω

and

∗ UΩ f := v ∈ −H (Ω) : lim sup v ≤ f, sup v < ∞ . ∂Ω

Ω The families U f

and U Ω f will be called, respectively, the family of the upper functions and of the lower functions related to f and Ω.

The function u ≡ ∞ (v ≡ −∞, respectively) is an upper function (lower funcΩ tion, respectively). Therefore, U f = ∅ (U Ω f = ∅, respectively). Definition 6.7.2 (Upper and lower solutions). With the hypotheses and notation of the previous definition, the real extended functions Ω

Ω

H f := inf U f ,

Ω HΩ f := sup U f

will be called the upper solution and the lower solution, respectively, to the problem (H-D). 2 The notation

lim inf u ≥ f ∂Ω

means lim inf u(x) ≥ f (y) x→y

Analogous meaning for lim sup∂Ω u ≤ f .

∀ y ∈ ∂Ω.

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6 Abstract Harmonic Spaces

It has to be noticed that

Ω

HΩ f = −H −f , Ω

since U Ω f = −U −f . We also have the following proposition, an easy consequence of the minimum principle for the H-hyperharmonic functions. Proposition 6.7.3. In the hypotheses of Definition 6.7.1, for every f : ∂Ω → [−∞, ∞], one has Ω (6.16) HΩ f ≤ Hf . Ω

∗ Proof. For any u ∈ U f and v ∈ U Ω f , one has u − v ∈ H (Ω) and

lim inf(u − v) ≥ 0. ∂Ω

Then, from Theorem 6.4.8 (the minimum principle), u − v ≥ 0 in Ω. Thus u≥v

Ω

∀ u ∈ Uf ,

∀ v ∈ UΩ f ,

and (6.16) readily follows. The next proposition is a straightforward consequence of the definitions of upper and lower solution. Proposition 6.7.4. Let f, g : ∂Ω → [−∞, ∞], α ∈ R, α > 0. Then: Ω

Ω

Ω (i) f ≤ g ⇒ H f ≤ H g , H Ω f ≤ Hg , Ω

Ω

Ω

Ω

Ω

Ω Ω (ii) H f +g ≤ H f + H g , H Ω f +g ≥ H f + H g , whenever the sums are defined. Ω

Ω Ω (iii) H αf = αH f , H Ω αf = αH f , H −αf = −αH f , Ω

(iv) f ≥ 0 ⇒ H f ≥ 0, H Ω f ≥ 0. Proof. It is left as an exercise. (Hint: to prove (iv) use the minimum principle and the fact that u ≡ 0 is H-harmonic, hence both H-hyperharmonic and H-hypoharmo nic.) In Section 6.9, we will use the following Beppo Levi-type property of the map Ω f →

H f , which we leave here as an exercise. Exercise 6.7.5. Let Ω ⊆ E be a relatively compact open set, and let (fn ) be a sequence of real extended functions fn : ∂Ω → [−∞, ∞] such that fn f Then

and

Ω

H fn ∈ H(Ω) Ω

∀n ∈ N.

Ω

H fn H f . Ω

Hint: h = supn H fn ∈ H∗ (Ω), see Theorem 6.3.8. For any fixed x ∈ Ω and ε > 0 Ω

Ω

choose un ∈ U fn such that un (x) < H fn (x) +

ε 2n .

Then

6.7 The Perron–Wiener–Brelot Operator

u=h+

361

∞ Ω Ω (un − H fn ) ∈ U f . n=1

Definition 6.7.6 (Resolutive function). Let (E, H) be a harmonic space, and let Ω ⊆ E be an open set with compact closure and non-empty boundary. A real extended function f : ∂Ω → [−∞, ∞] will be said resolutive if: Ω

(i) H f = H Ω f , Ω

(ii) H f ∈ H(Ω). In this case, we set Ω

HfΩ := H f (= H Ω f ), and we say that HfΩ is the generalized solution, in the sense of Perron–Wiener– Brelot, to the problem (H-D). We also call HfΩ the PWB function related to Ω and f . The set of the resolutive functions f : ∂Ω → [−∞, ∞] will be denoted by R(∂Ω), R(∂Ω) := {f : ∂Ω → [−∞, ∞] | f is resolutive}. (6.17) The connection between HfΩ and the H-Dirichlet problem (H-D) is showed by the following proposition. Proposition 6.7.7. Let the hypotheses of Definition 6.7.6 hold. Let f : ∂Ω → [−∞, ∞] be a bounded function. Then the following statements are equivalent: (i) f is resolutive and limx→y HfΩ (x) = f (y) for every y ∈ ∂Ω. (ii) There exists u ∈ H(Ω) such that limx→y u(x) = f (y) for every y ∈ ∂Ω. In this latter case, u = HfΩ . Proof. (i) ⇒ (ii). This is trivial. (ii) ⇒ (i). Just remark that u is bounded from above and below, and that u ∈ Ω Uf ∩ UΩ f . We would like to explicitly remark that R(∂Ω) is not empty, since it contains the null function. From Proposition 6.7.4 it follows that: (i) if f, g ∈ R(∂Ω) and f + g is defined, then f + g ∈ R(∂Ω) and HfΩ+g = HfΩ + HgΩ , Ω = cH Ω . (ii) if f ∈ R(∂Ω) and c ∈ R, then c f ∈ R(∂Ω) and Hcf f In particular,

R∞ (∂Ω) := R(∂Ω) ∩ L∞ (∂Ω)

is a real vector space, and the map R∞ (∂Ω) f → HfΩ ∈ C(Ω, R) is linear and monotone.

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We want to show that R(∂Ω) contains the restrictions to the boundary of Ω of the continuous superharmonic functions. We first show a proposition having an independent interest, an easy consequence of the fundamental theorem on the Perron families (Theorem 6.5.8, page 357). Proposition 6.7.8. Let Ω ⊆ E be an open set with compact closure and non-empty boundary. Let f : ∂Ω → [−∞, ∞]. Assume that there exist Ω

u0 ∈ U f ∩ S(Ω) Then

and

v0 ∈ U Ω f ∩ S(Ω).

(6.18)

Ω

Hf ,HΩ f ∈ H(Ω). Ω

Proof. Using (6.18), we recognize that U f is a Perron family. Hence, by Theorem 6.5.8, Ω H f ∈ H(Ω). Ω

From (6.18) it also follows that U −f is a Perron family, thus Ω

Ω HΩ f = sup U f = − inf U −f ∈ H(Ω).

This ends the proof.

Theorem 6.7.9 (Resolutivity. I). Let (E, H) be a harmonic space, and let Ω ⊆ E be an open set with compact closure and non-empty boundary. If u ∈ C(Ω, R) and u|Ω ∈ S(Ω), then f := u|∂Ω ∈ R(∂Ω). Proof. By the positivity axiom (A1), there exists h0 ∈ S(Ω) such that infΩ h0 > 0. Denote max{0, −u} . λ := infΩ h0 Then u ≥ −λh0 in Ω and −λh0 ∈ S(Ω) ∩ U Ω f . On the other hand, Ω

u|Ω ∈ S(Ω) ∩ U f . By Proposition 6.7.8, Ω

Hf ,HΩ f ∈ H(Ω). Ω

Moreover, H f ≤ u|Ω so that, since u is continuous up to ∂Ω, Ω

Hf ∈ UΩ f .

6.8 S-harmonic Spaces: Wiener Resolutivity Theorem Ω

Thus H f

363

≤ HΩ f . The opposite inequality always being true, this implies

Ω

Hf = HΩ f . Summing up, Ω

Hf = HΩ f ∈ H(Ω), that is, f is resolutive. This last theorem would achieve a full strength if we knew that the family of continuous subharmonic functions is sufficiently rich. This happens in the S-harmonic spaces introduced in the following section.

6.8 S-harmonic Spaces: Wiener Resolutivity Theorem Definition 6.8.1 (S-harmonic space). A harmonic space (E, H) will be said Sharmonic if the family +

S c (E) := {u ∈ S(E) ∩ C(E, R) : u ≥ 0} separates the points of E, that is, for every x, y ∈ E, x = y, there exists +

u, v ∈ S c (E) such that u(x)v(y) = u(y)v(x).

(6.19)

In a S-harmonic space, the positivity property of the axiom (A1) takes a stronger form. Proposition 6.8.2 (Positivity axiom in a S-harmonic space). Let (E, H) be a S+ harmonic space. For every compact set K ⊆ E, there exists w ∈ S c (E) such that inf w > 0. K

+

Proof. Since S c (E) separates the points of E, for every x ∈ K, there exists + ux ∈ S c (E) such that ux (x) > 0. By the lower continuity of u, we can find an open neighborhood Vx of x such that infV ux > 0. Since K is compact, there exist x1 , . . . , xp ∈ K such that p K⊆ Vxj . j =1

Then, if we define w :=

p j =1

+

we have w ∈ S c (E) and infK w > 0.

uxj ,

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6 Abstract Harmonic Spaces

A crucial property of the S-harmonic spaces is given by the next proposition. Proposition 6.8.3. Let (E, H) be a S-harmonic space. For every compact set K ⊆ + E, for every f ∈ C(K, R) and for every ε > 0, there exist u, v ∈ S c (E) such that sup |f − (u − v)| < ε. K

Proof. Define

+

A := {u − v : u, v ∈ S c (E)}. It is immediate to recognize that A is a linear subspace of C(E, R). Moreover, since (E, H) is S-harmonic, A separates the points of E. Finally, if p = u − v ∈ A, then also p + := max{p, 0} belongs to A, since p + = u − min{u, v}. Then, by the Stone–Weierstrass theorem,3 for every f ∈ C(K, R) and ε > 0, there exists p ∈ A such that sup |f − p| < ε. K

This ends the proof.

We are now ready to prove the resolutivity of any continuous function. Theorem 6.8.4 (Wiener resolutivity theorem). Let (E, H) be a S-harmonic space, and let Ω ⊆ E be an open set with compact closure and non-empty boundary. Every continuous function f : ∂Ω → R is resolutive. +

Proof. By Theorem 6.7.9, if u ∈ S c (E), then u|∂Ω is resolutive. As a consequence, every function + p = (u − v)|∂Ω , u, v ∈ S c (E), +

is resolutive. By Proposition 6.8.2, for every ε > 0, there exist uε , vε ∈ S c (E) such that (6.20a) |f − p| < ε on ∂Ω, p = pε := (uε − vε )|∂Ω . +

By Proposition 6.8.2, there exists w ∈ S c (E) such that inf∂Ω w > 1. Then, from (6.20a) we have p − εw < f < p + εw on ∂Ω, so that, letting v := w|∂Ω , we have Ω

Ω

Ω HΩ p−εv ≤ H f ≤ H f ≤ H p+εv .

On the other hand, we know that p − εv, p + εv ∈ R(∂Ω). It follows that 3 See the Appendix to this section.

(6.20b)

6.8 S-harmonic Spaces: Wiener Resolutivity Theorem

365

Ω Ω Ω HΩ p−εv = Hp−εv = Hp − εHv , Ω

Ω H p+εv = Hp+εv = HpΩ + εHvΩ .

Using these identities in (6.20b), we obtain Ω

Ω 0 ≤ Hf − HΩ f ≤ 2εHv ≤ 2εw,

(6.20c)

where, in the last inequality, we used the fact that w is an upper function related to w|∂Ω and Ω. Letting ε tend to zero in (6.20c), we get Ω

Hf = HΩ f .

(6.20d)

Inequalities (6.20b) also give, Ω

Ω Ω Ω 0 ≤ H f − Hp−εv ≤ Hp+εv − Hp−εv = 2εHvΩ ≤ 2εw ≤ 2ε sup w. Ω

Then, as ε → 0, Ω

Ω = HpΩε −εv → H f Hp−εv

uniformly on Ω.

Ω ∈ H(Ω), from this limit and Corollary 6.3.5 we obtain Since Hp−εv Ω

H f ∈ H(Ω). Together with (6.20d), this implies the resolutivity of f .

+

Exercise 6.8.5. Let Ω ⊆ E be open with Ω compact and ∂Ω = ∅. Let w ∈ S c (E) be such that infΩ w =: λ > 0. Then, for every bounded function f : ∂Ω → R, we have Ω −λ sup |f | ≤ H Ω f ≤ H f ≤ λ sup |f |. ∂Ω

∂Ω

In particular, if f is resolutive, sup |HfΩ | ≤ λ sup |f |. Ω

∂Ω

Ω

(Hint: Note that w supΩ |f | ∈ U f .) Exercise 6.8.6 (R(Ω) = R(Ω)). Let Ω ⊆ E be an open set as in the previous exercise. Let (fn )n∈N be a sequence of bounded resolutive functions such that sup |fn − f | → 0

as n → ∞.

∂Ω

Then f is resolutive. Exercise 6.8.7. Let Ω ⊆ E be an open set as in the previous exercise, and let f : ∂Ω → R be a bounded function. Then Ω

Hf ,HΩ f ∈ H(Ω).

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6 Abstract Harmonic Spaces

6.8.1 Appendix: The Stone–Weierstrass Theorem The aim of this section is to prove the following well-known density result. Theorem 6.8.8 (The Stone–Weierstrass theorem). Let (Y, T ) be a compact topological space, and let A ⊆ C(Y, R) satisfy the following conditions: (i) A is a real vector space, (ii) if p ∈ A, then p + := max{0, p} ∈ A, (iii) A separates the points of E, that is, for every x, y ∈ Y, x = y, there exists p, q ∈ A such that p(x)q(y) = p(y)q(x). Then, for every f ∈ C(Y, R) and for every ε > 0, there exists p ∈ A such that sup |f − p| < ε. Y

Proof. Assumptions (i) and (ii) imply |u| = u+ + (−u)+ ∈ A

for every u ∈ A.

As a consequence, max{u, v} = u + v + |u − v| and min{u, v} = u + v − |u − v| belong to A for any u, v ∈ A. Let us now fix u ∈ C(Y, R). For every x, y ∈ Y , due to assumption (iii), there exist α, β ∈ R and u, v ∈ A such that αu(x) + βv(x) = f (x), (6.21) αu(y) + βv(y) = f (y). Obviously, ux,y := αu + βv ∈ A, and, for every ε > 0, there exists a neighborhood Vy of y such that ∀z ∈ Vy . ux,y (z) < f (z) + ε p Since Y is compact, we can choose y1 , . . . , yp ∈ Y such that j =1 Vyj = Y . Letting ux := min{ux,y1 , . . . , ux,yp }, we have ux ∈ A and ux (z) < f (z) + ε

∀z ∈ Y.

(6.22)

On the other hand, by the first identity in (6.21), ux (x) = f (x) for every x ∈ Y . Then, since ux is continuous, for every x ∈ Y , there exists a neighborhood Wx of x such that ux (z) > f (z) − ε ∀ z ∈ Wx . q Since Y is compact, there exist Wx1 , . . . , Wxq such that j =1 Wxj = Y . As a consequence, letting u := max{ux1 , . . . , uxq }, we get u ∈ A and u(z) > f (z) − ε for every y ∈ Y . On the other hand, keeping in mind (6.22), we also have u = max{ux1 , . . . , uxq } < f + ε This completes the proof.

in Y.

6.9 H-harmonic Measures for Relatively Compact Open Sets

367

6.9 H-harmonic Measures for General Relatively Compact Open Sets. A Characterization of the Resolutive Functions As in the previous section, Ω will denote a relatively compact (i.e. with compact closure in E) open set with non-empty boundary, contained in a S-harmonic space (E, H). Let us fix a point x ∈ Ω. By the Wiener resolutivity theorem (Theorem 6.8.4) and Proposition 6.7.4, the map C(∂Ω, R) f → HfΩ (x) ∈ R is well defined, linear and positive. Then by the Riesz representation theorem, the following definition is well posed. Definition 6.9.1 (H-harmonic measure. II). Let Ω be a relatively compact open set with non-empty boundary, contained in the S-harmonic space (E, H). There exists a unique Radon measure, denoted by μΩ x , such that HfΩ (x) = f dμΩ for every f ∈ C(∂Ω, R). x ∂Ω

We shall call μΩ x the H-harmonic measure related to Ω and x. Exercise 6.9.2. For every relatively compact open set Ω ⊆ E, one has μΩ x (∂Ω) < Ω ∞ for every x ∈ Ω. (Hint: Note that μΩ x (∂Ω) = H1 (x) and use the a priori estimate of Exercise 6.8.5.) When Ω is an H-regular open set, this definition gives back the one given in Definition 6.2.3. Our aim here is to show the following theorem. Theorem 6.9.3 (Characterization of resolutivity). Let (E, H) be a S-harmonic space and let Ω ⊆ E be a relatively compact open set with non-empty boundary. Given f : ∂Ω → [−∞, ∞], the following statements are equivalent: (i) f ∈ R(∂Ω), (ii) f ∈ L1 (∂Ω, μΩ x ) for every x ∈ Ω. In this case

HfΩ (x)

= ∂Ω

f dμΩ x

for every x ∈ Ω.

The proof of this theorem requires some preliminary results having an independent interest. Notation. Given u : Ω → ]−∞, ∞], lower semicontinuous and bounded below, we set u(x) if x ∈ Ω, ∗ ∗ u : Ω → ]−∞, ∞], u (x) := if x ∈ ∂Ω. lim infy→x u(y)

368

6 Abstract Harmonic Spaces

It is quite easy to recognize that u∗ is lower semicontinuous. Similarly, given u : Ω → [−∞, ∞[, upper semicontinuous and bounded above, we set u(x) if x ∈ Ω, u∗ : Ω → [−∞, ∞[, u∗ (x) := if x ∈ ∂Ω. lim supy→x u(y) The function u∗ is upper semicontinuous. Lemma 6.9.4. With the previous notation, if u ∈ H∗ (Ω) and infΩ u > −∞, then:

(i) u(x) ≥ ∂Ω u∗ dμΩ x > −∞ for every x ∈ Ω,

∗ Ω (ii) x → ∂Ω u dμx belongs to H∗ (Ω). An analogous result also holds if u ∈ H∗ (Ω) and supΩ u < ∞. Proof. For every φ ∈ C(∂Ω, R), φ ≤ u∗ |∂Ω , we have HφΩ ≤ u, Ω

since u ∈ U φ . Then u(x) ≥

sup HφΩ (x) φ

= sup φ

∂Ω

φ dμΩ x

= ∂Ω

u∗ dμΩ x .

(The suprema are taken with respect to φ ∈ C(∂Ω, R), φ ≤ u∗ |∂Ω .) On the other hand, since inf∂Ω u∗ > −∞ and μx (Ω) < ∞ for every x ∈ Ω (see Exercise 12 at the end of the chapter), we have u∗ dμΩ x > −∞. ∂Ω

This proves (i). To obtain (ii), we just have to use Theorem 6.3.8 keeping in mind that HφΩ ∈ H(Ω). Lemma 6.9.5. Let f : ∂Ω → ]−∞, ∞] be a l.s.c. function. Then: (i) for every x ∈ Ω, we have Ω H f (x)

= ∂Ω

f dμΩ x ,

Ω

(ii) H f ∈ H(Ω) if and only if Ω

H f (x) < ∞

for every x ∈ Ω.

6.9 H-harmonic Measures for Relatively Compact Open Sets

369

Proof. (i) Let (fn ) be a sequence of continuous functions on ∂Ω such that fn f . Ω Since H fn = HfΩn ∈ H(Ω) for every n ∈ N, we have (see Exercise 6.7.5, page 360) Ω

HfΩn H f ,

(6.23)

so that, by the Beppo Levi monotone convergence theorem, Ω H f (x) = lim HfΩn (x) = lim fn dμΩ = x n→∞

n→∞ ∂Ω

∂Ω

f dμΩ x

for every x ∈ Ω. (ii) The only if part is trivial. The if part follows from (6.23) and Theorem 6.2.8, page 343. Proof (of Theorem 6.9.3). (i) ⇒ (ii). Let f ∈ R(∂Ω). Then, for every x ∈ Ω and Ω ε > 0, there exist u ∈ U f and v ∈ U Ω f such that u(x) − ε < HfΩ (x) < v(x) + ε. By Lemma 6.9.4, −∞ < ∂Ω

and we have

Ω u∗ dμΩ x − ε < Hf (x) <

∂Ω

∂Ω

v∗ dμΩ x + ε < ∞,

(u∗ − v∗ ) dμΩ x < 2ε,

so that, since v∗ ≤ f ≤ u∗ , u∗ is l.s.c., v∗ is u.s.c., and ε is an arbitrary positive number, ∀x∈Ω f ∈ L1 (∂Ω, dμΩ x )

and HfΩ (x) =

∂Ω

f dμΩ x

for every x ∈ Ω.

(ii) ⇒ (i). It is not restrictive to assume f ≥ 0. Define F := g : ∂Ω → [−∞, ∞[ | g u.s.c., 0 ≤ g ≤ f and

F := h : ∂Ω → ]−∞, ∞] | h l.s.c., h ≥ f .

Let us fix x ∈ Ω and ε > 0. By Vitali–Carathéodory’s theorem, there exist g ∈ F and h ∈ F such that (h − g) dμΩ x < ε. ∂Ω

Then h, g ∈ L1 (∂Ω, μΩ x ) and, by Lemma 6.9.5-(i),

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6 Abstract Harmonic Spaces

Ω

−∞ <

Ω Ω g dμΩ x = H g (x) ≤ H f (x) ≤ H f (x) ∂Ω Ω h dμx ≤ g dμΩ = x + ε < ∞. ∂Ω

∂Ω

Ω

This proves that −∞ < H Ω f (x) ≤ H f (x) < ∞ for every x ∈ Ω and that Ω

Ω

Ω sup H Ω g = H f = H f = inf H h . h∈F

g∈F

(6.24)

On the other hand, by Lemma 6.9.5-(ii), HΩ g ∈ H(Ω)

for every g ∈ F,

since g ≥ 0. Hence H Ω g > −∞ at any point of Ω. Thus, from (6.24) and Theorem 6.3.6 we finally obtain Ω

HΩ f = H f ∈ H(Ω). This completes the proof. Exercise 6.9.6. Prove that if f : ∂Ω → R is l.s.c. and bounded from above, then f is resolutive.

6.10 S∗ -harmonic Spaces: Bouligand’s Theorem We now give a definition introducing a new property which is not usually assumed in the abstract potential theory. However, as we will see in Chapter 7, this assumption does not affect at all the possibility to apply our theory to the sub-Laplacians. Definition 6.10.1 (S∗ -harmonic space). A S-harmonic space (E, H) will be said S∗ -harmonic if the following property holds: (A*) For every x0 ∈ E there exists sx0 ∈ S + c (E) such that sx0 (x0 ) = 0 and inf sx0 > 0

E\V

for every neighborhood V of x0 . (See Definition 6.8.1 for the definition of S+ c (E).) This property will make transparent the rôle of the barrier functions in studying the continuity up to the boundary of the Perron–Wiener–Brelot functions. Convention. Throughout this section, Ω will denote an open set in a S∗ -harmonic space (E, H). We will always assume, without any further comments, that Ω is compact and ∂Ω = ∅.

6.10 S∗ -harmonic Spaces: Bouligand’s Theorem

371

We know from the Wiener resolutivity theorem that every continuous function f : ∂Ω → R is resolutive, so that the Perron–Wiener–Brelot function HfΩ is Hharmonic in Ω. However, in general, we cannot expect a “good” behavior of HfΩ at the boundary points of Ω. Definition 6.10.2 (H-regular point). A point y ∈ ∂Ω will be called H-regular if lim HfΩ (x) = f (y)

Ωx→y

∀ f ∈ C(∂Ω, R).

Obviously, the function HfΩ is the solution (the unique solution, thanks to the minimum principle!) of the H-Dirichlet problem u ∈ H(Ω), (H-D) limx→y u(x) = f (y) ∀ y ∈ ∂Ω, for every f ∈ C(∂Ω, R), if and only if all the boundary points of Ω are H-regular points. Unfortunately, as we said before, we have to expect that, in general, ∂Ω contains boundary points which are not H-regular.4 The notion of H-barrier function will allow us to give a necessary and sufficient condition for the H-regularity. Definition 6.10.3 (H-barrier function). Let y ∈ ∂Ω. A H-barrier function for Ω at y is a function w defined in Ω ∩ V , being V a suitable open neighborhood of y, such that (see also Fig. 6.1): (i) w ∈ S(Ω ∩ V ), (ii) w(x) > 0 for every x ∈ Ω ∩ V , (iii) limx→y w(x) = 0. The link between H-regularity and H-barrier functions is given by the following theorem. Theorem 6.10.4 (Bouligand’s theorem). Let (E, H) be a S∗ -harmonic space, and let Ω ⊆ E be a relatively compact open set with non-empty boundary. A point x0 ∈ ∂Ω is H-regular for Ω if and only if there exists an H-barrier function for Ω at x0 . The proof of this theorem is quite long. It is convenient to premise some lemmas. Lemma 6.10.5. Let V ⊆ E be a regular open set, and let U be a relatively open subset of ∂V . For every x0 ∈ V and ε > 0, there exists a compact set K ⊆ U and a non-negative function h ∈ H(V ) such that: 4 See, for instance, Exercise 6.10.7 (page 375) related to the classical S. Zaremba’s example

of non-regular boundary points.

372

6 Abstract Harmonic Spaces

Fig. 6.1. A H-barrier function

(i) h(x0 ) < ε, (ii) lim infV x→y h(x) ≥ 1 for every y ∈ U \ K. Proof. Let K ⊆ U be such that μVx0 (U \ K) < ε, and let f : ∂V → R be the characteristic function of U \ K. Since f is bounded and l.s.c., it is resolutive (see Ex. 6.9.6, page 370). Let us put h := HfV . Then h ∈ H(V ), h ≥ 0, since f ≥ 0 and h(x0 ) = ∂V

f dμVx0 = μVx0 (U \ K) < ε.

We now choose a sequence (fn ) of continuous functions on ∂V such that 0 ≤ fn ≤ f and fn f . Since V is regular, for every y ∈ U \ K and n ∈ N, we have lim inf h(x) ≥ lim HfVn (x) = fn (y). x→y

x→y

By taking the supremum with respect to n at the last right-hand side, we get lim inf h(x) ≥ 1 x→y

for every y ∈ U \ K.

This completes the proof. Lemma 6.10.6. Let Ω ⊆ E be a relatively compact open set, and let x0 ∈ ∂Ω. Define f : ∂Ω → R, f (x) = sx0 (x), where sx0 is the continuous subharmonic function of assumption (A∗ ). Define sxΩ0 := HfΩ . Then: (i) sxΩ0 is harmonic in Ω, (ii) infΩ\U sxΩ0 > 0 for every neighborhood U of x0 , (iii) limΩx→x0 sxΩ0 (x) = 0 if x0 is H-regular for Ω, (iv) limΩx→x0 sxΩ0 (x) = 0 if there exists a barrier function for Ω at x0 .

6.10 S∗ -harmonic Spaces: Bouligand’s Theorem

373

Proof. (i) This is obvious, since f is continuous, hence resolutive. Ω (ii) Since sx0 |Ω ∈ U Ω f , one has sx0 |Ω ≤ Hf . Thus, the assertion follows from the positivity property of sx0 . (iii) Since sxΩ0 = HfΩ , if x0 is H-regular for Ω, then one has lim s Ω (x) x→x0 x0

= f (x0 ) = 0.

(iv) Suppose that there exists a barrier function b at x0 . Then there exists a neighborhood B of x0 such that b ∈ S(Ω ∩ B) and b(x) → 0 as x → x0 . + Let w ∈ S c (E) be such that w ≥ 1 in Ω (see Proposition 6.8.2). The maximum principle of Ex. 6.8.5 implies the existence of a positive constant M such that 0 ≤ sxΩ0 ≤ M

in Ω.

Let ε > 0 be fixed. Choose a regular neighborhood V of x0 such that V ⊆B

sup f ≤ ε.

and

V ∩∂Ω

Let K ⊆ Ω ∩ ∂V be compact, and let h ∈ H(V ) be such that h ≥ 0, h(x0 ) ≤

ε M

and

lim inf h(x) ≥ 1 x→y

∀y ∈ Ω ∩ ∂V \ K.

(see the previous lemma). Since b > 0 in B ∩ Ω and B ⊇ V , we have m := infK b > 0. For a given function u ∈ U Ω f , define u0 := u − εw − M

b +h m

in Ω ∩ V . Then u0 ∈ S(Ω), and lim sup∂(Ω∩V ) u0 ≤ 0. Indeed, since u ≤ HfΩ ≤ M, we have y ∈ K ⇒ lim sup u0 (x) ≤ sup u − x→y

Ω

M inf b ≤ M − M = 0, m K

(6.25a)

y ∈ Ω ∩ ∂V \ K ⇒ lim sup u0 (x) ≤ sup u − M lim inf h(x) ≤ M − M = 0, x→y

x→y

Ω

y ∈ V ∩ ∂Ω ⇒ lim sup u0 (x) ≤ lim sup u(x) − εw(y) ≤ f (y) − ε ≤ 0. x→y

(6.25b)

(6.25c)

x→y

The maximum principle for subharmonic functions implies u0 ≤ 0 in Ω ∩ V , that is, b u ≤ εw + M +h in Ω ∩ V . m Ω

This inequality holds for every u ∈ U f , so that

374

6 Abstract Harmonic Spaces

sxΩ0 ≡ HfΩ ≤ εw + M

b +h m

in Ω ∩ V .

Then, since b(x) → 0 as x → x0 , and Mh(x0 ) ≤ ε, 0 ≤ lim sup sxΩ0 (x) ≤ εw(x0 ) + ε

for every ε > 0.

x→x0

This completes the proof. We are now ready to prove Bouligand’s theorem. Proof (of Theorem 6.10.4). If x0 is H-regular for Ω, then sxΩ0 is a barrier function for Ω at x0 , see Lemma 6.10.6-(i)–(iii). Vice versa, suppose that there exists a barrier function for Ω at x0 . Let us prove that lim

Ωx→x0

HϕΩ (x) = ϕ(x0 )

(6.26)

for every ϕ ∈ C(∂Ω, R). Then sxΩ0 has properties (i)–(ii) and (iv) in Lemma 6.10.6. Let w ∈ Sc+ (Ω) be such that infΩ w ≥ 1 (see Proposition 6.8.2), and let V be a neighborhood of x0 . Define ωϕ (V ) := sup |ϕ(z) w(x0 ) − ϕ(x0 ) w(z)| : z ∈ V ∩ ∂Ω , Mϕ := sup |ϕ(z) w(x0 ) − ϕ(x0 ) w(z)| : z ∈ ∂Ω , m := inf sxΩ0 (x) : x ∈ Ω \ V , and

u :=

sxΩ ϕ(x0 ) + ωϕ (V ) w + Mϕ 0 . w(x0 ) m

Then u ∈ S(Ω). Moreover, lim inf u(x) ≥ ϕ(z)

Ωx→z

∀ z ∈ ∂Ω.

Indeed, if z ∈ V ∩ ∂Ω, the very definition of ωϕ (V ) and the inequality w ≥ 1 in Ω imply ϕ(x0 ) + ωϕ (V ) w(z) ≥ ϕ(z). lim inf u(x) ≥ x→z w(x0 ) On the other hand, if z ∈ ∂Ω \ V , keeping in mind the definitions of m and Mϕ , one has ϕ(x0 ) w(z) + Mϕ ≥ ϕ(z). lim inf u(x) ≥ x→z w(x0 ) Ω

We also have infΩ u > −∞. Thus, u ∈ U ϕ . Hence HϕΩ ≤ u and lim sup HϕΩ (x) ≤

Ωx→x0

lim

Ωx→x0

u(x) = ϕ(x0 ) + ωϕ (V )w(x0 ),

6.11 Reduced Functions and Balayage

375

since sxΩ0 (x) → 0 as x → x0 . By taking the infimum with respect to V at the last right-hand side, we get (6.27) lim sup HϕΩ (x) ≤ ϕ(x0 ) Ωx→x0

for every ϕ ∈ C(∂Ω, R). As a consequence, Ω lim inf HϕΩ (x) = − lim sup H−ϕ (x) ≥ ϕ(x0 ). x→x0

Ωx→x0

This inequality, together with (6.27), implies (6.26). Exercise 6.10.7 (S. Zaremba’s counterexample). Let x0 ∈ Ω. Consider the set = Ω \ {x0 }. Assume there exists u ∈ S + (Ω) such that Ω u(x0 ) = ∞,

0 ≤ u(x) < ∞

Define → R, f : ∂Ω Then

HfΩ

f (x) =

∀ x ∈ Ω.

0 if x ∈ ∂Ω, 1 if x = x0 .

(Hint: the function ε u ≡ 0. In particular, x0 is not H-regular for Ω. Ω

belongs to U f for all ε > 0.)

6.11 Reduced Functions and Balayage The notions of reduced function and balayage are crucial in the potential theory. Before giving the basic definitions, we need a general preliminary result. Theorem 6.11.1 (Envelopes in S(Ω)). Let (E, H) be a harmonic space and let Ω ⊆ E be open. Given F ⊆ S(Ω), assume that u0 := inf F is locally bounded from below. Then u0 ∈ S(Ω). Remark 6.11.2. In the setting of the harmonic spaces related to the sub-Laplacians, the statement of Theorem 6.11.1 can be strengthened (see Theorem 9.5.6, page 449). Proof. The function u0 is l.s.c. in Ω and > −∞ at any point. Moreover, for a given H-regular set V ⊆ V ⊆ Ω, we have u0 dμVx ≤ u dμVx ≤ u(x) (6.28) ∂V

∂V

at any point x ∈ V for every u ∈ F. Then

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6 Abstract Harmonic Spaces

∂V

u0 dμVx ≤ u0 (x)

∀ x ∈ V.

(6.29)

On the other hand, since u ∈ S(Ω), x → ∂V u dμVx is H-harmonic in Ω, and the first inequality in (6.28) implies that also u0 dμVx x → ∂V

is H-harmonic, hence continuous, in V (see Exercise 6.3.9). Therefore, from inequality (6.29) we obtain ∂V

so that u0 ∈ S(Ω).

u0 dμVx ≤ u0 (x),

Exercise 6.11.3. Let F ⊆ H∗ (Ω), for an open set Ω ⊆ E. Assume that u0 := inf F is locally bounded from below. Then u0 ∈ H∗ (Ω). (Hint: Apply the previous theorem to the families Fn := {min{u, n} : u ∈ F},

n ∈ N.

Then use Theorem 6.3.8.) Definition 6.11.4 (H-reduced function and H-balayage). Let f : E → [0, ∞] be a non-negative real extended function, and let A ⊆ E. Define f RA := inf v ∈ H∗ (E) : v ≥ 0 in E, v ≥ f in A and

f f RA := (RA ).

RA and RA are called, respectively, the H-reduced function and the regularized Hreduced function of f on A. f RA is also called the H-balayage of f on A. f

f

We, obviously, have

f f 0≤ RA ≤ RA .

Moreover, by Exercise 6.11.3, f RA ∈ H∗ (E).

If the function f ∈ H∗ (E), f ≥ 0, from the very definition of reduced function we f get RA ≤ f . So that, in this case, f f 0≤ RA ≤ RA ≤ f,

and RA is H-superharmonic if f is H-superharmonic. Moreover, f

f f RA (x) = RA (x) = f (x)

for every x ∈ Int(A).

(6.30)

6.11 Reduced Functions and Balayage

377

Exercise 6.11.5. Prove the claimed (6.30). Prove also that if f ∈ S(E), f ≥ 0, then f RE = inf v ∈ S(E) : v ≥ 0 in E, v ≥ f in A , f

and RE = f in A.

An application of the fundamental theorem on Perron families gives the following result. Theorem 6.11.6 (Properties of H-reduction and H-balayage. I). Let (E, H) be a harmonic space. Let f ∈ S(E), f ≥ 0, and let A ⊆ E. Then (i) RA is H-superharmonic in E, f f (ii) RA = f in A and RA = f in int(A), f f (iii) RA and RA are equal and H-harmonic in Ω := E \ A. f

An improvement of (iii) will be given in the setting of sub-Laplacians. Proof. We already know that (i) and (ii) hold true. To prove (iii), consider the family F := v|Ω : v ∈ S(E), v ≥ 0 in E, v ≥ f in A . It is easy to see that F is a Perron family in Ω and that f and g ≡ 0 are, respectively, an H-superharmonic majorant and an H-subharmonic minorant of F. Then, by Theorem 6.5.8, f (RA )|Ω = inf F ∈ H(Ω). f

In particular, RA is continuous in Ω, hence f f RA = RA

in Ω.

Then (iii) follows. The following theorem states some properties of the reduced function and the balayage. Their easy proofs are left as an exercise. Theorem 6.11.7 (Properties of H-reduction and H-balayage. II). Let (E, H) be a harmonic space, and let A, B ⊆ E and f, g ∈ S(E), f, g ≥ 0. Then f f RA ≤ RB if A ⊆ B, f RA λf RA

≤ =

g RA if f ≤ g, f λ RA if λ ≥ 0.

The same assertions hold for the operator R.

(6.31a) (6.31b) (6.31c)

378

6 Abstract Harmonic Spaces

Some other crucial properties of the reduced function and the balayage will be proved in the context of the harmonic spaces associated to the sub-Laplacians. The use of the average operators Mr and Mr will make the proofs much easier than in the general abstract setting. Bibliographical Notes. In this chapter, we presented some topics from the theory of abstract harmonic spaces mainly inspired by the ones developed by H. Bauer in [Bau66] and C. Costantinescu and A. Cornea in [CC72]. See also J.L. Doob [Doo01].

6.12 Exercises of Chapter 6 Unlike the other chapters, several exercises of Chapter 6 are distributed throughout the text. For the reading convenience, we gather below the list of these exercises. Ex. 1) Prove property (P1) in Section 6.1. Ex. 2) Prove property (P2) in Section 6.1 by recognizing that u : A → [−∞, ∞] is l.s.c. if and only if it is continuous from the topological space (A, T |A ) and ([−∞, ∞], I), where I is the topology on [−∞, ∞] generated by the intervals ]t, ∞], t ∈ R. Ex. 3) Prove property (P3) in Section 6.1. (Hint: Use (P2).) Ex. 4) Prove property (P4) in Section 6.1. (Hint: Use Ex. 2, first recognizing that K ⊆ ]−∞, ∞] is compact w.r.t. the topology I if and only if K has real minimum.) Ex. 5) Prove property (P5) in Section 6.1. (Hint: For every a ∈ A and ε > 0, there exists Va,ε ∈ Ua such that u(a) − ε < u(x) for every x ∈ Va,ε . Let m := minA u. Urysohn’s lemma implies the existence of a continuous function fa,ε : A → R such that, m − ε ≤ f ≤ u(a) − ε, f (a) = u(a) − ε and f (x) = m − ε for every x ∈ A \ Va,ε . Since {fa,ε : a ∈ A, ε > 0} ⊆ F := {f ∈ C(A, R) : f ≤ u}, one has supf ∈F f = u. To conclude the proof, use Proposition 6.1.1.) Ex. 6) Solve Ex. 6.3.4, page 346, and prove Proposition 6.2.6, page 342. Ex. 7) Solve Ex. 6.2.5, page 342. Ex. 8) Solve Exercises 6.3.9 and 6.3.1, page 348. Ex. 9) Solve Ex. 6.4.11, page 353. Ex. 10) Prove Proposition 6.5.4, page 355. Ex. 11) Prove Proposition 6.7.4, page 360. Ex. 12) Solve Exercises 6.8.5, 6.8.6, 6.8.7, page 365. Ex. 13) Solve Ex. 6.10.7, page 375. Ex. 14) Solve Ex. 6.11.3, page 376.

6.12 Exercises of Chapter 6

379

Ex. 15) Solve Ex. 6.11.5, page 377. Ex. 16) Prove Theorem 6.11.7. Ex. 17) Prove directly that if f is any function, then fis l.s.c. Moreover, if g is l.s.c. and g ≤ f , then g ≤ f. Ex. 18) Let Ω be a relatively compact open set in a harmonic space (E, H). Let u ∈ S(U ), where U is an open set containing Ω. Show that the function x → u dμΩ x =: h(x) ∂Ω

is H-harmonic in Ω. (Hint: F = {HϕΩ : ϕ ∈ C(∂Ω, R), ϕ ≤ u|∂Ω } is an up directed family of harmonic functions such that sup F = h and sup F ≤ u|Ω .) Ex. 19) Let (E, H) be a S∗ -harmonic space, and let Ω ⊆ E be open and relatively compact. Let z ∈ ∂Ω, and let U be an open neighborhood of z. Prove that z is H-regular for Ω ⇔ z is H-regular for Ω ∩ U. Ex. 20) Let (E, H) be a S∗ -harmonic space, and let Ω1 ⊆ Ω2 ⊆ E be open and relatively compact. Let z ∈ ∂Ω1 ∩ ∂Ω2 . Assume z is H-regular for Ω2 . Prove that z is H-regular for Ω1 . Ex. 21) Let Ω be a relatively compact open set in a S-harmonic space (E, H). Let (fn ) be a sequence of resolutive functions from ∂Ω into [−∞, ∞]. Suppose that Ω for every x ∈ Ω. sup H |fn | (x) < ∞ n

Assume (fn ) is pointwise convergent to f : ∂Ω → [−∞, ∞]. Prove that f is resolutive. Ex. 22) Let Ω be a relatively compact open set in a S-harmonic space (E, H). Let f : ∂Ω → [−∞, ∞[ be bounded above. Prove that if z ∈ ∂Ω is H-regular, one has Ω lim sup H f (x) ≤ lim sup f (y). Ωx→z

∂Ωy→z

(Hint: For every fixed β ∈ R, β > lim sup∂Ωy→z f (y), choose a function Ω

Ω

ϕ ∈ C(∂Ω, R) such that ϕ ≥ f and ϕ(z) = β. Then H f ≤ H ϕ = HϕΩ .) Ex. 23) Let (E, H) be a harmonic space such that the constant functions are Hharmonic. Let ϕ : R → R be a convex function. Prove the following statements: (i) if Ω ⊆ E is open and u ∈ H(Ω), then ϕ(u) ∈ S(Ω), (ii) if we also assume ϕ monotone increasing, then ϕ(u) ∈ S(Ω) whenever u ∈ S(Ω). (Hint: Since 1 ∈ H(E), μVx (∂V ) = 1 for every H-regular set V and every x ∈ V . Use Jensen’s inequality.) Ex. 24) Let Ω1 and Ω be open subsets of E with Ω1 ⊆ Ω. Given u ∈ H∗ (Ω) and v ∈ H∗ (Ω1 ) define

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6 Abstract Harmonic Spaces

w : Ω → ]−∞, ∞],

w=

u in Ω \ Ω1 , min{u, v} in Ω1 .

Prove that, if w is l.s.c., then w ∈ H∗ (Ω). Moreover, w ∈ S(Ω) if u ∈ S(Ω). Ex. 25) Let Ω1 and Ω be open subsets of E with Ω1 ⊆ Ω. Given u ∈ S(Ω) and v ∈ S(Ω1 ) define w as in the previous exercise. Assume that lim inf v(x) ≥ u(y)

Ω1 x→y

for every y ∈ Ω ∩ ∂Ω1 .

Then w ∈ S(Ω1 ). Ex. 26) Let Ω be a relatively compact open set in a S∗ -harmonic space (E, H) and let x0 ∈ ∂Ω. Prove that x0 is H-regular for Ω iff x0 is H-regular for every connected component O of Ω such that x0 ∈ ∂O.

7 The L-harmonic Space

The aim of this short chapter is to prove that the theory of abstract harmonic spaces developed in Chapter 6 can be applied to the setting of (homogeneous) Carnot groups G. More precisely, fixed a sub-Laplacian L on G, the L-harmonic functions provide a harmonic sheaf L H on G. We shall prove that (G, L H) turns out to be a S∗ -harmonic space. The ingredients we use are the structure of the fundamental solution for L and the Harnack-type theorem established in Chapter 5. The more involved task is to show the existence of a basis of the topology of RN formed by L-regular sets, i.e. by sets for which the Dirichlet problem for L is solvable in the usual classical sense. For reading convenience, in the second part of the chapter, we recall some basic definitions and results from the abstract theory of Chapter 6, just to show how they read in the sub-Laplacian setting. Moreover, we prove a criterion of Lsubharmonicity for C 2 -functions, we show that the gauge balls are L-regular sets and we write an explicit formula for the density of the L-harmonic measures of the gauge balls at their center. Convention. The same convention as in the incipit of Chapter 5 applies. In other words, the potential theory developed in this chapter can be straightforwardly generalized and rephrased in the context of the abstract stratified groups. It suffices to consider the notion of L-harmonic function on a stratified group in Definition 2.2.27 on page 146 and to keep in mind Remark 2.2.28. That remark ensures that, if H is an abstract stratified group, G is a “homogeneous copy” of H and Ψ : G → H the Lie group isomorphism in Remark 2.2.26 (page 145) the harmonic (hence the hyperharmonic, subharmonic, etc.) functions in H simply “pull back” via Ψ to the harmonic (hyperharmonic, subharmonic, etc.) functions in G.

7.1 The L-harmonic Space Let G = (RN , ◦, δλ ) be a homogeneous Carnot group in RN and let L be one of its sub-Laplacians. We know that

382

7 The L-harmonic Space

L = div(A(x)∇) =

N

ai,j (x) ∂xi ,xj +

i,j =1

N

bi (x) ∂xi ,

i=1

where A(x) = (ai,j )i,j ≤N is a symmetric non-negative matrix with smooth entries and constant a1,1 > 0. For every open set Ω ⊆ G denote L H(Ω) := u ∈ C ∞ (Ω, R) : Lu = 0 . A function u ∈ L H(Ω) will be called L-harmonic in Ω. The map L

H : Ω → L H(Ω)

is a harmonic sheaf. The aim of this section is to prove that (RN , L H) is a S∗ -harmonic space. We split the proof in four steps, corresponding to the axioms we have to verify. (A1) (Positivity). This axiom is trivially satisfied for the constant functions are Lharmonic. (A2) (Convergence). This readily follows from the Harnack theorem proved in Theorem 5.7.10, page 268. We postpone to the end of the section the proof of (A3), which is our main task. Now we proceed with (A4) and (A∗ ). (A4) (Separation). We shall show that Sc+ (G), the family of continuous nonnegative L H-superharmonic functions, separates the points of G. Since L is left translation invariant and the constant functions are L-harmonic, it is enough to prove the following statement: for every x0 = 0, there exists u ∈ Sc+ (G) : u(x0 ) = u(0). Let Γ be the fundamental solution of L with pole at the origin. Since Γ is L-harmonic in G \ {0} and lim Γ (x) = ∞ =: Γ (0),

x→0

one has Γ ∈ S(G) (see Exercise 7.3). For k ∈ N, define Γk = min{Γ, k}. +

Obviously, Γk ∈ S c (G). Moreover, Γk (0) ∞ as k ∞ while, as x0 = 0, Γk (x0 ) Γ (x0 ) ∈ ]0, ∞[. Then there exists k ∈ N such that Γk (x0 ) = Γk (0), and we are done.

7.1 The L-harmonic Space

383

(A∗ ) (The function s x0 ). For every fixed x0 ∈ G, let us define sx0 (x) = d(x0−1 ◦ x),

x ∈ G,

where d is an L-gauge function (see Definition 5.4.1). Then sx0 is continuous in G, sx0 (x0 ) = 0 and inf sx0 > 0 G\V

for every neighborhood V of x0 . Moreover, in G \ {x0 }, Lsx0 =

(Q − 1) |∇L d|2 ≥ 0 d

(see (5.34)). Then sx0 ∈ S(G \ {x0 }). Since sx0 ≥ 0 and sx0 (x0 ) = 0, it follows that sx0 ∈ S(G) (see Ex. 7.3.4). (A3) (Regularity). This is the most difficult part of our task. We shall use some results from the theory of linear second order partial differential equations of elliptic type. From now on, for a fixed ε > 0, we shall denote Lε := L + εΔ,

(7.1)

2 where Δ = N j =1 ∂xj is the Laplace operator. The characteristic form of Lε qLε (x, ξ ) := A(x)ξ, ξ + ε |ξ |2 ,

ξ ∈ RN ,

is strictly positive definite at any point of G. Moreover, for every compact set K ⊆ G there exists λ = λ(K) > 0 such that λ−1 |ξ |2 ≤ qLε (x, ξ ) ≤ λ|ξ |2

∀ x ∈ K, ξ ∈ RN .

This means that Lε is uniformly elliptic on every bounded open set Ω, so that if ∂Ω also satisfies some mild regularity condition, the Dirichlet problem Lε u = 0 in Ω, (7.2) u|∂Ω = ϕ, ϕ ∈ C(∂Ω, R), has a solution1

uε ∈ C ∞ (Ω) ∩ C(Ω)

such that uε |∂Ω = ϕ. A condition ensuring this existence result is the following one2 : (R) For every x0 ∈ ∂Ω, there exists a function h(x0 , ·) of class C 2 in an open set containing Ω such that: 1 L is hypoelliptic for it is elliptic and its coefficients are smooth. Then u ∈ C ∞ if L u = 0. ε ε 2 See the monograph by D. Gilbarg and N.S. Trudinger [GT77, Chapter 6].

7 The L-harmonic Space

384

(a) h(x0 , ·) > 0 in Ω \ {x0 }, h(x0 , x0 ) = 0. (b) Lε h(x0 , ·) ≤ 0 in Ω. A geometric condition ensuring (R) is a particular form of the Poincaré exterior ball condition. Definition 7.1.1 (Non-characteristic exterior ball condition). We say that ∂Ω satisfies the non-characteristic exterior ball condition at a point x0 ∈ ∂Ω if there exists z ∈ G \ Ω such that (i) D(z, r) ∩ Ω = {x0 }, r = |x0 − z|, (ii) qL (x, z − x) := A(x)(z − x), z − x > 0 ∀ x ∈ Ω. Here D(z, r) denotes the standard Euclidean ball with center z and radius r. The standard Euclidean norm is denoted by | · |. Note 7.1.2. If (i) and (ii) hold, then the boundary of Ω is non-characteristic at x0 , since ν := z − x0 is orthogonal to ∂Ω at x0 and qL (x0 , ν) = A(x0 ) ν, ν > 0. Lemma 7.1.3. Let Ω ⊆ G be a bonded open set satisfying the non-characteristic exterior ball condition at a point x0 ∈ ∂Ω. Define h(x0 , x) = exp(−μ r 2 ) − exp(−μ |z − x|2 ), where

x ∈ G,

|b(x), z − x| μ = max :x∈Ω x∈Ω qL (x, z − x)

(as before, | · | stands for the usual Euclidean norm). Then h ∈ C ∞ (G, R) and satisfies (a) and (b) of condition (R). Proof. The smoothness of h is obvious. Condition (R)-(a) readily follows from (i) in Definition 7.1.1. Moreover, by (ii) in Definition 7.1.1 and the very definition of μ, Lε h(x0 , x) equals − exp(−μ |x − x0 |2 )(4μ2 qL (x, z − x) + 2μb(x), z − x + 4εN ), which is non-negative in Ω.

We now prove another crucial lemma for our purposes. Lemma 7.1.4. Let Ω and h be as in Lemma 7.1.3, and let uε be the solution of the Dirichlet problem (7.2). Then |uε (x) − ϕ(x0 )| ≤ 2 hϕ (x0 , x)

∀ x ∈ Ω.

(7.3)

7.1 The L-harmonic Space

385

The function hϕ is defined as follows. For t > 0, we let ωϕ (x0 , t) := t + sup{|ϕ(x) − ϕ(x0 )| : x ∈ ∂Ω, |x − x0 | ≤ t}, and m(x0 , t) := inf{h(x0 , x) : |x − x0 | ≥ t, |x − z| ≥ r}. The function t → ωϕ (x0 , t)m(x0 , t) is continuous and strictly increasing. Denote by Tϕ (x0 , ·) its inverse function. We define hϕ (x0 , x) := Tϕ x0 , sup |ϕ|h(x0 , x) . ∂Ω

It should be noted that hϕ is independent of ε. Moreover, lim hϕ (x, x0 ) = 0.

x→x0

Proof (of Lemma 7.1.4). Let t > 0 be fixed. Consider the function w(x) := ϕ(x0 ) + ωϕ (x0 , t) + sup |ϕ| ∂Ω

h(x0 , x) . m(x0 , t)

One readily verifies that lim w(x) ≥ ϕ(y) = lim uε (x)

x→y

x→y

∀ y ∈ ∂Ω.

Then, since Le w ≤ 0, by the maximum principle we get uε ≤ w in Ω, that is uε (x) − ϕ(x0 ) ≤ ωϕ (x0 , t) + sup |ϕ| ∂Ω

h(x0 , x) m(x0 , t)

for every x ∈ Ω and t > 0. The same inequality holds by replacing in it uε with −uε and ϕ with −ϕ. Then |u(x) − ϕ(x0 )| ≤ ωϕ (x0 , t) + sup |ϕ| ∂Ω

h(x0 , x) , m(t)

x ∈ Ω,

(7.4)

for every t > 0. Now we choose t > 0 such that ω(x0 , t) = sup |ϕ| ∂Ω

h(x0 , x) , m(t)

that is t = T (x0 , sup∂Ω |ϕ|h(x0 , x)). With this choice of t, from (7.4) we obtain (7.3). This completes the proof. From Lemma 7.1.3 and Lemma 7.1.4 we obtain a resolutivity result for L. Proposition 7.1.5. Let Ω ⊆ G be a bounded open set satisfying the non-characteristic exterior ball condition at any point of ∂Ω. Then Ω is L H-regular, that is: for every ϕ ∈ C(∂Ω, R) there exists a unique function u ∈ C ∞ (Ω, R) such that Lu = 0 in Ω, limx→y u(x) = ϕ(y)

∀ y ∈ ∂Ω.

7 The L-harmonic Space

386

Moreover, u ≥ 0 if ϕ ≥ 0. Proof. First of all we prove the existence. Let ϕ ∈ C(∂Ω, R) and let uε , 0 < ε < 1, be the solution of the Dirichlet problem (7.2). By the maximum principle sup |uε | ≤ sup |ϕ|, Ω

∂Ω

so that

|uε |2 dx < ∞.

sup 0<ε<1 Ω

Then there exists u ∈ L2 (Ω) and a sequence εj 0 such that lim uεj = u

j →∞

in the weak topology of L2 (Ω). As a consequence, for every test function ψ ∈ C0∞ (Ω),

uL∗ ψ dx = lim uεj L∗ ψ dx j →∞ Ω Ω

uεj L∗ε ψ dx − εj uεj Δψ dx = lim j →∞ Ω

Ω ψLε uεj dx = 0. = lim j →∞ Ω

Therefore Lu = 0 in the weak sense of distributions. Since L is hypoelliptic, we may assume u ∈ C ∞ (Ω, R) and Lu = 0 in Ω. On the other hand, by Lemma 7.1.4, for every x0 ∈ ∂Ω we have ϕ(x0 ) − 2 hϕ (x0 , x) ≤ uεj (x) ≤ ϕ(x0 ) + 2 hϕ (x0 , x)

∀ x ∈ ∂Ω

(7.5)

and for every j ∈ N. Then, since the weak limits in L2 (Ω) of non-negative functions are non-negative, inequalities (7.5) imply |uε (x) − ϕ(x0 )| ≤ hϕ (x0 , x)

∀ x ∈ Ω, x0 ∈ ∂Ω.

Then, keeping in mind that hϕ (x0 , x) → 0 as x → x0 , lim u(x) = ϕ(x0 )

x→x0

∀ x0 ∈ ∂Ω.

This completes the proof of the existence. The uniqueness and the positivity of u if ϕ ≥ 0 straightforwardly follow from the weak maximum principle of Theorem 5.13.4 (page 295).

7.1 The L-harmonic Space

387

Finally, we are ready to complete the proof of (A3). We will construct a basis (Vλ )λ>0 of L-regular neighborhoods of the origin. It will follow that, for every x0 ∈ G, {x0 ◦ Vλ }λ>0 is a basis of L-regular open sets containing x0 . Let e1 = (1, 0, . . . , 0) be the first element of the canonical basis of G. Then qL (0, e1 ) = A(0) e1 , e1 = a1,1 > 0. As a consequence, there exists ρ > 0 such that qL (x, ν) > 0 for every x, ν ∈ G, |x| < ρ, |ν − e1 | < ρ. Therefore, for R > 0 sufficiently large and ε > 0 sufficiently small, V1 := D(R e1 , R + ε) ∩ D(−R e1 , R + ε)

(7.6)

satisfies the non-characteristic exterior ball condition at any point of its boundary. (See also Fig. 7.1.)

Fig. 7.1. Figure of (7.6)

Then, by Proposition 7.1.5, V1 is L H-regular. Since L commutes with the dilations δλ , λ > 0, if we put Vλ := δλ ◦ V , the family {Vλ }λ>0 is a basis of L H-regular neighborhoods of the origin. This completes the proof of (A3).

388

7 The L-harmonic Space

7.2 Some Basic Definitions and Selecta of Properties Throughout this section, Ω will be an open subset of a homogeneous Carnot group G. Notation. For the sake of brevity, we shall simply denote by H the harmonic sheaf L H introduced in the previous section. In Section 7.1, we have shown that (G, H) is a S∗ -harmonic space. Hence the abstract potential theory developed throughout Chapter 6 applies to the present setting. Some of the basic definitions and results from that theory will be recalled along this section, for reading convenience. A bounded open set V ⊂ G will be called L-regular if the boundary value prob lem Lu = 0 in V , (7.7) u|∂V = ϕ has a (unique) solution u := HϕV for every continuous function ϕ : ∂V → R. We say that u solves (7.7) if u is L-harmonic in V and lim u(x) = ϕ(y)

x→y

∀ y ∈ ∂V .

The weak maximum principle of Theorem 5.13.4 implies the uniqueness of the solution to the boundary value problem (7.7). Moreover, HϕV ≥ 0 in V whenever ϕ ≥ 0 on ∂V . Then, if V is L-regular, for every fixed x ∈ V the map C(∂V , R) ϕ → HϕV (x) ∈ R defines a linear positive functional on C(∂V , R). As a consequence (by the classical Riesz representation theorem for linear positive functionals), there exists a Radon measure μVx supported in ∂V such that

V Hϕ (x) = ϕ(y) dμVx (y) ∀ ϕ ∈ C(∂V , R). ∂V

the L-harmonic measure related to V and x. We shall call The family of L-regular sets is not empty. Indeed, in Section 7.1 we have proved the following proposition. μVx

Proposition 7.2.1. The family of the L-regular open sets is a basis of the Euclidean topology of G. The following definition of L-subharmonic function is equivalent to the abstract one given in Definition 6.5.1 (page 353), in force of Theorem 6.5.2 and the definition of H-hyperharmonic function given in Definition 6.2.4 (page 341).

7.2 Some Basic Definitions and Selecta of Properties

389

Definition 7.2.2 (L-subharmonic function). Let Ω be an open subset of G. A function u : Ω → [−∞, ∞[ will be called L-subharmonic in Ω if: (i) u is upper semi-continuous and u > −∞ in a dense subset of Ω, (ii) for every L-regular open set V with closure V ⊂ Ω and for every x ∈ V ,

u(y) dμVx (y). (7.8) u(x) ≤ ∂V

The family of the L-subharmonic functions in Ω will be denoted by S(Ω). Remark 7.2.3. As we have already noticed in the abstract harmonic spaces (see Section 6.2.1), condition (ii) in Definition 7.2.2 is equivalent to the following one: (ii)’

HϕV ≥ u|V

∀ ϕ ∈ C(∂V , R): ϕ ≥ u|∂V ,

for every L-regular open set V such that V ⊂ Ω. We shall denote by S(Ω) the family of the L-superharmonic functions, i.e. the set of functions u : Ω → ]−∞, ∞] such that −u ∈ S(Ω). Then, keeping in mind the previous definition, u ∈ S(Ω) if and only if (i) u is lower semi-continuous and u < ∞ in a dense subset of Ω, (ii) for every L-regular open set V with closure V ⊂ Ω and for every x ∈ V ,

u(x) ≥ u(y) dμVx (y). (7.9) ∂V

Proposition 7.2.4. A function u : Ω → R is L-harmonic in Ω if and only if u ∈ S(Ω) ∩ S(Ω). In other words, H(Ω) = S(Ω) ∩ S(Ω). Proof. See Exercise 6.3.4 on page 346. By using the weak maximum principle of Theorem 5.13.4 (page 295), we obtain the following criterion of L-subharmonicity for functions of class C 2 . Proposition 7.2.5 (Smooth L-subharmonic functions). Let u be a function in C 2 (Ω, R). Then u is L-subharmonic if and only if Lu ≥ 0 in Ω. Proof. Let V ⊂ V ⊂ Ω be an L-regular open set and let ϕ ∈ C(∂V , R), ϕ ≥ u. One has

390

7 The L-harmonic Space

L(u − HϕV ) ≥ 0

in V ,

lim supx→y (u(x) − HϕV (x))

≤ u(y) − ϕ(y) ≤ 0 for every y ∈ ∂V .

Then, by the maximum principle, u − HϕV ≤ 0 in V , so that u is L-subharmonic. Vice versa, let u ∈ S(Ω) and assume, by contradiction, Lu(x0 ) < 0 at some point x0 ∈ Ω. It follows that Lu < 0 in a neighborhood Ω0 of x0 . Then, by the first part of the proof, u ∈ S(Ω0 ). On the other hand, u ∈ S(Ω0 ) since u ∈ S(Ω). By Proposition 7.2.4, it follows that u ∈ H(Ω0 ), i.e. Lu = 0 in Ω0 , a contradiction. Remark 7.2.6. The previous proof can be easily adapted to prove the following statement. Let Ω ⊆ G be open and let x0 ∈ Ω. Let u ∈ C 2 (Ω \ {x0 }, R) be such that Lu ≥ 0 in Ω \ {x0 },

lim inf u(x) = −∞. x→x0

Then, if we extend u at x0 letting u(x0 ) := −∞, we have u ∈ S(Ω). Indeed, let V ⊂ V ⊂ Ω be a L-regular open set and let ϕ ∈ C(∂V , R), ϕ ≥ u|∂V . Proceeding as in the first part of the previous proof, we see that u|V ≤ (HϕV )|V

if x0 ∈ /V

and u|V \{x0 } ≤ (HϕV )|V \{x0 }

if x0 ∈ V .

Then, in any case, u|V ≤ (HϕV )|V , so that u ∈ S(Ω).

Example 7.2.7. Let d be an L-gauge and let x0 ∈ G be fixed. Define (d(x0−1 ◦ x))2−Q if x = x0 , γx0 (x) := ∞ if x = x0 . Then, γx0 is L-superharmonic in G. Indeed, since d is an L-gauge, γx0 is L-harmonic (hence, L-superharmonic) in G \ {x0 }. Let Ω ⊂ G be a bounded open set. By the Wiener resolutivity Theorem 6.8.4 in S-harmonic spaces (page 364), we know that any function ϕ ∈ C(∂Ω, R) is resolutive. Then there exists the Perron–Wiener–Brelot generalized solution HϕΩ ∈ H(Ω) to the boundary value problem Lu = 0 in Ω, (7.10) u|∂Ω = ϕ. As it is well-known, even in the classical case of Laplace operator, we cannot expect that lim HϕΩ (x) = ϕ(y) (7.11) x→y

for every y ∈ ∂Ω. If (7.11) holds for every ϕ ∈ C(∂Ω, R), we say that y is a L-regular point of ∂Ω.

7.2 Some Basic Definitions and Selecta of Properties

391

Obviously, problem (7.10) is solvable for every continuous boundary datum ϕ (i.e. Ω is an L-regular set) if and only if every point of ∂Ω is L-regular. Bouligand’s Theorem 6.10.4 in S∗ -harmonic spaces (page 371) implies that a point y ∈ ∂Ω is L-regular if there exists an L-barrier function at y in Ω, i.e. a function w ∈ S(Ω) such that w(x) > 0 for every x ∈ Ω and w(x) → 0 as x → y. By using this result, one easily proves the following proposition. Proposition 7.2.8 (The d-balls are L-regular). Let d be an L-gauge. Then, the dballs Bd (x0 , r), x0 ∈ G, r > 0 are L-regular open sets. Proof. The function v(x) :=

(d(x0−1 ◦ x))2−Q − r 2−Q ∞

if x = x0 , if x = x0

is L-superharmonic in Bd (x0 , r) (see Example 7.2.7). Moreover, v > 0 in Bd (x0 , r) and v(x) → 0 as x → y for every y ∈ ∂Bd (x0 , r). Then v is an L-barrier function for Bd (x0 , r) at any point of its boundary. Bouligand’s Theorem implies that Bd (x0 , r) is L-regular. The surface mean value formula (5.45) in Theorem 5.5.4 (page 256) provides an explicit “density” of the L-harmonic measures related to Bd (x0 , r) and x0 : Theorem 7.2.9 (Density). Let d be an L-gauge. Given x0 ∈ G and r > 0, we have d (x0 ,r) (y) = dμB x0

βd (Q − 2) KL (x0 , y) · dσ (y), r Q−1

where KL =

(7.12)

|∇L d|2 (x −1 ◦ y) |∇(d(x −1 ◦ ·))|(y)

and dσ denotes the Hausdorff (N − 1)-dimensional measure on ∂Bd (x0 , r). Proof. Let 0 < ρ < r and ϕ ∈ C(∂Bd (x0 , r), R). Since h := HϕBd (x0 ,r) is L-harmonic in Bd (x0 , r), by using the surface mean value formula (5.45) (page 256), we have

d (x0 ,r) (y) = h(x ) = M (h)(x ). ϕ(y) dμB 0 ρ 0 x0 ∂Bd (x0 ,r)

7 The L-harmonic Space

392

On the other hand, since h(y) → ϕ(z) as y → z for every z ∈ ∂Bd (x0 , r), lim Mρ (h)(x0 ) = Mr (h)(x0 )

ρ→r

=

βd (Q − 2) rQ

Then

∂Bd (x0 ,r)

d (x0 ,r) (y) = ϕ(y) dμB x0

∂Bd (x0 ,r)

βd (Q − 2) rQ

ϕ(y) KL (x, y) dσ (y).

∂Bd (x0 ,r)

ϕ(y) KL (x0 , y) dσ (y)

for every ϕ ∈ C(∂Bd (x0 , r), R) and the assertion follows. We refer the interested reader to the papers [UL97,UL02] for some estimates of L-Poisson kernels on more general domains of Carnot groups. Bibliographical Notes. In showing the existence of a basis of the topology of G formed by L-regular sets, we followed an idea by J.-M. Bony [Bon69]. For other presentations of potential theory, linear and non-linear and in subRiemannian settings, see [BT03,Bir95,HKM93,Kil94,TW02b,VM96]. For some topics in potential theory in an elliptic degenerate context and estimates of the Poisson kernel, see [BAKS84,CGN02,HH87,UL97,UL02].

7.3 Exercises of Chapter 7 Ex. 1) Prove that Γ ∈ S(G). (Hint: Use Ex. 4.) Ex. 2) Complete the proof of the following result, which is a consequence of Harnack’s Theorem for sub-Laplacians. Theorem 7.3.1. Let Ω ⊆ G be a bounded, open and connected set such that 0 ∈ Ω. For all x ∈ Ω, there exists hx ∈ L1 (∂Ω, μΩ 0 ) such that Ω dμΩ x (ξ ) = hx (ξ ) dμ0 (ξ ).

Moreover, if K is a compact subset of Ω, there exists a positive constant c(K) depending only on K such that c(K)−1 ≤ hx ; L∞ (∂Ω, μΩ 0 ) ≤ c(K)

for every x ∈ K.

Proof (Sketch). In this proof, for brevity, for any given x ∈ Ω we agree to set μx := μΩ x . Let E ⊆ ∂Ω be closed. Then χE is u.s.c., and we have (why?)

μx (E) = χE dμx = HχΩE (x). ∂Ω

7.3 Exercises of Chapter 7

393

Now, HχΩE is a non-negative L-harmonic function on the connected open set Ω, so that (why?) for every compact set K ⊂ Ω there exists c = c(K) > 0 such that HχΩE (x) ≤ cHχΩE (y)

for every x, y ∈ K.

Hence, μx (E) ≤ c μy (E) for every closed E ⊆ ∂Ω. Consequently, we obtain (why?) c−1 μy (E) ≤ μx (E) ≤ c μy (E)

for every Borel set E ⊆ ∂Ω.

This proves that μy μx μy for every couple of points x, y ∈ Ω. Hence, by Lebesgue decomposition theorem, we infer the existence of hx,y ∈ L1 (∂Ω, μy ) as in the assertion. We claim that c−1 ≤ hx,y (·); L∞ (∂Ω, μy ) ≤ c. Indeed, suppose first to the contrary that there exists a Borel set E ⊆ ∂Ω such that μy (E) > 0 and hx,y (ξ ) > c, μy -almost-everywhere on E. This would give the contradiction

dμy (ξ ) 1 μx (E) = = c. c≥ hx,y (ξ ) dμy (ξ ) c E μy (E) μy (E) E μy (E) Then take y = 0 and set hx := hx,0 .

Ex. 3) Let f ∈ Lp (∂Bd (0, 1), μ), 1 ≤ p < ∞, where μ := μ0Bd (0,1) is the L-harmonic measure of Bd (0, 1) at x0 = 0. Then: (i) f is resolutive, B (0,1) (ii) letting u := Hf d , we have

|u(δλ (x))|p dμ(x) = f Lp (Bd (0,1)) . sup 0≤λ≤1 ∂Bd (0,1)

Ex. 4) Let Ω ⊆ G be open and let x0 ∈ Ω. Suppose we are given a function u ∈ C(Ω, R) ∩ S(Ω \ {x0 }) such that u ≥ 0 and u(x0 ) = 0. Then u ∈ S(Ω). (Hint: For every y ∈ Ω, there exists a basis of regular neighborhoods of y such that

u(y) ≤ u dμVy . ∂V

Then use Corollary 6.4.9.)

394

7 The L-harmonic Space

Ex. 5) (Asymptotic Koebe Theorem). Let Ω ⊆ G be open and let u ∈ C(Ω, R). Assume that ALu = 0 in Ω

or ALu = 0

in Ω.

Prove that u ∈ C ∞ (Ω, R) and Lu = 0 in Ω. AL and AL denote the asymptotic L Laplacians introduced in Ex. 8 of Chapter 5. (Hint: Use the maximum principles of Ex. 10 of Chapter 5 to show that V for every L-regular open set V ⊆ V ⊆ Ω.) u|V = Hu| ∂V Ex. 6) (M. Picone’s Maximum Principle). Consider the following differential operator N N ai,j (x) ∂xi ,xj + bj (x) ∂xj , L= i,j =1

j =1

where ai,j = aj,i , bj are real functions in a bounded open subset Ω of G. Assume the following facts hold: N (i) qL (x, ξ ) := N i,j =1 ai,j (x)ξi ξj ≥ 0 ∀ x ∈ Ω, ∀ ξ ∈ R , 2 (ii) there exists h ∈ C (Ω, R) such that Lh < 0 and h > 0 in Ω. Then L satisfies the following weak maximum principle: if u ∈ C 2 (Ω, R) verifies Lu ≤ 0 in Ω, lim supx→y u(x) ≤ 0 ∀ y ∈ ∂Ω then u ≤ 0 in Ω. (Hint: See the proof of Theorem 5.13.4.) Ex. 7) Let L be the differential operator introduced in Ex. 6. Together with the hypotheses (i) and (ii) assumed in that exercise, suppose that (iii) L is hypoelliptic in Ω, (iv) for every x0 ∈ Ω, there exists ξ0 ∈ RN such that qL (x0 , ξ0 ) > 0, (v) for every x0 ∈ Ω, there exists a positive real function γx0 of class C 2 in Ω \ {x0 }) such that Lγx0 = 0 in Ω \ {x0 },

lim γx0 (x) = ∞.

x→x0

For every open set U ⊆ Ω define L H(U ) = u ∈ C ∞ (U, R) : Lu = 0 in U . Prove that (Ω, L H) is a S∗ -harmonic space. (Hint: Just follow the lines of Section 7.1. Take as sx0 the function 1/γx0 .) Ex. 8) Let L be a sub-Laplacian on a homogeneous Carnot group G. Let Ω ⊆ G be an L-regular connected open set. Show that supp μΩ x = ∂Ω

7.3 Exercises of Chapter 7

395

for every x ∈ Ω (Hint: Use the strong maximum principle.) Note: This result shows that the L-harmonic space on G is an “elliptic space in the sense of Brelot” (see [CC72]). Ex. 9) (The Exterior L-Ball Regularity Condition). Let G be a homogeneous Carnot group and let d be a L-gauge, L being a sub-Laplacian on G. In the following statement, we say that Ω has at y the property of the exterior L-ball if there exists a d-ball Bd (z, ρ) such that G \ Ω ⊇ Bd (z, ρ)

and y ∈ ∂Bd (z, ρ).

Prove the following result: Let Ω ⊆ G be a bounded open set and let y ∈ ∂Ω. Assume Ω has at y the property of the exterior L-gauge ball. Then y is an L-regular point for Ω.

8 L-subharmonic Functions

1 2 Throughout the chapter, G = (RN , ◦, δλ ), L = N j =1 Xj and d will denote, respectively, a homogeneous Carnot group, a sub-Laplacian on G and an L-gauge on G. Our main task is to provide some characterizations of the L-subharmonic functions u in terms of suitable sub-mean properties w.r.t. the mean value operator Mr u and Mr u (introduced in Chapter 5), in terms of the monotonicity w.r.t. the radius of these operators and in terms of the sign of Lu in the weak sense of distributions. We also provide some maximum principles for L-subharmonic functions and smoothing approximation theorems. Finally, in Section 8.3, we furnish a brief investigation on the continuous convex functions on G. Throughout the chapter S(Ω) and S(Ω) will denote the sheafs of the L-subharmonic and the L-superharmonic functions, respectively, in the open set Ω ⊆ G.

8.1 Sub-mean Functions In this section, we shall denote by Mr and Mr the average operators defined in (5.46) and (5.50f), respectively (pages 256 and 259). Definition 8.1.1 (Solid and surface sub-mean function). If Ω ⊆ G is an open set, we say that an u.s.c. function u : Ω → [−∞, ∞[ satisfies the local surface (local solid) sub-mean property if, for every x ∈ Ω, there exists rx > 0 such that u(x) ≤ Mr (u)(x) u(x) ≤ Mr (u)(x) for 0 < r < rx . (8.1) If (8.1) holds for any r > 0 such that Bd (x, r) ⊂ Ω, we shall say that u satisfies the global surface (global solid) sub-mean property. The next theorem shows that solid sub-mean functions satisfy weak and strong maximum principles.

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8 L-subharmonic Functions

Theorem 8.1.2 (Maximum principles for sub-mean functions). Let Ω ⊆ G be an open set. Let u : Ω → [−∞, ∞[ be an u.s.c. function satisfying the local solid sub-mean property. The following statements hold: (i) if Ω is connected and there exists x0 ∈ Ω such that u(x0 ) = maxΩ u, then u ≡ u(x0 ) in Ω, (ii) if Ω is bounded and lim supΩ y→x u(y) ≤ 0 for every x ∈ ∂Ω, then u(y) ≤ 0 in Ω. Proof. (i) Let x0 ∈ Ω be such that u(x0 ) is the maximum of u in Ω. We may suppose that u(x0 ) = −∞. Since u is L-sub-mean, we have md 0≤ Q ΨL (x0−1 ◦ y) u(y) − u(x0 ) dy (8.2) r Bd (x0 ,r) for some r = rx0 > 0. Thus, since ΨL ≥ 0 and u(y) ≤ u(x0 ), ΨL (x0−1 ◦ y) u(y) − u(x0 ) = 0 almost everywhere in Bd (x0 , r). On the other hand, ΨL > 0 in a dense open subset of Bd (x0 , r) (see the first step in the proof of Lemma 5.7.1, page 262) and u is u.s.c. This implies u = u(x0 ) on the whole Bd (x0 , r). The assertion follows from a connectedness argument. (ii) Let x0 ∈ Ω be such that u = sup u ∀ r > 0.

sup

Ω

Bd (x0 ,r)∩Ω

If x0 ∈ ∂Ω, by the hypothesis and the boundary behavior of u, we have 0 ≥ lim sup u = inf Ω y→x0

sup

r>0 Bd (x0 ,r)∩Ω

u = sup u, Ω

whence u ≤ 0 on Ω. If x0 ∈ Ω, by the upper semicontinuity of u, we have u(x0 ) ≥ inf

sup

r>0 Bd (x0 ,r)∩Ω

u = inf sup u = sup u. r>0 Ω

Ω

Hence u(x0 ) = maxΩ u. From (i) it follows that u ≡ u(x0 ) on the connected component of Ω containing x0 , so that, for some x ∈ ∂Ω, 0 ≥ lim sup u(y) ≥ u(x0 ) = max u. y→x

Ω

The assertion is proved. Theorems 8.1.2 and 7.2.9 (page 391) allow us to show that the solid and surface sub-mean properties are equivalent.

8.1 Sub-mean Functions

399

Theorem 8.1.3 (Equivalence of sub-mean properties). Let u : Ω → [−∞, ∞[ be an u.s.c. function. Then the following statements are equivalent: (1) u satisfies the local solid sub-mean property, (2) u satisfies the global solid sub-mean property, (3) u satisfies the local surface sub-mean property, (4) u satisfies the global surface sub-mean property. Proof. (1) ⇒ (4). Let Bd (x, r) ⊂ Ω and ϕ ∈ C(∂Bd (x, r), R) be such that u ≤ ϕ B (x,r) is a solid sub-mean function, by the maximum in ∂Bd (x, r). Since u − Hϕ d B (x,r) ≤ 0 in Bd (x, r). In particular, principle in Theorem 8.1.2, we have u − Hϕ d d (x,r) (y) ϕ(y) dμB u(x) ≤ HϕBd (x,r) (x) = x ∂Bd (x,r) βd (Q − 2) = (by Theorem 7.2.9) ϕ(y) KL (x, y) dσ (y). r Q−1 ∂Bd (x,r) Then, taking the infimum with respect to the continuous functions ϕ ≥ u|∂Ω , we get βd (Q − 2) u(y) KL (x, y) dσ (y) = Mr (u)(x), u(x) ≤ r Q−1 ∂Bd (x,r) and (4) is proved. (4) ⇒ (3). This is trivial. (3) ⇒ (1). Suppose u(x) ≤ Mr (u)(x) for 0 < r < rx . Then Q r Q−1 ρ Mρ (u)(x) dρ = Mr (u)(x). u(x) ≤ Q r 0 (4) ⇒ (2). The same proof as the previous one. (2) ⇒ (1). This is trivial. From now on, we shall call sub-mean any function satisfying one of the properties (1)–(4) in Theorem 8.1.3. Theorem 8.1.4 (L1loc of sub-mean functions). Let Ω be a connected open subset of G, and let u : Ω → [−∞, ∞[ be a sub-mean u.s.c. function. Suppose that u(x0 ) > −∞ for some x0 ∈ Ω. Then u ∈ L1loc (Ω). Proof. We first prove that u > −∞ in a dense subset of Ω. Since Ω is connected and there exists x0 ∈ Ω such that u(x0 ) > −∞, it is enough to show that u > −∞ in a dense subset of Bd (x, r) for any Bd (x, r) ⊂ Ω such that u(x) > −∞. This last statement follows from md −∞ < u(x) ≤ Mr (u)(x) = Q ΨL (x −1 ◦ y) u(y) dy r Bd (x,r) and the property ΨL > 0 in an open dense set.

400

8 L-subharmonic Functions

Let us now fix z0 ∈ Ω and R > 0 such that Bd (z0 , R) ⊂ Ω. We shall prove the existence of a constant λ > 0 such that u(z) dz > −∞. Bd (z0 ,λ R)

From this statement our theorem will follow. Let us choose three positive constants λ0 , λ, Λ such that c (λ0 + λ) ≤ Λ,

c (λ0 + Λ) ≤ 1,

(8.3)

where c is the constant appearing in the pseudo-triangle inequality for the gauge d (see (5.55), page 262). Since the set {x ∈ Ω | ΨL (x −1 ◦ z0 ) > 0 } is an open and dense subset of Ω and {x ∈ Ω | u(x) > −∞ } is dense in Ω too, there exists x0 ∈ Bd (z0 , λ0 R) such that x0 = z0 , u(x0 ) > −∞ and ΨL (x0−1 ◦ z0 ) > 0. Due to the smoothness of ΨL out of the origin, we may refine the choice of λ in such a way that, for a suitable m0 > 0, we get ΨL (x0−1 ◦ z) ≥ m0 for every z ∈ Bd (z0 , λ R). On the other hand, inequalities (8.3) imply the inclusions Bd (z0 , λ R) ⊆ Bd (x0 , Λ R) ⊆ Bd (z0 , R). Then, if we put U := max u (∈ R), Bd (z0 ,R)

we have (denoting by meas the Lebesgue measure in G) u(z) dz Bd (z0 ,λ R) = u(z) − U dz + U meas Bd (z0 , λ R) Bd (z0 ,λ R) 1 ≥ ΨL (x0−1 ◦ z) u(z) − U dz + U meas Bd (z0 , λ R) m0 Bd (z0 ,λ R) 1 ≥ ΨL (x0−1 ◦ z) u(z) − U dz + U meas Bd (z0 , λ R) m0 Bd (x0 ,Λ R) (Λ R)Q = MΛ R (u − U )(x0 ) + U meas Bd (z0 , λ R) md m0 (Λ R)Q ≥ u(x0 ) − U + U meas Bd (z0 , λ R) > −∞. md m0 This completes the proof.

8.2 Some Characterizations of L-subharmonic Functions

401

We close this section by proving that every (ε, G)-mollifier preserves the submean properties (see Definition 5.3.6 on page 239). Theorem 8.1.5 (Mollifier of a sub-mean function). Let u : Ω → [−∞, ∞[ be u.s.c. and u ∈ L1loc (Ω). Then, if u is sub-mean in Ω, uε is sub-mean in Ωε too. Proof. For x ∈ Ωε and r > 0 such that Bd (x, r) ⊆ Ωε , we have md −1 −1 Jε (z) Q ΨL (x ◦ y) u(z ◦ y) dy dz Mr (uε )(x) = r B (0,ε) Bd (x,r) d = Jε (z) Mr (u)(z−1 ◦ x) dz Bd (0,ε) ≥ Jε (z) u(z−1 ◦ x) dz = uε (x), Bd (0,ε)

and the assertion follows.

8.2 Some Characterizations of L-subharmonic Functions We begin by proving the following theorem. Theorem 8.2.1 (L-hypoharmonicity and sub-mean property). Let Ω be an open subset of G and u : Ω → [−∞, ∞[ be an u.s.c. function. Then u is L-hypoharmonic if and only if u is sub-mean. Proof. Let u be a sub-mean function. By using the maximum principle in Theorem 8.1.2 and arguing as in the first part of the proof of Theorem 8.1.3, it is easy to prove that u is L-hypoharmonic. Vice versa, let us suppose that u is L-hypoharmonic and prove that u satisfies the global surface sub-mean property. Let Bd (x, r) ⊂ Ω. Since Bd (x, r) is an L-regular open set, for all ϕ ∈ C(∂Bd (x, r), R) such that ϕ ≥ u on ∂Bd (x, r), we have Bd (x,r) d (x,r) (y). (x) = ϕ(y) dμB u(x) ≤ Hϕ x ∂Bd (x,r)

On the other hand, from Theorem 7.2.9 (page 391) we have βd (Q − 2) Bd (x,r) ϕ(y) dμx (y) = ϕ(y) KL (x, y) dσ (y). r Q−1 ∂Bd (x,r) ∂Bd (x,r) Hence βd (Q − 2) ϕ(y) KL (x, y) dσ (y) ≥ u(x) r Q−1 ∂Bd (x,r) so that βd (Q − 2) Mr (u)(x) = r Q−1

∀ ϕ ∈ C(∂Bd (x, r), R),

∂Bd (x,r)

u(y) KL (x, y) dσ (y) ≥ u(x).

Thus, u has the surface sub-mean property.

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8 L-subharmonic Functions

Corollary 8.2.2. Let Ω be an open subset of G, and let u : Ω → [−∞, ∞[ be an u.s.c. function, finite in a dense subset of Ω. Then u ∈ S(Ω) if and only if u is sub-mean. The following result will be used very often. Corollary 8.2.3 (Characterization of S(Ω)). Let Ω be a connected open subset of G, and let u : Ω → [−∞, ∞[ be an u.s.c. function. Then u ∈ S(Ω) if and only if u is sub-mean and u > −∞ in at least one point of Ω. Proof. If u ∈ S(Ω), then u > −∞ in a dense subset of Ω, and u is sub-mean by Theorem 8.2.1. Vice versa, if u is sub-mean and u > −∞ in at least one point of Ω, by Theorem 8.1.4, u ∈ L1loc (Ω). In particular, u > −∞ in a dense subset of Ω. Then, by Theorem 8.2.1, u ∈ S(Ω). Corollary 8.2.4. If u ∈ S(Ω), then u ∈ L1loc (Ω). Corollary 8.2.5 (A Brelot-type convergence result). Let {un }n∈N be a sequence of L-subharmonic functions in a connected open set Ω ⊆ G. Assume {un }n∈N is monotone decreasing. Then, if we set u := limn→∞ un , we have u ∈ S(Ω) provided there exists x0 ∈ Ω such that u(x0 ) > −∞. Proof. The function u is u.s.c. and sub-mean, since so are the un ’s, and un ≥ un+1 for every n ∈ N. Then, the assertion follows from Corollary 8.2.3. Corollary 8.2.6. Let Ω ⊆ G be open, and let x0 ∈ Ω. Let u ∈ S(Ω \ {x0 }) be such that lim u(x) = −∞. x→x0

Then, if we continue u at x0 by letting u(x0 ) = −∞, we have u ∈ S(Ω). Proof. The continued function u is u.s.c. and finite in a dense subset of Ω. Moreover, for every x ∈ Ω \ {x0 } and r > 0 such that Bd (x, r) ⊂ Ω \ {x0 }, we have u(x) ≤ Mr (u)(x), since u is L-subharmonic (hence sub-mean) in Ω \ {x0 }. On the other hand, u(x0 ) = −∞ ≤ Mr (u)(x0 )

∀ r > 0 : Bd (x0 , r) ⊂ Ω.

Thus, u is locally sub-mean in Ω, hence u ∈ S(Ω).

From Corollary 8.2.3 and the abstract results on directed families of hyperharmonic functions in Section 6.3.1 (page 347), we obtain the following assertion.

8.2 Some Characterizations of L-subharmonic Functions

403

Theorem 8.2.7 (Down directed families in S(Ω)). Let F be a down directed family of L-subharmonic functions in an open connected set Ω ⊆ G. The function u := inf F is L-subharmonic in Ω if and only if there exists a point x0 ∈ Ω such that u(x0 ) > −∞. Proof. By Theorem 6.3.8 (page 348), u is L-hypoharmonic in Ω. In particular, u is u.s.c. and sub-mean in Ω. Then, by Corollary 8.2.3, u is L-subharmonic if and only if there exists a point x0 ∈ Ω such that u(x0 ) > −∞. Corollary 8.2.8. Let F be a down directed family of L-subharmonic functions in an open set Ω ⊆ G. Then, on every connected component of Ω, the function u := inf F is L-subharmonic or identically equal to −∞. The L-subharmonicity can be characterized in terms of the monotonicity with respect to r of the averaging operators Mr and Mr . We first prove the following lemma. Lemma 8.2.9. Let m : [0, ∞[→ R be a strictly increasing function. Let f : [0, ∞[→ R be an increasing function, integrable w.r.t. m, in the sense of Riemann-Stieltjes. If r ∈ ]0, ∞[, we set r r 1 M(r) := dm(ρ), F (r) := f (ρ) dm(ρ). M(r) 0 0 Then F ≤ f and F is non-decreasing. Proof. From the monotonicity of f we have r r 1 1 F (r) = f (ρ) dm(ρ) ≤ f (r) dm(ρ) = f (r). M(r) 0 M(r) 0 Finally, if 0 < r1 < r2 , F (r2 ) − F (r1 ) r1 r2 1 1 1 − f (ρ) dm(ρ) + f (ρ) dm(ρ) = M(r2 ) M(r1 ) M(r2 ) r1 0 1 1 1 ≥ − M(r1 ) f (r1 ) + M(r2 ) − M(r1 ) f (r1 ) = 0. M(r2 ) M(r1 ) M(r2 ) This completes the proof.

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8 L-subharmonic Functions

In order to state the next theorem, it is convenient to introduce the following definition: if Ω ⊆ G is an open set and x ∈ Ω, we set R(x) := d(x, Ω)

(= sup{r > 0 : Bd (x, r) ⊆ Ω}).

Theorem 8.2.10 (S(Ω) and the monotonicity of Mr and Mr ). Let Ω be an open subset of G, and let u : Ω → [−∞, ∞[ be an u.s.c. function finite in a dense subset of Ω. Then the following statements are equivalent: (i) u ∈ S(Ω), (ii) r → Mr (u)(x) is monotone increasing for 0 < r < R(x) and u(x) = lim Mr (u)(x), r→0+

(iii) r → Mr (u)(x) is monotone increasing for 0 < r < R(x) and u(x) = lim Mr (u)(x). r→0+

Proof. (i) ⇒ (ii). Let Bd (x, s) ⊂ Ω and 0 < r < s. Define F := {ϕ ∈ C(∂Bd (x, s), R) : ϕ ≥ u on ∂Bd (x, s)}. Since Bd (x, s) is an L-regular open set, we have HϕBd (x,s) (y) ≥ u(y)

∀ y ∈ Bd (x, s),

so that, by the surface mean value Theorem 5.5.4, Mr (u)(x) ≤ Mr (HϕBd (x,s) )(x) = HϕBd (x,s) (x) = Ms (HϕBd (x,s) )(x) βd (Q − 2) = Ms (ϕ)(x) = KL (x, y) ϕ(y) dσ (y). r Q−1 ∂Bd (x,s) Hence

βd (Q − 2) KL (x, y) ϕ(y) dσ (y) ϕ∈F r Q−1 ∂Bd (x,s) βd (Q − 2) = KL (x, y) u(y) dσ (y) = Ms (u)(x). r Q−1 ∂Bd (x,s)

Mr (u)(x) ≤ inf

This proves the first part of the statement. In order to prove the second one, we just have to note that, by the upper semicontinuity of u, for every λ > u(x) there exists r > 0 such that Mr (u)(x) < λ for 0 < r < r. On the other hand, u(x) ≤ Mr (u)(x) for every r < R(x). Then u(x) ≤ Mr (u)(x) < λ ∀ r ∈ ]0, r[.

8.2 Some Characterizations of L-subharmonic Functions

(ii) ⇒ (iii). Since Mr (u)(x) =

Q rQ

r

ρ Q−1 Mρ (u)(x) dρ

405

∀ r ∈ ]0, R(x)[

0

and ρ → Mρ (u)(x) is monotone increasing, by Lemma 8.2.9, r → Mr (u)(x) is monotone increasing too. Moreover, by arguing as in the previous step, u(x) = limr→0+ Mr (u)(x). (iii) ⇒ (i). Since r → Mr (u)(x) is a monotone increasing function and u(x) = limr→0+ Mr (u)(x), we have u(x) ≤ Mr (u)(x)

if 0 < r < R(x).

Then u satisfies the global solid sub-mean property, so that, by Theorem 8.2.1, u ∈ S(Ω). We know that a smooth function u is L-subharmonic if and only if Lu ≥ 0. A similar characterization holds for non-smooth L-subharmonic functions. More precisely, we have the following result. Theorem 8.2.11 (S(Ω) and the inequality Lu ≥ 0 in D (Ω). I). Let Ω be an open subset of G, and let u : Ω → [−∞, ∞[ be an u.s.c. function. Then the following statements are equivalent: (i) u ∈ S(Ω), (ii) u ∈ L1loc (Ω), u(x) = limr→0+ Mr (u)(x) for every x ∈ Ω, and Lu ≥ 0 in Ω in the weak sense of distributions.1 Remark 8.2.12. We remark that in condition (ii) of Theorem 8.2.11 the hypothesis u(x) = limr→0+ Mr (u)(x) cannot be removed. Consider, for example, the function u = χ{0} (the characteristic function of the set {0}). Then, it is immediately seen that u is u.s.c., u ∈ L1loc (G), Lu = 0 in G (in the weak sense of distributions), but u∈ / S(G) for it is not sub-mean at the origin. On the other hand, if, in addition, u is continuous then u(x) = limr→0+ Mr (u)(x) always holds true. In order to prove the theorem, we need the following real analysis lemma. Lemma 8.2.13. Let f : ]0, ∞[→ R be a L1loc function such that ∞ f (t) h (t) dt ≤ 0 ∀ h ∈ C01 (]0, ∞[), h ≥ 0. 0

1 The inequality Lu ≥ 0 in the weak sense of distributions means

Ω

u Lφ ≥ 0

∀ φ ∈ C0∞ (Ω), φ ≥ 0.

(8.4)

406

8 L-subharmonic Functions

Let α > 1. Suppose that α F (t) := α t

t

s α−1 f (s) ds

0

is finite for 0 < t < T . Then t → F (t) is monotone increasing in ]0, T [. Proof. Let t1 , t2 be Lebesgue points of f such that 0 < t1 < t2 . For a fixed ε > 0, 1 1 ε < t2 −t 2 , we choose a sequence {hn }n of positive C0 (]0, ∞[) functions such that supp hn ⊆ [t1 , t2 ], supn supt |hn (t)| < ∞, and, as n → ∞, ⎧ if t1 < t < t1 + ε, ⎨ 1/ε hn (t) → −1/ε if t2 − ε < t < t2 , ⎩ 0 if t ∈ / [t1 , t1 + ε] ∪ [t2 − ε, t2 ]. Replacing hn in (8.4) and letting n tend to infinity, we get 1 t1 +ε 1 t2 f (t) dt − f (t) dt ≤ 0. ε t1 ε t2 −ε From this inequality, letting ε → 0+ , we obtain f (t1 ) ≤ f (t2 ), since t1 and t2 are Lebesgue points for f . Since almost every point of ]0, ∞[ is a Lebesgue point for f , we can use the same argument as in the proof of Lemma 8.2.9 in order to show that F (t1 ) ≤ F (t2 ) for every pair of Lebesgue points t1 , t2 ∈ ]0, T [, with t1 ≤ t2 . This result, together with the continuity of F , proves the lemma. We are now in the position to prove Theorem 8.2.11. Proof (of Theorem 8.2.11). (i) ⇒ (ii). If u ∈ S(Ω), then, by Corollary 8.2.4, u ∈ L1loc (Ω). Moreover, by Theorem 8.2.10, u(x) = limr→0+ Mr (u)(x). By Theorem 8.2.1, we also have that u is sub-mean in Ω, so that, by Theorem 8.1.5, uε is sub-mean in Ωε . As a consequence, by Theorem 8.2.1, uε ∈ S(Ωε ), so that, since uε ∈ C ∞ (Ωε ), Luε ≥ 0 in Ωε . Since uε → u in L1loc (Ω), as ε 0, we get Lu ≥ 0 in the weak sense of distributions. (ii) ⇒ (i). The inequality Lu ≥ 0 in the weak sense of distributions means u Lφ ≥ 0 ∀ φ ∈ C0∞ (Ω), φ ≥ 0. (8.5) Ω

We now choose a suitable test function φ. Let us fix a d-ball Bd (x0 , R) ⊆ Ω and a non-negative function g ∈ C ∞ ([0, R[, R), constant in a right neighborhood of 0. Define φ(x) := g(d(x0−1 ◦ x)). Then φ ∈ C0∞ (Bd (x0 , R)) and, by Proposition 5.4.3 (page 247), Q−1 2 Lφ = |∇L d| g (d) + g (d) . d

8.2 Some Characterizations of L-subharmonic Functions

407

Replacing the right-hand side in (8.5), using Federer’s coarea formula, and keeping in mind the definition of the average operator Mt , we get ∞ Q−1 g (t) u KL dσ dt 0≤ g (t) + t 0 {d=t} ∞ Q−1 1 = t g (t) Mt (u) dt. βd (Q − 2) 0 Then, if we choose

g(t) := t

∞

h(s) ds, s Q−1

where h is any non-negative C0∞ (]0, R[; R) function, the previous inequality can be written as ∞ h (t) Mt (u)(x0 ) dt ≤ 0.

0

Then, by Lemma 8.2.13, t → Mt (u)(x0 ) is monotone increasing in ]0, R[. On the other hand, by hypothesis, limr→0+ Mr (u)(x) = u(x). Thus, by Theorem 8.2.10, u ∈ S(Ω). Remark 8.2.14. From the previous proof, we have that t → Mt (u)(x),

0 ≤ t < R(x), x ∈ Ω,

is monotone increasing whenever u ∈ L1loc (Ω) and Lu ≥ 0 in the weak sense of distributions. Here, we denoted R(x) := sup{r > 0 : Bd (x, r) ⊂ Ω}. On the other hand, if u is just a L1loc (Ω) function, very standard arguments show that: (i) (x, t) → Mt (u)(x) is continuous in {(x, t) : x ∈ Ω, 0 ≤ t < R(x)}, (ii) limj →∞ Mtj (u)(x) = u(x) almost everywhere in Ω, for a suitable sequence tj ↓ 0 as j ↑ ∞. The above remark, together with Theorems 8.2.10 and 8.2.11, leads to the next theorem. Theorem 8.2.15 (S(Ω) and the inequality Lu ≥ 0 in D (Ω). II). Let u ∈ L1loc (Ω), with open Ω ⊆ G. Then the following statements are equivalent: (1) Lu ≥ 0 in Ω, in the weak sense of distributions, (2) t → Mt (u)(x) is monotone increasing on ]0, R(x)[ for every x ∈ Ω, u(x) almost everywhere in Ω. (3) there exists a function u ∈ S(Ω) such that u(x) = The function in (3) is unique. It is given by u(x) = lim Mt (u)(x), t↓0

x ∈ Ω.

408

8 L-subharmonic Functions

Proof. (1) ⇒ (2). This follows from the previous remark. (2) ⇒ (3). The monotonicity of t → Mt (u)(x) implies the existence of u(x) := lim Mt (u)(x) t↓0

∀ x ∈ Ω.

Remark 8.2.14-(ii) implies u(x) = u(x) almost everywhere in Ω. Then, if 0 < t < τ , we have Mt ( u)(x) = Mt (u)(x) ≤ Mτ (u)(x) = Mτ ( u)(x). Hence, t → Mt ( u)(x) is monotone increasing. Let us now show that u is u.s.c. Since u(x) = lim Mt (u)(x) = lim Mt ( u)(x), t↓0

t↓0

by Theorem 8.2.10, the assertion will follow. Let λ ∈ R and x ∈ Ω be such that u(x) < λ. Then, by the very definition of u, we have Mt (u)(x) < λ for every t small enough. The continuity of (t, y) → Mt (u)(y) now implies Mt (u)(y) < λ for every t sufficiently small and for any y close to x. As a consequence, being u ≤ Mt (u), u(y) < λ for every y in a neighborhood of x. Thus, u is u.s.c., and this part of the proof is complete. (3) ⇒ (1). By Theorem 8.2.11, if u ∈ S(Ω), then u ∈ L1loc (Ω) and L u ≥ 0 in the weak sense of distributions. Since u = u almost everywhere in Ω, these properties hold for u too. Theorem 8.2.16 (Caccioppoli–Weyl’s lemma for sub-Laplacians). Let u L1loc (Ω) be a weak solution to

∈

Lu = 0 in Ω. Then there exists a smooth L-harmonic u in Ω such that u(x) = u(x) almost everywhere in Ω. Proof. By the previous theorem, there exists u1 ∈ S(Ω) and u2 ∈ S(Ω) such that u2 (x) almost everywhere in Ω. Moreover, u1 (x) = u(x) = u1 (x) = lim Mt (u)(x) = u2 (x). t↓0

u2 ∈ S(Ω)∩S(Ω), so that, by Proposition 7.2.4, u is L-harmonic Hence, u := u1 = in Ω. We also obviously have u(x) = u(x) almost everywhere in Ω. This completes the proof.

8.2 Some Characterizations of L-subharmonic Functions

409

From Theorem 8.2.11 and Theorem 8.2.15 we also obtain the following characterization of L-subharmonic functions. Theorem 8.2.17 (S(Ω) and the inequality Lu ≥ 0 in D (Ω). III). Let Ω be an open subset of G. Then the following statements are equivalent: (i) u ∈ S(Ω), (ii) u ∈ L1loc (Ω), Lu ≥ 0 in Ω in the weak sense of distributions, and2 u(x) = ess lim sup u(y) y→x

∀ x ∈ Ω.

(8.6)

We first prove the following lemma. Lemma 8.2.18. Let Ω be an open subset of G. If u ∈ S(Ω), then (8.6) holds. Proof. Since u is u.s.c., we have u := ess lim sup u ≤ lim sup u ≤ u. Assume by contradiction that u = u. Then there exists x ∈ Ω such that u(x) < λ < u(x) (for some λ ∈ R). In particular u ≤ λ a.e. in a neighborhood of x, so that Mr (u)(x) ≤ λ for any small r > 0. Recall that, by Theorem 8.2.11, we have u(x) = limr→0+ Mr (u)(x). Hence we obtain u(x) ≤ λ, which gives a contradiction. Proof (of Theorem 8.2.17). (i) ⇒ (ii) follows from Theorem 8.2.11 and Lemma 8.2.18. (ii) ⇒ (i). According to Theorem 8.2.15, there exists v ∈ S(Ω) coinciding almost everywhere with u. From Lemma 8.2.18 we infer that v = ess lim sup v. In order to obtain u = v, we now only need to observe that v = u almost everywhere implies ess lim sup v = ess lim sup u and to recall that ess lim sup u = u by hypothesis. Further, we explicit state some maximum principles for L-subharmonic functions, which immediately follow from Theorems 8.1.2 and 8.2.1. Theorem 8.2.19 (The strong maximum principle for S(Ω)). Let Ω ⊆ G be an open set, and let u ∈ S(Ω). The following statements hold: (i) if Ω is connected and there exists x0 ∈ Ω such that u(x0 ) = maxΩ u, then u ≡ u(x0 ) in Ω, (ii) if lim supΩ y→x u(y) ≤ 0 for every x ∈ ∂Ω, and lim supΩ y→∞ u(y) ≤ 0 if Ω is unbounded, then u(y) ≤ 0 in Ω. 2 Here we use the following notation

ess lim sup u(y) = inf ess sup u : V is a neighborhood of x , y→x

V

where ess sup u = inf{m ∈ R ∪ {∞} : m ≥ u(x) for almost every x ∈ V }. V

410

8 L-subharmonic Functions

Proof. Due to Theorems 8.1.2 and 8.2.1, we only have to prove (ii) in the case when Ω is unbounded. From the conditions at the boundary of Ω and at infinity we obtain: for every ε > 0, there exists R > 0 such that u(x) ≤ ε

lim sup

∀ y ∈ ∂(Ω ∩ Bd (0, R))

Ω∩Bd (0,R) x→y

and ∀ x ∈ Ω \ Bd (0, R).

u(x) < ε

Then, by the maximum principle on bounded domains, u(x) < ε for every x ∈ Ω. Letting ε vanish, we get u ≤ 0 in Ω. We close the section by giving a theorem on superposition of superharmonic functions. Theorem 8.2.20. Let Λ ⊆ Rν and Ω ⊆ G be connected open sets. Let u : Λ × Ω → ]−∞, ∞] be a l.s.c. function. Finally, let μ be a Radon measure with compact support contained in Λ. Assume that u(z, ·) ∈ S(Ω) for every fixed z ∈ Λ. Then u(z, x) dμ(z), x ∈ Ω, U (x) := Λ

is L-superharmonic in Ω if there exists x0 ∈ Ω such that U (x0 ) < ∞.

(8.7)

Proof. Since u is l.s.c., we have lim inf U (x) ≥ x→y

lim inf u(z, x) dμ(z) Λ

≥

x→y

u(z, y) dμ(z)

∀ y ∈ Ω.

Λ

Thus, U is l.s.c. in Ω. Let x ∈ Ω and r > 0 be such that Bd (x, r) ⊂ Ω. By Fubini–Tonelli’s theorem, we obtain Mr (U )(x) = Mr (u(z, ·))(x) dμ(z) Λ

(since u(z, ·) is super-mean) ≥ u(z, x) dμ(z) = U (x).

(8.8)

Λ

Thus, U is super-mean. From Corollary 8.2.3 and condition (8.7) it follows that U ∈ S(Ω). This completes the proof.

8.3 Continuous Convex Functions on G

411

Remark 8.2.21. The previous theorem still holds if we replace the condition supp(μ) compact and contained in Λ with the following one: for every compact set K ⊂ Ω, there exists a function ϕK ∈ L1 (Λ, dμ) such that u(z, x) ≥ ϕK (z)

∀ z ∈ Λ ∀ x ∈ K.

(8.9)

Indeed, if this condition is satisfied, we can still apply Fubini–Tonelli’s theorem in (8.8). For example, we explicitly remark that (8.9) holds with ϕK ≡ 0 if u is non-negative.

8.3 Continuous Convex Functions on G Throughout this section, we shall follow two different approaches in defining convex functions in Carnot groups. Our main references will be G. Lu, J.J. Manfredi and B. Stroffolini [LMS04] (see also P. Juutinen and the same authors, [JLMS07]) and D. Danielli, N. Garofalo and D.-M. Nhieu [DGN03]. If an extra hypothesis is added (namely, continuity) these two notions are equivalent. Since the aim of this section is only to give an overview on convexity, we decided to restrict our attention to continuous convex functions on Carnot groups, since the dropping of the continuity hypothesis would lead us beyond our scopes. The reader will be referred to suitable references for more general results (see Note 8.3.18). A convex function in RN is subharmonic with respect to every constant coefficient elliptic operator, and vice versa (see, e.g. [Hor94]). This makes quite natural the following (first) extension of the notion of convexity to Carnot groups. To this end, we introduce the following notation. We shall denote by Sm + the cone of the real m × m symmetric and strictly positive definite matrices. Let G be an abstract stratified Lie group with the Lie algebra g. Let also V1 ⊕ V2 ⊕ · · · ⊕ Vr be a stratification of g according to Definition 2.2.3 (page 122). We set V = (V1 , . . . , Vr ). Let finally X = {X1 , . . . , Xm } be a basis of the first layer V1 of the stratification (the so-called horizontal layer). Given a matrix A = (aj,k )j,k ∈ Sm +, we denote by LA the sub-Laplacian3 LA =

m

aj,k Xj Xk .

(8.10)

j,k=1

Definition 8.3.1 (v-convex function). Let G be a stratified group, and let V = (V1 , . . . , Vr ) be a given stratification of the algebra of G. Let {X1 , . . . , Xm } be a basis of the first layer V1 . A continuous real-valued function u in an open set Ω ⊆ G is v-convex if it is LA -subharmonic in Ω for every A ∈ Sm + , being LA as in (8.10). 3 See Remark 8.3.3 below.

412

8 L-subharmonic Functions

Note 8.3.2. Obviously, the notion of v-convexity is in general well-given for upper semi-continuous functions. The above definition is one of the equivalent forms of the notion of v-convexity in [LMS04], convexity in the viscosity sense. Our choice to add the extra hypothesis of continuity has already been justified by the sake of simplicity. We explicitly remark the following fact: the notion of v-convexity depends on the stratification V = (V1 , . . . , Vr ) of g or, more precisely, by the horizontal layer V1 (since V1 determines all the Vi ’s). Hence, another possible notation should be “H-convex function” (where “H” stands for “horizontal”). Remark 8.3.3. As we shall see in Proposition 16.1.1 in Chapter 16 (page 623), any sub-Laplacian on G is of the form (8.10) for a suitable symmetric positive definite matrix LA in that form is a sub-Laplacian m A, 2and vice versa, many operator 1/2 ) Y with Y = (A X LA = k k,j j . This immediately proves the folk=1 k j =1 lowing proposition. Proposition 8.3.4 (Equivalent definition of v-convexity). A continuous real-valued function u in an open set Ω ⊆ G is v-convex if and only if it is L-subharmonic in Ω for every sub-Laplacian L (related to V ). More explicitly, denoted by V1 the first layer of the stratification V , a continuous function u is v-convex in Ω if and only if it is L-subharmonic in Ω w.r.t. every L = m 2 , being (Y , . . . , Y ) any basis of V . Y 1 m 1 k=1 k Note 8.3.5. Observe that the above Proposition 8.3.4 highlights the fact that the notion of v-convexity in Definition 8.3.1 does not depend on the particular basis (X1 , . . . , Xm ) of V1 . Remark 8.3.6. We know from Section 8.2 that the LA -subharmonicity of u is equivalent to the inequality in D (Ω), (8.11) LA u ≥ 0 i.e. in the weak sense of distributions. It follows that the v-convexity is invariant with respect to left translations and dilations on G, and that locally uniform limits of vconvex functions are v-convex. Indeed, such properties hold for the solutions to the inequality (8.11). For a function u of class C 2 , the v-convexity can be characterized in terms of its horizontal Hessian related to the family X , i.e. of the matrix Xj Xk u(x) + Xk Xj u(x) X 2 u(x) := . 2 j,k=1,...,m Proposition 8.3.7 (v-convexity and the horizontal Hessian). With all the above notation, a function u ∈ C 2 (Ω, R) is v-convex if and only if X 2 u(x) ≥ 0

for every x ∈ Ω.

8.3 Continuous Convex Functions on G

413

Proof. Since u ∈ C 2 , the distributional inequality (8.11) is equivalent to LA u(x) ≥ 0 for every x ∈ Ω, so that u is v-convex if and only if m

aj,k Xj Xk u(x) ≥ 0

for every A = (aj,k )j,k in Sm + and every x ∈ Ω.

j,k=1

Then the assertion follows from the following linear algebra lemma. Lemma 8.3.8. Let B = (bi,j )i,j be an m × m real matrix. Then m

aj,k bj,k ≥ 0

(8.12)

j,k=1

for every A = (aj,k )j,k in Sm + if and only if the symmetric part of B is non-negative definite, i.e. bj,k + bk,j sym ≥ 0. B := 2 j,k≤m Proof. Let ξ = (ξ1 , . . . , ξm )T ∈ Rm and ε > 0. Consider the matrix Aε := ξ · ξ T + ε Im = (ξj ξk )j,k≤m + ε Im , where Im denotes the m × m identity matrix. Clearly, Aε ∈ Sm + since it is symmetric and Aε η, η = ηT · ξ · ξ T · η + ε |η|2 = (ηT · ξ )(ξ T · η) + ε |η|2 = ξ, η2 + ε |η|2 , for every (column-vector) η ∈ Rm . Then, we can use Aε in (8.12) obtaining 0≤

m

bj,k ξj ξk + ε

j,k=1

m

bj,j .

j =1

Letting ε → 0, we get 0≤

m j,k=1

bj,k ξj ξk =

m bj,k + bk,j ξj ξk 2

∀ ξ ∈ Rm .

j,k=1

This proves the “only if” part of the lemma. To prove the “if” part, we first remark that inequality (8.12) rewrites as trace(A · B sym ) ≥ 0. m 2 Then, for given A ∈ Sm + taking a matrix C ∈ S+ such that C = A, we have

trace(A · B sym ) = trace(C 2 · B sym ) = trace(C · B sym · C) ≥ 0, since the matrix C · B sym · C = C · B sym · C T is non-negative definite if B sym is. This completes the proof.

414

8 L-subharmonic Functions

The following result holds. Corollary 8.3.9 (v-convexity for C 2 -functions). A C 2 (Ω, R) function u is v-convex if and only if one of the following equivalent statements hold: (1) X 2 u ≥ 0 on Ω for one basis X = (X1 , . . . , Xm ) of V1 ; (2) X 2 u ≥ 0 on Ω for every basis X = (X1 , . . . , Xm ) of V1 ; (3) Lu ≥ 0 on Ω for every sub-Laplacian L (related to the stratification V ). Proof. (1) is a restatement of Proposition 8.3.7. (2) follows from Proposition 8.3.7 and the independence of the notion of vconvexity w.r.t. the basis X of V1 (see Note 8.3.5). (3) follows from Proposition 8.3.4 and the characterization of L-subharmonicity for C 2 -functions. The smooth v-convex functions are dense in the set of the continuous v-convex functions. Indeed, the following result holds (see the definition of mollifier in Definition 5.3.6, page 239; precisely, see the particular case in Example 5.3.7 on page 239). Theorem 8.3.10 (Smoothing of a v-convex function). Let G be a homogeneous Carnot group. Let V be a fixed stratification of the algebra of G. Let also d be any fixed homogeneous norm on G. Suppose u ∈ C(Ω, R) is a v-convex function. For ε > 0, let Ωε be the set {y ∈ G : Bd (y, ε) ⊆ Ω}. For x ∈ Ωε , define as usual uε (x) = u(y) Jε (x ◦ y −1 ) dy, (8.13) Ω

where J ∈ C0∞ (Bd (0, 1)) (i) uε ∈ C ∞ (Ωε ) and uε

and Jε (z) = ε −Q J (δ1/ε (z)). Then: → u as ε → 0 uniformly on the compact sets in Ω,

(ii) uε is v-convex in Ωε . Proof. (i) is a standard property of the mollifiers. (ii). Since u is v-convex, u is LA -subharmonic for every A ∈ Sm + . It follows from Theorem 8.1.5 that uε is LA -subharmonic in its domain. Since this holds for every A ∈ Sm + , we infer that uε is v-convex in Ωε . The characterization of v-convexity in terms of the horizontal Hessian implies the convexity in the usual sense along the V -horizontal segments and vice versa. To make this statement precise, we first introduce the notion of V -horizontal segment and of V -horizontal subspace. Definition 8.3.11 (V -horizontal segment and subspace). Let (G, ◦) be a stratified group with the Lie algebra g. Let also V = (V1 , . . . , Vr ) be a given stratification of g, and let Exp : g → G be the usual exponential map. We set V := Exp (V1 ), and we say that V is the V -horizontal subspace of G.

8.3 Continuous Convex Functions on G

415

Let x ∈ G be fixed. If h is any element of V, we say that [x ◦ h−1 , x ◦ h] := x ◦ Exp (t Log (h)) : −1 ≤ t ≤ 1 is a V -horizontal segment (along h and) centered at x. Finally, for a fixed x ∈ G, we say that Vx := x ◦ V = {x ◦ Exp (X) : X ∈ V1 } is the V -horizontal subspace through x. Remark 8.3.12. Notice that V = Ve , being e the identity of G. We explicitly remark that V is a subset of G closed under inversion (for (Exp (X))−1 = Exp (−X)), it contains the identity of G, but V is not in general a subgroup of G. Indeed 1 Exp (X) ◦ Exp (Y ) = Exp (X Y ) = Exp X + Y + [X, Y ] + · · · , 2 and, if X, Y ∈ V1 , it does not generally hold X Y ∈ V1 . We remark that V is a submanifold of G of dimension m = dim(V1 ). Theorem 8.3.13 (v-convexity and the horizontal segments). Let u ∈ C 2 (Ω, R) with open Ω ⊆ G. Then u is v-convex in Ω if and only if, for every x ∈ Ω and every h ∈ V such that [x ◦ h−1 , x ◦ h] ⊂ Ω, the function (−1, 1) t → u x ◦ Exp (t X) (being X = Log (h)) is convex in the classical sense. Proof. We split the proof in two parts. Sufficiency. Let u be v-convex. Take any h ∈ V such that the segment along h centered at x is contained in Ω. Let X := Log (h). Hence, the function (−1, 1) t → f (t) := u x ◦ Exp (t X) is well posed and, thanks to the C 2 -regularity assumption on u, f ∈ C 2 (−1, 1). By (2.36a), page 117, we have f (t) = (X 2 u) x ◦ Exp (t X) . (8.14) Recalling that X ∈ V1 by definition of V, we can suppose that X is the first element of a basis of V1 , say X . By Proposition 8.3.7, X 2 u(z) ≥ 0 for every z ∈ Ω. Since z := x ◦ Exp (t X) ∈ Ω, for [x ◦ h−1 , x ◦ h] ⊂ Ω, this gives (X 2 u)(z) = X 2 u(z) e1 , e1 ≥ 0 (being e1 = (1, 0, . . . , 0) ∈ Rm ). Consequently, (8.14) gives f (t) ≥ 0. Being t ∈ (−1, 1) arbitrary, this proves that f is convex.

416

8 L-subharmonic Functions

Necessity. By the C 2 -regularity of u and thanks to Corollary 8.3.9-(3), we have 2 to prove that, given a basis X = (X1 , . . . , Xm ) for V1 , it holds m j =1 Xj u(x) ≥ 0 for every x ∈ Ω. Fix any j ∈ {1, . . . , m} and any x ∈ Ω. Since Ω is open, there exists ε > 0 so small that x ◦ Exp (t εXj ) ∈ Ω

for every t ∈ [−1, 1].

This means that setting h := Exp (ε Xj ), we have [x ◦ h−1 , x ◦ h] ⊂ Ω. Since h clearly belongs to V, the necessity assumption ensures that the function (−1, 1) t → f (t) := u x ◦ Exp (t ε Xj ) is a convex function in the classical sense. In particular, f (0) ≥ 0. On the other hand, again by (2.36a), page 117, we have 0 ≤ f (0) = ε 2 (Xj2 u)(x), whence (Xj2 u)(x) ≥ 0. The arbitrariness of j and x proves that negative on Ω, and the proof is complete.

m

2 j =1 Xj u

is non-

We explicitly remark that, though Definition 8.3.1 of v-convexity needs some regularity assumption on u to be well posed (for example, upper semicontinuity or a L1loc assumption), the characterization of v-convexity as stated in Theorem 8.3.13 suggests another notion of convexity, which is free from any a priori assumption on u. We hence give the following definition. Definition 8.3.14 (Horizontally convex function). Let (G, ◦) be a stratified group with a fixed stratification V = (V1 , . . . , Vr ). Let Ω ⊆ G be open. A function u : Ω → R will be called horizontally convex (or, in short, H-convex) in Ω if the function [−1, 1] t → u(x ◦ Exp (t X)) is convex (in the classical sense) for every X ∈ V1 such that (set h = Exp (X)) the horizontal segment [x ◦ h−1 , x ◦ h] is contained in Ω. Note 8.3.15. The above notion of H-convex function is equivalent (see Proposition 8.3.17) to that of “weakly H-convex” (weakly horizontally convex) function introduced by D. Danielli, N. Garofalo and D.-M. Nhieu [DGN03]. The same notion is also referred to as “CC-convexity”. L.A. Caffarelli’s 1997 NSF Proposal is cited in [DGN03] as seemingly the first reference for a notion of convexity in the Heisenberg groups. The simpler name “H-convex” is now commonly used. We now focus our attention on homogeneous Carnot groups. See Remark 8.3.20 for the details on the link between the notions of convexity in the abstract and in the homogeneous setting. It turns out that for continuous functions v-convexity and horizontal convexity are equivalent. This will be easily seen first noticing the following facts:

8.3 Continuous Convex Functions on G

417

(i) Pointwise limits of horizontally convex functions are horizontally convex; (ii) If u ∈ C(Ω, R) is horizontally convex, then so is its mollifier uε defined in (8.13). Indeed, the change of variable x ◦ y −1 = z in the integral at the right-hand side of (8.13) gives uε (x) = u(z−1 ◦ x)Jε (z) dz. Ω

Then, for every X ∈ V1 ,

t → uε (x ◦ Exp (t X)) =

u z−1 ◦ x ◦ Exp (t X) Jε (z) dz

Ω

is convex since t → u((z−1 ◦ x) ◦ Exp (t X)) is convex for every z ∈ Bd (0, ε) with ε > 0 small enough. We are now in the position to easily prove the equivalence between v- and H- convexity for continuous functions on homogeneous Carnot groups. Theorem 8.3.16 (v- and H-convexity for continuous functions). Let G be a homogeneous Carnot group. Let u ∈ C(Ω, R) with open Ω ⊆ G. Then u is v-convex in Ω if and only if it is horizontally convex in Ω. Proof. If u is v-convex, then by Theorem 8.3.10, the function uε in (8.13) is smooth and v-convex in Ωε . Then, by Theorem 8.3.13, uε is horizontally convex. On the other hand, uε converges to u as ε → 0, uniformly on the compact subsets of Ω. The previous remark (i) implies that u is horizontally convex. Vice versa, let us assume that u is horizontally convex. The previous remark (ii) tells us that uε has the same property, and Theorem 8.3.13 ensures that uε is vconvex. As ε → 0, we get the v-convexity of u (see Remark 8.3.6). This completes the proof. In the following result, we compare Definition 8.3.14 to the notion of H-convex function in the recent literature. For example, 4) in the proposition below is [DGN03, Definition 5.5], whereas 5) is [Mag06, Definition 3.4]. For the sake of simplicity, we consider the case Ω = G. For the general case, see V. Magnani [Mag06, Proposition 3.9]. Proposition 8.3.17 (Characterizations of H-convexity). Let (G, ∗) be a stratified group, and let V = (V1 , . . . , Vr ) be a given stratification of g, the Lie algebra of G. For any λ > 0, let Δλ : g → g be the linear map such that (for every i = 1, . . . , r) Δλ acts on Vi as the multiplication times λi . Let δλ := Exp ◦ Δλ ◦ Log . Finally, let u : G → R be a function. Then the following statements are equivalent: (1) u is horizontally convex on G; (2) for every x ∈ G, X ∈ V1 and t ∈ (0, 1), it holds u x ∗ Exp (t X) ≤ (1 − t) u(x) + t u x ∗ Exp (X) ;

(8.15a)

418

8 L-subharmonic Functions

(3) for every x ∈ G, h ∈ V and λ ∈ (0, 1), it holds u x ∗ δλ (h) ≤ (1 − λ) u(x) + λ u(x ∗ h);

(8.15b)

(4) for every x ∈ G, x ∈ Vx and λ ∈ (0, 1), it holds u x ∗ δλ (x −1 ∗ x ) ≤ (1 − λ) u(x) + λ u(x );

(8.15c)

(5) for every x, x ∈ G satisfying the geometrical constraint x −1 ∗ x ∈ V and every λ ∈ (0, 1), it holds (8.15c). Proof. Obviously, (4) and (5) are equivalent, since the condition x −1 ∗ x ∈ V is equivalent to x ∈ x ∗ V = Vx . Moreover, (3) is clearly equivalent to (4), by setting h = x −1 ∗ x . We immediately notice that (2) is a restatement of (3), since the following facts hold: V = Exp (V1 ); δλ ◦ Exp = Exp ◦ Δλ ; for every X ∈ V1 , we have δλ (Exp (X)) = Exp (Δλ (X)) = Exp (λ X). We are then left with the proof of the equivalence of (1) and (2). Suppose first that (1) holds. Then the function ()

[−1, 1] t → f (t) := u(x ∗ Exp (t X))

is convex for every x ∈ G and every X ∈ V1 . In particular, we have u(x ∗ Exp (t X)) = f (t) = f t · 1 + (1 − t) · 0 ≤ t f (1) + (1 − t) f (0) = t u(x ∗ Exp (X)) + (1 − t) u(x), and (8.15a) follows. Vice versa, suppose that (2) holds. We have to prove that the function f in () is convex. To this end, fix t1 , t2 ∈ [−1, 1] and θ ∈ [0, 1]. We have to demonstrate that f (θ t2 + (1 − θ )t1 ) ≤ θ f (t2 ) + (1 − θ ) f (t1 ), i.e. u x ∗ Exp θ t2 X + (1 − θ )t1 X ≤ θ u(x ∗ Exp (t2 X)) + (1 − θ ) u(x ∗ Exp (t1 X)). (8.16) Observe that x ∗ Exp θ t2 X + (1 − θ )t1 X = x ∗ Exp t1 X + θ (t2 − t1 ) X = x ∗ Exp (t1 X) θ (t2 − t1 ) X = x ∗ Exp (t1 X) ∗ Exp θ (t2 − t1 ) X .

(8.17)

Here we used the Campbell–Hausdorff formula Exp (A B) = Exp (A) ∗ Exp (B) (for every A, B ∈ g) jointly with the obvious fact (λ X) (μ X) = λ X + μ X (for every λ, μ ∈ R), since it holds

8.3 Continuous Convex Functions on G

(λ X) (μ X) = λ X + μ X +

419

1 [λ X, μ X] + · · · = λ X + μ X. 2

Now, taking into account (8.17) and applying (8.15a) with x, t and X respectively replaced by x ∗ Exp (t1 X), θ and (t2 − t1 ) X (which clearly belongs to V1 ), we infer u x ∗ Exp θ t2 X + (1 − θ )t1 X ≤ (1 − θ ) u(x ∗ Exp (t1 X)) + θ u x ∗ Exp (t1 X) ∗ Exp (t2 − t1 )X = (1 − θ ) u(x ∗ Exp (t1 X)) + θ u(x ∗ Exp (t2 X)), for arguing as above, Exp (t1 X) ∗ Exp (t2 − t1 )X = Exp (t1 X) (t2 − t1 )X = Exp (t1 X) + (t2 − t1 )X = Exp (t2 X). This ends the proof. Note 8.3.18 (On the continuity of convex functions on Carnot groups). We record some references from the recent literature. D. Danielli, N. Garofalo and D.-M. Nhieu [DGN03] proved that locally bounded H-convex functions on the Heisenberg groups are locally Lipschitz-continuous w.r.t. the quasi-distance generated by a homogeneous norm. Z.M. Balogh and M. Rickly [BR02] have improved this result, removing the hypothesis of local bound. For general Carnot groups, V. Magnani [Mag06] has proved that the H-convex functions which are locally bounded from above are locally Lipschitz-continuous (in the above sense). Finally, M. Sun and X. Yang [SY06] have recently proved that, in general Carnot groups of step two, the H-convex functions are locally bounded, so that (by the cited result in [Mag06]) they are locally Lipschitz-continuous. Note 8.3.19 (Some other references). We collect some recent references on convexity for Carnot groups. Besides the already mentioned D. Danielli, N. Garofalo, D.M. Nhieu [DGN03] and G. Lu, J.J. Manfredi and B. Stroffolini (and P. Juutinen) [LMS04,JLMS07] and the references in the previous Note 8.3.18, the reader is also referred to C.E. Gutiérrez and A. Montanari [GM04a,GM04b], M. Rickly [Ric06], C. Wang [Wan05]. Remark 8.3.20 (Convexity in the homogeneous Carnot group setting). By Proposition 2.2.22 on page 139, given any abstract stratified group (H, ∗) and any of its stratifications V = (V1 , . . . , Vr ), there exists a homogeneous Carnot group (G, •) with the following properties: (1) (G, •) is isomorphic to (H, ∗) via a Lie-group isomorphism Ψ : G → H, whence (denoted by g and h the relevant Lie algebras) dΨ : g → h is a Lie algebra iso-

8 L-subharmonic Functions

420

morphism; Ψ is completely determined by a fixed choice of E = (E1 , . . . , EN ), a basis of h adapted to the stratification V ; = (V 1 , . . . , V r ) is a i := (dΨ )−1 (Vi ) for every i = 1, . . . , r, then V (2) Set V stratification of g; more explicitly, V1 is formed by the first m elements of the Jacobian basis of g, where m = dim(V1 ); (3) Denoting by Exp H : h → H, Exp G : g → G the relevant exponential maps and 1 ), we have Ψ ( V = Exp G (V V) = V. setting V = Exp H (V1 ), (4) With the above notation, V has a very explicit expression w.r.t. the usual coordinates on G, V = (h1 , . . . , hm , 0, . . . , 0) : h1 , . . . , hm ∈ R . The statements in (3) and (4) deserve explicit proofs: (3). By Theorem 2.1.59 (page 119), we know that ()

Exp H ◦ dΨ = Ψ ◦ Exp G ,

whence 1 )) = Ψ Exp G ((dΨ )−1 (V1 )) = Exp H (V1 ) = V. Ψ ( V) = Ψ (Exp G (V (4). Let us denote by ei (i = 1, . . . , N) the i-th element of the standard basis of RN ≡ G. Also, let us denote by Z1 , . . . , ZN the Jacobian basis of g. By Proposition 2.2.22-(2), we know that Exp G is a linear map and that Exp G (Zi ) = ei ,

dΨ (Zi ) = Ei ,

i = 1, . . . , N.

As a consequence, it holds (use () and notice that V1 = span{E1 , . . . , Em }) V = Ψ −1 (V) = Ψ −1 Exp H (V1 ) = Exp G ◦ (dΨ )−1 (V1 ) (notice that Exp G ◦ (dΨ )−1 is linear) = span Exp G ◦ (dΨ )−1 (Ei ) : i = 1, . . . , m = span Exp G (Zi ) = ei : i = 1, . . . , m = (h1 , . . . , hm , 0, . . . , 0) : h1 , . . . , hm ∈ R . Let Ω be an open subset of H, and let us denote by V -Convv (Ω),

V -ConvH (Ω),

respectively, the v-convex functions and the horizontally convex functions on Ω is an open subset of G, we use the (w.r.t. the stratification V ). Analogously, if Ω notation -Convv (Ω), -ConvH (Ω) V V

8.3 Continuous Convex Functions on G

421

). We aim to prove that the following equalities (this time w.r.t. the stratification V hold: V -Convv (Ω) V -ConvH (Ω) -Convv (Ω) V -ConvH (Ω) V

-Convv (Ψ −1 (Ω))}, = { u ◦ Ψ −1 : u∈V -ConvH (Ψ −1 (Ω))}, = { u ◦ Ψ −1 : u∈V = {u ◦ Ψ : u ∈ V -Convv (Ψ (Ω))}, = {u ◦ Ψ : u ∈ V -ConvH (Ψ (Ω))}.

The first (hence the third) equality holds true, since the notion of v-convexity only involve sub-harmonicity w.r.t. the relevant sub-Laplacians (and we know that the sub are Ψ -related). The fourth equality Laplacians related to V and to those related to V (hence the second) follows by the arguments below. ∈V 1 be such that ⊆ G be open, let x ∈ Ω and X Let Ω ∈Ω for every t ∈ [−1, 1]. x • Exp G (t X) ∈ Ψ (Ω) for every t ∈ [−1, 1]. But we have This is equivalent to Ψ (x • Exp G (t X)) (see () above) = Ψ (x) ∗ Ψ Exp G (t X) = Ψ (x) ∗ Exp H t dΨ (X) . Ψ x • Exp G (t X) by definition of H-convexity, we have that So, if u ∈ V -ConvH (Ψ (Ω)), [−1, 1] t → f (t) := u Ψ (x) ∗ Exp H (t X) and we noticed that, by definition of V 1 = is convex (here we let X := dΨ (X) −1 (dΨ ) (V1 ), it holds X ∈ V1 ). On the other hand, by the above arguments, , f (t) = (u ◦ Ψ ) x • Exp G (t X) ∈V 1 ) proves that u ◦ Ψ ∈ and the convexity of f (jointly with the arbitrariness of X V -ConvH (Ω). The above remarks demonstrate that the study of v- and H- convexity can be entirely carried out in the homogeneous setting, without lack of generality. The (more) manageable structure of homogeneous Carnot groups (see, e.g. property (4) above) may be useful in studying convexity in a simpler way. Bibliographical Notes. A comprehensive introduction to the classical potential theory and to the study of sub-harmonicity can be found, e.g. in the following monographs: D.H. Armitage and S.J. Gardiner [AG01] and in W.K. Hayman and P.B. Kennedy [HK76]. See also H. Aikawa and M. Essén [AE96], S. Axler, P. Bourdon, W. Ramey [ABR92], N. du Plessis [duP70], L.L. Helms [Helm69], N.S. Landkof [Lan72]. For a bibliography concerning with the notions of convexity in Carnot groups, see the references within Section 8.3. Some of the topics presented in this chapter also appear in [BL03].

422

8 L-subharmonic Functions

8.4 Exercises of Chapter 8 Ex. 1) Prove that if u is a continuous function which coincides with an L-subharmonic function almost everywhere, then u is L-subharmonic. (Hint: Use Theorem 8.2.11.) Ex. 2) Prove that two L-superharmonic functions on an open set Ω which are equal a.e. in Ω coincide everywhere. (Hint: Use Theorem 8.2.11.) Ex. 3) Prove that LA in (8.10) is indeed a sub-Laplacian on G. Ex. 4) Let Ω be a connected bounded open set in G and let u : Ω → [−∞, ∞[ be an L-hypoharmonic function. Then u is L-subharmonic in Ω if and only if there exists a point x0 ∈ Ω such that u(x0 ) > −∞. Ex. 5) Show that if u ∈ S(Ω), then u(x) = lim sup u(y). Ω y→x

Ex. 6) Let F be a family of L-subharmonic functions in an open set Ω ⊆ G. Assume that u(x) := sup{v(x) : v ∈ F} is u.s.c. and belongs to L1loc (Ω). Then u ∈ S(Ω). Ex. 7) Let Γ be the fundamental solution of L, and let A be a closed subset of G. Define u(x) := sup{Γ (y −1 ◦ x) : y ∈ A}. Then u ∈ S(G \ A) and it is locally Lipschitz-continuous with respect to d. Ex. 8) There exists a function u ∈ S(G) ∩ C(G) such that Ut := {x : u(x) ≤ t} is compact for every t > 0 and Ut = G t>0

(Hint: Use suitable powers of a gauge function.) Ex. 9) Let Ω be any open set in G. Then there exists a function u ∈ S(Ω) ∩ C(Ω) such that Ut := {x : u(x) ≤ t} is compact for every t > 0 and Ut = Ω. t>0

(Hint: Use the previous exercises.)

8.4 Exercises of Chapter 8

423

Ex. 10) Let u be an L-superharmonic function in an open set Ω ⊆ G. Show that Mr (u)(x) < ∞ for every r > 0 such that Bd (x, r) ⊆ Ω. (Hint: Mr (u)(x) < ∞ since u ∈ L1loc . Then use the integral representation of Mr in terms of Mρ , and the monotonicity of Mρ , 0 < ρ ≤ r.) Ex. 11) Let Ω ⊆ G be open, u : Ω → ]−∞, ∞] be L-superharmonic and α ∈ G. Show that x → u(α ◦ x) is L-superharmonic in α −1 ◦ Ω. Ex. 12) Let Ω be a connected bounded open set in G, and let f : ∂Ω → [−∞, ∞]. Ω Ω Ω Assume H f ≡ ∞ and H Ω f ≡ −∞. Then H f and H f are L-harmonic Ω

Ω

in Ω. (Hint: If H f ≡ ∞, then U f contains an L-superharmonic function. Ω Similarly, if H Ω f ≡ −∞, then U f contains an L-subharmonic function.) Ex. 13) Let T be as in (5.150) of Ex. 18, Chapter 5 (page 326), and let u ∈ S(Ω). Prove the following statements: (i) T (u) ≤ u, (ii) T (k) (u) ≥ T (k+1) (u) for every k ∈ N, where T (k) = T · · ◦ T . ◦ · k times

Ex. 14) Let T be as in the previous exercise, and let u ∈ S(Ω) and v ∈ S(Ω) be such that v ≤ u. Then: (i) v ≤ T (k) (v) ≤ T (k) (u) ≤ u for every k ∈ N, (ii) T (k) (v) ↑ v ∗ , T (k) (u) ↓ u∗ for suitable v ∗ , u∗ ∈ L1loc (Ω). Moreover, T (v ∗ ) = v ∗ ≤ u∗ = T (u∗ ). Ex. 15) Let Ω ⊆ G be a bounded open set, and let T be as in the previous exercises. Let u ∈ S(Ω) ∩ C(Ω) and f = u|∂Ω . Then (T (k) (u))k≥1 is monotone decreasing and lim T (k) (u) = HfΩ . k→∞

(Hint: If h := limk→∞ T (k) (u), then h = T (h). It follows that v ≤ h ≤ w Ω for every v ∈ U Ω f and w ∈ U f .) Ex. 16) We keep the notation of the previous exercise. Let F ∈ C(Ω) and f = F |∂Ω . Prove that the sequence (T (k) (F ))k≥1 is convergent and lim T (k) (F ) = HfΩ .

k→∞

(Hint: For every n ∈ N, there exists un ∈ S(G) ∩ C(G) and vn ∈ S(G) ∩ C(G) such that vn − n1 < F < un + n1 in Ω. Then use the previous exercises.)

424

8 L-subharmonic Functions

Ex. 17) Let L be a sub-Laplacian, and let d be an L-gauge. Prove that d α ∈ S(G)

⇐⇒

α ≥ 0.

Prove also that d α ∈ S(G)

⇐⇒

2 − Q ≤ α ≤ 0.

(Hint: Compute L(d α ) in G \ {0}. Remind that if d α ∈ S(G), then d α < ∞ at any point.) Ex. 18) Let Ω ⊆ G be open, and let u ∈ C(Ω, R). Prove the equivalence of the following statements: • u ∈ S(Ω), • ALu ≥ 0 in Ω, • ALu ≥ 0 in Ω. Here AL and AL denote the asymptotic L sub-Laplacians introduced in Exercise 8 of Chapter 5.

9 Representation Theorems

The aim of this chapter is mainly to establish some representation theorems of Riesz-type for L-subharmonic functions, being L a sub-Laplacian on a homogeneous Carnot group G. We start by introducing the L-Green function GΩ , first for an L-regular domain Ω, then for general open sets. In order to prove the symmetry of GΩ in the latter case, we show that any open set can be approximated by L-regular domains. We then come to the core of the chapter, by introducing the L-Green potentials and proving several Riesz-type representation theorems for L-subharmonic functions. In the rest of the chapter, we give some applications of the above results. In Section 9.5, we prove the Poisson–Jensen formula for L-regular domains. We shall turn back to this formula in Chapter 11, where we shall prove the Poisson–Jensen formula for arbitrary domains (see Theorem 11.7.6 on page 518), by using the theory of L-polar sets. In Section 9.7, we prove a monotone approximation theorem for L-subharmonic functions via smooth L-subharmonic functions, by using the smoothing operators constructed in Section 5.6 (page 257) by superposition of surface mean operators. In Section 9.8, we study isolated singularities for L-harmonic functions and we prove some Bôcher-type theorems. As an application, we prove that the kernel appearing in the solid mean value formula of Theorem 5.6.1 (page 259) is constant on G \ {0} if and only if G is the Euclidean group.

9.1 L-Green Function for L-regular Domains Definition 9.1.1 (L-Green function for an L-regular domain). Let Ω be a bounded L-regular open subset of G. We call L-Green function of Ω with pole at x ∈ Ω, the function GΩ (x, ·) : Ω → ]−∞, ∞] defined as follows GΩ (x, y) := Γ (x −1 ◦ y) − hx (y), where Γ is the fundamental solution for L, and hx denotes the solution to the boundary value problem

426

9 Representation Theorems

Lh = 0 h(z) = Γ (x −1 ◦ z)

in Ω, for every z ∈ ∂Ω.

With the above definition, we have GΩ (x, ·) is L-harmonic in Ω \ {x}, GΩ (x, y) −→ 0 as y → z, for every z ∈ ∂Ω, and GΩ (x, y) = Γ (x

−1

◦ y) − ∂Ω

Γ (x −1 ◦ z) dμΩ y (z),

(9.1a) (9.1b)

x, y ∈ Ω.

(9.1c)

We recall that μΩ y denotes the L-harmonic measure related to (the L-regular open set) Ω and the point y (see Section 7.2, page 388). The following theorem states some other important properties of the L-Green function. Theorem 9.1.2. For every x, y ∈ Ω, x = y, we have: (i) GΩ (x, y) ≥ 0, (ii) GΩ (x, y) > 0 iff x and y belong to the same connected component of Ω, (iii) GΩ (x, y) = GΩ (y, x). Proof. (i). Since Γ (x −1 ◦ z) → ∞ as z → x and hx is smooth in Ω, there exists r > 0 such that GΩ (x, z) > 0 for every z ∈ Bd (x, r). Moreover, in Ω \ Bd (x, r), LGΩ (x, ·) = 0 limz→ζ GΩ (x, z) ≥ 0 for every ζ ∈ ∂(Ω \ Bd (x, r)), so that, by the maximum principle, GΩ (x, z) ≥ 0 in Ω \Bd (x, r). Thus, GΩ (x, z) ≥ 0 for any z ∈ Ω. In particular, GΩ (x, y) ≥ 0. (ii). Suppose x, y ∈ Ω0 , with Ω0 ⊆ Ω open and connected. Assume by contradiction GΩ (x, y) = 0. Then, since GΩ (x, ·) is non-negative and L-harmonic in Ω0 \ {x}, by the strong maximum principle of Theorem 5.13.8 (page 296), GΩ (x, z) = 0 for every z ∈ Ω0 \ {x}. This is impossible, because GΩ (x, z) → ∞ as z → x. Let us now suppose y ∈ Ω1 , being Ω1 a connected component of Ω not containing x. Then z → Γ (x −1 ◦ z) is L-harmonic in an open set containing Ω1 , so that hx (z) = Γ (x −1 ◦ z) for every z ∈ Ω1 . It follows that GΩ (x, ·) = 0 in Ω1 . In particular, GΩ (x, y) = 0. (iii). Let y ∈ Ω be fixed. Denote by gy the Γ -potential of μΩ y , i.e. gy (z) :=

Γ (ζ ∂Ω

−1

◦ z) dμΩ y (ζ )

= ∂Ω

Γ (z−1 ◦ ζ ) dμΩ y (ζ ),

z ∈ G.

9.2 L-Green Function for General Domains

427

The function gy is L-harmonic in Ω since μΩ y is supported on ∂Ω (see Theorem 9.3.5, page 433). On the other hand, (9.1c) together with the positivity of GΩ , gives gy (z) ≤ Γ (z−1 ◦ y) ∀ z ∈ Ω. It follows that lim supz→ζ gy (z) ≤ Γ (ζ −1 ◦ y) for every ζ ∈ ∂Ω. The definition of hy and the maximum principle imply gy (z) ≤ hy (z) for every z ∈ Ω. In particular, gy (x) ≤ hy (x). Then Γ (x −1 ◦ y) − GΩ (x, y) = gy (x) ≤ hy (x) = Γ (y −1 ◦ x) − GΩ (y, x), so that, since Γ (x −1 ◦ y) = Γ (y −1 ◦ x), GΩ (x, y) ≥ GΩ (y, x). By interchanging the rôles of x and y, we also get GΩ (y, x) ≥ GΩ (x, y). Hence,

GΩ (x, y) = GΩ (y, x). Example 9.1.3. We know that the d-ball Bd (x, r) is an L-regular domain. Since Γ (x −1 ◦ y) = Γ (r) if y ∈ ∂Bd (x, r), we have hx ≡ Γ (r). Then GBd (x,r) (x, y) = Γ (x −1 ◦ y) − Γ (r).

(9.2)

9.2 L-Green Function for General Domains In this section, we extend the notion of L-Green function to general open sets. Definition 9.2.1 (L-Green function for a general domain). Let Ω ⊆ G be open, and let x ∈ Ω. The function y → Γ (x −1 ◦ y) is L-superharmonic and non-negative in Ω. Then it has a greatest L-harmonic minorant in Ω. Let us denote it by hx . The function Ω × Ω (x, y) → GΩ (x, y) := Γ (x −1 ◦ y) − hx (y) ∈ [0, ∞] is the L-Green function for Ω. We explicitly remark that hx and GΩ (x, ·) are L-harmonic, respectively, in Ω and in Ω \ {x}. Moreover, GΩ (x, ·) is L-superharmonic in Ω and hx = sup{v ∈ S(Ω)|v ≤ Γ (x −1 ◦ ·)}. As a consequence, 0 ≤ hx ≤ Γ (x −1 ◦ ·) and GΩ ≥ 0. For future references it is worth stating the following proposition. Proposition 9.2.2. Let x ∈ Ω, and let v ∈ S(Ω) be such that v ≤ GΩ (x, ·)

in Ω.

Then v ≤ 0. Hence, the null function is the greatest L-harmonic minorant of the function Ω x → GΩ (x, ·).

428

9 Representation Theorems

Proof. The hypothesis implies v + hx ≤ Γ (x −1 ◦ ·). Then, since v + hx ∈ S(Ω), we infer v + hx ≤ hx , that is v ≤ 0. The second part of the proposition trivially follows from the first one.

Example 9.2.3. The L-Green function for G is GG (x, y) = Γ (x −1 ◦ y),

x, y ∈ G.

Indeed, since 0 ≤ hx ≤ Γ (x −1 ◦ ·) and Γ (x −1 ◦ y) → 0 as y → ∞, hx ≡ 0 and GG (x, ·) = Γ (x −1 ◦ ·). When Ω is bounded, GΩ can be expressed in terms of the PWB operator. Indeed, the following theorem holds. Theorem 9.2.4. Let Ω ⊆ G be open and bounded. Then, for every x ∈ Ω, the greatest L-harmonic minorant in Ω of the map x → Γ (x −1 ◦ y) is the PWB solution to the Dirichlet problem Lh = 0 in Ω, h|∂Ω = Γ (x −1 ◦ ·). Remark 9.2.5. From this theorem it follows that Γ x −1 ◦ z dμΩ GΩ (x, y) = Γ (x −1 ◦ y) − y (z),

x, y ∈ Ω,

(9.3)

∂Ω

where, as usual, μΩ y denotes the L-harmonic measure related to Ω and y. When Ω is L-regular, this formula gives back (9.1c) of Section 9.1. Proof (of Theorem 9.2.4). Let x ∈ Ω be fixed, and let ϕ := Γ (x −1 ◦ ·)|∂Ω . Let v ∈ S(Ω). From the maximum principle in Theorem 8.2.19 (page 409) we obtain lim sup v ≤ ϕ

iff v ≤ Γ (x −1 ◦ ·)

in Ω.

∂Ω

Then, since ϕ is resolutive, Ω Ω hx = sup{v ∈ S(Ω) : v ≤ Γ (x −1 ◦ ·)} = sup U Ω ϕ = H ϕ = Hϕ ,

as we aimed to prove.

The L-Green function GΩ is non-negative in Ω × Ω and such that, for every fixed x ∈ Ω, GΩ (x, ·) is the sum of Γ (x −1 ◦ ·) plus an L-harmonic function on Ω. We now show that Γ (x −1 ◦ ·) does not exceed any other function sharing the same property. Proposition 9.2.6. Let x ∈ Ω, and let u ∈ S(Ω), u ≥ 0, be such that u = Γ (x −1 ◦ ·) + v with v ∈ S(Ω). Then

u ≥ GΩ (x, ·).

9.2 L-Green Function for General Domains

429

Proof. The condition u ≥ 0 implies −v ≤ Γ (x −1 ◦ ·), so that (since −v ∈ S(Ω))

−v ≤ hx . Hence Γ (x −1 ◦ ·) − u ≤ hx , and the assertion follows. Corollary 9.2.7. Let Ω1 ⊆ Ω2 ⊆ G. Then GΩ1 ≤ GΩ2

in Ω1 × Ω1 .

Proof. Let x ∈ Ω1 . The function GΩ2 (x, ·)|Ω1 is L-superharmonic and non-negative in Ω1 and is the sum of Γ (x −1 ◦ ·) plus an L-harmonic function. Proposition 9.2.6

implies GΩ1 (x, ·) ≤ GΩ2 (x, ·)|Ω1 . The following approximation theorem also holds. Theorem 9.2.8. Let (Ωn )n∈N be a monotone increasing sequence of open sets, and let Ωn . Ω := n∈N

Then1 lim GΩn = GΩ .

(9.4)

n→∞

Proof. Since Ωn ⊆ Ωn+1 ⊆ Ω, Corollary 9.2.7 gives GΩn ≤ GΩn+1 ≤ GΩ . Hence the limit in (9.4) exists and is ≤ GΩ . To prove the opposite inequality, we fix x ∈ Ω and consider n ∈ N such that Ωn x. Then Γ (x −1 ◦ ·) − GΩn (x, ·) = hn ,

hn := hΩn ,x

and hn ≥ hn+1 ≥ 0

in Ωn .

By the Harnack convergence Theorem 5.7.10, page 268, there exists a L-harmonic function h ∈ H(Ω) such that hn ↓ h. It follows that U := Γ (x −1 ◦ ·) − h is non-negative in Ω since Γ (x −1 ◦ ·) − hn ≥ 0 in Ωn . Then, by Proposition 9.2.6, GΩ (x, ·) ≤ U = lim GΩn (x, ·). n→∞

This completes the proof.

1 We agree that lim n→∞ GΩn = GΩ means

x, y ∈ Ωp ⇒ lim GΩp+n (x, y) = GΩ (x, y). n→∞

430

9 Representation Theorems

In order to prove the symmetry of the L-Green function for general domains, we need a result which is of independent interest. Lemma 9.2.9 (Approximation from the inside by L-regular sets). Given an open set Ω ⊆ G, there exists a monotone increasing sequence of bounded L-regular open sets (Ωn )n∈N such that Ωn = Ω. n∈N

Proof. We first assume that Ω is bounded. For every n ∈ N, let us cover ∂Ω by a finite family of L-gauge balls Bd (xjn , rjn ) j =1,...,p with 0 < rjn < 1/n. n

We choose the balls in a such a way that Bd (xjn+1 , rjn+1 ) ⊆

pn

Bd (xin , rin ).

i=1

Then, defining Ωn := Ω \

pn

Bd (xin , rin ),

i=1

we have Ωn ⊆ Ωn+1 and n Ωn = Ω. The open sets Ωn are L-regular. Indeed, if z0 ∈ ∂Ωn , there exists j ∈ {1, . . . , pn } such that z0 ∈ ∂Bd (xjn , rjn ). The function w = Γ (rjn ) − Γ (xjn )−1 ◦ · is L-harmonic and strictly positive in the complement of Bd (xjn , rjn ), hence in Ωn . Moreover, limz→z0 w(z) = 0. Thus w is an L-barrier for Ωn at z0 , i.e. z0 is L-regular for Ωn . Let us now suppose that Ω is unbounded and put On := Ω ∩ Bd (0, n),

n ∈ N.

can find an increasing sequence of bounded L-regular open Since On is bounded, we

sets (Onk )k∈N such that k∈N Onk = On . Using the first part of the proof, we can k for every n and for every k. It follows that Onk ⊆ choose Onk such that Onk ⊆ On+1 m Om if k, n ≤ m. Let us now put (see also Fig. 9.1) Ωn = Onn . Then Ωn is bounded and L-regular. Moreover, On = Onk ⊆ Onn ⊆ Ω. Ω= n

Hence Ω =

n Ωn ,

n

k

and the proof is complete.

n

9.2 L-Green Function for General Domains

431

Fig. 9.1. Approximation of an open set by L-regular open sets

This lemma, together with Theorem 9.1.2, page 426, immediately proves the following proposition. Proposition 9.2.10 (Symmetry of the L-Green function). Let Ω ⊆ G be an open set. Then for every x, y ∈ Ω. GΩ (x, y) = GΩ (y, x) In particular, for every fixed y ∈ Ω, the function x → GΩ (x, y) is L-harmonic in Ω \ {y}. Remark 9.2.11. From (9.3) and the above Proposition 9.2.10 we immediately get the following formula: −1 GΩ (x, y) = Γ (y ◦ x) − Γ (y −1 ◦ η) dμΩ ∀ x, y ∈ Ω. x (η) ∂Ω

By collecting together some of the results of this section, we have the following characterizations of the L-Green function, which we state for future reference and for the reading convenience. Proposition 9.2.12. Let Ω ⊆ G be open, and let x ∈ Ω be fixed. Let us denote by hx the greatest L-harmonic minorant of Γ (x −1 ◦ ·) in Ω. Then the following facts hold: (i) The L-Green function GΩ (x, y) = Γ (x −1 ◦ y) − hx (y) is a symmetric function, i.e. GΩ (x, y) = GΩ (y, x) for all x, y ∈ Ω. Moreover, GΩ is continuous (in the extended sense) on Ω × Ω, and the greatest L-harmonic minorant of GΩ (x, ·) in Ω is the null function. (ii) It holds hx = sup u ∈ S(Ω) : u ≤ Γ (x −1 ◦ ·) on Ω .

432

9 Representation Theorems

(iii) If Ω is a bounded domain, then hx = HΓΩ(x −1 ◦·) , in the sense of Perron-WienerBrelot, or equivalently

hx = sup u ∈ S(Ω) : lim sup u(z) ≤ Γ (x −1 ◦ ζ ) for all ζ ∈ ∂Ω . z→ζ

(iv) If Ω is a L-regular domain, then hx is the solution (in the classical sense) to Lu = 0 in Ω

and u = Γ (x −1 ◦ ·) on ∂Ω.

(9.5)

(v) An equivalent definition of the L-Green function is the following one: GΩ is a non-negative function on Ω × Ω such that (for every x ∈ Ω) the function GΩ (x, ·) is the sum of Γ (x −1 ◦·) plus a L-harmonic function on Ω and, moreover, GΩ (x, ·) does not exceed any other non-negative L-superharmonic function on Ω which is the sum of Γ (x −1 ◦ ·) plus a L-superharmonic function on Ω.

9.3 Potentials of Radon Measures Definition 9.3.1 (GΩ -potential of a Radon measure). Let Ω ⊆ G be open and let GΩ be its L-Green function. Let μ be a Radon measure in Ω. The function GΩ (x, y) dμ(y) GΩ ∗ μ : Ω → [0, ∞], (GΩ ∗ μ)(x) := Ω

is well defined and l.s.c. It is called the GΩ -potential of μ. The following theorem holds. Theorem 9.3.2 (Subharmonicity of a GΩ -potential). Suppose Ω is connected. Then GΩ ∗ μ ∈ S(Ω) if and only if there exists x0 ∈ Ω such that (GΩ ∗ μ)(x0 ) < ∞. Proof. The “only if” part is trivial: actually, if GΩ ∗ μ ∈ S(Ω), then GΩ ∗ μ < ∞ in a dense subset of Ω. To prove the “if” part, by Corollary 8.2.3 (page 402), it is enough to check that GΩ ∗ μ is super-mean. Given x ∈ Ω and r > 0 such that Bd (x, r) ⊆ Ω, we have md Mr (GΩ ∗ μ)(x) = Q ΨL (x −1 ◦ y) (GΩ ∗ μ)(y) dy r B (x,r) d md −1 = Q ΨL (x ◦ y) GΩ (y, z) dμ(z) dy r (x,r) Ω Bd md −1 = Q ΨL (x ◦ y) GΩ (y, z) dy dμ(z) r Ω Bd (x,r) (since y → GΩ (y, z) is L-superharmonic, hence super-mean, see Example 7.2.7, page 390) ≤ GΩ (x, z) dμ(z) = (GΩ ∗ μ)(x). Ω

Then GΩ ∗ μ is super-mean, and the proof is complete.

9.3 Potentials of Radon Measures

433

Corollary 9.3.3. Let μ be a Radon measure in Ω such that μ(Ω) < ∞. Then GΩ ∗ μ ∈ S(Ω). Proof. Let Bd (x0 , r) ⊆ Ω. Since (GΩ ∗ μ)(x) dx = Bd (x0 ,r)

Ω

GΩ (x, y) dx dμ(y)

Bd (x0 ,r)

≤ μ(Ω) sup

GΩ (x, y) dx < ∞,

y∈Ω Bd (x0 ,r)

then GΩ ∗ μ < ∞ almost everywhere in Bd (x0 , r). Then the assertion follows from the previous theorem.

By using the Harnack inequality, Corollary 9.3.3 can be improved as follows. Corollary 9.3.4. Let Ω ⊆ G be open and connected, and let μ be a Radon measure in Ω such that K := supp(μ) is a compact subset of Ω. For every fixed y0 ∈ K and every bounded and connected open set U such that K ⊆ U,

U ⊆ Ω,

there exists a positive constant C = C(U, y0 ) such that (GΩ ∗ μ)(x) ≤ C GΩ (x, y0 )

∀ x ∈ Ω \ U.

Proof. For every x ∈ Ω \ U , the function y → GΩ (x, y) is L-harmonic and nonnegative in U . Then, by the Harnack inequality (Corollary 5.7.3, page 265), there exists a positive constant C independent of x, such that GΩ (x, y) ≤ C GΩ (x, y0 ) for every y ∈ K. As a consequence, (GΩ ∗ μ)(x) = GΩ (x, y) dμ(y) ≤ C μ(K) GΩ (x, y0 ) K

for every x ∈ Ω \ U .

Theorem 9.3.5 (Sub-Laplacian in D of a GΩ -potential). Under the hypothesis of Theorem 9.3.2, we have L(GΩ ∗ μ) = −μ in the weak sense of distributions. In particular, GΩ ∗ μ is L-harmonic in Ω \ supp(μ). Proof. By Theorem 9.3.2 and Corollary 8.2.4 (page 402), GΩ ∗ μ ∈ L1loc (Ω). Moreover, for every ϕ ∈ C0∞ (Ω),

434

9 Representation Theorems

(Γ (y −1 ◦ x) − hy (x)) Lϕ(x) dx dμ(y) Ω Ω Γ (y −1 ◦ x) Lϕ(x) dx dμ(y) (since Lhy = 0) = Ω Ω (by Theorem 5.3.3) = − ϕ(y) dμ(y).

(GΩ ∗ μ)(x) Lϕ(x) dx = Ω

Ω

This proves the first part of the theorem. The second one follows from the hypoellipticity of L.

From this theorem and Corollary 9.3.4 we obtain a corollary that will be used very soon. Corollary 9.3.6. Let μ be a compactly supported Radon measure in Ω. Then: (i) GΩ ∗ μ ∈ S(Ω), (ii) GΩ ∗ μ is L-harmonic in Ω \ supp(μ), (iii) if v ∈ S(Ω) and v ≤ GΩ ∗ μ, then v ≤ 0. Proof. (i) and (ii) directly follow from Corollary 9.3.3 and Theorem 9.3.5. To prove (iii), we assume that Ω is connected (this is not restrictive) and use Corollary 9.3.4. First of all, if v ∈ S(Ω) and v ≤ GΩ ∗ μ, there exists h ∈ H(Ω) such that v ≤ h ≤ GΩ ∗ μ (h is the greatest L-harmonic minorant of GΩ ∗ μ). For a fixed y0 ∈ K := supp(μ) and a connected bounded open set U ⊇ K, U ⊆ Ω, we have ∀ x ∈ Ω \ U. h(x) ≤ (GΩ ∗ μ)(x) ≤ GΩ (x, y0 ) Since GΩ (·, y0 ) is L-harmonic, hence continuous, in Ω \ {y0 }, we have lim (GΩ (x, y0 ) − h(x)) = GΩ (ξ, y0 ) − h(ξ ) ≥ 0

U x→ξ

for every ξ ∈ ∂U . Moreover, lim

U \{y0 }x→y0

(GΩ (x, y0 ) − h(x)) = ∞.

Then, by the maximum principle, GΩ (·, y0 ) ≥ h in U . Summing up: h ≤ GΩ (·, y0 )

in Ω

so that, by the very definition of GΩ , we get h ≤ 0. Hence v ≤ 0, and the proof is complete.

If a GΩ -potential is L-superharmonic, then its L-harmonic minorants are nonpositive constant functions. Indeed, the following theorem holds. Theorem 9.3.7 (L-harmonic minorants of a GΩ -potential). Let μ be a Radon measure in an open and connected set Ω such that (GΩ ∗ μ)(x0 ) < ∞ for some x0 ∈ Ω. Let h be a L-harmonic function in Ω such that h(x) ≤ (GΩ ∗ μ)(x) Then h ≤ 0.

∀ x ∈ Ω.

(9.6)

9.3 Potentials of Radon Measures

435

Proof. Let {Kn } be a sequence of compact subsets of Ω such that Kn ⊆ Kn+1 , Kn = Ω. n

For every n ∈ N, we have h ≤ GΩ ∗ μ = GΩ ∗ (μ|Kn ) + GΩ ∗ (μ|Ω\Kn ) =: vn + wn . The functions vn and wn are non-negative and L-superharmonic in Ω (see Theorem 9.3.2). Moreover, the greatest L-harmonic minorant of vn is the zero function (see Corollary 9.3.6). Then, by Proposition 6.6.3 (page 358), h is less than the greatest L-harmonic minorant of wn . In particular, h ≤ wn

in Ω

∀ n ∈ N.

(9.7)

On the other hand, by the monotone convergence theorem, we infer vn ↑ GΩ ∗ μ, so that wn = GΩ ∗ μ − vn ↓ 0,

as n → ∞.

This, together with (9.7), implies h ≤ 0 and completes the proof.

9.3.1 The Potentials Related to the Average Operators for L This section provides detailed computations of the L-potentials of the measures naturally related to the mean integral operators Mr and Mr introduced in Sections 5.5 and 5.6 (pages 251 and 257, respectively, see in particular Theorems 5.5.4 and 5.6.1). For the sake of brevity, we rewrite our mean value formulas from those sections with a different notation. Indeed, in order to normalize the mean value formulas, we make the following choice: fixed the sub-Laplacian L and its fundamental solution Γ , we take d := Γ 1/(2−Q) . Then, d is an L-gauge (see Proposition 5.4.2, page 247), and by Theorem 5.5.6, page 256, the constant βd appearing in the surface mean value formula (see (5.43n), page 256) equals 1. Moreover, it is immediately seen that with this choice of d, for every y ∈ ∂Bd (x, r), we have (Q − 2) βd |∇L Γ |2 (x −1 ◦ y) . K (x, y) = L r Q−1 |∇Γ (x −1 ◦ ·)(y) As a consequence, the mean value formulas rewrite as follows: If u ∈ C 2 (Ω), d := Γ 1/(2−Q) (where Γ is the fundamental solution for L) and B d (x, r) ⊂ Ω, it holds

436

9 Representation Theorems

Γ (x −1 ◦ y) − r 2−Q Lu(y) dH N (y),

u(x) = mr [u](x) − Bd (x,r)

u(x) = Mr [u](x) Q r Q−1 Γ (x −1 ◦ y) − ρ 2−Q Lu(y) dH N (y) dρ, ρ − Q r Bd (x,ρ) 0 where k(x, y) := |∇L Γ |2 (x −1 ◦ y)/|∇Γ (x −1 ◦ ·)(y), K := |∇L d|2 , and mr [u](x) = u(y) k(x, y) dH N −1 (y), ∂Bd (x,r)

Q(Q − 2) Mr [u](x) = rQ

(9.8) u(y) K(x

−1

◦ y) dH (y). N

Bd (x,r)

We next give the notation which will be used in this section. If x ∈ G and r > 0 are fixed, we introduce the measures λx,r and Λx,r defined by dλx,r (y) := χ∂Bd (x,r) (y) k(x, y) dH N −1 (y), Q(Q − 2) dΛx,r (y) := χBd (x,r) (y) K(x, y) dH N (y), rQ

(9.9)

with k and K as above. We explicitly point out that, with this notations, mr [u](x) = u dλx,r and Mr [u](x) = u dΛx,r . We are interested in computing the Γ -potentials Γ ∗ λx,r and Γ ∗ Λx,r . We first need two lemmas. B (x,r)

Lemma 9.3.8. With the above notation, λx,r is equal to μx d measure of Bd (x, r) at x) prolonged to zero outside ∂Bd (x, r).

(the L-harmonic

Proof. This is just Theorem 7.2.9, page 391.

Lemma 9.3.9. Let Ω be L-regular and such that ∂Ω has vanishing Lebesgue measure. Then, for every x ∈ Ω, we have Γ (z−1 ◦ x) if z ∈ / Ω, Ω (Γ ∗ μx )(z) = Ω HΓ (z−1 ◦·) (x) if z ∈ Ω, Ω where μΩ x denotes the measure obtained by prolonging μx to 0 outside ∂Ω.

The same assertion holds for a general L-regular set (without the assumption that ∂Ω has vanishing Lebesgue measure). A proof in this case may be obtained by arguing as in (9.19) in the proof of Theorem 9.5.1 (page 445).

9.3 Potentials of Radon Measures

437

Proof. Let x ∈ Ω be fixed. With the notation in the assertion, for any z ∈ G we aim to consider Ω (Γ ∗ μx )(z) = Γ (z−1 ◦ y) dμΩ x (y). ∂Ω

Γ (z−1 ◦ y)

If z ∈ / Ω, the map y → of L-harmonic measure, we have

is L-harmonic about Ω. Hence, by the definition

Γ (z−1 ◦ x) =

∂Ω

Γ (z−1 ◦ y) dμΩ x (y).

Let now z ∈ Ω. Then the map y → Γ (z−1 ◦y) is continuous on ∂Ω, and the equality of Ω Γ (z−1 ◦ y) dμΩ x (y) = HΓ (z−1 ◦·) (x) ∂Ω

is the very definition of the latter. We end by considering a fixed z0 ∈ ∂Ω. We set J = Γ ∗ μΩ x . Since J ∈ S(G), by Theorem 8.2.10 (page 404) we have Mr [J ](z0 ) −→ J (z0 )

as r → 0.

Since the function GΩ (z, x) = Γ (z−1 ◦ x) − HΓΩ(z−1 ◦·) (x) and Γ are symmetric functions, we have HΓΩ(z−1 ◦·) (x) = HΓΩ(x −1 ◦·) (z),

whenever x, z ∈ Ω.

Then, since Bd (z0 , r) ∩ ∂Ω has zero H N -measure, we have (by applying what we proved above) Mr [J ](z0 ) = Mr [J χΩ ](z0 ) + Mr [J χG\Ω ](z0 ) = Mr HΓΩ(x −1 ◦·) χΩ (z0 ) + Mr [Γ (x −1 ◦ ·) χG\Ω ](z0 ) = Mr [Γ (x −1 ◦ ·)](z0 ) − Mr [GΩ (x, ·)χΩ ](z0 ). Clearly, we have Mr [Γ (x −1 ◦ ·)](z0 ) −→ Γ (x −1 ◦ z0 ) = Γ (z0−1 ◦ x)

as r → 0.

On the other hand, since GΩ (x, z) vanishes with continuity as Ω z → z0 ∈ ∂Ω, and Mr [1] ≡ 1, we obtain Mr [GΩ (x, ·)χΩ ](z0 ) −→ 0 as r → 0. This completes the proof.

438

9 Representation Theorems

Theorem 9.3.10. For every x ∈ G and r > 0, we have (see Fig. 9.2) (Γ ∗ λx,r )(z) = mr [Γ (z−1 ◦ ·)](x) / Bd (x, r), Γ (z−1 ◦ x) if z ∈ = 2−Q r if z ∈ Bd (x, r). (Γ ∗ Λx,r )(z) = Mr [Γ (z−1 ◦ ·)](x) ⎧ ⎨ Γ (z−1 ◦ x) = 2 − Q d 2 (x, z) Q ⎩ + r 2−Q 2 rQ 2

(9.10a)

if z ∈ / Bd (x, r), if z ∈ Bd (x, r).

(9.10b)

In particular, the Γ -potentials Γ ∗ λx,r , Γ ∗ Λx,r are continuous on G. Moreover, (Γ ∗ λx,r )(z), mr [Γ (z−1 ◦ ·)](x),

(Γ ∗ Λx,r )(z), Mr [Γ (z−1 ◦ ·)](x)

are symmetric functions in x and z.

Fig. 9.2. The functions in Theorem 9.3.10

B (x,r)

Proof. By Lemma 9.3.8, λx,r is the trivial prolongation of μx d outside ∂Bd (x, r). Then, we can apply Lemma 9.3.9 for Ω = Bd (x, r) in order to compute Γ ∗ λx,r . We have to prove that B (x,r)

2−Q d HΓ (z −1 ◦·) (x) = r

for every z ∈ Bd (x, r).

By the symmetry of the L-Green function for Bd (x, r), we have B (x,r)

B (x,r)

d d HΓ (z −1 ◦·) (x) = HΓ (x −1 ◦·) (z).

9.3 Potentials of Radon Measures

439

Now, the function of z in the above right-hand side is the L-harmonic function on Bd (x, r) which equals Γ (x −1 ◦ ·) on ∂Bd (x, r). Since Γ (x −1 ◦ ·) ≡ r 2−Q on ∂Bd (x, r) (and the constants are L-harmonic), (9.10a) follows. Finally, we have Q r Q−1 −1 (Γ ∗ Λx,r )(z) = Mr [Γ (z ◦ ·)](x) = Q ρ mρ [Γ (z−1 ◦ ·)](x) dρ. r 0 / Bd (x, ρ) also for any 0 < ρ < r, whence, by (9.10a), If z ∈ / Bd (x, r), then z ∈ Q r Q−1 (Γ ∗ Λx,r )(z) = Q ρ Γ (z−1 ◦ x) dρ = Γ (z−1 ◦ x). r 0 On the other hand, suppose z ∈ Bd (x, r). Arguing as above, we have Q r Q−1 ρ mρ [Γ (z−1 ◦ ·)](x) dρ rQ 0 Q d(x,z) Q−1 Q r = Q ρ Γ (z−1 ◦ x) dρ + Q ρ Q−1 ρ 2−Q dρ r r 0 d(x,z) = Γ (z−1 ◦ x)

d Q (x, z) Q + Q (r 2 − d 2 (x, z)). rQ 2r

Thus, (9.10b) is proved. The last assertion of the theorem now follows from (9.10a)

and (9.10b), since Γ (z−1 ◦ x) = Γ (x −1 ◦ z) and d(x, z) = d(z, x). For an application of Theorem 9.3.10, see Exercise 5 at the end of the chapter. The next result gives another consequence of Theorem 9.3.10. Indeed, we prove that the mean integrals of a Γ -potential are L-superharmonic continuous functions. Theorem 9.3.11. Let μ be a Radon measure in G and r > 0. We suppose that Γ ∗ μ ≡ ∞. Then the map x → mr [Γ ∗ μ](x) belongs to S(G) ∩ C(G, R). Moreover, the map (x, r) → mr [Γ ∗ μ](x) is lower semicontinuous. The same assertion holds for Mr [Γ ∗ μ]. Proof. (i). First we prove that mr [u] and Mr [u] are finite-valued and continuous for every u ∈ S(G). We fix x ∈ G. Then |Mr [u](x)| ≤ K∞ |u(y)| dH N (y) < ∞, Bd (x,r)

since u ∈ L1loc (G). Let R > r and set Ω = Bd (x, R). Set also (see Definition 9.4.1 in the next section) μ := (μ[u])|Ω .

440

9 Representation Theorems

Then, by Theorem 9.3.5, we have L(u − Γ ∗ μ) = 0 in the distributional sense in Ω, so that2 (by the hypoellipticity of L) u = h + Γ ∗ μ on Ω for a suitable h ∈ H(Ω). Hence (being ∂Bd (x, r) ⊂ Bd (x, R)) mr [u](x) = mr [h](x) + mr [Γ ∗ μ](x) = h(x) + mr [Γ (z−1 ◦ ·)](x) dμ(z) Ω (by (9.10a)) = h(x) + min{Γ (z−1 ◦ x), r 2−Q } dμ(z), Ω

which is finite, since μ is a Radon measure and Ω is bounded. Moreover, the above representation of mr [u](x) also proves that mr [u] is continuous. r Another argument of dominated convergence then shows that Mr [u](x) = rQQ 0 ρ Q−1 mρ [u](x) dρ is continuous at x. (ii). We now observe a simple fact (see also Theorem 8.2.20, page 410): let (A, λ) be an arbitrary measure space, and let {fx }x∈G be a family of non-negative λmeasurable functions on A such that, for every fixed z ∈ A, the function x → fx (z) is in S(G). Then the integral function F (x) = fx (z) dλ(z) A

is in S(G) too (unless it is ≡ ∞). Indeed, by Theorem 8.1.3, it is enough to prove that F is L-supermean. By Tonelli’s theorem we have mr [F ](x) = mr [y → fy (z)](x) dλ(z) ≤ fx (z) dλ(z) = F (x) A

A

(since y → fy (z) is L-supermean), i.e. F is L-supermean. (iii). Tonelli’s theorem also ensures that mr [Γ ∗ μ](x) = mr [Γ (z−1 ◦ ·)](x) dμ(z)

G

[Γ (z−1 ◦·)](x) dμ(z). Consequently, by (ii), the theorem

and Mr [Γ ∗μ](x) = G Mr is proved if we show that the maps

x → mr [Γ (z−1 ◦ ·)](x),

Mr [Γ (z−1 ◦ ·)](x)

are in S(G) for any z ∈ G. (iv). Now, the L-superharmonicity of mr [Γ (z−1 ◦ ·)] follows from (9.10a). Finally, we have Q r Q−1 −1 Mr [Γ (z ◦ ·)](x) = Q ρ mρ [Γ (z−1 ◦ ·)](x) dρ. r 0 Consequently, the L-superharmonicity of Mr [Γ (z−1 ◦ ·)] follows by an application of (ii) for A = [0, r] and dλ(ρ) = rQQ ρ Q−1 dρ, and making use of what we proved at the beginning of step (iv). The semicontinuity of (x, r) → mr [Γ ∗ μ](x) is a standard consequence of Fatou’s lemma.

2 The reader has certainly realized that we have just proved a Riesz representation type result:

we investigate the topic in details in Section 9.4.

9.4 Riesz Representation Theorems for L-subharmonic Functions

441

Corollary 9.3.12. Let u ∈ S(G) and r > 0. Then mr [u], Mr [u] belong to S(G) ∩ C(G, R). Proof. See (i) in the proof of Theorem 9.3.11.

9.4 Riesz Representation Theorems for L-subharmonic Functions Definition 9.4.1 (L-Riesz measure). Let u be an L-subharmonic function in an open set Ω ⊆ G. By Theorem 8.2.11 (page 405), u ∈ L1loc (Ω) and Lu ≥ 0 in the weak sense of distributions. Then3 there exists a Radon measure μ in Ω such that Lu = μ

(9.11)

in the weak sense of distributions. The measure μ will be called the L-Riesz measure of u. If u is L-superharmonic in Ω, the L-Riesz measure related to −u will be referred to as the L-Riesz measure of u. In this case, it holds Lu = −μ, in the weak sense of distributions. With reference to the above definition, we shall sometimes also write μ[u] or μu instead of μ. Example 9.4.2. If u = Γ , then μ[Γ ] = Dirac0 , the Dirac mass supported at {0}. Indeed, L(−Γ ) = Dirac0 , for (see (5.14), page 236) (−Γ ) Lϕ = ϕ(0) = Dirac0 ϕ ∀ ϕ ∈ C0∞ (G). G

G

When μ[u] is compactly supported, a representation theorem easily follows. Theorem 9.4.3 (Riesz representation. I). Let Ω ⊆ G be open, and let u ∈ S(Ω). Let μ be the L-Riesz measure of u. Assume that supp(μ) is a compact subset of Ω. Then there exists an L-harmonic function h in Ω satisfying the identity u = h − GΩ ∗ μ

in Ω.

(9.12)

Proof. By Corollary 9.3.3 and Theorem 9.3.5, v := GΩ ∗ μ is L-superharmonic in Ω, and Lv = −μ in the weak sense of distributions. It follows that L(u + v) = 0 in Ω in the weak sense of distributions. Since L is hypoelliptic, there exists a function h, L-harmonic in Ω, such that h(x) = u(x) + v(x) almost everywhere in Ω. As a consequence, for every x ∈ Ω and for every r < distd (x, ∂Ω) 3 Here we used the following result. Given a linear map L : C ∞ (Ω) → R such that L(ϕ) ≥ 0

0 whenever ϕ ≥ 0, there exists a Radon measure μ on Ω such that L(ϕ) = ϕ dμ for every ϕ ∈ C0∞ (Ω). For a proof of this result, it suffices to rerun the proof of the classical Riesz representation theorem of positive functionals on C0 as presented, e.g. in [Rud87].

442

9 Representation Theorems

Mr (u)(x) = −Mr (v)(x) + h(x), where Mr is the solid average operator (5.50f) (page 259). Here we used the L-harmonicity of h to write h in place of Mr (h). Letting r tend to zero in the last identity and using Theorem 8.2.11-(ii) (page 405), we get u(x) = −v(x) + h(x)

∀ x ∈ Ω.

This completes the proof.

For the future reference, we explicitly show the following theorem, which can be proved as Theorem 9.4.3. Theorem 9.4.4 (Riesz representation. II). Let Ω ⊆ G be open, and let u ∈ S(Ω). Let μ be the L-Riesz measure of u. Then, for every bounded open set Ω1 such that Ω1 ⊂ Ω, there exists a function h, L-harmonic in Ω1 , satisfying the identity Γ (y −1 ◦ x) dμ(y) + h(x) ∀ x ∈ Ω1 . (9.13) u(x) = − Ω1

Proof. The function

Γ (y −1 ◦ x) dμ(y),

v(x) := −

x ∈ G,

Ω1

is L-subharmonic in G and satisfies Lv = μ|Ω1 in the weak sense of distributions. Therefore, L(u − v) = 0 in D (Ω1 ). Then, just proceeding as in the proof of the previous theorem, we show the existence of an L-harmonic function in Ω1 such that

u = v + h in Ω1 . Lemma 9.4.5. Let u ∈ S(Ω), let μ be its L-Riesz measure and let K ⊆ Ω be compact. There exists w ∈ S(Ω) such that u = GΩ ∗ (μ|K ) + w

in Ω.

Proof. Let H ⊆ Ω be compact and such that K ⊆ Int(H ). Let k ∈ H(Int(K)) and h ∈ H(Int(H )) be such that u = h + GΩ ∗ (μ|H ) Define

w : Ω → ]−∞, ∞], w :=

in Int(H ).

h + GΩ ∗ (μ|H \K ) u − GΩ ∗ (μ|K )

in Int(H ), in Ω \ K.

This definition is well-posed since the function GΩ ∗ (μ|K ) is L-harmonic in Int(H ) \ K, hence real-valued, and h + GΩ ∗ (μ|H \K ) = h + GΩ ∗ (μ|H ) − GΩ ∗ (μ|K ) = u − GΩ ∗ (μ|K ). Moreover, w is L-superharmonic in Ω since it is L-superharmonic in Int(H ) and in Ω \ K, and Ω = Int(H ) ∪ (Ω \ K).

9.4 Riesz Representation Theorems for L-subharmonic Functions

443

Theorem 9.4.6 (Superharmonicity of GΩ ∗ Lu). Let u ∈ S(Ω), and let μ be its L-Riesz measure. Then (9.14) GΩ ∗ μ ∈ S(Ω) if and only if there exists v ∈ S(Ω) such that v ≤ u. Proof. We first prove the “if” part and assume u ≥ v in Ω with v ∈ S(Ω). Let (Kn )n∈N be an increasing sequence of compact sets such that ∪n Kn = Ω. Let us put μn = μ|Kn . By Lemma 9.4.5, there exists wn ∈ S(Ω) such that u = wn + GΩ ∗ μn . The hypothesis implies 0 ≤ u − v = (wn − v) + GΩ ∗ μn in Ω, so that GΩ ∗ μn ≥ v − wn . By Corollary 9.3.6, we have v − wn ≤ 0 in Ω. Hence u − v ≥ GΩ ∗ μn

in Ω

∀ n ∈ N.

Letting n tend to infinity, we get u − v ≥ GΩ ∗ μ. Since u − v ∈ S(Ω), this implies (9.14). Vice versa, assume (9.14) is true. Then, by Theorem 9.3.5, L(GΩ ∗ μ − u) = 0 in Ω, in the weak sense of distributions. Since L is hypoelliptic, there exists a function h ∈ H(Ω) such that GΩ ∗ μ = u + h a.e. in Ω, hence everywhere (see Exercise 2 of Chapter 8, page 422). Since

GΩ ∗ μ ≥ 0, we have u ≥ −h, and the proof is complete. Theorem 9.4.7 (The Riesz representation). Let u ∈ S(Ω), and let μ be the L-Riesz measure of u. The following statements are equivalent: (i) there exists h ∈ H(Ω) such that u = GΩ ∗ μ + h

in Ω,

(9.15)

(ii) there exists v ∈ S(Ω) such that v ≤ u in Ω, (iii) every connected component of Ω contains a point x0 such that (GΩ ∗ μ)(x0 ) < ∞. Moreover, (9.15) holds with h ∈ H(Ω) if and only if h is the greatest L-harmonic minorant of u in Ω. Proof. (i) ⇒ (ii). If (9.15) holds, then h is an L-harmonic minorant of u (hence an L-subharmonic function), since GΩ ∗ μ ≥ 0. (ii) ⇔ (iii). This follows from Theorems 9.4.6 and 9.3.2. (iii) ⇒ (i). It is not restrictive to assume that Ω is connected. Let x0 ∈ Ω be such that (GΩ ∗ μ)(x0 ) < ∞, and let {Ωn }n∈N be a sequence of bounded open sets such that

444

9 Representation Theorems

Ωn ⊂ Ωn+1 , Ωn+1 ⊂ Ω,

Ωn = Ω.

n∈N

Define μn := μ|Ωn ,

n ∈ N.

Then, by the representation Theorem 9.4.3, there exists an L-harmonic function hn such that (9.16) u(x) = (GΩ ∗ μn )(x) + hn (x) ∀ x ∈ Ωn , ∀ n ∈ N. Since GΩ ∗ μn GΩ ∗ μ, we have hn (x) ≥ hn+k (x) ≥ u(x) − (GΩ ∗ μ)(x)

∀ x ∈ Ωn , ∀ n, k ∈ N.

On the other hand, by Theorem 9.3.2, GΩ ∗ μ ∈ S(Ω). Then, keeping in mind that u(x) > −∞ for every x ∈ Ω, for any n ∈ N, we have inf hn+k > −∞ in a dense subset of Ωn . k

By using Theorem 5.7.10 (page 268), we infer the existence of a function h : Ω → R, L-harmonic in Ω, such that h(x) = lim hn+k (x) k→∞

∀ x ∈ Ωn , ∀ n ∈ N.

Then (9.15) follows from (9.16). We are left with the proof of the second part of the theorem. Assume (9.15) holds with h ∈ H(Ω). Then, if k ∈ H(Ω) and k ≤ u, we have k − h ≤ GΩ ∗ μ, so that, by Theorem 9.3.7, we have k − h ≤ 0, i.e. k ≤ h. Vice versa, assume h is the greatest L-harmonic minorant of u. Then, by (ii), there exists k ∈ H(Ω) such that u = GΩ ∗ μ + k. This implies that k is the greatest L-harmonic minorant of u, i.e.

k = h. Thus u = GΩ ∗ μ + h. The proof is complete. Corollary 9.4.8 (Riesz representation in space). Let u ∈ S(G) be such that U := sup u < ∞. Then u = −Γ ∗ μ + U, where μ is the L-Riesz measure of u. Proof. Since U is an L-harmonic majorant of u, by the previous theorem we have u = −Γ ∗ μ + h, where h is the least L-harmonic majorant of u. Then h ≤ U so that, by the Liouville Theorem 5.8.1 (page 269) h ≡ U0 , a real constant. Hence u = −Γ ∗ μ + U0 . As a consequence U = sup u ≤ U0 ≤ U , that is, U = U0 . The proof is complete.

We remark that a stronger version of the previous corollary will be provided in Theorem 9.6.1, page 451. Corollary 9.4.9 (Riesz representation. III). Let u ∈ S(Ω) be such that Lu = 0 outside a compact set K ⊂ Ω. Then there exists an L-harmonic function h in Ω such that u = −GΩ ∗ μ + h in Ω.

9.5 The Poisson–Jensen Formula

445

Proof. Since supp(μ) ⊆ K, we have GΩ ∗ μ ∈ S(Ω). Then GΩ ∗ μ < ∞ in a dense subset of Ω, and the assertion follows from Theorem 9.4.7.

For the future references, we explicitly write the following consequence of Theorems 9.3.2 and 9.3.5. Corollary 9.4.10 (Riesz representation. IV). Let u ∈ S(Ω), and let μ be the LRiesz measure of u. Assume (Γ ∗ μ)(x0 ) < ∞ at some point x0 ∈ G. Then there exists an L-harmonic function h in Ω such that u(x) = −(Γ ∗ μ)(x) + h(x)

∀ x ∈ Ω.

(9.17)

Proof. It is sufficient to note that Γ ∗ μ ∈ SG and that L(u + Γ ∗ μ) = 0 in Ω.

9.5 The Poisson–Jensen Formula The next theorem, when L = Δ is the classical Laplace operator, will give back the classical Poisson–Jensen formula (see, e.g. [HK76, Theorem 3.14]). An improved version of Theorem 9.5.1 (removing the hypothesis of L-regularity of Ω) will be given in Section 11 (see Theorem 11.7.6, page 518). Theorem 9.5.1 (Poisson–Jensen’s formula). Let U, Ω be open subsets of G, Ω ⊂ U and Ω be L-regular. Let u ∈ S(U ) and μ = Lu be its L-Riesz measure. Then Ω u(x) = u(y) dμx (y) − GΩ (y, x) dμ(y), x ∈ Ω. (9.18) ∂Ω

Ω

Here GΩ is the L-Green function of Ω and μΩ x is the L-harmonic measure related to Ω. Proof. Let O be a bounded open set such that Ω ⊂ O ⊂ O ⊂ U . By the Riesz representation Theorem 9.4.4, there exists an L-harmonic function h in O such that Γ (y −1 ◦ x) dμ(y) + h(x) =: v(x) + h(x) ∀ x ∈ O. u(x) = − O

We have v ∈ S(G) and Lv = μ|O . Then, since h(x) = h(y) dμΩ x (y)

∀ x ∈ Ω,

∂Ω

it suffices to prove (9.18) with u replaced by v. We can also suppose that v(x) > −∞. Indeed, if v(x) = −∞, then u(x) = −∞ and

446

9 Representation Theorems

GΩ (y, x) dμ(y) ≥ Ω

Γ (y −1 ◦ x) dμ(y) = −u(x) − h(x) = ∞.

Ω

Moreover, since u ∈ S(U ), the function x → ∂Ω u(y) dμΩ x (y) is L-harmonic, hence real-valued, in Ω (see Exercise 18 of Chapter 6). Thus, in this case, (9.18) trivially holds. Let us fix x ∈ Ω. We have Ω −1 Ω v(y) dμx (y) = − Γ (z ◦ y) dμx (y) dμ(z). ∂Ω

O

∂Ω

The crucial part of the proof is to show that Γ (z−1 ◦ y) dμΩ x (y) ∂Ω −1 = Γ (z−1 ◦ x), Γ (z ◦ x) − GΩ (z, x),

z ∈ O \ Ω, z ∈ Ω.

(9.19)

With (9.19) at hand, and keeping in mind the assumption v(x) = −∞, which implies that z → Γ (z−1 ◦ x) is μ-summable, we get the assertion. Indeed, Ω −1 v(y) dμx (y) = − Γ (z ◦ x) dμ(z) + GΩ (z, x) dμ(z) ∂Ω O Ω GΩ (z, x) dμ(z). = v(x) + Ω

Then, it remains to prove (9.19). If z ∈ O \ Ω, the function Γ (z−1 ◦ ·) is harmonic in O \ {z}, so that −1 Γ (z−1 ◦ y) dμΩ Γ (z ◦ x) = x (y). ∂Ω

If z ∈ Ω, the function Γ (z−1 ◦ ·) is continuous in ∂Ω, hence the solution hz to the Dirichlet problem Lh(x) = 0, x ∈ Ω, h(y) = Γ (z−1 ◦ y), y ∈ ∂Ω is given by

hz (x) := ∂Ω

Γ (z−1 ◦ y) dμΩ x (y).

Then, by the definition of the L-Green function, −1 Γ (z−1 ◦ y) dμΩ ◦ x) − GΩ (z, x). x (y) = hz (x) = Γ (z ∂Ω

Finally, we fix z0 ∈ ∂Ω. Let us prove that −1 Γ (y −1 ◦ z0 ) dμΩ ◦ z0 ). x (y) = Γ (x ∂Ω

(9.20)

9.5 The Poisson–Jensen Formula

447

Since Γ (x −1 ◦ z0 ) = Γ (z0−1 ◦ x), this will give (9.19) in the case z0 ∈ ∂Ω. Let us define w(z) := Γ (x −1 ◦ z) − Γ (y −1 ◦ z) dμΩ z ∈ G \ {x}. x (y), ∂Ω

The function w is L-subharmonic in G \ {x} and lim sup w(z) = 0 ∀ ζ ∈ ∂Ω.

(9.21)

z∈∂Ω, / z→ζ

Indeed, (9.19) holds in O \ ∂Ω and GΩ (z, x) → 0 as z → ζ from inside of Ω, since Ω is L-regular and GΩ is symmetric. In order to prove (9.20), we have to show that w ≡ 0 on ∂Ω. First of all, we observe that w(ζ ) ≥ lim sup w(z) ≥ lim sup w(z) = 0 ∀ ζ ∈ ∂Ω. z∈∂Ω, / z→ζ

z→ζ

Suppose, by contradiction, that w > 0 somewhere in ∂Ω. Then we have max∂Ω w > 0. From (9.21) it follows that there exists an open set V ⊆ O such that ∂Ω ⊂ V , x ∈ / V and maxV w = max∂Ω w. Let ζ0 ∈ ∂Ω be such that w(ζ0 ) = maxV w. Since w is L-subharmonic in V , by Theorems 8.1.2 and 8.2.1 (pages 398 and 401), w ≡ w(ζ0 ) in the connected component of V containing ζ0 . Thus, lim sup w(z) = w(ζ0 ) = max w > 0, z∈∂Ω, / z→ζ0

V

in contradiction with (9.21). This completes the proof of (9.19).

If, in the previous theorem, we take Ω = Bd (x, r), we obtain an extension of the mean value formulas (5.45) and (5.51) (pages 256 and 259) to the L-subharmonic functions. Theorem 9.5.2 (Mean value formulas for L-subharmonic functions). Let Ω ⊆ G be open, and let u ∈ S(Ω). Then, for every x ∈ Ω and r > 0 such that Bd (x, r) ⊂ Ω, we have

u(x) = Mr (u)(x) −

Γ (x −1 ◦ y) − Γ (r) dμ(y)

(9.22)

Bd (x,r)

and Q u(x) = Mr (u)(x) − Q r

r

ρ 0

Q−1

(Γ (x

−1

◦ y) − Γ (ρ)) dμ(y) dρ,

Bd (x,ρ)

where μ := Lu is the L-Riesz measure of u and Γ (ρ) := βd ρ 2−Q .

(9.23)

448

9 Representation Theorems

Proof. Take Ω = Bd (x, r) in Poisson–Jensen’s Theorem 9.5.1. By Theorem 7.2.9 (page 391), we get d (x,r) = M (u)(x). u(y) dμB r x ∂Bd (x,r)

Moreover, if GΩ denotes the L-Green function of Bd (x, r), then GΩ (x, y) = Γ (x −1 ◦ y) − Γ (r).

(9.24)

Then (9.22) follows from (9.18). Identity (9.23) follows from (9.22) keeping in mind that r Q ρ Q−1 Mρ (u)(x) dρ. Mr (u)(x) = Q r 0 This completes the proof.

Remark 9.5.3. In Chapter 11, Theorem 9.5.1 will be extended to arbitrary bounded open sets Ω. From Theorem 9.5.2 it is easy to derive the following left-continuity result of the surface average. Proposition 9.5.4 (Left-continuity of the surface operator). Let v ∈ S(Ω), and let Bd (x, r) ⊆ Ω. Then lim Mρ (u)(x) = Mr (u)(x).

(9.25)

ρ↑r

Proof. We first assume that v(x) > −∞, and we write the Poisson–Jensen formula (9.22) as follows Γ (x −1 ◦ y) − Γ (ρ) dμ(y), 0 < ρ ≤ r. (9.26) Mρ (v)(x) = v(x) + Bd (x,ρ)

Since Bd (x, ρ) ↑ Bd (x, r) and, as ρ ↑ r, Γ (x −1 ◦ y) − Γ (ρ) ↑ Γ (x −1 ◦ y) − Γ (r) , one obtains (9.25) from (9.26) by using the monotone convergence theorem. If v(x) = −∞, we replace v by its Perron-regularized w := vBd (x,r/2) , and we get w ∈ S(Ω), w(x) > −∞ and Mρ (u)(x) = Mρ (w)(x) for

r 2

< ρ ≤ r. Then, from what we have already proved, lim Mρ (u)(x) = lim Mρ (w)(x) = Mr (w)(x) = Mr (u)(x).

ρr

ρr

9.5 The Poisson–Jensen Formula

449

The previous proposition is the needed tool to prove the following result. Proposition 9.5.5. Let v ∈ S(Ω), and let Bd (x0 , r) ⊆ Ω. Denote B := Bd (x0 , r). Then the function h : B → R,

h(x) = ∂B

v dμB x

is the least harmonic majorant of v on B. Proof. Since v is bounded from above on B, by Theorem 6.6.1, v has a least harmonic majorant h0 on B. Then h0 ≤ h, since h is harmonic on B and h ≥ v|B . On the other hand, for 0 < ρ < r, h0 (x0 ) = Mρ (h0 )(x0 ) ≥ Mρ (u)(x0 ). As ρ ↑ r, by using the previous proposition and Theorem 7.2.9, we get h0 (x0 ) ≥ Mr (v)(x0 ) = h(x0 ). Therefore, h0 − h ≤ 0 in B and (h0 − h)(x0 ) = 0. The strong maximum principle

of Theorem 8.2.19 implies h0 ≡ h. We are now in the position to show that, in the present setting, Theorem 6.11.1 on page 375 can be improved. Theorem 9.5.6 (Envelopes in S(Ω)). Let Ω ⊆ G be open, and let F ⊆ S(Ω). u0 ∈ S(Ω) and Assume u0 := inf F is locally bounded from below. Then u0 = u0

a.e. in Ω.

Proof. We already know, from Theorem 6.11.1, that u0 ∈ S(Ω). Then, we only u0 a.e. in Ω. By Choquet’s lemma (Proposition 6.1.2), have to prove that u0 = v = u0 , where v = infn∈N vn . Defining there exists a sequence (vn ) in F such that wn := min{v1 , . . . , vn }, n ∈ N, we obtain a decreasing sequence {wn }n∈N of Lsubharmonic functions in Ω such that wn ↓ v. For every L-gauge ball B with B ⊆ Ω, we have u0 dμB ≤ wn dμB ∀ y ∈ B, y y ≤ wn (y) ∂B

∂B

so that, letting n tend to infinity, u0 dμB ≤ y ∂B

∂B

v dμB y ≤ v(y)

∀ y ∈ B,

and y → ∂B v dμB y is L-harmonic in B (it is the limit of a decreasing sequence of L-harmonic functions bounded from below by an L-harmonic function). Then the previous inequalities extends to the following ones u0 dμB ≤ v dμB v (y) = u0 (y) ∀ y ∈ B. (9.27) y y ≤ ∂B

∂B

450

9 Representation Theorems

On the other hand, by Proposition 9.5.5, y → ∂B u0 dμB y is the greatest harmonic minorant of u0 in B. Thus, the first inequality in (9.27) actually is an equality, i.e. u0 dμB = v dμB ∀ y ∈ B. y y ∂B

∂B

In particular, if B = Bd (x0 , r), we have u0 )(x0 ) = Mr (v)(x0 ). Mr ( Since this identity holds for every gauge ball Bd (x0 , r) with Bd (x0 , r) ⊆ Ω, we also have u0 )(x0 ) = Mr (v)(x0 ) Mr ( u0 ≤ for every x0 ∈ Ω and r > 0 such that Bd (x0 , r) ⊆ Ω. Keeping in mind that

u0 ≤ v, the assertion follows from the next lemma. Lemma 9.5.7. Let Ω ⊆ G be open, and let f : Ω → [0, ∞] be a Lebesgue measurable function such that (9.28) Mr (f )(x0 ) = 0 for every x0 ∈ Ω and r > 0 such that Bd (x0 , r) ⊆ Ω. Then f = 0 a.e. in Ω. Proof. Let K be the kernel appearing in the average operator Mr (see (5.50f), page 259). Hypothesis (9.28) implies that f = 0 a.e. in Bd+ (x0 , r) := x ∈ Bd (x0 , r) : K(x0−1 ◦ x) > 0 for every x0 ∈ Ω and r > 0 such that Bd (x0 , r) ⊆ Ω.

+ Bd (x0 , r), Then, since Bd+ (x0 , r) is open, it is enough to show that Ω = where the union is taken over the family of the L-gauge balls with the closure contained in Ω. Let x ∈ Ω. Since K is smooth out of the origin and δλ -homogeneous of degree zero, there exists z ∈ G such that K(z) > 0 and d(z) < r :=

1 distd (x, ∂Ω). 2

Then, setting x0 = x ◦ z−1 , we have x0−1 ◦ x = z, so that x ∈ Bd (x0 , r) and K(x0−1 ◦ x) = K(z) > 0. This means that x ∈ Bd+ (x0 , r), and we are done, since Bd (x0 , r) ⊆ Ω.

Corollary 9.5.8. Let F and u0 be as in Theorem 9.5.6. Then u0 (x) = lim Mr (u0 )(x) r→0

for every x ∈ Ω.

Proof. From Theorem 8.2.11 we have u0 (x) = lim Mr ( u0 )(x) r→0

The assertion follows, since u0 = u0 a.e. in Ω.

for every x ∈ Ω.

9.6 Bounded-above L-subharmonic Functions in G

451

9.6 Bounded-above L-subharmonic Functions in G This section is devoted to the proof of the following sharp Riesz representation theorem for L-subharmonic functions bounded from above in G. We recall that we are always assuming that G is equipped with a homogeneous Carnot group structure G such that the homogeneous dimension Q of G is strictly greater than 2. G)). Let μ be a Theorem 9.6.1 (The L-Riesz measure of a bounded-above u ∈ S(G Radon measure in G, and let x0 ∈ G. Then μ is the L-Riesz measure of a boundedabove L-subharmonic function u in G with u(x0 ) > −∞ if and only if the following condition holds: ∞ μ(Bd (x0 , t)) dt < ∞. (9.29) t Q−1 0 If this condition is satisfied, then there is a unique L-subharmonic function u in G having the L-Riesz measure μ, the least upper bound U < ∞ and such that u(x0 ) > −∞. It is given by U − Γ ∗ μ, i.e. u(x) = U − Γ (y −1 ◦ x) dμ(y), x ∈ G. (9.30) G

Proof. It is not restrictive to assume x0 = 0. For the sake of brevity, let us put n(t) := μ(Bd (0, t)). The proof is split into the “sufficiency” and the “necessity” parts. • Sufficiency. Let μ be a Radon measure satisfying (9.29) with x0 = 0, and let U ∈ R. Consider the function u(x) := U − Γ (y −1 ◦ x) dμ(y), x ∈ G. G

We shall show that u(0) > −∞, sup u = U,

(9.31a) (9.31b)

G

(v ∈ S(G) : Lv = μ, sup v = U ) G

⇒

v = u.

(9.31c)

Since (9.31a) implies u ∈ S(G) and Lu = μ (see Theorems 9.3.2 and 9.3.5), this will prove the sufficiency part. Proof (of (9.31a)). We put Γ (t) := βd t 2−Q . Since n(t) is monotone increasing and Q > 2, condition (9.29) implies μ({0}) = lim n(t) = 0. t↓0

Then u(0) − U = −

G\{0}

Γ (y

−1

) dμ(y) = − lim

λ↓0 λ

1/λ

Γ (t) dn(t).

(9.32)

452

9 Representation Theorems

On the other hand, we have 1/λ 1/λ Γ (t) dn(t) = Γ (1/λ) n(1/λ) − Γ (λ) n(λ) − Γ (t) n(t) dt λ ∞λ n(t) dt ≤ Γ (1/λ) n(1/λ) + βd (Q − 2) t Q−1 λ ∞ n(t) dt. (9.33) ≤ 2 βd (Q − 2) t Q−1 λ Here, being n(λ) monotone increasing and Γ ≤ 0, we have used the fact that ∞ ∞ Γ (λ) n(λ) = −n(λ) Γ (t) dt ≤ − Γ (t) n(t) dt λ λ ∞ n(t) dt ∀ λ > 0. = βd (Q − 2) Q−1 t λ Identity (9.32), together with (9.33) and condition (9.29), implies (9.31a).

Proof (of (9.31b)). For a fixed R > 0, let us split u − U as follows u − U = uR + u∞ R,

where

(9.34)

Γ (y −1 ◦ x) dμ(y).

uR (x) := − Bd (0,R)

Then lim uR (x) = 0.

x→∞

(9.35)

On the other hand, since u∞ R is the Γ -potential of the measure μ|G\Bd (0,R) and

u∞ R (0) ≥ u(0) − U > −∞,

∞ by Theorem 9.3.2, u∞ R is L-subharmonic in G. Hence, uR is sub-mean. Thus, for every r > 0, keeping in mind (9.33), we have ∞ ∞ ∞ Mr (uR )(0) ≥ uR (0) ≥ − Γ (t) dn(t) R ∞ n(t) dt. ≥ −2 βd (Q − 2) Q−1 t R

This implies the existence of at least one point y(r, R) ∈ ∂Bd (0, r) such that ∞ n(t) ∞ uR (y(r, R)) ≥ −2 βd (Q − 2) dt. (9.36) Q−1 R t Since d(y(r, R)) = r for every R > 0, (9.31b) follows from (9.34)–(9.36) and condition (9.29).

9.6 Bounded-above L-subharmonic Functions in G

453

Proof (of (9.31c)). Let v ∈ S(G) be such that Lv = μ and sup v = U . Since v is bounded above by the L-harmonic function U , by Theorem 9.4.7 there exists an L-harmonic function h in G such that v = h − Γ ∗ μ. Since v ≤ U , we have h − U ≤ Γ ∗ μ in G. Theorem 9.3.7 and Liouville Theorem 5.8.1 (page 269) imply h − U ≡ c with c ∈ R, c ≤ 0. As a consequence, v = U + c − Γ ∗ μ = u + c. But v and u have the same upper bound. Hence c = 0 and v ≡ u.

• Necessity. Let u ∈ S(G) be such that u(0) > −∞, sup = U < ∞. G

Since u is sub-mean, we have −∞ < u(0) ≤ Mr (u)(0) ≤ U, so that, by the Poisson–Jensen formula (9.22), (Γ (y) − Γ (r)) dμ(y) ≤ U − u(0)

∀ r > 0.

(9.37)

Bd (0,r)

On the other hand, since Γ (y) − Γ (r) ≥ 1 − (1/2)Q−2 Γ (y)

if 0 < d(y) ≤ t < r/2,

we have

Γ (y) dμ(y) n(t) Γ (t) = μ(Bd (0, t)) Γ (t) ≤ Bd (0,t) ≤c (Γ (y) − Γ (r)) dμ(y) Bd (0,t) (Γ (y) − Γ (r)) dμ(y), c= ≤c Bd (0,r)

1 1 − ( 12 )Q−2

Then, n(t) is bounded for 0 < t < r/2. In particular, μ({0}) = n(0+ ) = lim n(t) = 0. t↓0

Therefore, for 0 < ε < r, we have U − u(0) ≥ (Γ (y) − Γ (r)) dμ(y) {ε
.

454

9 Representation Theorems

Using the boundedness of {Γ (ε) n(ε) : 0 < ε < r/2} and letting ε tend to zero and r tend to ∞, we get ∞ n(t) βd (Q − 2) dt ≤ U − u(0) < ∞. Q−1 t 0 The proof is complete.

Example 9.6.2. The function u = −Γ is L-subharmonic and bounded from above in G, for U := sup u = 0. The L-Riesz measure of u is μ := Dirac0 , the Dirac mass supported at {0} (see Example 9.4.2, page 441). For every x0 ∈ G we have 1 if d(x0 ) < t, μ(Bd (x0 , t)) = 0 if d(x0 ) ≥ t, so that

0

∞

μ(Bd (x0 , t)) dt = t Q−1

∞

d(x0 )

1 dt. t Q−1

Since Q ≥ 3, this integral is finite if and only if d(x0 ) > 0, i.e. if and only if x0 = 0. The representation formula (9.30) obviously holds true for U − Γ ∗ μ = −Γ ∗ Dirac0 = −Γ = u.

In Theorem 9.6.1, we can remove the explicit mention of the point x0 . Precisely, the following theorem holds. G). II). A Radon Theorem 9.6.3 (L-Riesz measure of a bounded-above u ∈ S(G measure μ in G is the L-Riesz measure of a bounded-above L-superharmonic function in G if and only if ∞ μ(Bd (0, t)) dt < ∞. (9.38) t Q−1 1 If this condition is satisfied and U ∈ R, the function u(x) := U − Γ (y −1 ◦ x) dμ(y), RN

x ∈ G,

is the unique L-subharmonic function whose L-Riesz measure is μ and the least upper bound is U . Proof. Since every L-subharmonic function is finite in a dense subset of its domain, by Theorem 9.6.1 it remains only to prove the first part of the theorem. Let u ∈ S(G). Denote by μ := Lu its L-Riesz measure. Let v := uB ,

where B := uBd (0,1/2)

is the Perron-regularization of u related to the d-ball B. Then v ∈ S(G), v(0) > −∞ and v = u in G \ Bd (0, 1/2). Let μv := Lv be the L-Riesz measure of v. Then, by Theorem 9.6.1,

9.7 Smoothing of L-subharmonic Functions

∞

0

μv (Bd (0, t)) dt < ∞. t Q−1

455

(9.39)

Moreover, for every t > 1, μ(Bd (0, t)) = μ(Bd (0, 1)) − μv (Bd (0, 1)) + μv (Bd (0, t)). By using this identity in (9.39), we obtain (9.38). Vice versa, assume (9.38) is satisfied and define μ1 := μ|Bd (0,2) and μ2 := μ|G\Bd (0,2) . Since 0

∞

μ2 (Bd (0, t)) dt = t Q−1

1

∞

μ2 (Bd (0, t)) dt ≤ t Q−1

∞ 1

μ(Bd (0, t)) dt < ∞, t Q−1

by Theorem 9.6.1 there exists a function u2 ∈ S(G) bounded from above and such that Lu2 = μ2 . Then, letting u = −Γ ∗ μ1 + u2 , we get u ∈ S(G), sup u ≤ sup u2 < ∞ and Lu = μ1 + μ2 = μ.

9.7 L-subharmonic Smoothing of L-subharmonic Functions Let be a homogeneous norm on G (not necessarily an L-gauge for some subLaplacian L). Given a L1loc -function u : Ω → [−∞, ∞], Ω ⊆ G open, we know that the (ε, G)-mollifier of u (see Example 5.3.7, page 239) uε (x) = (u ∗G Jε )(x) := u(y −1 ◦ x) Jε (y) dy B (0,ε)

is well defined and smooth in Ωε := {x ∈ Ω : dist (x, ∂Ω) > ε}. We recall thatJε (x) = ε −Q J (δ1/ε (x)) and J is a smooth function supported in Bd (0, 1) with G J = 1. We also know (see Theorem 8.1.5, page 401) that the mapping u → u ∗G Jε preserves the L-sub-mean property for every sub-Laplacian L on G. As a consequence, by Theorem 8.2.1 (page 401), the (ε, G)-mollifiers preserve the L-subharmonicity for every sub-Laplacian L on G. Note that the notion of (ε, G)-mollification depends only on the structure of (G, ◦, δλ ) and on the homogeneous norm and does not depend on the sub-Laplacian L. The following theorem holds.

456

9 Representation Theorems

Theorem 9.7.1 (Smoothing of L-subharmonic functions). Let Ω be an open subset of G, let L be any sub-Laplacian on G and let u : Ω → [−∞, ∞[ be an L-subharmonic function. Then: (i) uε ∈ C ∞ (Ωε , R), (ii) uε −→ u in L1loc (Ω) as ε → 0, (iii) uε is L-subharmonic in Ωε . By using the smoothing operators constructed by superposition of the surface mean operators (see Section 5.6, page 257), one can find monotone sequences of Lsmooth and L-subharmonic functions approximating a given L-subharmonic function. However, the smooth approximating functions constructed via this method are L-subharmonic only with respect to the given sub-Laplacian L. The starting point of this construction is the following crucial lemma. Lemma 9.7.2. Let Γ be the fundamental solution for L and, for any fixed z ∈ G, let us set Γz (x) := Γ (z−1 ◦ x). Then, for every r > 0, Mr (Γz )(x) = min{Γ (z−1 ◦ x), Γ (r)},

x ∈ G.

(9.40)

As a consequence, the function x → Mr (Γz )(x) is L-superharmonic in G. Moreover, the map (x, z, r) → Mr (Γz )(x) is lower semicontinuous. Proof. (9.40) is (9.10a), page 438. The L-superharmonicity of Mr (Γz ) also follows from Proposition 6.5.4-(iii) (page 355). The last part of the assertion is a standard consequence of Fatou’s lemma. An alternative proof which makes use of the Poisson–Jensen formula is the following one. Since Γz ∈ S(G) and LΓz = −Diracz (the Dirac mass supported at {z}) the Poisson–Jensen formula (9.22) gives Γ (z−1 ◦ x) = Γz (x) = Mr (Γz )(x) + wz (x),

where wz (x) = Then

0 Γ (x −1 ◦ z) − Γ (r)

if d(z−1 ◦ x) ≥ r, if d(z−1 ◦ x) < r.

wz (x) = max{Γ (x −1 ◦ z) − Γ (r), 0}

and (9.40) follows from (9.41) and the symmetry of Γ .

(9.41)

9.7 Smoothing of L-subharmonic Functions

457

From Theorem 9.3.11 (page 439), one can easily obtain the approximation Theorem 9.7.3 below. For the reading convenience, we recall that in (5.50b) on page 258, we introduced the following integral operator Φr : ∞ Φr (u)(x) := ϕr (ρ) Mρ (u)(x) dρ, 0 1 t 1 , ϕ ∈ C0∞ (]0, 1[, R) and where ϕr (t) := ϕ ϕ = 1. (9.42) r r 0 Theorem 9.7.3 (Approximation I. Γ -potentials). Let u ∈ S(G) be the Γ -potential of a Radon measure μ on G. Then, for every r > 0, Φr (u) ∈ S(G) ∩ C ∞ (G),

(9.43)

where Φr is as in (9.42). Moreover, the following assertions hold: u(x) ≥ Φρ (u)(x) ≥ Φr (u)(x) ∀ x ∈ G, 0 < ρ ≤ r, lim Φr (u)(x) = u(x) ∀ x ∈ G. r→0

(9.44a) (9.44b)

Proof. Property (9.43) follows from (9.42), Theorem 9.3.11 (page 439) and Theorem 8.2.20, page 410 (in this last remark, we take Λ = (0, ∞), z = ρ and dμ(ρ) = ϕr (ρ) dρ). To prove (9.44a), let us assume 0 < ρ ≤ r. By Theorem 8.2.10 (page 404), ρ → Mρ (u) is monotone decreasing, so that ∞ Φρ (u)(x) = ϕ(t) Mρ t (u)(x) dt 0 ∞ ≥ ϕ(t) Mr t (u)(x) dt = Φr (u)(x). 0

Moreover, since u is super-mean, ∞ Φρ (u)(x) = ϕ(t) Mρ t (u)(x) dt ≤ 0

∞

ϕ(t) dt u(x) = u(x).

0

Thus, (9.44a) holds. Finally, (9.44b) follows from the inequality u(x) ≥ Φr (u)(x) and the lower semicontinuity of u.

If we consider the integral operator Φr related to a non-smooth function ϕ, Theorem 9.7.3 still holds with (9.43) replaced by Φr (u) ∈ S(G). In particular, by taking ϕ(t) = we obtain the following theorem.

Q t Q−1 0

if 0 < t < 1, otherwise,

(9.43)

458

9 Representation Theorems

Theorem 9.7.4 (Approximation II. Γ -potentials). Let u ∈ S(G) be the Γ -potential of a Radon measure μ in G. Then: (i) x → Mr (u)(x) is L-superharmonic in G for every r > 0, (ii) Mr (u)(x) ≤ Mρ (u)(x) ≤ u(x) for every x ∈ G and 0 < ρ ≤ r < ∞, (iii) limr↓0 Mr (u)(x) = u(x) for every x ∈ G. Mr is the solid average operator defined in (5.50f), page 259. By using the Riesz representation Theorem 9.4.4, Theorem 9.7.3 extends to L-subharmonic functions on arbitrary open subsets of G: Theorem 9.7.5 (L-subharmonic-monotone-smooth approximation). Let u ∈ S(Ω) and Ω ⊆ G be open. For every r > 0, let us define ur (x) := Φr (u)(x),

x ∈ Ωr := {y ∈ Ω : distd (y, ∂Ω) > r},

where Φr is the smoothing operator of Theorem 9.7.3. Then: (i) ur ∈ S(Ωr ) ∩ C ∞ (Ωr ) for every r > 0, (ii) u ≤ uρ ≤ ur for every 0 < ρ ≤ r < ∞, (iii) limr↓0 ur (x) = u(x) for every x ∈ Ω. Proof. Let x ∈ Ωr , and let O be a bounded open set such that Bd (y, r) ⊂ O ⊂ O ⊂ Ω for every y ∈ V , where V is a suitable neighborhood of x. By the Riesz-type representation Theorem 9.4.4, there exists a Radon measure μ supported in O and a function h, L-harmonic in O, such that u(y) = h(y) − (Γ ∗ μ)(y)

∀ y ∈ O.

As a consequence, ur (y) = hr (y) − (Γ ∗ μ)r (y) = h(y) − (Γ ∗ μ)r (y) Then, by (9.43),

∀ y ∈ V.

(9.45)

(ur )|V ∈ S(V ) ∩ C ∞ (V ),

and (i) is proved. Now, (ii) and (iii) directly follow from (9.45) with y = x and, respectively, (9.44a) and (9.44b).

9.8 Isolated Singularities—Bôcher-type Theorems Given an open set Ω ⊆ G, a point x0 ∈ Ω and an L-harmonic function u in Ω \ {x0 }, we say that x0 is a removable singularity of u if there exists an L-harmonic function v in Ω such that u(x) = v(x) ∀ x ∈ Ω \ {x0 }.

9.8 Isolated Singularities—Bôcher-type Theorems

459

A first result on removable singularities will easily follow from the next lemma, in which we shall use the notation B˙ d (x0 , r) := Bd (x0 , r) \ {x0 }. We recall that Q (the homogeneous dimension of G) is always supposed > 2. Lemma 9.8.1. Let h : B˙ d (x0 , r) → R be an L-harmonic function such that: (i) limx→y h(x) = 0 for every y ∈ ∂Bd (x0 , r), Q−2 (ii) lim supx→x0 h(x) d(x0−1 ◦ x) = 0. Then h ≡ 0. Proof. Let ε > 0. Define 2−Q , hε (x) := ±h(x) − ε d(x0−1 ◦ x)

x ∈ B˙ d (x0 , r).

Since d is an L-gauge and hε is L-harmonic, he is L-harmonic in B˙ d (x0 , r). Moreover, conditions (i) and (ii) imply lim sup hε (x) ≤ 0 ∀ y ∈ ∂ B˙ d (x0 , r). x→y

Then, by the maximum principle, hε ≤ 0 in B˙ d (x0 , r). Letting ε tend to zero, we obtain h ≤ 0. The same inequality holds for −h. Thus h = 0.

Theorem 9.8.2 (Characterization of removable singularities). Let Ω be an open subset of G. A point x0 ∈ Ω is a removable singularity of the L-harmonic function u : Ω \ {x0 } → R if and only if Q−2 = 0. (9.46) lim sup u(x) d(x0−1 ◦ x) x→x0

Proof. The “only if” part is trivial. We turn to the “if” part. Assume that (9.46) is satisfied and r > 0 is such that Bd (x0 , r) ⊂ Ω. Let w be the solution to the boundary value problem Lw = 0 in Bd (x0 , r), w|∂Bd (x0 ,r) = u|∂Bd (x0 ,r) . Then h := u − w satisfies the hypothesis of Lemma 9.8.1. It follows that h = 0 in B˙ d (x0 , r), i.e. u = w in B˙ d (x0 , r). Thus, the function u(x) in Ω \ Bd (x0 , r), v : Ω → R, v(x) := w(x) in Bd (x0 , r) is L-harmonic in Ω and equals u in Ω \ {x0 }. This completes the proof.

Remark 9.8.3. The function 2−Q x → Γ (x0−1 ◦ x) = d(x0−1 ◦ x) is L-harmonic in G \ {x0 } with a non-removable singularity at x = x0 .

460

9 Representation Theorems

Non-negative L-harmonic functions in G \ {x0 } can be characterized in terms of the function x → Γ (x0−1 ◦ x). We first show a local version of this statement, which extends the classical Bôcher theorem to the sub-Laplacian setting. Theorem 9.8.4 (Bôcher-type theorem on punctured balls). Let u be L-harmonic and non-negative in the punctured ball B˙ d (x0 , R). Then there exist a non-negative constant a and an L-harmonic function h in Bd (x0 , R) such that u = aΓ +h

in B˙ d (x0 , R).

(9.47)

Proof. It is not restrictive to assume x0 = 0. If u were bounded near zero, then (9.47) would follow, with a = 0, from Theorem 9.8.2. Assume that lim sup u(x) = ∞.

(9.48)

x→0

For 0 < r ≤ R/2, define and s(r) := inf u(x).

S(r) := sup u(x)

d(x)=r

d(x)=r

The Harnack inequality on rings of Corollary 5.7.7 (page 267) implies S(r) ≤ c s(r),

0 < r ≤ R/2,

(9.49)

with a positive constant c independent of r. On the other hand, by the maximum principle, max{S(r), S(R/2)} =

max

r≤d(x)≤R/2

u(x),

0 < r ≤ R/2,

so that, by (9.48), lim S(r) = ∞. r↓0

(9.50)

Using (9.50) and (9.49), we get lim infx→0 u(x) = ∞. Hence lim u(x) = ∞.

x→0

If we extend u at 0 letting u(0) = ∞, by Corollary 8.2.6 (page 402), u ∈ S(Bd (0, R)). Let us now observe that the L-Riesz measure of u μ := −Lu is supported at {0}, since u is L-harmonic in B˙ d (0, r). Therefore, μ = a Dirac0 , for a suitable a > 0 (here Dirac0 is the Dirac mass supported at the origin). Now, the Riesz representation result of Corollary 9.4.9 implies u = Γ ∗ μ + h = a Γ + h, where h is L-harmonic in Bd (0, R). The proof is complete.

9.8 Isolated Singularities—Bôcher-type Theorems

461

Theorem 9.8.5 (Bôcher-type theorem for L). Let Ω ⊆ G be open, and let x0 ∈ Ω. Let u : Ω \ {x0 } → R be L-harmonic and non-negative. Then there exist a nonnegative constant a and an L-harmonic function v in Ω such that u = aΓ +v

in Ω \ {x0 }.

(9.51)

Proof. Let R > 0 be such that Bd (x0 , R) ⊂ Ω. By Theorem 9.8.4, there exist a constant a ≥ 0 and an L-harmonic function h in Bd (x0 , R) such that u = a Γ + h in B˙ d (x0 , R). Thus, the function

v : Ω \ {x0 } → R,

v :=

h u−aΓ

in Bd (x0 , R), in Ω \ Bd (x0 , R)

is an L-harmonic function in Ω satisfying (9.51).

This theorem, together with Liouville Theorem 5.8.1 (page 269), implies the following corollary. Corollary 9.8.6 (Non-negative L-harmonic functions in G \ {0}). Let u : G \ {0} → R be a non-negative L-harmonic function. Then there exist two non-negative constants a and b such that u = aΓ +b

in G \ {0}.

u = aΓ +v

in G \ {0},

Proof. By Theorem 9.8.5, where a is a non-negative constant and v is L-harmonic in G. Since u ≥ 0 and Γ (x) → 0 as x → ∞, the function v is bounded from below and lim infx→∞ v(x) ≥ 0. Then, by Liouville Theorem 5.8.1 (page 269), v is constant, and this constant is non-negative.

With this corollary at hand, we can trivially characterize the δλ -homogeneous non-negative L-harmonic functions in G \ {0}. ˙ )). Suppose u : Corollary 9.8.7 (Non-negative δλ -homogeneous functions in H(G G \ {0} → R is non-negative, L-harmonic and δλ -homogeneous of degree m. If u is not identically zero, then m = 0 or m = 2 − Q. Moreover: (i) u = constant if m = 0, (ii) u = a Γ for a suitable constant a > 0 if m = 2 − Q. As a consequence, we immediately obtain the following improvement of Theorem 5.5.6 on page 256. Corollary 9.8.8 (“Uniqueness” of the L-gauges. II). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let d be a homogeneous norm on G, smooth out of the origin and such that L(d α ) = 0 in G \ {0}, for a suitable α ∈ R, α = 0. Then α = 2 − Q and d is an L-gauge on G.

462

9 Representation Theorems

9.8.1 An Application of Bôcher’s Theorem The aim of this section is to prove the following Proposition 9.8.9, as a consequence of Bôcher’s Theorem 9.8.5 above and of the Liouville-type Theorem 5.8.4 (page 270). We recall that if d is an L-gauge on G, then ΨL = |∇L d|2 is the kernel appearing in the relevant solid mean value formula of Theorem 5.6.1 (page 259). Proposition 9.8.9. Suppose L is a sub-Laplacian on the (homogeneous) Carnot group G and d is an L-gauge on G such that |∇L d|2 is constant on G \ {0}. Then G has step 1, i.e. G = (G, +) is the Euclidean group. ˙ := G \ {0}. Let d be an L-gauge on G. Then, by Proof. Let us set, for brevity, G Definition 5.4.1 (page 247), ˙ ∩ C(G) d ∈ C ∞ (G) ˙ (here Q is the homogeneous dimension of G). From and d 2−Q is L-harmonic on G the formula L(α(u)) = α (u) |∇L u|2 + α (u) Lu (for suitably regular functions u : G → R, α : R → R), we have 0 = L(d 2−Q ) = (2 − Q)(1 − Q) d −Q |∇L d|2 + (2 − Q)d 1−Q Ld ˙ i.e. on G,

d Ld = (Q − 1) |∇L d|2

Analogously, we have

L(d 2 )

= 2 |∇L

d|2

(9.52)

+ 2 d Lu. Hence, by (9.52),

L(d 2 ) = 2 Q |∇L d|2 Consequently, (9.53) proves that ˙ |∇L d|2 = constant on G

˙ on G.

⇒

˙ on G.

˙ . L(d 2 ) = constant on G

(9.53)

(9.54)

We claim that

˙ |∇L d|2 = constant on G

⇒

2 d ∈ C ∞ (G) .

(9.55)

From this claim the proposition follows. Indeed, by continuity, the claimed (9.55) and (9.54) prove that L(d 2 ) = constant on G. Then, thanks to the Liouville-type Theorem 5.8.4 (page 270), d 2 is a polynomial function of G-degree at most 2. Denoting, as usual, by x = (x (1) , x (2) , . . . , x (r) ) the “stratification” of the variables of the (homogeneous) group G, we have d 2 (x) = p2 (x (1) ) + p1 (x (2) ),

9.9 Exercises of Chapter 9

463

where pi is a polynomial function of ordinary degree i (i = 1, 2). Due to the nonnegativity of d 2 , this gives p1 ≡ 0, so that d 2 (x) = p2 (x (1) ). But since d is positive outside 0 (see the definition of homogeneous norm in Section 5.1, page 229), if G had step r ≥ 2, fixed x (2) = 0, we would have the contradiction 0 < d 2 (0(1) , x (2) , 0(3) , . . . , 0(r) ) = p2 (0(1) ) = 0. (Recall that p2 is a homogeneous polynomial of degree 2.) It remains to prove (9.55). Suppose |∇L d|2 = c ∈ R and take a non-negative cut-off function ϕ ∈ C0∞ (G) such that ϕ ≡ 1 on a neighborhood U of the origin, say U = Bd (0, ε) with ε small. Denote by Γ the fundamental solution of L. From Theorem 5.3.5 (page 238) the function u(y) := Γ (y −1 ◦ x) ϕ(x) dx G

is non-negative, smooth on G and satisfies Lu = −ϕ on G. In particular, Lu = −1 on U .

(9.56)

By (9.53) and (9.56), we have L(d 2 + 2 Q c u) = 0

on U \ {0}.

Hence, d 2 + 2 Q c u is L-harmonic and non-negative on U \ {0}. Furthermore, it is bounded there, for d and u are continuous on G. By Theorem 9.8.5, there exists h ∈ H(U ) such that d 2 + 2 Q c u = h on U \ {0}. By continuity, this holds on the whole U , thus proving (9.55).

Bibliographical Notes. The topics developed in this chapter within the classical theory of Laplace’s operator can be found in all the main monographs devoted to the potential theory; see the references in the Bibliographical Notes of Chapter 8. For Bôcher-type theorems, see [TW02b]. Some of the topics presented in this chapter also appear in [BL03,BL07].

9.9 Exercises of Chapter 9 Ex. 1) Prove that if u > 0 is L-harmonic in an open set Ω, then log(u) is Lsuperharmonic in Ω.

464

9 Representation Theorems

Ex. 2) Prove that any L-superharmonic function u in an open set Ω belongs to p Lloc (Ω) for every Q 1≤p< . Q−2 (Hint: Use Theorem 9.4.4, in order to restrict the proof to the case where u = Γ ∗ μ with μ compactly supported. Then use the Hölder inequality and the summability properties of Γ .) Ex. 3) Prove that there exists an L-superharmonic function u in G such that u does p Q . (Hint: u = Γ .) not belong to Lloc (Ω) with p = Q−2 Ex. 4) Prove that if u is an L-superharmonic function in an open set Ω, then ∇L u is well-defined in the weak sense of distributions, and it holds p

|∇L u| ∈ Lloc (Ω) for every 1≤p<

Q . Q−1

(Hint: Use Theorem 9.4.4, to restrict to the case where u = Γ ∗ μ with μ compactly supported. Then use the summability properties of ∇L Γ .) Ex. 5) Complete the sketched proof of the following proposition. Proposition 9.9.1. Let u ∈ S(G) be non-negative. Then the following assertions hold: a) u is a Γ -potential iff limR→∞ mR [u](0) = 0 or iff infG u = 0. b) u is a Γ -potential iff u is bounded from above by a Γ -potential. Proof (Sketch). Let u = Γ ∗ μ, where μ is a Radon measure in G. Then (why?) mR [u](0) = k(0, y)Γ (z−1 ◦ y) dH N −1 (y) dμ(z) G ∂Bd (0,R) = (Γ ∗ λ0,R )(z) dμ(z) = min{Γ (z), R 2−Q } dμ(z). G

G

By monotone convergence derive limR→∞ mR [u](0) = 0. Vice versa (why?), u = infG u + Γ ∗ (μu ). Hence mR [u](0) = inf u + mR [Γ ∗ (μu )](0). G

Then, if limR→∞ mR [u](0) = 0, we obtain (why?) infG u = 0, i.e. u = Γ ∗ (μu ). This argument also proves the second “iff” in (1). Finally, derive (2) from (1).

Ex. 6) Prove the assertion in Example 9.1.3, page 427.

9.9 Exercises of Chapter 9

465

Ex. 7) Let u be an L-harmonic function in the punctured ball B˙ d (0, 1) := Bd (0, 1) \ {0}. Assume that lim inf u(x) d(x)2−Q > −∞. |x|→0

Then there exists b ∈ R and v, L-harmonic in Bd (0, 1), such that in B˙ d (0, 1).

u = bΓ + v

Ex. 8) Let Ω be an open set of G. Denote p p b (Ω) := u L-harmonic in Ω : |u| < ∞ , Ω

for 1 ≤ p < ∞. Prove that bp (Ω) is a closed subspace of Lp (Ω). Ex. 9) Let bp (Ω) be as in the previous exercise. Let Ω = Bd (0, 1) \ {0}, and let u ∈ bp (Ω), p ≥ Q/(Q − 2). Prove that 0 is a removable singularity of u. Ex. 10) Let 1 ≤ p < Q/(Q − 2). Prove the existence of a function u ∈ bp (Bd (0, 1) \ {0}) such that 0 is not a removable singularity of u. Ex. 11) Let u : G \ {0} → R be an L-harmonic function. Assume that, for some p ∈ [1, ∞[, G\{0}

|u|p < ∞.

Prove that: (i) −Q

|u(x)| ≤ c|x| p

G\{0}

|u|p

∀x = 0,

where c > 0 is independent of u and x, (ii) u ≡ 0 in G provided that p ≥ Q/(Q − 2). Ex. 12) Prove the following result, as a consequence of Corollary 9.8.6. Corollary 9.9.2. Let be any homogeneous norm on G. Suppose there exists a non-constant function α : (0, ∞) → (0, ∞) such that α ◦ is L˙ Then is an L-gauge function and α(t) = a t 2−Q + b, for harmonic in G. some constants c, b ≥ 0. ˙ Then, by Corol(Hint: α ◦ is a non-negative L-harmonic function on G. lary 9.8.6, we have (here d is the L-gauge function such that d 2−Q is the fundamental solution for L) α((x)) = a d 2−Q (x) + b

˙ for every x ∈ G.

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9 Representation Theorems

Since d and are both δλ -homogeneous of degree 1, this gives, for every ξ ∈ G such that (ξ ) = 1, α(t) = α(t (ξ )) = α((δt ξ )) = a d 2−Q ((δt ξ )) + b = a d 2−Q (ξ ) t 2−Q + b.

(9.57)

When t = 1, this proves that d is constant (say, a) on the level set of {ξ : (ξ ) = 1}, so that d(x) = d δ(x) δ1/(x) (x) = (x) d δ1/(x) (x) = (x) a, whence = a−1 d is an L-gauge function. Finally, (9.57) also shows that a = a a2−Q .) α(t) = a t 2−Q + b, with ˙ We recall: if there exists Ex. 13) Let d be a homogeneous norm on G, smooth in G. a real constant γ = 0 such that ˙ d γ is L-harmonic in G,

(9.58)

then we say that d is an L-gauge. As we know, a remarkable homogeneous norm on G is given by dC (x) := dX (x, 0),

x ∈ G,

(9.59)

where dX denotes the Carnot–Carathéodory distance related to the family of vector fields X = {X1 , . . . , Xm } (see Proposition 5.2.8). The norm dC in (9.59) will be referred to as the LC -norm on G. In studying surface measures in Carnot–Carathéodory spaces, R. Monti and F. Serra-Cassano proved that dC satisfies the L-Eikonal equation, i.e. |∇L dC | = 1

almost everywhere in G

(see [MSC01, Theorem 3.1]). Prove the following assertion. Corollary 9.9.3. Let G be a Carnot group different from an Euclidean group, and let L be a sub-Laplacian on G. Then the LC -norm (9.59) is not an L-gauge. More explicitly, for every γ ∈ R \ {0}, γ ˙ dC is not L-harmonic in G.

Moreover, suppose there exists a function α : (0, ∞) → (0, ∞) such that ˙ Then t → α(t) is a constant function. α ◦ dC is L-harmonic in G. (Hint: Use the cited result in [MSC01, Theorem 3.1] and our Theorem 9.8.9. Then use Corollary 9.9.2.) By using Proposition 9.8.9, prove also the following assertion. If L is a sub-Laplacian on the (homogeneous) Carnot group G and d is an L-gauge on G satisfying the L-Eikonal equation |∇L d| = 1 on G \ {0}, then G has step 1, i.e. G = (G, +) is the Euclidean group.

9.9 Exercises of Chapter 9

467

Ex. 14) Prove the following result, by making use of the proof of Theorem 9.8.9. Proposition 9.9.4. Let u : G → R be a (smooth) solution to Lu = c

in G,

where c is a real constant. Assume u is coercive, i.e. u(x) −→ ∞ as x → ∞. Then G is the Euclidean group. Ex. 15) When d is an L-gauge function, we already know that the kernel ΨL = |∇L d|2 appears in the (solid) average formula of Theorem 5.6.1 (see also (5.42) on page 252). For the sake of brevity, we introduce a new notation: ΨL , mr (u) := K := ΨL , K := u K dH N −1 (9.60) |∇d| ∂Bd (0,r) (here, ∇ is the standard gradient vector in RN , and H N −1 denotes the (N − 1)-Hausdorff measure in G). Compare to our previous notation (see Theorem 5.5.4 and (5.42)) and notice that K = KL (0, ·),

mr (u) =

1 r Q−1 Mr (u)(0). (Q − 2)βd

Prove the following result. Theorem 9.9.5. Let d be an L-gauge, and let γ be the related exponent as in (9.58). If u is L-harmonic in the unit punctured ball B˙ = Bd (0, 1) \ {0}, then there exist real constants a, b such that r γ −1 mr (u) = a r γ + b

for every r ∈ ]0, 1[.

(9.61)

(Hint: We have proved in Section 5.5 (page 251) that if the sub-Laplacian L is written in the divergence form L = div(A(x) ∇), then we have g A∇f, ν − f A∇g, ν dH N −1 (9.62) (g Lf − f Lg) = Ar,R

∂Ar,R

˙ and 0 < r < R < 1, where Ar,R := {x : r < for each f, g ∈ C 2 (B) d(x) < R} and ν is the normal outer vector on ∂Ar,R . If f = u and g ≡ 1, then N −1 A∇u, ν dH = A∇u, ν dH N −1 . (9.63) ∂Bd (0,R)

∂Bd (0,r)

Hence, ∂Bd (0,R) A∇u, ν dH N −1 is constant w.r.t. R, say α. We now choose f = u and g = d γ in (9.62). By noticing that on ∂Bd (0, ρ) we have

468

9 Representation Theorems

ν = ∇d/|∇d|, g = ρ γ and A∇g, ν = γ ρ γ −1 |∇L d|2 /|∇d|. Then (9.62) becomes (see the definition of mr in (9.60) and use (9.63)) 0 = α R γ − γ R γ −1 mR (u) − α r γ + γ r γ −1 mr (u). In other words, the function ]0, 1[ r → α r γ − γ r γ −1 mr (u) is constant, say β. This is exactly (9.61) with a = α/γ and b = −β/γ .) Ex. 16) Re-derive Theorem 5.5.6 on page 256 as a consequence of the above Theorem 9.9.5. In other words, prove the following assertion. Corollary 9.9.6. Let d be an L-gauge function, and let γ be the related exponent as in (9.58). Then γ = 2 − Q,

and Γ = βd d 2−Q

(9.64)

is the fundamental solution (with pole at the origin) for L. Here Q is the homogeneous dimension of G and −1 K. (9.65) βd = Q(Q − 2) Bd (0,1)

Hint: By means of the coarea formula, we have r dH N −1 (x) K(x) K(x) dx = |∇d(x)| d(x)=ρ d(x)
Bd (0,1)

By differentiating both sides w.r.t. r, we get dH N −1 (x) = r Q−1 Q ωd K(x) |∇d(x)| d(x)=r

for all r > 0.

(9.66)

Now, recalling (9.60), the above identity rewrites as mr (1) = r Q−1 Q ωd .

(9.67)

On the other hand, by Theorem 9.9.5, there exist constants a, b such that mr (1) = a r + b r 1−γ , whence (9.67) gives Qωd r Q−2 = a + b r −γ which is possible iff (recall that Q > 2) a = 0 and γ = 2 − Q. Finally, let us set Γ (x) := βd d 2−Q (x) (βd as in (9.65)), and let us show that Γ is the fundamental solution for L. First, show that Q < ∞. Γ (x) dx = βd H N (Bd (0, 1)) 2 Bd (0,1) Hence Γ ∈ L1loc (G). Moreover, for 2 − Q < 0, Γ (x) vanishes when x → ∞. Finally, let us check that LΓ = −Dirac0 in the weak sense of

9.9 Exercises of Chapter 9

469

distributions. Let ϕ ∈ C0∞ (G) be fixed. By the divergence theorem, show that d 2−Q Lϕ Γ Lϕ = lim ε→0 d>ε dH N −1 = lim − d 2−Q ∇L ϕ, ∇L d ε→0 |∇d| d(x)=ε − ∇L (d 2−Q ), ∇L ϕ d>ε

dH N −1 ϕ (2 − Q) d 1−Q K = lim −I1 (ε) + ε→0 |∇d| d=ε + L(d 2−Q ) ϕ d>ε

= lim (−I1 (ε) + I2 (ε) + 0) = −ϕ(0) βd−1 . ε→0

Indeed, prove that |I1 (ε)| ≤ ε 2−Q sup |∇L ϕ| |∇L d| On the other hand, limε→0 I2 (ε) =

d(x)=ε

−ϕ(0) βd−1 ,

dH N −1 = c ε. |∇d|

since (recall (9.67))

I2 (ε) = −βd−1 ϕ(0) + I3 (ε), and (exploiting again (9.67)) |I3 (ε)| ≤ Q(Q − 2)ωd sup |ϕ(x) − ϕ(0)| → 0 as ε → 0. d(x)=ε

Ex. 17) Let mr be as in (9.60). Consider our usual surface average operator Mr (u)(0), briefly denoted by Mr (u). Following an idea exploited by Axler, Bourdon and Ramey [ABR92] in the classical harmonic setting, given a smooth function u : B˙ → R, we define S(u)(x) := Md(x) (u),

˙ x ∈ B.

(9.68)

If u is L-harmonic in B˙ and d is an L-gauge, by Theorem 9.9.5 and Corollary 9.9.6, we have S(u)(x) = a d 2−Q (x) + b,

˙ x ∈ B,

for suitable a, b ∈ R. Thus, S(u) is L-harmonic in B˙ and radially symmetric with respect to d, i.e. S(u)(x) = S(u)(y) if d(x) = d(y). Moreover, since Mr (1) = 1 and d is constant on ∂Bd (0, 1), we have S(S(u)) = S(a d 2−Q + b) = a d 2−Q + b = S(u). Prove the following result.

(9.69)

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9 Representation Theorems

Lemma 9.9.7. If u is L-harmonic on B˙ and d is an L-gauge, there exist a, b ∈ R such that Mr (u)(0) = a r 2−Q + b

for every r ∈ ]0, 1[,

S(u)(x) = a d 2−Q (x) + b

˙ for every x ∈ B.

(9.70)

Moreover, for every n ∈ N, we have S n (u) = S(u), where S n denotes the n-th iterate of the operator S. (Hint: Use Theorem 9.9.5, Corollary 9.9.6 and Mr (1) = 1.) The operator S can be used to prove the following radial-symmetry result for non-negative L-harmonic functions in a punctured L-gauge ball, vanishing on the boundary. Denote B = Bd (0, 1). Theorem 9.9.8. Let d be an L-gauge, and let w be a non-negative ˙ Assume that w is continuous up to the boundL-harmonic function in B. ary of B and w(x) = 0 for every x ∈ ∂B. Then w is d-radially symmetric. ˙ so that, for a suitable More precisely, w(x) = S(w)(x) for every x ∈ B, real constant a > 0, ˙ w(x) = a d 2−Q (x) − 1 for every x ∈ B. Proof. Following an idea in [ABR92], we consider the Harnack inequality on d-spheres in Theorem 5.16.5 (page 327). Rewrite it to derive the existence of a constant c∈(0, 1) such that c h(z) ≤ h(x)

whenever 0 < d(z) = d(x) ≤

We claim that ˙ ˙ ∩ C(B), h ∈ H(B) ⇒ h ≡ 0 on ∂B and h ≥ 0 on B˙

1 2

˙ h > 0. and h ∈ H(D), (9.71)

˙ h − c S(h) ≥ 0 on B,

(9.72)

˙ ∩ C(B \ {0}) is nonwhere B˙ = Bd (0, 1) \ {0}. Suppose that w ∈ H(B) negative on B˙ and null on ∂B. Set cn := 1 − (1 − c)n ,

n ∈ N ∪ {0}.

(9.73)

Let us prove by induction that w − cn S(w) ≥ 0 on B˙ for all n ∈ N ∪ {0}.

(9.74)

The case n = 0 is obvious. Suppose (9.74) holds, and let us prove it for n+1. The function h := w − cn S(w) satisfies the hypothesis of (9.72). Indeed, w|∂B = 0 implies S(w)|∂B = 0. Moreover, both w and S(w) belong to ˙ since (see also Lemma 9.9.7, b = −a) ˙ ∩ C(B), H(B)

9.9 Exercises of Chapter 9

S(w)(x) = Md(x) (w)(0) = a d 2−Q (x) − a.

471

(9.75)

Consequently, by (9.72), we have on B˙ 0 ≤ h − c S(h) = w − cn S(w) − c S w − cn S(w) = w − cn+1 S(w). (Use cn + c − c cn = cn+1 and Lemma 9.9.7.) Thus (9.74) is proved by ˙ On the induction. Letting n → ∞ in it, we infer that w ≥ S(w) on B. other hand, the inequality w(x) > S(w)(x) for some x ∈ B˙ is impossible, otherwise S(w)(x) > S(w)(x). Consequently, by using (9.75), w(x) = ˙ S(w)(x) = a(d 2−Q (x) − 1) for x ∈ B. It remains to prove (9.72). Let h be as in (9.72). From Lemma 9.9.7 we have ˙ so that S(h)(x) = a d 2−Q (x) + b on B, ˙ ˙ ∩ C(B). H := h − c S(h) ∈ H(B) Furthermore, H = 0 on ∂B. If we multiply the inequality in (9.71) times (Q − 2)βd d 1−Q (x) K(z) and then integrate w.r.t. z ∈ Bd (0, d(x)), we get H (x) ≥ 0 for every x ∈ Bd (0, 1/2). Then the weak maximum principle ˙ ensures that H ≥ 0 on B.

Ex. 18) Derive Theorem 9.8.5 from Theorem 9.9.8. (Hint: Let Bd (x0 , ε) ⊂ Ω. Let h be the solution to the Dirichlet problem Lh = 0 in Bd (x0 , ε), h = u on ∂Bd (x0 , ε). Consider the function w(x) := (u − h)(x0 ∗ δε (x)) + Γ (x) − 1. Prove, by the weak maximum principle, that ˙ Hence, by Theorem 9.9.8, there exists c ∈ R such that u = h+ w ≥ 0 on B. −1 c Γ (x0 ∗·) for x ∈ B˙ d (x0 , ε). Then c ≥ 0, otherwise limx→x0 u(x) = −∞. Thus, v := u − Γ (x0−1 ∗ ·) extends L-harmonically through x0 .) Ex. 19) Let x0 ∈ Ω ⊆ G, where Ω is open, and let Ω˙ = Ω \ {x0 }. Suppose w : Ω˙ → R satisfies the following conditions: ˙ and w, ∇L w are bounded on Ω, ˙ (i) w ∈ C ∞ (Ω) ∞ ˙ (ii) Lw = f in Ω, where f ∈ C (Ω). Then w extends to a C ∞ function on the whole Ω. (Hint: Suppose x0 = 0. It is enough to prove that Lw = f in Ω, in the weak sense of distributions. Let ϕ ∈ C0∞ (Ω), and let d be a homogeneous norm ˙ Argue as in the proof of Corollary 9.9.6, to show that on G, smooth on G. dH N −1 − w Lϕ = lim − w ∇L ϕ, ∇L d ∇L w, ∇L ϕ ε→0 |∇d| Ω d(x)=ε d>ε N −1 dH + ϕ ∇L w, ∇L d ϕ Lw = lim −I1 (ε) + ε→0 |∇d| d=ε d>ε fϕ = f ϕ. = lim −I1 (ε) + I2 (ε) + ε→0

d>ε

Ω

Indeed (see also Ex. 24, Chapter 5), |I1 (ε)|, |I2 (ε)| ≤ c ε Q−1 .)

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9 Representation Theorems

Ex. 20) Let Ω ⊆ G be open, and let u ∈ S(Ω). Show that, for every x ∈ Ω and r > 0 such that Bd (x, r) ⊆ Ω, one has r μ(Bd (x, t)) u(x) = Mr (u)(x) − βd (Q − 2) dt, t Q−1 0 where μ := Lu is the L-Riesz measure of u. (Hint: Write r

Γ (d) − Γ (r) =

Γ (t) dt

d

and use Fubini’s theorem in the Poisson–Jensen formula (9.22).) Ex. 21) Assume the same hypotheses as in the previous exercise. Show that r μ(Bd (x, t)) dt, Mr (u)(x) − Mρ (u)(x) = βd (Q − 2) t Q−1 ρ for 0 < ρ < r. (Hint: First, assume u(x) > −∞ and use the Poisson–Jensen formula (9.22). Then complete the proof by replacing u with its Perron-regularized uBd (x,ρ/2) .) Ex. 22) Let Ω ⊆ G be open, and let u ∈ S(Ω). Show that, for every x ∈ Ω, μ({x}) =

1 lim ρ Q−2 Mρ (u)(x), βd ρ!0

where μ := −Lu is the L-Riesz measure of u. (Hint: Use the previous exercise.) Ex. 23) Assume the same hypotheses as in the previous exercise. Show that μ({x}) = 0

if u(x) < ∞.

(Hint: Use the previous exercise.) Ex. 24) Let f : G → R ∪ {∞} be defined as follows ⎧ −2 if 0 < d(x) < 1, ⎪ ⎨ (d(x)) f (x) = ∞ if x = 0, ⎪ ⎩ 0 if d(x) ≥ 1. Show that: (i) f ∈ L1 (G), (ii) x → u(x) := RN Γ (y −1 ◦ x)f (y) dy is L-superharmonic in G, (iii) u(0) = ∞, (iv) μ({0}) = 0, where μ is the L-Riesz measure of u.

10 Maximum Principle on Unbounded Domains

The aim of this chapter is to study the following version of the maximum principle for sub-Laplacians L on Carnot groups G. An open set Ω ⊆ G is called a maximum principle set for L (MP set, in short) if every bounded from above L-subharmonic function u is ≤ 0 in Ω whenever lim sup u(x) ≤ 0 x→y

for every y ∈ ∂Ω.

We already know that bounded domains are MP sets. Here, we introduce the notion of L-thin set at infinity, and we prove that Ω is an MP set if and only if G \ Ω is not L-thin at infinity. Then, we show some geometrical conditions for non-Lthinness at infinity, based on the notion of q-set. We prove that a sufficient condition for F to be not L-thin at infinity is that F is not a q-set, for some q > Q − 2. As usual, Q is the homogeneous dimension of G. To this end, a crucial tool will be Theorem 9.6.1 (page 451), the representation theorem for bounded from above L-subharmonic functions on G. It will follow, in particular, that open sets Ω satisfying an exterior G-cone condition are MP sets, since G \ Ω is not a Q-set, hence not L-thin at infinity. Some other explicit criteria are given. As usual, G = (RN , ◦, δλ ) will denote a homogeneous Carnot group and L a fixed sub-Laplacian on G.

10.1 MP Sets and L-thinness at Infinity The maximum principle of Theorem 5.13.4 (page 295) does not hold, in general, if the open set Ω is unbounded.1 The aim of this section is to provide sufficient conditions on unbounded Ω’s for an analogue of the maximum principle to hold for a given sub-Laplacian L. 1 Consider, for example, the classical case of the ordinary Laplace operator on R2 , Ω = {(x, y) ∈ R2 : y > 0}, u(x, y) = exp(x) sin(y). Then u satisfies Δu(x, y) = 0 in Ω,

u = 0 on ∂Ω, but u is not identically zero.

474

10 Maximum Principle on Unbounded Domains

To begin with, we introduce some notation and definitions. If Ω ⊆ G is open, S b (Ω) will denote the (cone of the) bounded from above L-subharmonic functions in Ω. For example, −Γ ∈ S b (G). Definition 10.1.1 (MP set for L). We say that Ω is a maximum principle set for L (MP set for L, in short) if the following assertion holds: u ∈ S b (Ω) ⇒ u ≤ 0 in Ω. (10.1) lim supx→y u(x) ≤ 0 ∀ y ∈ ∂Ω By Theorem 8.2.19-(ii), page 409, it follows that every bounded open set is an MP set for L. We would like to remark that an u.s.c. function u : Ω → [−∞, ∞[ satisfying the boundary condition in (10.1) is bounded from above if Ω is bounded.2 Then, if Ω is bounded, in (10.1), S b (Ω) may be replaced by S(Ω). The maximum principle is deeply related to the notion of L-thinness at infinity, which we now introduce. Definition 10.1.2 (L-thinness at infinity). A subset F of G will be said L-thin at infinity if there exists a bounded from above L-subharmonic function u in G such that3 lim sup u(x) lim sup u(x). (10.2) x→∞, x∈F

x→∞, x∈G

From this definition it follows that a set F ⊆ G is not L-thin at infinity iff for every u ∈ S b (G) lim sup u(x) = lim sup u(x). x→∞, x∈F

x→∞, x∈G

The following theorem holds (see also Fig. 10.1). Theorem 10.1.3 (Characterization of MP set). An open set Ω ⊆ G is an MP set for L if and only if G \ Ω is not L-thin at infinity. The proof of this result will easily follow from the next two lemmas. Lemma 10.1.4. Let u : Ω → [−∞, ∞[ be an L-subharmonic function in Ω satisfying (10.3) lim sup u(x) ≤ 0 ∀ y ∈ ∂Ω. x→y

Then the function

v : G → [−∞, ∞[,

v=

max{u, 0} 0

is L-subharmonic in G. 2 See Exercise 1 at the end of the Chapter. 3 We agree to let

lim sup u(x) = −∞, x→∞, x∈F

if F is bounded.

in Ω, in G \ Ω

10.1 MP Sets and L-thinness at Infinity

475

Fig. 10.1. A MP set Ω

Proof. Condition (10.3) implies that v is upper semicontinuous in G. Moreover, since max{u, 0} ∈ S(Ω), v is locally sub-mean in Ω. On the other hand, since v ≥ 0, for every x ∈ G \ Ω. v(x) = 0 ≤ Mr (v)(x) Then, v is everywhere locally sub-mean, hence L-subharmonic in G.

Lemma 10.1.5. Let F ⊆ G and let u ∈ S b (G). Then lim sup u(x) = lim sup u(x) x→∞, x∈F

(10.4a)

x→∞, x∈G

if and only if sup u = sup u.

(10.4b)

G

F

Proof. We may assume that u is not constant. Then, by the maximum principle of Theorem 8.2.19 (page 409), we have ∀ x ∈ G.

u(x) < sup u G

(10.5)

As a consequence, sup u = lim sup u(x). G

(10.6)

x→∞, x∈G

Then lim sup u(x) ≤ sup u ≤ sup u = lim sup u(x). x→∞, x∈F

G

F

x→∞, x∈G

This shows that (10.4a) implies (10.4b). Let us now assume (10.4b). Then sup F ∩Bd (0,R)

u<

sup F \Bd (0,R)

Indeed, if (10.7) were false, we would have

u

∀ R > 0.

(10.7)

476

10 Maximum Principle on Unbounded Domains

u≥

sup F ∩Bd (0,R)

sup

u,

F \Bd (0,R)

for a certain R > 0, so that, for a suitable x0 in the closure if F ∩ Bd (0, R), u(x0 ) =

sup

u = sup u

F ∩Bd (0,R)

F

(by (10.4b)) = sup u, G

thus contradicting (10.5). From (10.7), we obtain lim sup u(x) = sup u x→∞, x∈F

F

(by (10.4b)) = sup u G

(by (10.6)) = lim sup u(x), x→∞, x∈G

and (10.4a) follows.

From Lemma 10.1.5 and the very definition of L-thin set, we have the following assertion. Corollary 10.1.6 (Characterization of L-thinness at infinity). Let F ⊆ G. Then F is L-thin at infinity iff there exists u ∈ S b (G) such that supF u < supG u. Vice versa, F is not L-thin at infinity iff supF u = supG u for all u ∈ S b (G). We are now ready to prove Theorem 10.1.3. Proof (of Theorem 10.1.3). Assuming that Ω is an MP set for L, we shall prove that G \ Ω is not L-thin at infinity. Let u ∈ S b (G). Put U := sup u.

(10.8)

G\Ω

Then lim sup u(x) ≤ u(y) ≤ U

∀ y ∈ ∂Ω,

Ωx→y

so that, since Ω is an MP set, u = U in Ω. Definition (10.8) now implies sup u = sup u,

G\Ω

G

hence, by Lemma 10.1.5, G \ Ω is not L-thin at infinity. Vice versa, assuming that G \ Ω is not L-thin at infinity, we shall prove that Ω is an MP set. Let u ∈ S b (Ω) be such that lim sup u(x) ≤ 0 ∀ y ∈ ∂Ω. x→y

10.2 q-sets and the Maximum Principle

477

By Lemma 10.1.4, the function v : G → [−∞, ∞[,

v=

max{u, 0} 0

in Ω, in G \ Ω

is L-subharmonic in G. Then, being G \ Ω not L-thin at infinity, sup v = sup v = 0. G

G\Ω

In particular, v ≤ 0 in Ω. Hence, u ≤ 0 in Ω, and Ω is an MP set.

From Theorem 10.1.3 and Corollary 10.1.6 we obtain the following equivalent characterization of MP sets. Corollary 10.1.7 (Characterization of MP set. II). An open set Ω ⊆ G is an MP set for L if and only if supG\Ω u = supG u for all u ∈ S b (G).

10.2 q-sets and the Maximum Principle This section is devoted to some geometrical conditions for non-thinness (w.r.t. L) at infinity. Our criteria will be based on the notion of q-set. Throughout the sequel, L will denote a fixed sub-Laplacian on G, Γ its fundamental solution and d any L-gauge for L. We adopt the usual notation Γ = βd d 2−Q (see Theorem 5.5.6, page 256). We recall that Q denotes the homogeneous dimension of G, and βd is a suitable positive constant. Definition 10.2.1 (q-set for a sub-Laplacian). Let q be a real positive number. A subset F of G will be called a q-set (w.r.t. L) if there exists a finite or countable family of d-balls {Bd (xj , rj )}j ∈J such that: (i) F ⊆ j ∈J Bd (xj , rj ), r (ii) j ∈J ( d(xjj ) )q < ∞. (See also Fig. 10.2.) The next theorem will play a crucial rôle in what follows. Theorem 10.2.2 (L-thinness at infinity and q-sets). Let F ⊆ G be L-thin at infinity. Then F is a q-set for every q > Q − 2. The proof of Theorem 10.2.2 rests on the following deep lemma. Lemma 10.2.3. Let u ∈ S b (G). Then, for every q > Q − 2, there exists a q-set F ⊆ G such that u = sup u. lim x→∞, x ∈F /

G

478

10 Maximum Principle on Unbounded Domains

Fig. 10.2. A q-set F

We prove this result in the Appendix of this chapter. Note that the assertion of the above lemma may fail to be true for q = Q − 2 (see Ex. 7 at the end of the chapter). We are now in the position to prove Theorem 10.2.2. Proof (of Theorem 10.2.2). Assume by contradiction that F is not a q0 -set, being q0 > Q − 2. Fixed u ∈ Sb (G), by Lemma 10.2.3, there exists a q0 -set F0 ⊆ G such that u = sup u. (10.9) lim x→∞, x ∈F / 0

G

On the other hand, F \ F0 is not a q0 -set, since F is not. In particular, F \ F0 is non-empty and unbounded. Then, by (10.9), lim

x→∞, x∈F \F0

u = sup u, G

so that lim sup u(x) = sup u = lim sup u(x). x→∞, x∈F

G

x→∞

This proves that F is not L-thin at infinity, in contradiction with the hypothesis.

For the future references, it is convenient to rephrase the previous theorem as follows. Theorem 10.2.4. Let F ⊆ G. Assume there exists q > Q − 2 such that F is not a q-set. Then F is not L-thin at infinity. Our more explicit geometric condition for non-L-thinness at infinity will follow from the next proposition. Proposition 10.2.5. Let F ⊆ G. Assume there exists a sequence of d-balls {Bd (zj , Rj )}j ∈N contained in F and such that

10.2 q-sets and the Maximum Principle

d(zj ) → ∞,

lim inf j →∞

479

Rj > 0. d(zj )

Then F is not a Q-set (hence, it is not L-thin at infinity). Proof. Let c be the constant in the pseudo-triangle inequality (see Proposition 5.1.7, page 231). Let us put M := 4c2 and choose a subsequence {zkj }j of {zj }j such that, for each j ∈ N, d(zkj +1 ) ≥ M 2 d(zkj ),

Rkj ≥ δ d(zkj ),

(10.10)

with a suitable δ ∈ ]0, 1/M[. Let us now define yj := zkj ,

ρj := δ d(yj ),

Bj := Bd (yj , ρj ),

F0 :=

Bj .

j ∈N

Since ρj ≤ Rkj , we have Bj ⊆ Bd (zkj , Rkj ) ⊆ F for every j ∈ N. Then, in order to show that F is not a Q-set, it is enough to prove that F0 is not a Q-set. We argue by contradiction and assume the existence of a family of d-balls {Bd (xk , rk )}k∈N covering F0 and such that Q ∞ rk < ∞. d(xk ) k=1

Then, if we fix ε ∈ ]0, 1/M[, there exists k ∈ N such that rk ≤ ε d(xk )

∀ k ≥ k.

Moreover, there exists j ∈ N such that the family {Dk }k≥k , is a covering of

Dk := Bd (xk , rk ),

Bj .

j ≥j

For every j ∈ N, j ≥ j , we finally define Kj := {k ∈ N | k ≥ k, Bj ∩ Dk = ∅}. We claim that 1 d(xk ) ≤ ≤M M d(yj )

∀ k ∈ Kj ,

∀ j ≥ j.

These inequalities, together with the first one in (10.10), imply

(10.11)

480

10 Maximum Principle on Unbounded Domains

Ki ∩ Kj = ∅ if i = j. By means of this last fact, we infer that rk Q rk Q ∞> ≥ d(xk ) d(xk ) j ≥j k∈Kj

k≥k

(by (10.11)) ≥

Q 1 1 Q (rk )Q M d(yj ) j ≥j

=

1 ωd M Q

j ≥j

k∈Kj

1 d(yj )

Q

meas(Dk )

k∈Kj

(since {Dk }k∈Kj is a covering of Bj ) 1 Q 1 meas(Bj ) ≥ ωd M Q d(yj ) =

j ≥j

Q ρj 1 Q = ∞, M d(yj ) j ≥j

since ρj = δ d(yj ) for every j ∈ N. This contradiction shows that F0 cannot be a Q-set. We are then left with the proof of (10.11). If k ∈ Kj , there exists x ∈ Bj ∩ Dk . Hence by the pseudo-triangle inequality,

d(xk ) ≤ c (d(yj ) + d(yj , xk )) ≤ c2 d(yj ) + d(yj , x ) + d(x , xk )

≤ c2 (d(yj ) + ρj + rk ) ≤ c2 (1 + δ) d(yj ) + ε d(xk ) 1 ≤ 2 c2 d(yj ) + d(xk ). 2 Then 1 d(xk ) ≤ 2 c2 d(yj ), 2 and the second inequality in (10.11) follows. With the same argument, one can prove the first one too.

Examples of subsets of G satisfying the hypothesis of Proposition 10.2.5 are the G-cones. The relevant definition is the following one. G-cone). A subset C of G will be said a G-cone with vertex at the Definition 10.2.6 (G origin if (see also Fig. 10.3) δλ (x) ∈ C

∀ x ∈ C,

∀ λ > 0.

If C is such a G-cone, we shall call z0 ◦ C = {z0 ◦ x | x ∈ C } a G-cone with vertex at z0 .

10.2 q-sets and the Maximum Principle

481

Fig. 10.3. G-cones

Proposition 10.2.7. Every open G-cone is not a Q-set. Proof. It is enough to prove the assertion for an open G-cone C with vertex at the origin. Let x0 ∈ C \ {0} and R0 > 0 be such that Bd (x0 , R0 ) ⊆ C. Then, since C is a G-cone, Bd (δλ (x0 ), λ R0 ) ⊆ C ∀ λ > 0. The assertion follows from Proposition 10.2.5, since zλ := δλ (x0 ) −→ ∞ as λ → ∞ and

λ R0 R0 = d(zλ ) d(x0 )

∀ λ > 0.

This completes the proof.

Corollary 10.2.8. Every half-space of G is not a Q-set. Proof. Let

Π := x = (x1 , . . . , xN ) ∈ G :

N

aj xj > α

j =1

be an open half-space of G (here, a1 , . . . , aN , α ∈ R), and let us consider

C0 := x = (x1 , . . . , xN ) ∈ Π : aj xj ≥ 0 ∀ j = 1, . . . , N . C0 is a non-empty open subset of Π with the following property: x ∈ C0 , λ ≥ 1

⇒

δλ (x) ∈ C0 .

Then, if we choose a point x0 ∈ C0 and a real number R0 > 0 such that Bd (x0 , R0 ) ⊆ C0 , we have Bd (δλ (x0 ), λ R0 ) ⊆ C0 for every λ ≥ 1. Since

482

10 Maximum Principle on Unbounded Domains

zλ := δλ (x0 ) −→ ∞ as λ → ∞ and

λ R0 R0 = ∀ λ ≥ 1, d(zλ ) d(x0 ) from Proposition 10.2.5 it follows that C0 , and hence Π, is not a Q-set.

10.3 The Maximum Principle on Unbounded Domains We collect Theorems 10.1.3, 10.2.2 and 10.2.4, Proposition 10.2.5 and Corollaries 10.2.7 and 10.2.8 in the following theorem. Theorem 10.3.1. The open set Ω ⊆ G is an MP set for L if one of the following (sufficient) conditions is satisfied: (1) G \ Ω is not L-thin at infinity (this condition is also necessary), (2) G \ Ω is not a q-set for some q > Q − 2, (3) G \ Ω contains a sequence of d-balls {Bd (zj , Rj )}j ∈N such that d(zj ) → ∞,

lim inf j →∞

Rj > 0, d(zj )

(4) G \ Ω contains an open G-cone, (5) G \ Ω contains a half-space or, equivalently, Ω is contained in a half-space. We close this section by giving an application of the previous theorem. Corollary 10.3.2 (A maximum principle on unbounded domains). Let Ω be an open subset of G satisfying one of the five conditions of Theorem 10.3.1. Let c : Ω → R, c ≤ 0, and let u be a bounded from above function on Ω of class C 2 satisfying Lu + c u ≥ 0 in Ω, (10.12) lim supx→y u(x) ≤ 0 for every y ∈ ∂Ω. Then u ≤ 0 in Ω. Proof. Define v := max{u, 0} and Ω0 := {x ∈ Ω | u(x) > 0}. We have to show that Ω0 = ∅. Assume, by contradiction, Ω0 = ∅. Then, since c ≤ 0 and v = u in Ω0 , the function v is of class C 2 in Ω0 and Lv ≥ 0 in Ω0 . It follows that v is L-subharmonic, hence L-submean, in Ω0 . On the other hand, for every x ∈ Ω \ Ω0 and r > 0 such that Bd (x, r) ⊂ Ω, v(x) = 0 ≤ Mr (v)(x). This shows that v is locally L-submean in Ω. Thus v ∈ S b (Ω). From the boundary condition in (10.12) we also get lim supx→y v(x) ≤ 0 for every y ∈ ∂Ω. Since Ω is an MP set (by Theorem 10.3.1), this implies v ≤ 0 in Ω, i.e. u ≤ 0 in Ω. Then

Ω0 = ∅, and the proof is complete.

10.4 The Proof of Lemma 10.2.3

483

10.4 Appendix to Chapter 10. The Proof of Lemma 10.2.3 The proof of Lemma 10.2.3 rests on the following deep lemma, an extension to the sub-Laplacian setting of a theorem by Cartan [Cart28] related to the classical Laplace operator (see also [HK76, pp. 131–134]). In the proof of the lemma below, c denotes, as usual, the constant appearing in the pseudo-triangle inequality for the L-gauge d (and Γ = βd d is the fundamental solution for L). Lemma 10.4.1 (Cartan-type covering lemma). Let μ be a Radon measure on G with finite total mass μ0 := μ(G) < ∞. Let q > Q − 2 (Q being the homogeneous dimension of G). Then, if h > 0, the set {x ∈ G | (Γ ∗ μ)(x) ≥ h} can be covered by a finite or countable family of closed d-balls Bd (xn , rn ) satisfying the following condition rn q < A (μ0 / h)q/(Q−2) . n

The constant A depends only on Q, q and c. Proof. We fix ν ∈ N and set rν := (μ0 / h)1/(Q−2) 2

2ν − q+Q−2

.

Let Dν := {D k,ν = Bd (xk,ν , rν /2) | k = 1, . . . , kν } be a maximal family of disjoint closed balls such that μ(D k,ν ) ≥

μ0 2ν

∀ k = 1, . . . , kν .

Since μ0 < ∞ and the balls are disjoint, then kν ≤ 2ν . Define F := Bd (xk,ν , c rν ). ν∈N k≤kν

We observe that if x ∈ / F then Bd (x, rν /2) does not intersect any ball of the maximal family Dν . Hence μ0 2ν

∀ ν ∈ N.

(10.13)

μ({x}) = lim μ(Bd (x, rν /2)) = 0.

(10.14)

μ(Bd (x, rν /2)) < In particular, we have ν→∞

484

10 Maximum Principle on Unbounded Domains

Hence, if x ∈ / F , by (10.14) we have (Γ ∗ μ)(x) = Γ (x −1 ◦ y) dμ(y) G\{x} ∞ + = r r d(x −1 ◦y)≥

1 2

≤ βd (r1 /2)

2−Q

ν=1 ∞

μ0 +

ν+1 ≤d(x −1 ◦y)< rν 2 2

≤ (by (10.13))

βd 2

2−Q

βd (rν+1 /2)

ν=1 Q−2

Γ (x −1 ◦ y) dμ(y)

μ0

∞

rν+1 rν −1 2 ≤d(x ◦y)< 2

dμ(y)

21−ν rν 2−Q

ν=1 ∞

= βd 2Q−1 h

2

Q−2−q Q−2+q

ν

=: A1 h,

ν=1

where A1 depends only on Q and q. Then {x ∈ G | (Γ ∗ μ)(x) > A1 h} ⊆

Bd (xk,ν , c rν ).

ν∈N k≤kν

Moreover, since kν ≤ 2ν , we have

(c rν ) ≤ A2 q

ν∈N k≤kν

μ0 h

q/(Q−2) ,

where A2 depends only on q, Q and c. This ends the proof.

We are now ready to prove Lemma 10.2.3. Proof (of Lemma 10.2.3). Let ν ∈ N. Define Cν := {x ∈ G | (2 c)ν < d(x) ≤ (2c)ν+1 }, where c denotes the constant appearing in the pseudo-triangle inequality for d (see Proposition 5.1.7, page 231). By Theorem 9.6.1 (page 451), if μ is the L-Riesz measure of u and U := supG u, we have U − u(x) = (Γ ∗ μ)(x) = I1 (x) + I2 (x) + I3 (x), where

I1 (x) :=

d(y)≤(2 c)ν−1

βd d(x −1 ◦ y)2−Q dμ(y) ;

I2 (x) :=

(2 c)ν−1

I3 (x) :=

d(y)≥(2 c)ν+2

βd d(x −1 ◦ y)2−Q dμ(y) ;

βd d(x −1 ◦ y)2−Q dμ(y).

10.4 The Proof of Lemma 10.2.3

485

From now on, we shall denote by C, C , . . . positive constants depending only on Q, c and βd . We also let n(t) := μ(Bd (0, t)). If x ∈ Cν , we have

I1 (x) ≤ βd (2 c)

(ν−1)(2−Q) d(y)≤(2 c)ν−1

2−Q

dμ(y) ≤ C · (2 c)ν n (2 c)ν

≤ (because n(t) is increasing) C

∞

(2 c)ν

n(t) dt. t Q−1

Analogously, ∞ d(y) 2−Q dμ(y) ≤ C t 2−Q dn(t) 2c d(y)≥(2 c)ν+2 (2 c)ν ∞ ∞ ∞ n(t) n(t) n(t) +C dt ≤ C dt. = C Q−2 Q−1 Q−1 ν ν t t t (2 c) (2 c) (2 c)ν

I3 (x) ≤ βd

The estimate of I2 (x), x ∈ Cν , is the crucial step of the proof. Let q > Q − 2 be fixed. Define

μν := μ {y ∈ G | (2 c)ν−1 < d(y) < (2 c)ν+2 } , ην := μν (2 c)(2−Q) ν , εν := ην1−(Q−2)/q . Then

∞

ην ≤ C

ν=1

∞

(2 c)ν+2

ν−1 ν=1 (2 c)

dn(t) ≤ C t Q−2

∞

1

dn(t) < ∞. t Q−2

(10.15)

This last inequality follows from (9.29). On the other hand, by Lemma 10.4.1, there exists a countable family of closed balls Bd (xk,ν , rk,ν ) k∈J such that ν

{x ∈ Cν | I2 (x) < εν } ⊇ Cν \

Bd (xk,ν , rk,ν )

k∈Jν

and

k∈Jν

q/(Q−2) μν (rk,ν ) < A . εν q

(10.16)

As a consequence, q/(Q−2) μν = A ην (2c)q ν , (rk,ν )q ≤ A εν so that rk,ν ≤ (A ην )1/q (2c)ν .

(10.17)

486

10 Maximum Principle on Unbounded Domains

We may also suppose Bd (xk,ν , rk,ν ) ∩ Cν = ∅ for every k ∈ Jν . This implies d(xk,ν ) ≥ (2 c)ν−2 . Indeed, 1 d(xk,ν ) ≥ (see (10.17)) (2c)ν − (A ην )1/q (2c)ν c 1 = (2c)ν − (A ην )1/q , c and, since ην → 0 as ν → ∞ (see (10.15)), the claim follows. As a consequence, from (10.15), (10.16), the choice of εν and the bound for d(xk,ν ) we get q ∞ ∞ ∞ rk,ν (2 c)ν q ≤A η = C ην < ∞. ν d(xk,ν ) (2 c)(ν−2)q ν=1 k∈Jν

ν=1

(10.18)

ν=1

Collecting the estimates for I1 , I2 and I3 , we finally obtain ∞ n(t) U − u(x) < C dt + ην1−(Q−2)/q Q−1 ν t (2 c)

(10.19)

for every x ∈ Cν \ k∈Jν Bd (xk,ν , rk,ν ) and for every ν ∈ N. By (9.29) and the positivity of the exponent of ην , the right-hand side of (10.19) tends to zero, as ν → ∞. Together with (10.18), this proves that the set F =

∞

Bd (xk,ν , rk,ν )

ν=1 k∈Jν

is a q-set satisfying lim

x→∞, x ∈F /

This ends the proof.

u = sup u. G

Bibliographical Notes. In the recent literature, a special attention has been paid to the maximum principle on unbounded domains. Such a principle plays a key rôle in looking for symmetry properties of solutions to semilinear Poisson equations (see [BN91,BCN97,BHM00]). In order to provide our analogue in the Carnot group setting, we have been inspired by some ideas of J. Deny published in 1947 (see Deny’s theorem [Den48, p. 142]; see also [HK76, Theorem 3.21]). We would like to stress that maximum principles in half-spaces are crucial tools in looking for monotonicity and symmetry properties of solutions to the semilinear equation Lu+f (u) = 0 in G. See [BHM00] and [BP99] for some recent noteworthy results in the cases L = Δ and L = ΔHN , respectively. See also [BL02,BL03,BP01, BP02]. Furthermore, we would like to note that Theorem 10.2.2 can be used as a starting point in studying the asymptotic behavior of L-subharmonic functions not bounded from above in G. Some of the topics presented in this chapter also appear in [BL02].

10.5 Exercises of Chapter 10

487

10.5 Exercises of Chapter 10 1) Suppose that Ω is a bounded open set. Let u : Ω → [−∞, ∞[ be u.s.c. Suppose that ∀ y ∈ ∂Ω. lim sup u(x) ≤ 0 x→y

2) 3) 4) 5) 6)

Then u is bounded from above. (Hint: Let B ⊆ Ω be an open set such that ∂Ω ⊂ B and u < 1 on B. Then write Ω = A ∪ B and observe that u is bounded from above on A which is a compact subset of Ω. Indeed, any u.s.c. function restricted to a compact set attains its maximum!) Does there exist any u ∈ S b (Ω), non-constant and bounded from below? In case, provide examples. (Hint: See also Theorem 9.3.10.) Let F0 be a q-set. Then, F \ F0 is not a q-set, whenever F is not. Complete the proof of (10.11) arguing as in the last paragraph of Proposition 10.2.5. Prove that every subset of a q-set is a q-set and every bounded set is a q-set (for every q > 0). Prove also that a finite union of q-sets is a q-set. Consider the subset of R3 S = {(x1 , x2 , x3 ) ∈ R3 : x1 ≤ −1, x2 = 0, x3 = 0}. Prove that S is not a q-set when q = 1 and the relevant d in Definition 10.2.1 is the Euclidean norm. (Hint: Show that if {Bj = B(cj , rj )}j ∈N is a sequence of ordinary balls in R3 , then there exists a constant M > 0 such that rj d x ≤M , (10.20) |cj | H x j ∈N

where

H = π1 S ∩

Bj

j

and π1 : R3 → R,

π1 (x1 , x2 , x3 ) = x1 .

Thus, if S were a 1-set and {Bj }j the relevant covering of S as in Definition 10.2.1, we would have H = ]−∞, −1], so that the left-hand side of (10.20) would diverge, whereas the right-hand side is finite by the definition of q-set when q = 1.)

488

10 Maximum Principle on Unbounded Domains

7) In this exercise, we show that the assertion of Lemma 10.2.3 may fail to be true for q = Q − 2. For instance, let G = (R3 , +) be the usual Euclidean group on R3 , and let L = Δ be the usual Laplace operator. Consider the function u(x1 , x2 , x3 ) = −

−1

−∞

((x1 − y)2 + x22 + x32 )−1/2 dy. 1−y

Show that: a) The function u coincides with the convolution − Γ (y −1 ◦ x) dμ(y), G

where μ is the measure on R3 supported on the set S of Ex. 6, and coinciding there with χ(−∞,−1)(y) dy. 1−y Here, dy denotes the usual Hausdorff 1-dimensional measure on R1 . b) u ∈ S(R3 ) satisfies all the conditions in Theorem 9.6.1 (page 451). Indeed, with the notation in that theorem, we have 0 if t ∈ ]0, 1] n(t) = μ(Bd (0, t)) = −1 1 −t 1−y dy = log(1 + t) − log 2 if t > 1. Thus

∞

1

n(t) dt < ∞. t2

c) Deduce that μ is the L-Riesz measure of u in R3 , u(0) > −∞ and the least upper bound of u is 0. Moreover, u = −∞ on S and is finite elsewhere. Note that, if Lemma 10.2.3 were true for q = Q − 2 = 1, there would exist a 1-set F ⊆ R3 such that lim

|x|→∞, x ∈F /

u = sup u = 0. R3

Hence, F would definitely cover S, so that S would be a 1-set. But this is false, as shown by Exercise 6.

11 L-capacity, L-polar Sets and Applications

The aim of this chapter is to provide an ad hoc theory of capacity and of polar sets for a sub-Laplacian L on a Carnot group G. The structure of the fundamental solution for L has a prominent rôle in the whole chapter. Our starting points are a continuity principle for L-potentials and a suitable version of the Maria–Frostman domination principle. Starting from these results, we develop a theory of capacity following two classical approaches: one based on the notion of L-energy, see Section 11.4, and one based on the balayage, see Section 11.5. We compare these approaches to each other, showing that they lead to the same concept of capacity. As an application of the above results we prove a Poisson–Jensen formula and the so-called “fundamental convergence theorem”. Notation. Throughout the chapter, we shall use the following notation and definitions. M denotes the set of Radon measures μ on G, i.e. of the Borel measures on G which are finite on compact sets. For μ ∈ M, we denote by supp(μ) the support of μ, i.e. the complement of the largest open set with μ-measure zero. We denote by M0 the subset of M of compactly supported Radon measures. Moreover, if E is any set, we denote by M(E) (respectively, M0 (E)) the set of measures μ ∈ M (respectively, μ ∈ M0 ) such that supp(μ) ⊆ E. If μ ∈ M and A is any set, by μ|A we mean the measure in M defined by μ|A (E) = μ(E ∩ A) for every Borel set E. + Moreover, we write A B whenever A is a compact subset of B. Finally, S (Ω) denotes the set of the non-negative L-superharmonic functions in Ω.

11.1 The Continuity Principle for L-potentials Theorem 11.1.1 (The continuity principle for L-potentials). Let E G be a compact set, and let μ be a Radon measure in G supported in E. Let x0 ∈ E, and let the function (Γ ∗ μ)|E be finite and continuous at x0 (as a function in E). Then Γ ∗ μ is also continuous at x0 (as a function in the whole G).

490

11 L-capacity, L-polar Sets and Applications

Proof. We can suppose that x0 ∈ ∂E, since otherwise there is nothing to prove. Let us set for brevity V0 := (Γ ∗ μ)(x0 ). We first observe that μ({x0 }) = 0 since +∞ > V0 ≥ Γ (y −1 ◦ x0 ) dμ(y) = Γ (0) · μ({x0 }) = ∞ · μ({x0 }). {x0 }

If x0 is an isolated point of E, then there exists an open neighborhood O of x0 such that O ∩ E = {x0 }. Since μ({x0 }) = 0 and supp μ ⊆ E, we get μ(O) = 0. Then, by Theorem 9.3.5, page 433 (see also Corollary 9.3.3), Γ ∗ μ is L-harmonic in O and hence continuous. We can then suppose that x0 ∈ ∂E is a limit point of E. By hypothesis, we know that limx→x0 (Γ ∗ μ)|E (x) = V0 , i.e. for fixed ε > 0, there exists ρ0 > 0 such that |(Γ ∗ μ)(ξ ) − V0 | < ε

∀ ξ ∈ Eρ0 .

(11.1)

Here and in the following we denote Er := E ∩ Bd (x0 , r). We have to prove that lim (Γ ∗ μ)|(G\E) (x) = V0 .

Since μ({x0 }) = 0 and

x→x0

G Γ (y

−1

(11.2)

◦ x0 ) dμ(y) = V0 ∈ R, we have

Γ (y −1 ◦ x0 ) dμ(y) → 0 as ρ → 0+ .

Eρ

Hence there exists ρ1 > 0 (ρ1 ≤ ρ0 ) such that Γ (y −1 ◦ x0 ) dμ(y) < ε

∀ ρ ≤ ρ1 .

(11.3)

Eρ

We now set μ1 = μ|E\Eρ1 , and we consider Γ ∗ μ1 . Such a function is L-harmonic outside the support of μ1 . In particular, it is continuous in a neighborhood of x0 . Therefore, the function Γ (y −1 ◦ ·) dμ(y) = Γ ∗ μ − Γ ∗ μ1 Eρ1

is continuous at x0 if restricted to E. In particular, recalling (11.3), there exists ρ2 > 0 (ρ2 < ρ1 ) such that −1 Γ (y ◦ ξ ) dμ(y) ≤ Γ (y −1 ◦ x0 ) dμ(y) + ε < 2ε ∀ ξ ∈ Eρ2 . (11.4) Eρ1

Eρ1

By the continuity of Γ ∗ μ1 at x0 , there also exists ρ3 > 0 (ρ3 ≤ ρ2 ) such that |(Γ ∗ μ1 )(η) − (Γ ∗ μ1 )(x0 )| < ε

∀ η ∈ B d (x0 , ρ3 ).

(11.5)

11.2 L-polar Sets

491

We can now prove (11.2). Let x ∈ G \ E be such that x ∈ Bd (x0 , (2c)−1 ρ3 ), where c≥1 is the constant in the pseudo-triangle inequality for d (see Proposition 5.1.8, page 231). Let x ∈ Eρ1 be a point that minimizes the d-distance between x and Eρ1 . We have |(Γ ∗ μ)(x) − V0 | ≤ |(Γ ∗ μ)(x) − (Γ ∗ μ)(x)| + |(Γ ∗ μ)(x) − V0 | ≤ |(Γ ∗ μ1 )(x) − (Γ ∗ μ1 )(x0 )| + |(Γ ∗ μ1 )(x0 ) − (Γ ∗ μ1 )(x)| −1 −1 Γ (y ◦ x) dμ(y) + Γ (y ◦ x) dμ(y) + |(Γ ∗ μ)(x) − V0 |. + Eρ1

Eρ1

Let us estimate the far right-hand side. The first two summands are smaller than ε by means of (11.5). Indeed, x ∈ Bd (x0 , (2c)−1 ρ3 ) ⊂ B d (x0 , ρ3 ) and x ∈ Bd (x0 , ρ3 ) since, recalling the choice of x, d(x, x0 ) ≤ c(d(x, x) + d(x, x0 )) ≤ 2c d(x, x0 ) < ρ3 . The third summand is smaller than 2ε by means of (11.4) (x ∈ Eρ2 , as we have just seen that d(x, x0 ) < ρ3 ≤ ρ2 ). Let us estimate the fourth summand. Again by the choice of x, for every y ∈ Eρ1 we have d(x, y) ≤ c(d(x, x) + d(x, y)) ≤ 2c d(x, y). This immediately gives Γ (y −1 ◦ x) = d 2−Q (x, y) ≤ (2c)Q−2 d 2−Q (x, y) = (2c)Q−2 Γ (y −1 ◦ x), and we thus derive −1 Q−2 Γ (y ◦ x) dμ(y) ≤ (2c) Eρ1

Γ (y −1 ◦ x) dμ(y) < (2c)Q−2 2ε

Eρ1

(here we have used the estimate of the third summand). Finally, the fifth summand is smaller than ε by means of (11.1). This proves (11.2) and completes the proof of the theorem.

11.2 L-polar Sets Definition 11.2.1 (L-polar set). A set E ⊆ G is called L-polar if there exist an open set Ω ⊇ E and a function u ∈ S(Ω) such that u ≡ ∞ in E. Exercise 11.2.2. (i) Any singleton {x0 } is L-polar. (ii) If E1 ⊆ E2 and E2 is L-polar, then also E1 is L-polar. of Gδ type (i.e. countable (iii) If E is L-polar, then E is contained in an L-polar set E intersection of open sets). (iv) If E is L-polar, then E has Lebesgue measure zero. (v) Any countable set is L-polar.

492

11 L-capacity, L-polar Sets and Applications

Theorem 11.2.3 (L-polar sets and L-potentials. I). Let E ⊆ G be an L-polar set, and let y ∈ / E. Then there exists μ ∈ M such that Γ ∗ μ ≡ ∞ in E and (Γ ∗ μ)(y) < ∞. In particular,1 Γ ∗ μ ∈ S(G). Proof. By the definition of L-polar set, there exists u ∈ S(Ω) (for some open set Ω ⊇ E) such that u ≡ ∞ in E. Moreover, we can suppose that u > 0 in Ω (recall that the lower semicontinuity of u implies that {x ∈ Ω | u(x) > 0} is open). Let Dk be a sequence of open sets such that Dk ⊆ D k ⊆ Dk+1 , Dk = Ω \ {y}. k∈N

By the Riesz representation Theorem 9.4.4, page 442, there exists hk ∈ H(Dk ) such that u(x) = hk (x) + Γ (y −1 ◦ x) dμu (y) ∀ x ∈ Dk . (11.6) Dk

Let us set, for brevity, vk (x) := Dk Γ (y −1 ◦ x) dμu (y). We remark that vk ∈ S(G), by means of Corollary 9.3.3, page 433. Since vk is L-harmonic outside D k (see Theorem 9.3.5, page 433), we have, in particular, vk (y) ∈ R for every k ∈ N. We then set 2−k vk (x)/(vk (y) + 1) ∀ x ∈ G. v(x) :=

k∈N

2−k

Since v(y) ≤ k∈N < ∞, we have v ∈ S(G) (see Corollary 8.2.8, page 403). Moreover, v ≥ 0. Hence we can use Corollary 9.4.8, page 444, and obtain v = Γ ∗ μv + inf v. G

It is now easy to see that μ := μv satisfies the assertion of the theorem. Indeed, v(y) < ∞ immediately gives (Γ ∗ μ)(y) < ∞. Moreover, for every x0 ∈ E, there exists k0 ∈ N such that x0 ∈ Dk0 . Since u ≡ ∞ in E, (11.6) implies that vk0 (x0 ) = ∞. As a consequence, (Γ ∗ μ)(x0 ) = v(x0 ) − infG v = ∞. This ends the proof. Example 11.2.4. [There exist finite-valued discontinuous L-superharmonic functions.] Indeed, let E ⊂ G be a non-closed L-polar set (let, e.g. E = {xn }n∈N with / E), and let x0 be a limit point for E not belonging to E. By xn → 0 and 0 ∈ Theorem 11.2.3, there exists v ∈ S(G) such that v ≡ ∞ on E and v(x0 ) < ∞. Let now u := min{v, v(x0 ) + 1}. Then u clearly satisfies the requisites of the assertion. Corollary 11.2.5 (L-polar sets and L-potentials. II). Let E ⊆ G be a bounded L-polar set, and let y ∈ / E. Then there exists ν ∈ M0 such that Γ ∗ ν ≡ ∞ in E and (Γ ∗ ν)(y) < ∞. 1 By Theorem 9.3.2, page 432.

11.2 L-polar Sets

493

Proof. By Theorem 11.2.3, there exists μ ∈ M such that Γ ∗ μ ≡ ∞ in E and (Γ ∗ μ)(y) < ∞. We have E ∪ {y} D(0, R) =: B, for some R > 0. Setting ν := μ|B , we have ν ∈ M0 . Moreover, by the Riesz representation Theorem 9.4.4, page 442 (see also Theorem 9.3.5, page 433), there exists h ∈ H(B) such that Γ ∗μ=h+Γ ∗ν in B. It is now immediate to recognize that Γ ∗ ν has the required properties. This completes the proof. Corollary 11.2.6. Let {Ej }j ∈N be a sequence of L-polar sets. Then j ∈N Ej is Lpolar. Proof. Using Theorem 11.2.3 and observing that L-polar sets have Lebesgue measure zero (since L-superharmonic functions are L1loc ), we can find y ∈ G and uj ∈ S(G), uj ≥ 0, such that uj |Ej ≡ ∞ and uj (y) < ∞ for every j ∈ N. We then set 2−k uk (x) ∀ x ∈ G. u(x) := (uk (y) + 1)

k∈N

2−k

Since u(y) ≤ k∈N < ∞, we have u ∈ S(G) (see Corollary 8.2.8, page 403). Moreover, u ≡ ∞ in j ∈N Ej . We end the section with an improvement of the maximum principle for L. Theorem 11.2.7 (The extended maximum principle for L). Let Ω ⊂ G be a bounded domain, and let u ∈ S(Ω) be bounded from above. If lim sup u(x) ≤ M

for every ξ ∈ ∂Ω

x→ξ

except for an L-polar set E ⊂ ∂Ω, then u ≤ M on Ω. Proof. Let x0 ∈ Ω. By Theorem 11.2.3, there exists ω ∈ S(G), ω < 0 and finite in x0 , such that ω|E ≡ −∞. Then, for ε > 0, the function uε = u + ε ω is L-subharmonic in Ω and has lim sup less than M on ∂Ω. Therefore, by the weak maximum principle (Theorem 8.2.19, page 409), we get u(x0 ) + ε ω(x0 ) ≤ M. Letting ε → 0+ and recalling that ω(x0 ) > −∞, we finally get u(x0 ) ≤ M. This ends the proof.

494

11 L-capacity, L-polar Sets and Applications

11.3 The Maria–Frostman Domination Principle Lemma 11.3.1. Let Ω ⊆ G be an open set, and let f, g ∈ S(Ω). If f ≤ g a.e. in Ω, then f ≤ g in Ω. In particular, if f = g a.e., then f ≡ g. Proof. By Theorem 8.2.11, page 405, we know that f (x) = limr→0+ Mr (f )(x) and g(x) = limr→0+ Mr (g)(x) for every x ∈ Ω. On the other hand, f ≤ g a.e. implies Mr (f ) ≤ Mr (g). Letting r → 0+, we get the assertion. The following result is a particular case of Exercise 24, Chapter 6. We provide the proof for the sake of completeness. Lemma 11.3.2. Let Ω2 ⊆ Ω1 ⊆ G be open sets, and let s ∈ S(Ω1 ), u ∈ S(Ω2 ) be such that lim supΩ2 x→y u(x) ≤ s(y) for every y ∈ (∂Ω2 ) ∩ Ω1 . Then

max{s(x), u(x)}, x ∈ Ω2 , v : Ω1 −→ [−∞, ∞[, v(x) := s(x), x ∈ Ω1 \ Ω 2 , is L-subharmonic in Ω1 . (See Fig. 11.1.)

Fig. 11.1. Figure of Lemma 11.3.2

Proof. Clearly we have v ∈ S(Ω1 \ Ω 2 ) and v ∈ S(Ω2 ) (recall Proposition 6.5.4, page 355). Let y ∈ (∂Ω2 ) ∩ Ω1 , and let us prove that lim supx→y v(x) ≤ v(y). Since s is u.s.c. in Ω1 , we have lim sup v(x) =

Ω1 \Ω2 x→y

lim sup s(x) ≤ s(y) = v(y),

Ω1 \Ω2 x→y

lim sup v(x) = lim sup (max{s(x), u(x)})

Ω2 x→y

Ω2 x→y

≤ max lim sup s(x), lim sup u(x) ≤ s(y) = v(y). Ω2 x→y

Ω2 x→y

This proves that v is u.s.c. in Ω1 . Moreover, v is finite in a dense subset of Ω1 , since s ≤ v has the same property. By Theorem 8.2.1, page 401, we only have to prove that

11.3 The Maria–Frostman Domination Principle

495

v is sub-mean. Let y ∈ (∂Ω2 ) ∩ Ω1 , B d (y, r) ⊂ Ω1 . Since v ≥ s and s ∈ S(Ω1 ), we have Mr (v)(y) ≥ Mr (s)(y) ≥ s(y) = v(y). Observing that v ∈ S(Ω1 \Ω 2 ) and v ∈ S(Ω2 ) imply that v is sub-mean in Ω1 \∂Ω2 , this completes the proof. Lemma 11.3.3 (Lusin-type theorem for potentials). Let μ ∈ M0 be such that Γ ∗ μ < ∞ in K = supp μ G. Then, for every ε > 0, there exists a compact set C ⊆ K such that: (i) μ(K \ C) < ε, (ii) Γ ∗ (μ|C ) ∈ C(G, R). Proof. We first observe that Γ ∗ μ < ∞ in G, since Γ ∗ μ < ∞ in K by hypothesis and Γ ∗ μ ∈ H(G \ supp μ) by Theorem 9.3.5, page 433. Moreover, Γ ∗ μ is l.s.c. Hence Γ ∗ μ is measurable, and we can apply the Lusin theorem (see, e.g. [Rud87]): for every ε > 0, there exists a compact set C ⊆ K such that μ(K \ C) < ε and (Γ ∗ μ)|C is continuous in C. Since Γ ∗ μ < ∞ in G, we can write Γ ∗ (μ|C ) = Γ ∗ μ − Γ ∗ (μ|K\C ). Since Γ ∗(μ|C ) is l.s.c., −Γ ∗(μ|K\C ) is u.s.c. and (Γ ∗μ)|C is continuous in C, we obtain that (Γ ∗(μ|C ))|C is continuous in C. We can now use the continuity principle for potentials (Theorem 11.1.1, page 489) and get that Γ ∗ (μ|C ) is continuous in C. On the other hand, Γ ∗ (μ|C ) is L-harmonic and hence continuous in G \ supp (μ|C ). Therefore, Γ ∗ (μ|C ) is continuous in G. This ends the proof. Theorem 11.3.4 (The Maria–Frostman domination principle). Let μ ∈ M be + such that Γ ∗ μ is finite in G. Let P be an L-polar set. If u ∈ S (G) and u ≥ Γ ∗ μ in (supp μ) \ P , then u ≥ Γ ∗ μ in G. Proof. Since P is L-polar, by Theorem 11.2.3 we can find ν ∈ M such that ω := Γ ∗ ν ∈ S(G),

ω ≡ ∞ in P .

Let ε > 0. Clearly, we have u + ε ω ≥ Γ ∗ μ in suppμ.

(11.7)

Let Kn G be such that Kn ⊆ Kn+1 and

n∈N Kn

= G.

Let us set μn := μ|Kn . Since we obviously have Γ ∗ μn ≤ Γ ∗ μ < ∞ in G, we can apply Lemma 11.3.3 and find

496

11 L-capacity, L-polar Sets and Applications

Ln,ε supp μn = Kn ∩ supp μ such that μn (Kn \Ln,ε ) < ε and vn,ε := Γ ∗(μn |Ln,ε ) ∈ C(G). We explicitly remark that in G. (11.8) vn,ε ≤ Γ ∗ μn ≤ Γ ∗ μ < ∞ We now set

s(x) :=

max{vn,ε (x) − u(x) − εω(x), 0}, 0,

x ∈ G \ Ln,ε , x ∈ Ln,ε .

Since vn,ε is L-harmonic outside Ln,ε (see Theorem 9.3.5, page 433), we have vn,ε − u − εω ∈ S(G \ Ln,ε ). Moreover, for every y ∈ ∂Ln,ε , we have lim sup (vn,ε (x) − u(x) − εω(x))

G\Ln,ε x→y

≤ (vn,ε ∈ C(G), −u, −εω ∈ S(G)) ≤ vn,ε (y) − u(y) − εω(y) ≤ 0, by recalling that vn,ε ≤ Γ ∗ μ ≤ u + εω

in supp(μ),

(11.9)

thanks to (11.8) and (11.7). We can then apply Lemma 11.3.2 with Ω1 = G and Ω2 = G \ Ln,ε and obtain that s ∈ S(G). Since 0 ≤ s ≤ vn,ε in G, s ∈ S(G), vn,ε ∈ S(G), by Theorem 6.6.1 (page 358) there exists h ∈ H(G) such that 0 ≤ s ≤ h ≤ vn,ε in G. We now use Theorem 9.3.7, page 434, and we obtain that h ≡ 0. This implies s ≡ 0 in G. By the definition of s, this means that vn,ε − u − εω ≤ 0 in G \ Ln,ε . Since also (11.9) holds, we obtain vn,ε ≤ u + ε ω

in G.

(11.10)

For every x ∈ G \ (Kn ∩ supp μ), we have Γ (y −1 ◦ x) dμn (y) (Kn ∩supp μ)\Ln,ε

≤ μn (Kn ∩ supp μ) \ Ln,ε · sup Γ (y −1 ◦ x) | y ∈ (Kn ∩ supp μ) \ Ln,ε ≤ ε sup Γ (y −1 ◦ x) | y ∈ (Kn ∩ supp μ) < ∞, and then, using (11.10), we obtain (Γ ∗ μn )(x) = vn,ε (x) +

(Kn ∩supp μ)\Ln,ε

Γ (y −1 ◦ x) dμn (y)

≤ u(x) + εω(x) +

(Kn ∩supp μ)\Ln,ε −1

≤ u(x) + εω(x) + ε sup Γ (y

Γ (y −1 ◦ x) dμn (y)

◦ x) | y ∈ (Kn ∩ supp μ) .

11.4 L-energy and L-equilibrium Potentials

497

Since ω ∈ S(G), we have ω < ∞ a.e. (see Corollary 8.2.4, page 402). Letting ε → 0, we then obtain Γ ∗ μn ≤ u

a.e. in G \ (Kn ∩ supp μ).

On the other hand, we know by hypothesis that Γ ∗ μn ≤ Γ ∗ μ ≤ u in supp μ \ P , hence a.e. in supp μ (P is L-polar and then has Lebesgue measure zero). Therefore, Γ ∗ μn ≤ u a.e. in G. Recalling that Γ ∗ μn , u ∈ S(G), by Lemma 11.3.1, we obtain Γ ∗ μn ≤ u in G. Since Γ ∗ μn → Γ ∗ μ pointwise in G, we finally get Γ ∗ μ ≤ u in G.

11.4 L-energy and L-equilibrium Potentials Definition 11.4.1 (L-energy and L-equilibrium value). Let E G be a compact set. For any Radon measure μ on G, supported in E, we set Γ (y −1 ◦ x) dμ(x) dμ(y). I (μ) := (Γ ∗ μ)(x) dμ(x) = G

EE

I (μ) is called the L-energy of μ. We then define the L-equilibrium value V (E) of E as V (E) := inf I (μ) | μ Radon measure, supp μ ⊆ E, μ(E) = 1 . The next theorem states that the above infimum is actually a minimum. Theorem 11.4.2 (L-equilibrium potential). Let E G be a compact set. Then there exists a Radon measure μ supported in E such that μ(E) = 1 and I (μ) = V (E) = min I (μ) | μ Radon measure, supp μ ⊆ E, μ(E) = 1 . Such a measure μ will be called an L-equilibrium distribution for E. The related potential Γ ∗ μ will be called an L-equilibrium potential for E. In Section 11.5, we shall prove that the L-equilibrium potential is unique if V (E) < ∞ (see Corollary 11.5.13). Proof. We can assume that V (E) < ∞, otherwise there is nothing to prove. Let us take a minimizing sequence {μn }n∈N of Radon measures such that supp μn ⊆ E, μn (E) = 1 and V (E) = limn→∞ I (μn ). Then2 there exists a subsequence {μnk }k∈N w∗

and a Radon measure μ, supp μ ⊆ E, such that μnk −→ μ as k → ∞, i.e. lim f dμnk = f dμ for every f ∈ C(E). k→∞ E

2 See, e.g. [Rud87].

E

498

11 L-capacity, L-polar Sets and Applications

In particular, taking f ≡ 1, we have μ(E) = 1. Then V (E) ≤ I (μ), by definition w∗

of V (E). On the other hand, μnk −→ μ also implies that3 w∗

μnk ⊗ μnk −→ μ ⊗ μ in E × E. Observing that (x, y) → Γ (y −1 ◦ x) is l.s.c. in E × E, we obtain −1 Γ (y ◦ x) dμnk (x) dμnk (y) V (E) = lim I (μnk ) = lim inf k→∞ k→∞ E E ≥ Γ (y −1 ◦ x) dμ(x) dμ(y) = I (μ). E E

We have here used a general property of weakly convergent measures.4 Therefore V (E) = I (μ) and the proof is complete. The main goal of this section is to prove Theorem 11.4.5 below as a consequence of the Maria–Frostman domination principle proved in Section 11.3. To this end we need the following definition. Definition 11.4.3 (L-polar∗ set). We shall say that a set E ⊂ G is L-polar∗ if there exists a countable family of compact sets Fn such that V (Fn ) = ∞ and E ⊆ n∈N Fn . In Section 11.5 (Corollary 11.5.12) we shall prove that any L-polar∗ set is an L-polar set according to our previous Definition 11.2.1, page 491. Lemma 11.4.4. Let E G be a compact set and let ν be a Radon measure supported in E such that ν(E) > 0 and I (ν) < ∞. Then, for any L-polar∗ set E1 , we have ν(E1 ) = 0. Proof. Let us prove the lemma in the case that E1 is a compact set such that V (E1 ) = ∞. The general case will follow immediately from the definition of Lμ := ν|E1 /ν(E1 ). Obvipolar∗ set. If by contradiction ν(E1 ) > 0, we can consider μ(E1 ) = 1. Moreover, ously, supp μ ⊆ E1 and −2 I ( μ) = ν (E1 ) Γ (y −1 ◦ x) dν(x) dν(y) ≤ ν −2 (E1 )I (ν) < ∞. E1 E1

This contradicts V (E1 ) = ∞ and completes the proof.

Theorem 11.4.5 (The fundamental theorem on L-equilibrium potentials). Let E G be a compact set such that V (E) < ∞. Let μ be an L-equilibrium distribution and UE = Γ ∗ μ an L-equilibrium potential for E. Then we have UE (x) ≤ V (E) for all x ∈ G, (11.11) UE (x) = V (E) for all x ∈ E \ P for a suitable L-polar∗ set P . 3 Ibidem. 4 Ibidem.

11.4 L-energy and L-equilibrium Potentials

499

Proof. We first observe that we only need to prove UE ≤ V (E) UE ≥ V (E)

in supp(μ), in E \ P for a suitable L-polar∗ set P .

(11.12) (11.13)

Then the assertion of the theorem will follow from the Maria–Frostman domination principle (Theorem 11.3.4), observing that UE = Γ ∗ μ is L-harmonic outside the support of μ and then it is finite in G if (11.12) also holds. We now want to prove (11.13) by showing that P := {x ∈ E | UE (x) < V (E)} is L-polar∗ . We have P = n∈N An , where An := {x ∈ E | UE (x) ≤ V (E) − 1/n} are compact sets, since UE is l.s.c. We have to show that V (An ) = ∞. Let us assume by contradiction that V (An0 ) < ∞ for some n0 ∈ N, i.e. that there exists a Radon measure σ such that supp σ ⊆ An0 , σ (An0 ) = 1 and I (σ ) < ∞. We set (ε) (ε) ε := (2n0 )−1 , and we observe that there exists a0 ∈ supp μ such that UE (a0 ) > V (E) − ε. Indeed, UE ≤ V (E) − ε in supp μ would give the contradiction V (E) = UE dμ ≤ (V (E) − ε) μ(supp μ) = V (E) − ε. Recalling again that UE is l.s.c., we can find a radius r (ε) > 0 such that (ε)

UE > V (E) − ε

in Bε := Bd (a0 , r (ε) ) (ε)

and Bε is at a positive distance from An0 . Moreover, a0 m(ε) := μ(Bε ) > 0. We also have

∈ supp μ implies that

m(ε) ≤ μ(supp μ) = 1. (ε)

We now set σ (ε) := m(ε) σ , and we define a (signed) measure σ1 in E by σ1(ε) (ε)

⎧ (ε) ⎨σ = −μ ⎩ 0

in An0 , in Bε , elsewhere.

(ε)

Then, for 0 < η < 1, μ1 := μ + η σ1 is a (positive) Radon measure supported in E and (ε)

μ1 (E) = μ(E) + η (σ (ε) (An0 ) − μ(Bε )) = 1. Moreover, it is easy to see that the integral

500

11 L-capacity, L-polar Sets and Applications (ε) I(σ1 ) :=

E E

Γ (y −1 ◦ x) dσ1 (x) dσ1 (y) (ε)

(ε)

is convergent, since I (σ ) < ∞, I (μ) < ∞ and Bε is at a positive distance from An0 . Now, recalling that Γ (x −1 ◦ y) = Γ (y −1 ◦ x), we have (ε) (ε) Γ (y −1 ◦ x) dμ(x) η dσ1 (y) I (μ1 ) − I (μ) = E E (ε) (ε) + Γ (y −1 ◦ x)η dσ1 (x) dμ(y) + η2 I(σ1 ) E E (ε) (ε) = 2η Γ (y −1 ◦ x) dμ(y) dσ1 (x) + η2 I(σ1 ) E E (ε) (ε) = 2η UE dσ1 + η2 I(σ1 ) E (ε) (ε) UE dσ − UE dμ + η2 I(σ1 ) = 2η An0

Bε

≤ 2η{(V (E) − 2ε)σ

(ε)

(An0 ) − (V (E) − ε)μ(Bε )} + η2 I(σ1 ) (ε)

= 2η (I(σ1(ε) )η/2 − εm(ε) ) < 0 if η is small enough. This contradicts the minimality property of I (μ) and proves (11.13). We now turn to the proof of (11.12). We argue by contradiction assuming that there exists x0 ∈ supp μ such that UE (x0 ) > V (E). Since UE is l.s.c., there exist (ε) ε > 0 and ρ (ε) > 0 such that UE > V (E) + ε in Bd (x0 , ρ (ε) ). Moreover, m0 := μ(Bd (x0 , ρ (ε) )) > 0 since x0 ∈ supp μ. We have already proved that the set E1 := {x ∈ supp μ | UE (x) < V (E)} is L-polar∗ . Then μ(E1 ) = 0 by Lemma 11.4.4. Therefore, setting E2 := (supp μ) ∩ Bd (x0 , ρ (ε) ), we have

V (E) =

UE dμ =

E3 := (supp μ) \ (E1 ∪ E2 ),

UE dμ +

E2

UE dμ E3

≥ (V (E) + ε) μ(E2 ) + V (E) μ(E3 ) (ε)

(ε)

(ε)

= (V (E) + ε) m0 + V (E) (1 − m0 ) = V (E) + ε m0 > V (E). This gives a contradiction and completes the proof of the theorem.

11.5 L-balayage and L-capacity In this section, we show some important properties of the reduced function and the balayage in the setting of the L-harmonic spaces. We will start from the general

11.5 L-balayage and L-capacity

501

abstract results proved in Section 6.11, page 375. Our main ingredients will be the characterizations of the L-superharmonicity in terms of the average operators Mr and Mr , proved in Chapter 8. To begin with, for reading convenience, we rewrite the definitions of reduced function and balayage in the L-harmonic space. Definition 11.5.1 (L-reduced function, L-balayage). Given A ⊆ G and u ∈ + S (G), the L-reduced function (or L-réduite) of u relative to A is RuA := inf{f | f ∈ ΦAu }, where

+

ΦAu := {f ∈ S (G) | f ≥ u in A }.

(11.14)

The L-balayage of u relative to A is the lower semicontinuous regularization RuA of RuA , i.e. for any x ∈ G, RuA (x) := lim inf RuA (y). y→x

Since all the results in Chapter 6 apply to the present setting, from Theorems 6.11.6 and 6.11.7 (page 377) we get a series of properties of the L-reduced function and the L-balayage that we list here for reading convenience. (I) (II) (III) (IV)

f f f f f RA ≤ RA in G, RA = RA in G \ ∂A, RA = f in Int(A), f RA is L-subharmonic in G and L-harmonic in G \ A, f g f g if 0 ≤ f ≤ g then RA ≤ RA , RA ≤ RA , f f f f if A ⊆ B ⊆ G then R ≤ R , R ≤ R . A

B

A

B

From Theorem 9.5.6 (page 449) we also have the following crucial result f f (V) RA = RA almost everywhere in G.

Moreover, by Corollary 9.5.8 (page 450), f f (VI) RA (x) = limr→0 Mr (RA )(x) for every x ∈ G,

where Mr is the mean value operator in (5.50f), page 259. Properties (V) and (VI) allow to give an easy proof of the following theorem. Theorem 11.5.2 (Further properties of L-reduction and L-balayage). Let A ⊆ G, + and let u, v ∈ S (G). One has: (i) Ru+v ≤ RuA + RvA , A u+v (ii) RA ≤ RuA + RvA . Proof. (i). If f ∈ ΦAu and g ∈ ΦAv then f + g ∈ ΦAu+v . As a consequence, Ru+v ≤ f + g. A By taking the infimum with respect to f ∈ ΦAu and g ∈ ΦAv , we obtain (i).

502

11 L-capacity, L-polar Sets and Applications

(ii). By using property (VI), from (i) we immediately obtain u+v u v Ru+v A (x) = lim Mr (RA )(x) ≤ lim Mr (RA )(x) + lim Mr (RA )(x) r→0

r→0

r→0

RvA (x) = RuA (x) + at any point x ∈ G.

When A ⊆ G is compact, the balayage takes the following form. +

Theorem 11.5.3. Let K G be a compact set, and let u ∈ S (G). Then there exists a Radon measure μ in G such that RuK = Γ ∗ μ. + Proof. Since RuK ∈ S (G) (see Theorem 6.11.6, page 377), by Corollary 9.4.8, page 444, we only need to prove that

inf RuK = 0. G

We first assume that u is bounded in K. Then there exists λ > 0 such that u ≤ λ Γ u . As a consequence, in K, so that λ Γ ∈ ΦK 0≤ RuK ≤ RuK ≤ λ Γ

in G,

which immediately gives infG RuK = 0. We now consider the general case. By the cited Corollary 9.4.8, we have u = v + h in G, where v = Γ ∗ μ is an L-potential and h ≡ infG u. Since infG v = 0 (see Theorem 9.3.7, page 434) and 0≤ RvK ≤ RvK ≤ v we have infG RvK = 0. Hence RvK is an L-potential (again by Corollary 9.4.8). On the other hand, since h is bounded in K, from the first part of the proof it follows RvK + RhK is an L-potential and thus it has that also RhK is an L-potential. Therefore, infimum = 0 (again by Theorem 9.3.7). We now use RuK ≤ RvK + RhK RuK = 0. This ends the proof. (see (6.31b), page 377), and we obtain infG We now give the definition of L-capacity. In Theorem 11.5.8 below, we shall see that this L-capacity turns out to be equal to the inverse of the L-equilibrium value defined in Section 11.4. Definition 11.5.4 (L-capacity for a compact set). Let K G be a compact set. We define R1K . WK := R1K , VK := By Theorem 11.5.3, VK is an L-potential called L-capacitary potential of K. Moreover, the L-Riesz measure μK of VK will be called the L-capacitary distribution for K (so that VK = Γ ∗ μK ). We define the L-capacity of K as C(K) := μK (K).

(11.15)

11.5 L-balayage and L-capacity

503

We shall use the following properties of the L-capacitary potential: 0 ≤ VK ≤ WK ≤ 1 in G, VK = WK in G \ ∂K, WK = 1 in K, (11.16a) supp μK ⊆ ∂K, VK ∈ H(G \ ∂K), lim VK (x) = 0. (11.16b) |x|→∞

The properties in (11.16a) and the inclusion VK ∈ H(G \ K) directly follow from properties (I) and (II) of the balayage listed in the previous pages. Let us complete the proof of (11.16b). Since the constant 1 is L-harmonic and VK = 1 in Int(K), we obtain that VK ∈ H(G \ ∂K). Hence, the inclusion supp μK ⊆ ∂K readily follows recalling that μK is the L-Riesz measure of VK . Finally, by the structure of the fundamental solution Γ = d 2−Q (being d a suitable L-gauge), we have VK (x) = Γ (y −1 ◦ x) dμK (y) ≤ d(x, K)2−Q μK (K) → 0 as |x| → ∞. K

Theorem 11.5.5 (On the L-capacitary potential). Let K G be a compact set and let μ ∈ M(K) be a Radon measure such that Γ ∗ μ ≤ 1 in G. Then Γ ∗ μ ≤ VK in G. +

1 , i.e. v ∈ S (G), v ≥ 1 in K. Since Proof. Let v ∈ ΦK

Γ ∗μ≤1≤v

in supp μ,

we have Γ ∗ μ ≤ v in G, in force of the Maria–Frostman domination principle (Theorem 11.3.4, page 495). Therefore, 1 Γ ∗ μ ≤ inf{v | v ∈ ΦK } = WK .

Recalling that Γ ∗ μ is l.s.c., we finally obtain Γ ∗ μ ≤ VK in G.

Theorem 11.5.6 (On the L-capacity. I). Let K G be a compact set. Then C(K) = max μ(K) | μ ∈ M(K), Γ ∗ μ ≤ 1 in G . (11.17) Proof. We first observe that the L-capacitary distribution μK for K has the properties μK ∈ M(K), Γ ∗ μK ≤ 1 in G, by (11.16a)-(11.16b). Let Ω be an open set such that K ⊂ Ω ⊂ Ω G, and let ν = μΩ be the L-capacitary distribution for Ω. Then VΩ = Γ ∗ ν ≡ 1 in Ω ⊃ K, by (11.16a). Let now μ ∈ M(K) be such that Γ ∗ μ ≤ 1 in G. By Theorem 11.5.5, we have Γ ∗ μ ≤ VK = Γ ∗ μK in G. Recalling that Γ (x) = Γ (x −1 ), we obtain

504

11 L-capacity, L-polar Sets and Applications

μ(K) = =

K G

1 dμ =

Γ ∗ ν dμ = Γ ∗ ν dμ = Γ ∗ μ dν ≤ Γ ∗ μK dν G G G Γ ∗ ν dμK = Γ ∗ ν dμK = 1 dμK = μK (K) = C(K). K

K

K

This completes the proof. The L-capacitary potentials have the following sub-additivity property. Theorem 11.5.7 (Sub-additivity of the L-capacitary potential). Let K1 and K2 be compact subsets of G. Then: (i) WK1 ∪K2 + WK1 ∩K2 ≤ WK1 + WK2 , (ii) VK1 ∪K2 + VK1 ∩K2 ≤ VK1 + VK2 . Proof. (i). The inequality trivially holds on K1 ∪ K2 . Indeed, if y ∈ K1 ∪ K2 , then y ∈ K1 ∩ K2 or y ∈ Ki \ Kj for i = j . In the first case, WK1 ∪K2 (y) = WK1 ∩K2 (y) = WK1 (y) = WK2 (y) = 1, while in the second case WK1 ∪K2 (y) = WK2 (y) = 1,

WK1 ∩K2 (y) ≤ WK2 (y).

1 , i = 1, 2. To show that (i) holds in Ω := G \ (K1 ∪ K2 ), let us consider ui ∈ ΦK i Since WK = VK in G \ K, for every compact K, and VK (x) → 0 as |x| → ∞, we have

lim inf w(x) := lim inf(u1 (x) + u2 (x) − (WK1 ∪K2 (x) + WK1 ∩K2 (x))) |x|→∞

|x|→∞

≥ lim inf(u1 (x) + u2 (x)) ≥ 0. |x|→∞

Moreover, if y ∈ ∂Ω = K1 ∪ K2 , then lim inf w(x) ≥ u1 (y) + u2 (y) − lim sup(WK1 ∪K2 (x) + WK1 ∩K2 (x)) ≥ 0.

Ωx→y

Ωx→y

Indeed, if y ∈ K1 ∩ K2 , we have u1 (y) + u2 (y) ≥ 2 ≥ sup(WK1 ∪K2 + WK1 ∩K2 ), G

while, if y ∈ Ki \ Kj , for i = j , ui (y) ≥ 1 ≥ sup WK1 ∪K2 G

and uj (y) ≥ WK1 ∩K2 (y) = lim WK1 ∩K2 (x), x→y

since WK1 ∩K2 is L-harmonic, hence continuous, in a neighborhood of y. Then, the minimum principle for L-superharmonic functions (Theorem 8.2.19-(ii)) implies w ≥ 0 in Ω. We have thus proved that

11.5 L-balayage and L-capacity

u1 + u2 ≥ WK1 ∪K2 + WK1 ∩K2

505

in G,

1 , i = 1, 2. From this inequality (i) immediately follows. for every ui ∈ ΦK i (ii). From (i) and property (VI) we have

VK1 ∪K2 + VK1 ∩K2 ≤ VK1 + VK2

a.e. in G.

This inequality extends on G in force of Corollary 9.5.8, page 450. Theorem 11.5.8 (On the L-capacity. II). Let K G be a compact set, and let C(K) and V (K) be the L-capacity and the L-equilibrium value of K. Then we have C(K) = (V (K))−1 . Moreover, if C(K) > 0 and μK is the L-capacitary distribution for K, then C(K)−1 μK is an L-equilibrium distribution for K. Proof. We first observe that, by definition, it is always C(K) < ∞ and V (K) > 0. If C(K) > 0, setting ν = C(K)−1 μK , we have ν(K) = 1. Moreover, supp ν ⊆ K (see (11.16b)). Hence, by Definition 11.4.1 of V (K) and recalling that Γ ∗ μK = VK ≤ 1 (see (11.16a)), we have V (K) ≤ I (ν) = C(K)−2 (Γ ∗ μK ) dμK ≤ C(K)−2 μK (K) = C(K)−1 < ∞. On the other hand, if V (K) < ∞ and μ is an L-equilibrium distribution for K, then, setting μ = V (K)−1 μ, we have Γ ∗ μ = V (K)−1 (Γ ∗ μ) ≤ 1 in G, in force of the fundamental Theorem 11.4.5 on L-equilibrium potentials. Therefore, from Theorem 11.5.6 it follows that C(K) ≥ μ(K) = V (K)−1 μ(K) = V (K)−1 > 0. This proves that C(K) > 0 iff V (K) < ∞ and that we have C(K) = V (K)−1 . The last statement of the theorem is an immediate consequence of the above arguments. Lemma 11.5.9. If {Kj }j is a decreasing sequence of compact sets, then WKj ↓ Wj Kj as j → ∞. Proof. Let us set K := WKj ↓. We set

j

Kj . From (6.31a) (page 377), it immediately follows that

1 g := lim WKj = inf WKj = inf v v ∈ ΦK . j j →∞

j

j ∈N

506

11 L-capacity, L-polar Sets and Applications

1 Using Proposition 6.5.4-(iii) (page 355), it is easily seen that the family j ΦK is j down directed. Moreover, it is bounded from below by zero. From Theorem 6.11.1, page 375, it follows that the lower semicontinuous regularization g of g is Lsuperharmonic in G. By the Corollary 9.4.8 (page 444) we know that g = Γ ∗ ν + inf g in G, G

where ν is the L-Riesz measure of g . Since 0 ≤ g ≤ WKj in G, we have 0≤ g (x) ≤ VKj (x) → 0 at infinity (see (11.16b)). Hence infG g = 0 and g =Γ ∗ν

in G.

For a fixed j0 , (WKj )j ≥j0 is a decreasing sequence of L-harmonic functions in G \ Kj0 (see (11.16a)–(11.16b)). Hence, by Theorem 5.7.10 (page 268), also the limit g is L-harmonic in G \ Kj0 . g = g in G \ K). Since j0 is arbitrary, g is L-harmonic in G \ K (in particular Hence the support of ν is contained in K. Moreover, WKj ≤ 1 gives g ≤ 1 and then Γ ∗ν = g ≤ 1 in G. Thus, by Theorem 11.5.5, we have g ≤ VK . On the other hand, since WKj ≥ WK ≥ VK for every j , (see (6.31a)), it is g ≥ VK , and then g ≥ VK . Therefore g = VK , and we obtain g = g = lim WKj ≥ WK = VK = g j →∞

in G \ K.

As a consequence, limj →∞ WKj = WK in G \ K. Observing that WKj = 1 = WK in K, the proof is complete. Proposition 11.5.10. Given a sequence {Kj }j of compact subsets of G, the following properties of the L-capacity hold: K1 ⊆ K2 Kj ↓ ⇒

⇒

C(K1 ) ≤ C(K2 ), C(Kj ) ↓ C Kj ,

(11.18) (11.19)

j

C(K1 ∪ K2 ) + C(K1 ∩ K2 ) ≤ C(K1 ) + C(K2 ).

(11.20)

Proof. Fix any Ki involved in any of the above statements. Let Ω be an open set such that Ki ⊂ Ω ⊂ Ω G, and let ν be the L-capacitary distribution for Ω. We have VΩ = Γ ∗ν ≡ 1 in Ω and supp ν ⊆ ∂Ω by (11.16a)–(11.16b). Let VKi and μKi be, respectively, the L-capacitary potential and the L-capacitary distribution for Ki . Then supp μKi ⊆ Ki and VKi ≡ WKi outside Ki , hence in ∂Ω (again by (11.16a)– (11.16b)).

11.5 L-balayage and L-capacity

507

(i). If K1 ⊆ K2 , then WK1 ≤ WK2 , see (6.31a), page 377. Recalling that Γ (x) = Γ (x −1 ), we obtain C(K1 ) = 1 dμK1 = Γ ∗ ν dμK1 = Γ ∗ ν dμK1 = Γ ∗ μK1 dν K1 K1 G G = VK1 dν = WK1 dν ≤ WK2 dν = VK2 dν ∂Ω ∂Ω ∂Ω ∂Ω Γ ∗ μK2 dν = Γ ∗ ν dμK2 = 1 dμK2 = C(K2 ). = G

G

K2

(ii). If Kj ↓ and K := j Kj , then WKj ↓ WK by Lemma 11.5.9. Recalling that VK ≡ WK outside K, hence in ∂Ω, and arguing as in (i), we obtain C(Kj ) = Γ ∗ ν dμKj = Γ ∗ μKj dν = VKj dν G G ∂Ω WKj dν WK dν = VK dν = ∂Ω ∂Ω ∂Ω Γ ∗ μK dν = Γ ∗ ν dμK = C(K). = G

G

(iii). By (6.31c) (page 377), we have WK1 ∪K2 + WK1 ∩K2 ≤ WK1 + WK2 . The same arguments as in (i), (ii) give C(K1 ∪ K2 ) + C(K1 ∩ K2 ) = Γ ∗ ν dμK1 ∪K2 + Γ ∗ ν dμK1 ∩K2 G G = Γ ∗ μK1 ∪K2 dν + Γ ∗ μK1 ∩K2 dν G G = VK1 ∪K2 dν + VK1 ∩K2 dν ∂Ω ∂Ω (WK1 ∪K2 + WK1 ∩K2 ) dν = ∂Ω (WK1 + WK2 ) dν = (VK1 + VK2 ) dν ≤ ∂Ω ∂Ω Γ ∗ ν dμK1 + Γ ∗ ν dμK2 = (Γ ∗ μK1 + Γ ∗ μK2 ) dν = G

= C(K1 ) + C(K2 ).

G

G

This ends the proof. Theorem 11.5.11 (Characterization of L-polarity for K G ). Let K G be a compact set. Then K is L-polar if and only if C(K) = 0. Proof. First, suppose K is L-polar. By Corollary 11.2.5, there exists μ ∈ M0 such that Γ ∗ μ ≡ ∞ in K. Recalling that Γ (x) = Γ (x −1 ) and that Γ ∗ μK = VK ≤ 1 (see (11.16a)), we obtain

508

11 L-capacity, L-polar Sets and Applications

∞ · C(K) = ∞ · μK (K) =

Γ ∗ μ dμK =

Γ ∗ μK dμ ≤ μ(G) < ∞.

Therefore, it must be C(K) = 0. Suppose now C(K) = 0. By (11.19), we can find a sequence of bounded open sets Ωj ⊃ K such that Kj +1 := Ω j +1 ⊂ Ωj

and C(Kj ) < 1/j 2 .

We now choose x0 ∈ / K1 , and we set m := maxy∈K1 Γ (y −1 ◦ x0 ). We have VKj (x0 ) = Γ (y −1 ◦ x0 ) dμKj (y) ≤ m μKj (Kj ) = m C(Kj ) < m/j 2 . Kj

We now define ω = j ∈N VKj . Then ω ∈ S(G), since ω(x0 ) < ∞ (see Corollary 8.2.8, page 403). Moreover, ω ≡ ∞ in K, since VKj ≡ 1 in Ωj ⊃ K (see (11.16a)). Therefore, K is L-polar. Corollary 11.5.12 (L-polar∗ ⇒ L-polar). Any L-polar∗ set is L-polar. Proof. Let E ⊂ G be an L-polar∗ set, i.e. there exists a countable family of compact sets Fn such that V (Fn ) = ∞ and E ⊆ n∈N Fn . From Theorem 11.5.8 and Theorem 11.5.11 it follows that Fn is L-polar for every n ∈ N. We now only need to recall that any countable union of L-polar sets is L-polar, by means of Corollary 11.2.6. Corollary 11.5.13 (On the L-equilibrium potential). Let K G be a compact set with L-equilibrium value V (K) < ∞ (or, equivalently, with L-capacity C(K) > 0). Then there exists a unique L-equilibrium potential of K given by UK = C(K)−1 VK , where VK is the L-capacitary potential of K. Proof. Let U1 = Γ ∗ μ1 , U2 = Γ ∗ μ2 be L-equilibrium potentials of K. From the fundamental Theorem 11.4.5 on L-equilibrium potentials it follows that U1 , U2 ≤ V (K) < ∞ in G and U1 = V (K) = U2 in K \ P , for a suitable L-polar∗ set P . Observing that P is also L-polar, by Corollary 11.5.12, and recalling that supp μi ⊆ K, we infer that U1 = U2 in G, in force of the Maria–Frostman domination principle (Theorem 11.3.4). In order to complete the proof, we only need to observe that by Theo rem 11.5.8, UK = C(K)−1 VK is an L-equilibrium potential of K. We now aim to extend the notion of L-capacity to a larger class of sets. Definition 11.5.14 (L-capacitable set and L-capacity). Given a non-empty set E ⊆ G, we define the interior L-capacity of E as C∗ (E) := sup{C(K) | K compact, K ⊆ E}.

11.5 L-balayage and L-capacity

509

We then define the exterior L-capacity of E as follows C ∗ (E) := inf{C∗ (Ω) | Ω open, Ω ⊇ E}. We also set C∗ (∅), C ∗ (∅) := 0. A set E ⊆ G is called L-capacitable if C∗ (E) = C ∗ (E). In this case, this common value is denoted by C(E) and it is called the Lcapacity of E. Proposition 11.5.15. Any Borel set is L-capacitable. Moreover, for compact sets the above definition of L-capacity agrees with the one given in Definition 11.5.4. Proof. With Proposition 11.5.10 at hand, one can follow verbatim the classical capacitability theory as presented, e.g. in [Helm69] or [AG01]. We omit further details. Theorem 11.5.16 (Characterization of L-polarity. I). A set E ⊆ G is L-polar if and only if E is capacitable and C(E) = 0. (see Proof. First, suppose E is L-polar. Then E is contained in a Borel L-polar set E Ex. 11.2.2). Any compact subset K of E is L-polar and then it has null L-capacity by = 0. Since E is a Borel set, it is capacitable Theorem 11.5.11. Thus, we have C∗ (E) by Proposition 11.5.15. Hence = C ∗ (E) ≥ C ∗ (E) ≥ C∗ (E) ≥ 0 0 = C∗ (E) (we remark that from (11.18) it readily follows the monotonicity of interior and exterior capacity). Therefore E is capacitable and C(E) = 0. We now want to prove the reverse implication. We first observe that we can assume that E is bounded. The general case follows by considering the sequence of bounded sets En := E ∩ D(0, n). From C(E) = 0 we get 0 = C ∗ (E) ≥ C ∗ (En ) ≥ C∗ (En ) ≥ 0, and thus E n is capacitable with C(En ) = 0, hence L-polar; then, by Corollary 11.2.6, also E = n En is L-polar. Suppose now that E is a bounded capacitable set and C(E) = 0. Then, for every j ∈ N, there exists an open set Ωj ⊇ E such that C(Ωj ) < 1/j 2 . Moreover, we can suppose that all the Ωj ’s are contained in a fixed bounded set. In particular, there exists x0 ∈ G such that Bd (x0 , 1) ∩ Ωj = ∅ for every j , so that Γ (ζ −1 ◦ x0 ) ≤ 1 ∀ζ ∈ Ωj . (11.21) j

For a fixed j ∈ N, we consider a sequence {Bi }i of open sets such that B i ⊂ Bi+1 , Bi = Ωj . i

We set Ki := B i . From (11.21) we obtain

510

11 L-capacity, L-polar Sets and Applications

VKi (x0 ) =

Γ (ζ Ki

−1

◦ x0 ) dμKi (ζ ) ≤

Ki

dμKi = C(Ki ) ≤ C(Ωj ) ≤ j −2 .

From Ki ⊆ Ki+1 we infer 0 ≤ VKi ≤ VKi+1 ≤ 1. If we set ωj = lim VKi , i→∞

we have ωj (x0 ) ≤ j −2 , 0 ≤ ωj ≤ 1. Moreover, ωj ∈ S(G) by Corollary 8.2.8, page 403. On the other hand, (11.16a) gives 1 ≥ ωj ≥ VKi = 1 on Bi for every i, so that ωj ≡ 1 on i Bi = Ωj ⊇ E. Finally, setting ω = j ∈N ωj , we have ω(x0 ) ∈ R, ω ≡ ∞ on E and ω ∈ S(G) (again by Corollary 8.2.8). This implies that E is L-polar.

11.6 The Fundamental Convergence Theorem The following theorem is sometimes referred to as the fundamental convergence theorem of potential theory. It generalizes to our sub-Laplacian setting a theorem due to Cartan [Cart45, Théorème 8.4] (see also [Helm69, Theorem 7.39] and [AG01, Theorem 5.7.1]). Theorem 11.6.1 (The fundamental convergence theorem). Let F be a family of L-superharmonic functions uniformly bounded from below ( for example, a family of L-potentials). v at Then, v = infu∈F u differs from its lower semicontinuous regularization most on an L-polar set. Proof. Since G is a countable union of d-balls, and a countable union of L-polar sets is L-polar, it is sufficient to prove that v = v in B := Bd (x0 , r) except at most on an L-polar subset of B. We can also assume that u ≥ 0 for every u ∈ F, since F is uniformly bounded from below. Moreover, replacing F by the collection of infimums of finitely many elements of F, if necessary, we can assume that F is down directed. Let us now consider the L-balayage Ru of u ∈ F. By Theorem 6.11.6, page 377, B Ru is we know that Ru is L-superharmonic and it is equal to u in B. Moreover, B B the potential of a measure supported in B, by Theorem 11.5.3 and Theorem 6.11.6. Observing also that the family u RB : u ∈ F is still down directed, it is then not restrictive to assume that any u ∈ F is the potential of a measure supported in B. Finally, recalling that we are working with a down directed family, we can obviously assume that there exists u0 ∈ F such that u ≤ u0 for every u ∈ F. Let now D be a d-ball such that B ⊂ D, and let us consider the L-capacitary potential

11.6 The Fundamental Convergence Theorem

511

VD = Γ ∗ μD ≡ 1 in D. For every u = Γ ∗ μ ∈ F (μ ∈ M(B)), we have μ(B) = Γ ∗ μD dμ = Γ ∗ μD dμ = Γ ∗ μ dμD B G G ≤ u0 dμD = Γ ∗ μ0 dμD = Γ ∗ μD dμ0 = μ0 (B), G

G

G

where u0 = Γ ∗ μ0 , μ0 ∈ M(B). Hence, the set μ(B) | u ∈ F, u = Γ ∗ μ is bounded. The assertion of the theorem now follows from Lemma 11.6.2 below: we find ν ∈ M such that Γ ∗ ν ≤ v in B with equality except possibly on an L-polar set P , and we deduce that v = Γ ∗ ν ≤ v ≤ v in B \ P . Lemma 11.6.2. Let {μi }i∈I be a family of Radon measures on G with the following properties: (i) there exists a compact set K such that supp (μi ) ⊆ K for every i ∈ I , (ii) the family {Γ ∗ μi | i ∈ I } is down directed, (iii) there exists m ∈ R such that μi (G) ≤ m for every i ∈ I . Then there exists μ ∈ M such that Γ ∗ μ ≤ inf Γ ∗ μi , i∈I

and there exists an L-polar set P such that Γ ∗ μ = inf Γ ∗ μi in G \ P . i∈I

Proof. By Proposition 6.1.2, page 339, there exists a countable set I0 ⊆ I such that if g is l.s.c. on G and g ≤ infi∈I0 (Γ ∗ μi ) then g ≤ infi∈I (Γ ∗ μi ). Since the family {Γ ∗ μi | i ∈ I } is down directed, it is not restrictive to assume that I0 = {in }n∈N and {Γ ∗ μin }n is decreasing. Now,5 up to extracting a subsequence, μin converges in the w ∗ -topology to some measure μ ∈ M(K), μ(G) ≤ m, and (Γ ∗ μ)(x) = Γ (y −1 ◦ x) dμ(y) ≤ lim inf Γ (y −1 ◦ x) dμin (y) G

n→∞

= lim inf(Γ ∗ μin )(x) n→∞

= inf (Γ ∗ μi )(x). i∈I0

Recalling that Γ ∗ μ is l.s.c. on G, it follows that 5 See, e.g. [Rud87].

G

11 L-capacity, L-polar Sets and Applications

512

Γ ∗ μ ≤ inf(Γ ∗ μi ). i∈I

Let us now consider the Borel set E := x ∈ G | (Γ ∗ μ)(x) < lim (Γ ∗ μin )(x) . n→∞

Since

x ∈ G | (Γ ∗ μ)(x) < inf(Γ ∗ μi )(x) ⊆ E, i∈I

it remains only to prove that E is L-polar. By means of Theorem 11.5.16 and Proposition 11.5.15, it suffices to show that C∗ (E) = 0. Assume by contradiction that there exists a compact set C ⊆ E such that C(C) > 0, and consider the L-capacitary distribution μC . Then μC (C) = C(C) > 0 and Γ ∗ μC ≤ 1. By Lemma 11.3.3, we infer that there exists a compact set C ⊆ C, μC (C ) > 0, such that setting ν = μC |C we have Γ ∗ ν ∈ C(G, R). Thus n→∞ Γ ∗ ν dμin −→ Γ ∗ ν dμ G

G

and, by Fatou’s lemma, lim Γ ∗ μin dν ≤ lim inf Γ ∗ μin dν n→∞ G G n→∞ = lim inf Γ ∗ ν dμin n→∞ G = Γ ∗ ν dμ = Γ ∗ μ dν < ∞ G

G

(Γ ∗ ν is continuous and μ is compactly supported). Therefore (Γ ∗ μ − lim Γ ∗ μin ) dν ≥ 0. C

n→∞

Since the integrand is strictly negative on C and ν(C) > 0, we get a contradiction. This completes the proof. For the reader’s convenience and for the future references, we collect some of the properties of the L-balayage that we know so far. + RuE is L-superharmonic on Ω. Lemma 11.6.3. Let E ⊆ G. Let u ∈ S (G). Then Moreover, the following properties hold:

RuE ≥ 0 on G, (i) u ≥ RuE ≥ u (ii) u = RE on E, (iii) u = RuE = RuE on the interior of E,

11.6 The Fundamental Convergence Theorem

513

(iv) RuE = RuE on G \ E, and they are L-harmonic on G \ E, u (v) RE differs from RuE on a (L-polar) subset of ∂E, (vi) E ⊆ F ⇒ RuE ≤ RuF , + RuE ≤ RvE , (vii) v ∈ S (G) and u ≤ v on E ⇒ (viii) λ > 0 ⇒ RλEu = λ RuE , u+v u v (ix) RE ≤ RE + RE . The following result holds. +

Proposition 11.6.4 (RuE and RuE . I). Let E be any subset of G and u ∈ S (G). Then u the L-reduced function RE and the L-balayage RuE coincide on G \ E. RuE may differ only Proof. To begin with, from Lemma 11.6.3 we know that RuE and u u on a subset of ∂E. From Theorem 11.6.1 we derive that RE = RE on an L-polar set, say P . Let Z := E ∩ P . Collecting the above remarks, we infer that Z is an L-polar subset of E ∩ ∂E. / E. In particular, x0 ∈ / Z, so that, by Theorem 11.2.3, there exists Let x0 ∈ + w ∈ S (G) such that w(x0 ) ∈ R and w ≡ ∞ on Z. Then, for every ε > 0, the function v := RuE + εw +

belongs to S (G) and is ≥ u on E (indeed, if z ∈ E∩P = Z then v(z) = ∞ ≥ u(z), whereas if z ∈ E \ P , by definition of P it holds v(z) ≥ RuE (z) = RuE (z) = u(z), see Lemma 11.6.3). Then, by the very Definition 11.5.1 of RuE , we have RuE + ε w ≤ RuE

on G.

In particular, RuE (x0 ) + ε w(x0 ) ≤ RuE (x0 ). Letting ε → 0+ (recall that w(x0 ) ∈ R), we infer RuE (x0 ) ≤ RuE (x0 ), which together with Lemma 11.6.3-(i), proves RuE (x0 ) = RuE (x0 ). Being x0 ∈ / E arbitrary, the assertion of the proposition follows. From Theorem 11.6.1 and Proposition 11.6.4, one immediately obtains the following remarkable result. + Theorem 11.6.5 (RuE and RuE . II). Let E ⊆ G and u ∈ S (G). Then the L-reduced function RuE differs from the L-balayage RuE at most on an L-polar subset of E ∩∂E.

In particular, we have the following assertion. +

Corollary 11.6.6. Let Ω ⊆ G be open and u ∈ S (G). Then RuΩ ≡ RuE . Proof. Apply Theorem 11.6.5 and note that Ω ∩ ∂Ω = ∅ since Ω is open.

514

11 L-capacity, L-polar Sets and Applications

11.7 The Extended Poisson–Jensen Formula The aim of this section is to prove Theorem 11.7.6, which extends the Poisson– Jensen formula of Theorem 9.5.1 (page 445) in the case when the open set Ω is not necessarily an L-regular set. The notion of extended L-Green function is also needed. Throughout this section, we say that Ω ⊆ G is a domain if it is an open and connected set. Lemma 11.7.1. Let Ω ⊂ G be a bounded domain, and let ζ0 ∈ ∂Ω. Suppose Ω Bd (x0 , r) and set E = B d (x0 , r) \ Ω. If UE (ζ0 ) = V (E) < ∞ (being UE the L-equilibrium potential for E and V (E) its L-equilibrium value), then there exists an L-barrier for Ω at ζ0 . Proof. To begin with, we have UE ∈ H(G \ E),

UE ≤ V (E) in G

(see Theorem 11.4.5). Since UE → 0 at infinity, from the strong maximum principle (Theorem 5.13.8, page 296) it follows that UE < V (E) in G \ E. Moreover, UE l.s.c. and UE (ζ0 ) = V (E) imply that there exists lim

G\Ex→ζ0

UE (x) = V (E).

Hence, the function w := V (E) − UE is an L-barrier for Ω at ζ0 .

The following result is of unquestionable importance. Theorem 11.7.2 (On the L-polarity of L-irregular points). Let Ω ⊂ G be a bounded domain. Then the subset of ∂Ω of the L-irregular points is an L-polar∗ set. In particular, it is L-polar. Proof. Let {Bd (xj , rj )}j ≤n be a cover of ∂Ω such that Ω Bd (xj , rj ) for every j ≤ n. We set Ej = Bd (xj , rj ) \ Ω and Fj = ∂Ω ∩ Bd (xj , rj ). We write ∂Ω = Fj ∪ Fj =: B1 ∪ B2 . j : C (Fj )=0

j : C (Fj )>0

Let us recall once for all that C(K) = V (K)−1 for any compact set K, by Theorem 11.5.8. In particular, we have that B1 is L-polar∗ . Suppose next that j ∈ {1, . . . , n} is such that C(Fj ) > 0. Since Fj ⊆ Ej , we also have C(Ej ) > 0 (see (11.18)). From Theorem 11.4.5 it follows that

11.7 The Extended Poisson–Jensen Formula

515

UEj (x) = V (Ej ) for all x ∈ Ej \ Pj , where Pj ⊆ Ej is L-polar∗ . Hence, for every ζ0 ∈ Fj \ Pj , we have UEj (ζ0 ) = V (Ej ) < ∞. We can then apply Lemma 11.7.1 and obtain that there exists an Lbarrier for Ω at ζ0 . Thus ζ0 is an L-regular point, by Bouligand’s theorem 6.10.4, page 371. We finally write ∂Ω = B1 ∪ (Pj ∩ Fj ) ∪ (Fj \ Pj ) j : C (Fj )>0

=:

B1 ∪ B2

j : C (Fj )>0

∪ B2 ,

and we remark that B1 ∪ B2 is L-polar∗ , whereas all the points of B2 are L-regular. This ends the proof. Theorem 11.7.3 (On the L-harmonic measure and L-polarity). Let Ω ⊂ G be a bounded domain. Let E be an L-polar subset of ∂Ω. Then, for every x ∈ Ω, we have μΩ x (E) = 0. Proof. Since μΩ x is regular, it is enough to prove the theorem when E is compact. By Theorem 6.9.3 (page 367), we know that the characteristic function χE of E is L-resolutive, and we have Ω Ω Ω μx (E) = dμx (y) = χE (y) dμΩ x (y) = HχE (x). E

∂Ω

In particular, HχΩE ∈ H(Ω) and HχΩE is bounded. We have to prove that HχΩE = 0 in Ω. We start by showing that lim

Ωx→ζ0

HχΩE (x) = 0 ∀ ζ0 ∈ ∂Ω \ F,

(11.22)

where F = E ∪ E0 and E0 is the set of the L-irregular points of ∂Ω. Consider a sequence δn ∈ C(∂Ω), δn χE , such that there exists n0 with δn0 (ζ0 ) = 0 (we can take δn (ζ ) := max{1 − n dist(ζ, E), 0}). We have 0 ≤ HχΩE (x) ≤ HδΩn (x)

for every x ∈ Ω

(see Proposition 6.7.4-(i), page 360). Moreover, since ζ0 is an L-regular point, lim

Ωx→ζ0

HδΩn (x) = δn0 (ζ0 ) = 0. 0

Thus, (11.22) follows. On the other hand, E0 is L-polar∗ by means of Theorem 11.7.2, hence L-polar by Corollary 11.5.12. Moreover, E is L-polar by hypothesis. Thus also F is L-polar (see Corollary 11.2.6). We can now use the extended maximum principle for L in Lemma 11.2.7, and we obtain HχΩE = 0 in Ω. This ends the proof.

516

11 L-capacity, L-polar Sets and Applications

Definition 11.7.4 (The extended L-Green function). Let Ω ⊂ G be a bounded domain, and let x ∈ Ω be fixed. We define the extended L-Green function GΩ (x, ·) for Ω by ⎧ if y ∈ Ω, ⎪ ⎨ GΩ (x, y) if y ∈ G \ Ω, (11.23) GΩ (x, y) := 0 ⎪ ⎩ lim sup Ωz→y GΩ (x, z) if y ∈ ∂Ω. In particular, GΩ (x, y) = 0 for every L-regular point y ∈ ∂Ω. We have the following remarkable result. Theorem 11.7.5 (On the extended L-Green function). Let Ω ⊂ G be a bounded domain, and let x ∈ Ω be fixed. Then the extended L-Green function for Ω is Lsubharmonic in G \ {x}. Moreover, for all y ∈ G,

GΩ (x, y) = Γ (y −1 ◦ x) − Γ y −1 ◦ η dμΩ (11.24) x (η). ∂Ω

Proof. Let x ∈ Ω be fixed. We define the function g Ω (x, ·) on G by

g Ω (x, ·) := Γ (x −1 ◦ ·) − Γ (·)−1 ◦ η dμΩ x (η).

(11.25)

∂Ω

(It is easy to see that no indeterminate forms ∞−∞ may occur.) From Theorem 9.3.2 (page 432) and the symmetry of Γ it follows that g Ω (x, ·) ∈ S(G \ {x}). We are left to prove that g Ω = GΩ . If y ∈ Ω, then from the symmetry of GΩ (see Proposition 9.2.10, page 431) it follows that g Ω (x, y) = GΩ (y, x) = GΩ (x, y). Let now y ∈ G \ Ω. Since Γ (y −1 ◦ ·) ∈ H(Ω) ∩ C(Ω), we have Ω −1 Γ (y −1 ◦ η) dμΩ ◦ x) x (η) = HΓ (y −1 ◦·) (x) = Γ (y ∂Ω

(see Proposition 6.7.7, page 361). In this case, g Ω (x, y) = 0. We now show that g Ω (x, ·) is non-negative on G. It suffices to prove it for y ∈ ∂Ω. For η ∈ ∂Ω, we set fn (η) := min{n, Γ (y −1 ◦ η)}. Then fn ∈ C(∂Ω, R) and fn (η) Γ (y −1 ◦ η) as n → ∞. Then, by monotone convergence, Ω Γ (y −1 ◦ η) dμΩ x (η) = lim Hfn (x). ∂Ω

n→∞

If we prove that HfΩn (x) ≤ Γ (y −1 ◦ x) for any x ∈ Ω, it follows g Ω (x, y) ≥ 0. By the extended maximum principle in Lemma 11.2.7, since the set ∂2 Ω of the Lirregular points of ∂Ω is an L-polar set (see Theorem 11.7.2 and Corollary 11.5.12), it is enough to show that

11.7 The Extended Poisson–Jensen Formula

lim sup HfΩn (x) − Γ (y −1 ◦ x) ≤ 0

517

for ζ ∈ ∂1 Ω,

Ωx→ζ

where ∂1 Ω is the set of the L-regular points of ∂Ω, and that HfΩn (x) − Γ (y −1 ◦ x) is bounded from above. The latter fact follows from HfΩn (x) − Γ (y −1 ◦ x) ≤ HfΩn (x) ≤ max fn < ∞. ∂Ω

We then prove the former claim. Since each ζ ∈ ∂1 Ω is an L-regular point,

lim sup HfΩn (x) − Γ (y −1 ◦ x) ≤ fn (ζ ) + lim sup(−Γ (y −1 ◦ x)) Ωx→ζ

Ωx→ζ

which is ≤ 0 by definition of fn . This proves that g Ω (x, y) ≥ 0. In order to prove that g Ω coincides with GΩ defined in (11.23), it remains to show that, for any y0 ∈ ∂Ω, it holds g Ω (x, y0 ) = lim sup GΩ (x, y). Ωy→y0

First, we suppose y0 ∈ ∂1 Ω. Since ∂2 Ω is L-polar, by Corollary 11.2.5 there exists h ∈ S(G), finite at y0 , h < 0 such that h|∂2 Ω ≡ −∞. For any ε > 0, let gε (y) := g Ω (x, y) + ε h(y),

y ∈ G \ {x}.

Since g Ω (x, y) = GΩ (x, y) for every y ∈ Ω, g Ω (x, y) vanishes as y approaches ∂1 Ω from the inside of Ω and remains bounded as y ∈ Ω approaches ∂2 Ω. Using also the fact that h is u.s.c., we have that, for any y1 ∈ ∂Ω, lim sup gε (y) < 0. Ωy→y1

Moreover, lim sup gε (y) < 0, G\Ωy→y1

since g Ω (x, y) = 0 for every y ∈ G \ Ω. Then there exists a compact neighborhood N of ∂Ω such that gε < 0 on N \ ∂Ω. Recalling that g Ω (x, ·) ∈ S(G \ {x}), from the maximum principle for L-subharmonic functions (Theorem 8.2.19, page 409) we then obtain gε ≤ 0 in N. In particular, gε (y0 ) ≤ 0 and, letting ε vanish, we get g Ω (x, y0 ) ≤ 0, which (jointly with g Ω (x, ·) ≥ 0) gives g Ω (x, y0 ) = 0. We have thus proved that lim GΩ (x, y) = 0 = g Ω (x, y0 ) Ωy→y0

if y0 ∈ ∂1 Ω. We now treat the case of y0 ∈ ∂2 Ω. Let us set for brevity

518

11 L-capacity, L-polar Sets and Applications

E := ∂2 Ω,

A := G \ {x},

u := g Ω (x, ·).

With this notation, we have E L-polar and u ∈ S(A). We now set u(x) := lim sup u(ξ ),

for x ∈ A ∩ E,

A\Eξ →x

u(x) := u(x),

for x ∈ A \ E.

Since we know that u ≥ 0 is u.s.c., we immediately get 0 ≤ u ≤ u in A. Moreover, since E has Lebesgue measure zero being L-polar, the following equality of the u). As a consequence, u is sub-mean in A, being solid means holds: Mr (u) = Mr ( u ∈ S(A), hence sub-mean by Theorem 8.2.1 (page 401). Furthermore, it is easy to verify that u is u.s.c. Therefore, again from Theorem 8.2.1, we obtain u ∈ S(A). We now use Lemma 11.3.1 and conclude that u ≡ u in A. In particular, we have g Ω (x, y0 ) = u(y0 ) = u(y0 ) =

lim sup

G\∂2 Ωy→y0

g Ω (x, y)

= lim sup GΩ (x, y). Ωy→y0

In the last equality we used the following facts: g Ω (x, ·) ≥ 0, g Ω (x, ·) ≡ 0 in G \ (Ω ∪ ∂2 Ω) and g Ω (x, ·) = GΩ (x, ·) in Ω. This completes the proof. Theorem 11.7.6 (The extended Poisson–Jensen formula). Let Ω be a bounded domain of G. Suppose u is L-subharmonic on a neighborhood of Ω. Then, for all x ∈ Ω, we have u(x) = u(y) dμΩ (y) − GΩ (x, y) dμu (y), (11.26) x ∂1 Ω

Ω∪∂2 Ω

where ∂1 Ω, ∂2 Ω denote, respectively, the subsets of ∂Ω of the regular and the irregular points for the Dirichlet problem related to L, μu is the L-Riesz measure of u, μΩ x is the L-harmonic measure for Ω at x, and GΩ is the extended L-Green function for Ω. Proof. Let O be a bounded domain such that Ω ⊂ O and such that u is Lsubharmonic on a neighborhood of O. By the Riesz representation Theorem 9.4.4 (page 442), there exists h ∈ H(O) such that u(x) = h(x) − Γ (y −1 ◦ x) dμu (y) ∀ x ∈ O. O

We certainly have h(x) =

h(η) dμΩ x (η). We now let

∂Ω

v(x) := −

Γ (y −1 ◦ x) dμu (y)

for x ∈ G.

O

Recalling Theorem 9.3.5, page 433, v ∈ S(G) and μv coincides with μu on O and vanishes on G \ O. We claim that it is enough to prove (11.26) with u replaced by v.

11.8 Further Results. A Miscellanea

519

Indeed, using the fact that μΩ x (∂2 Ω) = 0 for every x ∈ Ω (which is an immediate consequence of Theorem 11.7.2 and Theorem 11.7.3), this would give u(x) = h(x) + v(x) Ω = h(y) dμΩ (y) + v(y) dμ (y) − GΩ (x, y) dμv (y) x x ∂Ω ∂1 Ω ∂2 Ω∪Ω = (h(y) + v(y)) dμΩ (y) − GΩ (x, y) dμu (y) x ∂1 Ω ∂2 Ω∪Ω = u(y) dμΩ (y) − GΩ (x, y) dμu (y). x ∂1 Ω

∂2 Ω∪Ω

We are then left to prove (11.26) for v. We recall that GΩ (x, ·) ≡ 0 on ∂1 Ω and on G \ Ω (see the definition (11.23) of GΩ ). Using this fact, exploiting μΩ x (∂2 Ω) = 0, and using (11.24), we obtain that, for every x ∈ Ω, v(η) dμΩ (η) − GΩ (x, y) dμv (y) x ∂1 Ω ∂2 Ω∪Ω Γ (y −1 ◦ η) dμu (y) dμΩ (η) − GΩ (x, y) dμu (y) =− x ∂1 Ω O ∂2 Ω∪Ω −1 Ω =− Γ (y ◦ η) dμx (η) dμu (y) − GΩ (x, y) dμu (y) O ∂Ω O − Γ (y −1 ◦ η) dμΩ = x (η) − GΩ (x, y) dμu (y) O ∂Ω =− Γ (y −1 ◦ x) dμu (y) = v(x). O

This completes the proof.

11.8 Further Results. A Miscellanea The aim of this section is to collect a miscellanea of results concerning with the L-capacity. These assertions, besides completing the investigation of the previous sections, also provide useful results, some having an interest in its own. Theorem 11.8.1 (The L-balayage as a L-potential). Let E ⊂ G be bounded. Then the following assertions hold: +

(a) For every u ∈ S (G), the L-balayage RuE is an L-potential. In particular, if u ≡ 1, there exists a Radon measure νE such that R1E = Γ ∗ νE (i.e. νE is the 1 L-Riesz measure of RE ); (b) With the above notation, we have C ∗ (E) = νE (G).

520

11 L-capacity, L-polar Sets and Applications

Proof. (a). Let R 1 be such that E ⊂ Bd (0, R). Let K1 := Bd (0, R),

v := RuK1 .

K2 := Bd (0, R + 1),

Since v ∈ H(G \ K1 ) (see Lemma 11.6.3-(iv)), there exists a 1 such that a Γ ≥ v on ∂K2 . We set v(x) if x ∈ K2 , w(x) := min{a Γ (x), v(x)} if x ∈ G \ K2 . +

By Lemma 11.3.2, w ∈ S (G). Moreover,6 w ≥ u on E. Hence, by the definition of L-balayage, w ≥ RuE ≥ RuE . Then, on G \ K2 we have RuE ≤ w = min{a Γ, v} ≤ a Γ. As a consequence,

RuE ≤ inf ≤ inf a Γ = 0. inf G

G\K2

G\K2

We are therefore in a position to apply Proposition 9.9.1 (see Ex. 5, Chapter 9, page 464) and derive that RuE is an L-potential. This proves assertion (a). (b). Let us now take u = 1 and apply what we have proved above: there exists a R1E = Γ ∗νE . Our task is to prove that C ∗ (E) = νE (G). Radon measure νE such that The proof is split in two parts: first, we suppose that E is also open, then we consider the general case. (b’). Let E = Ω be open and bounded. Let {Kn }n be a sequence of compact subsets of G such that Kn = Ω. Kn ⊂ Int(Kn+1 ) ⊂ Kn+1 ⊂ Ω and n

By Ex. 15 at the end of the chapter, we infer (see also Corollary 11.6.6 and part (a) of the proof) R1Kn = R1Ω = lim R1Ω = Γ ∗ νΩ . (11.27) n→∞

As a consequence, we have C(Ω) = lim C(Kn ) = lim μKn (G) = νΩ (G). n→∞

n→∞

The first equality is a consequence of Ex. 16-(6) at the end of the chapter; the second is the definition of L-capacity for a compact set; the third equality follows from Ex. 14 at the end of the chapter.7 6 Indeed, since E is contained in the interior of K , we have w = v = RuK = RuK = u 1 1 1

on E, see Lemma 11.6.3.

7 We apply Ex. 14 with ν = ν , σ = μ , K = Ω. Indeed, note that Ω n Kn

R1Kn Γ ∗ νΩ Γ ∗ μKn = VKn = by (11.27) and property (vi) of Lemma 11.6.3.

11.8 Further Results. A Miscellanea

521

(b”). To begin with, we make some remarks on the definition of R1E . We recall that R1E = inf u u∈ΦE1

where

+

ΦE1 = {u ∈ S (G) | u ≥ 1 on E}. Since ΦE1 is obviously down directed and 1 ∈ ΦE1 , then8 inf u =

u∈ΦE1

inf

u∈ΦE1 , u≤1

u.

Furthermore, by a very general result on the lower semicontinuous regularization of the infimum of an arbitrary family of functions,9 there exists a sequence {un } in ΦE1 with un ≤ 1 such that R1E . inf un = inf u = n

u∈ΦE1

We can also suppose that u1 = 1. Let vn := min{u1 , u2 , . . . , un }. +

We have 0 ≤ vn ≤ 1, vn ∈ S (G), vn ≥ 1 on E and {vn }n∈N is non-increasing. Also, (11.28) lim vn = inf un . n→∞

n∈N

Let ε ∈ (0, 1). Let Ω be an open set such that E ⊆ Ω and C(Ω) < C ∗ (E) + ε.

(11.29)

We can suppose that Ω is bounded, by replacing it by Ω ∩ Bd (0, R) (with R 1 such that Bd (0, R) ⊃ E; note that C(Ω ∩ Bd (0, R)) ≤ C(Ω)). We set wn := RvΩn

and Ωn := {x ∈ Ω | wn (x) > 1 − ε}.

We collect some properties of wn and Ωn : (a) (b) (c) (d)

RvΩn , for Ω is open (see Corollary 11.6.6); wn := {wn }n is decreasing (see Lemma 11.6.3-(vii)); Ωn is open, for wn is l.s.c.; Ωn ⊇ E, for wn = RvΩn = vn ≥ 1 > 1 − ε on E;

8 If F is a down directed family of functions and u ∈ F , then inf{u | u ∈ F } = inf{u | u ∈ 0 F , u ≤ u0 }. Indeed, obviously inf{u | u ∈ F } ≤ inf{u | u ∈ F , u ≤ u0 }. Moreover, for every u ∈ F , there exists v ∈ F such that v ≤ min{u, u0 } ≤ u0 . Hence inf{u | u ∈ F , u ≤ u0 } ≤ v ≤ u. Taking the infimum over {u | u ∈ F } in the last inequality, the assertion

follows. 9 See Proposition 6.1.2, page 339.

522

11 L-capacity, L-polar Sets and Applications

(e) wn = RvΩn ∈ H(G \ Ωn ) (see Lemma 11.6.3-(iv)); R1E (indeed, wn ∈ ΦE1 by (d) above, whence wn ≥ R1E , and take the lower (f) wn ≥ semicontinuous regularization); RvΩn (g) there exists a Radon measure σn on G such that wn = Γ ∗ σn (since wn = and by the first part of the proof, being Ω bounded); R1Ωn = Γ ∗ νΩn (by the first part of the proof); (h) R1Ωn = (i) by (e) above, it follows that σn in (g) is supported in Ωn (hence compactly supported); the same is true for νΩn . As a consequence of (f),

vn ≥ RvΩn = wn ≥ R1E .

Letting n → ∞ (see also (11.28)), we get inf un ≥ W := lim wn ≥ R1E n→∞

n∈N

on G.

(11.30)

By (b) and (e) above, it follows that, on each component of G \ Ω, W is L-harmonic unless it is −∞. From (11.30) we infer that the latter is impossible. Hence, W = W on G\Ω whence, taking lower semicontinuous regularization in (11.30), one obtains 1 = ≥ R R1E = inf un ≥ W R1E E n

on G \ Ω. This proves W = R1E on G \ Ω. We claim that wn ≥ (1 − ε) R1Ωn .

(11.31) +

Indeed, the function w n := wn /(1 − ε) belongs to S (G) and, by the definition 1 , so that n > 1 on Ωn . Thus w n ∈ ΦΩ of Ωn , w n R1Ωn = inf v ≤ w n = wn /(1 − ε), 1 v∈ΦΩ n

which is the claimed (11.31). By (g) and (h) above, (11.31) rewrites as follows

Γ ∗ σn ≥ (1 − ε) Γ ∗ νΩn = Γ ∗ (1 − ε) νΩn . By Ex. 13 at the end of the chapter, the above inequality ensures that σn (G) ≥ (1 − ε) νΩn (G). The following facts hold: (i) supp(σn ) ⊆ Ω G for every n ∈ N, (ii) Γ ∗ σn = wn is decreasing, (iii) on G \ Ω, it holds wn → W = R1E = Γ ∗ νE (by the definition of νE ).

(11.32)

11.8 Further Results. A Miscellanea

523

Hence, we are in the position to apply Ex. 14 at the end of the chapter to derive that νE (G) = limn σn (G). By (11.32), we get νE (G) ≥ (1 − ε) lim sup νΩn (G).

(11.33)

n→∞

Moreover, from part (a) of the proof, E ⊆ Ω, Lemma 11.6.3-(vi) and the definition of νE , we get Γ ∗ νΩ = R1Ω ≥ R1E = Γ ∗ νE . Hence (by another application of Ex. 13 at the end of the chapter) νΩ (G) ≥ νE (G). This inequality, together with (11.33) implies νΩ (G) ≥ (1 − ε) lim sup νΩn (G).

(11.34)

n→∞

Finally, by (11.29), part (b’) of the proof (for Ω), (11.34), again part (b’) of the proof (this time for Ωn ) and Ωn ⊇ E (whence C(Ωn ) ≥ C ∗ (E)), we obtain C ∗ (E) + ε > C(Ω) = νΩ (G) ≥ νE (G) ≥ (1 − ε) lim sup νΩn (G) n→∞

= (1 − ε) lim sup C(Ωn ) ≥ (1 − ε) C ∗ (E). n→∞

Passing to the limit as ε → 0+ , we get C ∗ (E) = νE (G), which we aimed to prove. Proposition 11.8.2 (On the inner and outer L-capacity). Let E ⊂ G be a bounded set. Then we have: (i) C∗ (E) = sup{μ(E) | μ ∈ M(E), Γ ∗ μ ≤ 1 on G}, (ii) C ∗ (E) = inf{μ(G) |μ ∈ M(G), Γ ∗ μ ≥ 1 on E except for a L-polar set}. Proof. (i). Let μ ∈ M(G) with K := supp(μ) ⊆ E and Γ ∗ μ ≤ 1 on G. Let + also v ∈ S (G) be such that v ≥ 1 on K. From Maria–Frostman Theorem 11.3.4, 1 arbitrary, we get R1 ≥ Γ ∗ μ. it follows v ≥ Γ ∗ μ on G. Hence, being v ∈ ΦK K By passing to the lower semicontinuous regularization, we infer (denoting the Lcapacitary distribution of K by νK ) Γ ∗ νK = VK = R1K ≥ Γ ∗ μ on G. By Ex. 13 at the end of the chapter, this yields μ(G) ≤ νK (G), whence (being μ(G) = μ(E), νK (G) = C(K)) μ(E) = μ(K) ≤ νK (G) ≤ sup C(K) : K ⊆ E, K compact = C∗ (E). Passing to the supremum over the μ’s as above, we get the “≥” sign in assertion (i). On the other hand, for every compact subset K of E, it holds (see Theorem 11.5.6)

524

11 L-capacity, L-polar Sets and Applications

C(K) = max{μ(K) = μ(E) | μ ∈ M(K), Γ ∗ μ ≤ 1 in G} ≤ sup{μ(E) | μ ∈ M(E), Γ ∗ μ ≤ 1 on G}. Finally, passing to the supremum over all possible compact subsets K of E, by definition of C∗ (E), we get the remaining “≤” sign. Thus (i) is proved. (ii). Let μ ∈ M be such that Γ ∗ μ ≥ 1 on E \ F , where F ⊆ E is an L-polar set. By Corollary 11.2.5 (being F ⊆ E and E bounded), there exists μ0 ∈ M0 such that Γ ∗ μ0 ≡ ∞ on F . Let ε > 0. Consider the measure μ1 :=

ε μ0 . 2 μ0 (G)

Then μ1 (G) = ε/2, μ1 ∈ M0 and Γ ∗ μ1 =

ε Γ ∗ μ0 ≡ ∞ 2 μ0 (G)

on F .

Obviously (by the minimum principle for S(G)), we have Γ ∗ μ1 > 0 on G (otherwise μ1 ≡ 0, i.e. μ0 ≡ 0 contradicting Γ ∗ μ0 ≡ ∞ on F ). Let ξ ∈ E. Then we have 0+∞ if ξ ∈ F , > 1. (Γ ∗ μ)(ξ ) + (Γ ∗ μ1 )(ξ ) ≥ 1 + (Γ ∗ μ1 )(ξ ) if ξ ∈ E \ F , Consequently, setting μ2 := μ + μ1 , we have Γ ∗ μ2 = Γ ∗ μ + Γ ∗ μ1 > 1 on E. Being Γ ∗ μ2 l.s.c., for every ξ ∈ E there exists an open neighborhood Uξ of ξ such that Γ ∗ μ2 > 1 on Uξ . Then the set U := ξ ∈E Uξ is an open neighborhood of E such that Γ ∗ μ2 > 1 on U . It is also not restrictive to suppose that U is bounded (for E is). Then Γ ∗ μ2 ∈ ΦU1 , so that Γ ∗ μ2 ≥ R1U = R1U = Γ ∗ νU . Here, in the first equality we used Corollary 11.6.6, whereas in the second equality we used Theorem 11.8.1-(a) and the notation therein. So, we have Γ ∗ μ2 ≥ Γ ∗ νU , whence (see Ex. 13 at the end of the chapter) μ2 (G) ≥ νU (G).

(11.35)

Now, from Theorem 11.8.1-(b) we have νU (G) = C ∗ (U ) = C(U ) (since U is open), whereas (by the definition of μ2 and μ1 ) μ2 (G) < μ(G) + ε. Hence (11.35) yields C ∗ (E) ≤ C(U ) < μ(G) + ε. Due to the arbitrariness of ε, we get C ∗ (E) ≤ μ(G). Hence, we have proved the “≤” sign in assertion (ii). On the other hand, since E is a bounded set, it obviously holds C ∗ (E) = inf{C(Ω) | Ω open and bounded, Ω ⊇ E}.

(11.36)

11.8 Further Results. A Miscellanea

525

So, let Ω be open and bounded with Ω ⊇ E. By Theorem 11.8.1, there exists νΩ ∈ M such that R1Ω = Γ ∗ νΩ and C(Ω) = νΩ (G). Hence (see Corollary 11.6.6) 1 Γ ∗ νΩ = RΩ = 1 on U ⊇ E (see Lemma 11.6.3-(iii)), so that inf{μ(G) |μ ∈ M(G), Γ ∗ μ ≥ 1 on E except for a L-polar set} ≤ νΩ (G) = C(Ω). Passing to the infimum over the above Ω’s, from (11.36) we get the “≥” sign in assertion (ii). This ends the proof. +

Proposition 11.8.3. Let Z, E ⊆ G, and let Z be L-polar. Let also u ∈ S (G). Then RuE = RuE\Z . Proof. From Lemma 11.6.3-(vi), we have RuE ≥ RuE\Z . We are thus left to prove the +

reverse inequality. Let x0 ∈ / Z. By Theorem 11.2.3, there exists v ∈ S (G) such that + u (i.e. w ∈ S (G) and w ≥ u on E \ Z). v(x0 ) ∈ R and v ≡ ∞ on Z. Let w ∈ ΦE\Z +

The function w + ε v belongs to S (G) and is clearly ≥ u on E. Then RuE ≤ RuE ≤ w + ε v

on G.

Evaluating this at x0 and letting ε → 0+ , one gets RuE (x0 ) ≤ w(x0 ). Being x0 ∈ u G \ Z arbitrary, we infer RE ≤ w on G \ Z, so that10 we obtain RuE ≤ w on G. u , this gives Taking the infimum over w ∈ ΦE\Z RuE ≤ RuE\Z , then taking lower u u RE\Z . semicontinuous regularization, we get RE ≤ Proposition 11.8.4 (Characterization of L-polarity. II). Let E ⊆ G be any set. Then the following statements are equivalent: (a) E is L-polar, (b) There exists u ∈ S(G), u strictly positive on G, such that RuE ≡ 0, + RuE ≡ 0. (c) For every u ∈ S (G), it holds Proof. (a) ⇒ (c). See Ex. 7 at the end of the chapter. (c) ⇒ (b). Take u ≡ 1 and apply directly (c). (b) ⇒ (a). Let u be as in (b) above. Since, by Theorem 11.6.5, RuE differs from u RE at most on an L-polar set, they are equal almost everywhere on G. Hence, by hypothesis (b), RuE = 0 a.e. Let x0 ∈ G be such that RuE (x0 ) = 0. By the very + a sequence {vn }n∈N in S (G) such that vn ≥ u on definition of RuE (x0 ), there exists E and vn (x0 ) ≤ 2−n . Set v := n∈N vn . Being v(x0 ) ≤ 1 and vn ≥ u > 0, we + have v ∈ S (G) (see Theorem 8.2.7, page 403). Moreover, v ≡ ∞ on E, since, for every e ∈ E, we have vn (e) ≥ u(e) > 0. Hence, by definition, E is L-polar and (a) follows. 10 Recall that L-polar sets have zero Lebesgue measure, and any L-superharmonic function

f ∈ S(Ω) satisfies f (x) = lim infΩy→x f (y) for every x ∈ Ω.

526

11 L-capacity, L-polar Sets and Applications

We have the following result, concerning with the “smallness” of L-polar sets w.r.t. the L-harmonic measures (in the classical context of Laplace’s operator, see [AG01, Theorem 5.1.9]). Proposition 11.8.5. Let O ⊆ G be an open set, and let u ∈ S(O) be locally bounded. If μu denotes the L-Riesz measure of u on O, then μu (Z) = 0 for every Borel L-polar set Z ⊆ O. Proof. We only need to show that μu (Z ∩ B) = 0 for every d-ball B with B ⊂ O. First, we claim that there exists a compactly supported Radon measure ν on G such that GB ∗ ν = ∞ on Z ∩ B. Indeed, let w ∈ S(G) be such that w|Z∩B ≡ ∞. The lower semicontinuity of w proves that w is bounded from below on B. Hence, by Riesz representation Theorem 9.4.7 (page 443), we have w = GB ∗ μw + h on B, for a suitable h ∈ H(B). Since h is finite-valued, this proves the claim. Let α = infB u (α ∈ R since u is l.s.c.). Again from Riesz representation Theorem 9.4.7 applied to the function w = u − α (which is non-negative on B, hence endowed with an L-harmonic minorant on B), we infer the existence of h ∈ H(B) such that (11.37) u(x) − α = (GB ∗ μu )(x) + h(x) for every x ∈ B. (We used μw = μu .) Since h is the greatest L-harmonic minorant of w ≥ 0 on B, it holds h ≥ 0, so that (11.37) gives (GB ∗ μu )(x) ≤ u(x) − α

for every x ∈ B.

(11.38)

As a consequence, we have ∞ · μu (Z ∩ B) ≤ (GB ∗ ν)(y) dμu (y) = (GB ∗ μu )(y) dν(y) B B (see (11.38)) ≤ sup u − α · ν(G) < ∞ (by the local boundedness of u). B

(In the equality sign, we used Fubini–Tonelli’s theorem jointly with the symmetry of GB .) This gives μu (Z ∩ B) = 0, ending the proof. The following proposition shows some more “smallness” properties of the L-polar sets. Proposition 11.8.6. The following assertions hold: (i) If A ⊆ G is an open connected set and E ⊂ A is L-polar and relatively closed in A, then A \ E is connected; (ii) If A ⊆ G is open (and non-empty) and ∂A is L-polar, then A is connected and dense in G. Proof. (i). With the notation of the assertion, let A0 be a connected component of A \ E. We are done if we show that A0 = A \ E. Note that, since E is relatively closed in A, then A \ E is open, whence A0 is open too. Since E is L-polar, there exists u ∈ S(G) such that u ≡ ∞ on E. Set

11.9 Further Reading and the Quasi-continuity Property

v : A → (−∞, ∞],

v(x) =

u(x) ∞

527

if x ∈ A0 , if x ∈ A \ A0 .

:= (A \ E) \ A0 is an open set since it coincides with the union of the Note that A connected components of the open set A \ E, except for A0 . It is easily seen that v is lower semicontinuous, not identically ∞ and L-supermean on A, since u|A0 is L and this can happen superharmonic. Then v ∈ S(A). But v ≡ ∞ on the open set A, = ∅, for L-superharmonic functions are locally integrable. This proves that only if A A contains no connected components other than A0 , thus completing the proof of (i). (ii). Since ∂A is closed in G (connected), by (i) we have G \ ∂A connected. If by contradiction G \ A = ∅, then G \ ∂A = A ∪ (G \ A) is the union of two open disjoint and non-empty sets, i.e. G \ ∂A is not connected (contradicting the above argument). This proves that A = G. Moreover, the above argument also gives G \ ∂A = A ∪ (G \ A) = A, so that A is connected, for G \ ∂A is.

11.9 Further Reading and the Quasi-continuity Property The aim of this section is to sketch a brief investigation of the so-called quasicontinuity property of L-subharmonic functions. As we have seen in Section 8.3 following [JLMS07], a function u which is subharmonic w.r.t. every sub-Laplacian on G is v-convex (even under the assumption of upper semicontinuity instead of continuity in the definition of v-convexity) and has fine regularity properties. On the other hand, if a function is superharmonic with respect to only one sub-Laplacian L, it nonetheless possesses some (weaker) continuity property. Indeed, the following quasi-continuity property holds (see Theorem 11.9.1 below for the precise statement): (Q-C) Every L-superharmonic function is continuous, if restricted to the complement of an open set with arbitrarily small L-capacity. The quasi-continuity of superharmonic functions has important applications in the theory of PDE’s, where it is often desirable to know the accurate pointwise behavior of the relevant super- and sub-solutions and their fine properties. In the classical case of the Laplace operator on G, the (Q-C) property was proved by H. Cartan [Cart28]. Proofs of the (Q-C) property for more general operators are nowadays available: see [HKM93, Theorem 10.9] for a class of non-linear elliptic operators; see [MV97b, Theorem 14.3] for a class of subelliptic operators generalizing the latter; see [TW02b, Theorem 4.1] for a wide class of quasilinear elliptic and subelliptic operators. The last two results also cover the case of sub-Laplacians. Nonetheless, in the significant case of Carnot groups, a direct and simpler approach can be applied in proving (Q-C). See, for instance, [BC05]. In particular, the stratified context allows us to recover several fine results on the theory of the energy for L, i.e. of the integral

528

11 L-capacity, L-polar Sets and Applications

Γ (x, y) dμ(x)dμ(y)

(where μ is a Radon measure on G).

The proof of the (Q-C) property in [BC05] is in the spirit of this book and differs from those in [HKM93,MV97b,TW02b]. Indeed, the approach of these papers (due to the considered wider classes of operators) requires highly non-trivial techniques such as deep weak-convergence results, a suitable theory of the relevant Sobolev-type function spaces, some results from the theory of variational inequalities, a quantitative use of Caccioppoli-type and Harnack inequalities (both weak and strong), Hölder estimates for weak solutions, etc. In the remainder of this section, we sketch a line of the proof of the following result, as approached in the cited paper [BC05]. Theorem 11.9.1 (Quasi-continuity of L-superharmonic functions). Let u ∈ S(Ω). Then, for every ε > 0, there exists an open set Oε ⊆ Ω with C(Oε ) < ε and such that the restriction of u to Ω \ Oε is continuous on Ω \ Oε . Following Cartan’s theory of energy and the related ideas in the classical potential theory, we give the following definition (we recall that M denotes the set of the Radon measures on G). Definition 11.9.2 (Mutual L-energy). Let μ, ν ∈ M. The mutual L-energy of μ and ν is μ, ν = Γ ∗ μ dν. G

The quantity μ

2 :=

μ, μ will be called the L-energy of μ ∈ M. Finally, we set E + := {μ ∈ M : μ < ∞}.

We notice that μ on page 497.

2

= I (μ) if μ ∈ M0 , where I (μ) is as in Definition 11.4.1

Proposition 11.9.3. For every x ∈ G and r > 0, the measure λx,r introduced in (9.9) (page 436) belongs to E + and λx,r 2 = r 2−Q . Proof. Let x ∈ G and r > 0. Then, by (9.9) and (9.10a), we have λx,r 2 = Γ (z−1 ◦ x) dλx,r (z) + r 2−Q dλx,r (z) G\B d (x,r) B d (x,r) = 0 + r 2−Q k(x, z) dH N −1 (z) = r 2−Q mr [1](x) = r 2−Q . ∂Bd (x,r)

This completes the proof. The L-polarity is related to the set E + , as assertion (i) of the following theorem states (see [BC05, Lemma 6.1 and Theorem 6.1]).

11.9 Further Reading and the Quasi-continuity Property

529

Theorem 11.9.4. The following assertions hold: (i) Let E be a Borel set. Then E is L-polar iff μ(E) = 0 for every μ ∈ E + ; (ii) (L-energy principle) μ, ν ≤ μ · ν for every μ, ν ∈ M. We next give a new definition. We denote by E the set of formal11 differences μ − ν, μ, ν ∈ E + . By a slight abuse of notation, the mutual L-energy and the Lenergy are defined as follows μ1 − ν1 , μ2 − ν2 = μ1 , μ2 − μ1 , ν2 − ν1 , μ2 + ν1 , ν2 , μ − ν 2 = μ 2 − 2μ, ν + ν 2 . Thanks to the L-energy principle in Theorem 11.9.4-(ii), we have μ, ν ≤ μ · ν for every μ, ν ∈ E. It can be proved that ·, · defines an inner product on E. We need many preliminary results. Proposition 11.9.5. For every x ∈ G and every r > 0, τx,r := λx,r/2 − λx,r belongs to E and τx,r 2 = (r/2)2−Q − r 2−Q . Proof. We fix x ∈ G, r > 0. By Proposition 11.9.3, τx,r ∈ E. By (9.10a), λx,r/2 , λx,r = Γ (z−1 ◦ x) dλx,r (z) G\B d (x,r/2) 2−Q + (r/2) dλx,r (z) = r 2−Q , B d (x,r/2)

so that λx,r/2 − λx,r λx,r/2

2

2

equals

+ λx,r

2

− 2λx,r/2 , λx,r = (r/2)2−Q − r 2−Q .

This ends the proof. We now introduce a set of functions which play a prominent rôle in the theory of L-energy. We set T = {Tx,r := Γ ∗ λx,r/2 − Γ ∗ λx,r | x ∈ G, r > 0}.

(11.39)

In the following definition, we agree to denote by C0+ (G) the set of non-negative compactly supported functions on G. G)). A set T ⊆ C0+ (G) is called total if it has Definition 11.9.6 (Total set in C0+ (G the following property: for every C G, for every open set W ⊃ C, for all ε > 0 and for all v ∈ C0+ (G) supported in C, there exist β1 , . . . , βn > 0, f1 , . . . , fn ∈ T supported in W such that v − ni=1 βi fi ∞ < ε. 11 More precisely, E is the set of the couples (μ, ν) ∈ E + × E + modulo the equivalence

relation (μ1 , ν1 ) ∼ (μ2 , ν2 ) iff μ1 + ν2 = μ2 + ν1 .

530

11 L-capacity, L-polar Sets and Applications

We have an important result, a G-version of a theorem due to Cartan [Cart28]: Theorem 11.9.7 (Bonfiglioli and Cinti, [BC05], Theorem 4.3). Suppose T ⊆ C0+ (G) has the following properties: (1) for every f ∈ T and every α ∈ G, it holds f (α ◦ ·) ∈ T , (2) for every x ∈ G and every δ > 0, there exists f ∈ T , f ≡ 0, such that f vanishes outside Bd (x, δ). Then T is a total set (according to Definition 11.9.6). From Theorem 11.9.7 and Ex. 10 at the end of this chapter we deduce the following remarkable assertion. Corollary 11.9.8. The set T in (11.39) is a total set of functions in C0+ (G). We can now prove the theorem below. Theorem 11.9.9. The square-root · of the L-energy is a norm on E, and E + is a complete subset of the normed space (E, · ). Proof. We only prove the first part of the assertion. The second part is proved in [BC05, Theorem 6.3] by delicate results on the weak and weak∗ convergence in E. We explicitly remark that (E, · ) is not complete even in the classical Euclidean case of Laplace’s operator (see [Cart28, p. 87]). We must show that μ − ν ∼ 0 if μ − ν, μ, ν ∈ E + satisfy μ − ν = 0. We have |λ, μ − ν| ≤ λ · μ − ν = 0, i.e. λ, μ = λ, ν for every λ ∈ E. In particular, this holds for λ = λx,r/2 − λx,r (see Proposition 11.9.5). Thus, λx,r/2 , μ − λx,r , μ = λx,r/2 , ν − λx,r , ν, i.e. (Γ ∗ λx,r/2 − Γ ∗ λx,r ) dμ = (Γ ∗ λx,r/2 − Γ ∗ λx,r ) dν G

G

for every x ∈ G and every r > 0. By Corollary 11.9.8, the functions Γ ∗ λx,r/2 − Γ ∗ λx,r form a total set in C0+ (G), so that (invoking Ex. 11 at the end of this chapter) μ = ν, and the theorem is proved. In order to prove our (Q-C) property, we need the following lemma (whose proof follows by collecting together Lemmas 7.1, 7.2, 7.3 in [BC05]). Lemma 11.9.10. The following assertions hold: (i) Let μ ∈ E + . Then there exists a sequence μj ∈ E + converging to μ in the Lenergy norm with the following properties: supp(μj ) is compact, Γ ∗μj ≤ Γ ∗μ and Γ ∗ μj ∈ C(G, R) for every j ∈ N. (ii) Let μ, ν ∈ E + . Given ε > 0, we set E = {x ∈ G | (Γ ∗ μ)(x) > (Γ ∗ ν)(x) + ε}. Then C ∗ (E) ≤ ε −2 μ − ν 2 .

11.9 Further Reading and the Quasi-continuity Property

531

(iii) Let μ ∈ M be such that μ(G) ≤ m ∈ R. Then, for every ε > 0, the L-capacity of the set A = {x ∈ G | (Γ ∗ μ)(x) > ε} does not exceed m/ε. The key-rôle in the proof of the (Q-C) property is played by the following lemma. Lemma 11.9.11 (Quasi-continuity for Γ -potentials). Let μ ∈ M with Γ ∗ μ ≡ ∞. Then, for every ε > 0, there exists an open set O ⊆ Ω with C(O) < ε such that the restriction of Γ ∗ μ to Ω \ O is continuous. Proof. We split the proof in three steps. Throughout the proof, ε > 0 is fixed. (I) μ ∈ E + . For any n ∈ N, by Lemma 11.9.10-(i), there exists μn ∈ E + with compact support such that Γ ∗μn ≤ Γ ∗μ, Γ ∗μn ∈ C(G, R) and μn −μ 2 < ε/8n . We consider En := {x ∈ G | (Γ ∗ μ)(x) > (Γ ∗ μn )(x) + 1/2n }. En is an open set, hence L-capacitable. By Lemma 11.9.10-(ii), we have C(En ) ≤ 4n μ − μn 2 < ε/2n . As a consequence, the L-capacity of E = ∪n En does not exceed ε. Set F = G \ E. It is easily seen that Γ ∗ μn converges uniformly to Γ ∗ μ on F . Since Γ ∗ μn ∈ C(G, R), this proves that (Γ ∗ μ)|F is continuous on F . (II) μ ∈ M0 . Let p ∈ N be such that μ(G)/p < ε/2. Consider the function min{Γ ∗ μ, p}. By Proposition 9.9.1-(b) (page 464), we have min{Γ ∗ μ, p} = Γ ∗ ν for some ν ∈ M. Moreover, by applying Lemma 11.9.10-(iii) with m = μ(G), it follows C(A) < ε/2, where A = {x ∈ G | (Γ ∗μ)(x) > p}. Obviously, Γ ∗ν ≡ Γ ∗μ on G \ A. Now, ν ∈ E + . Indeed, ν 2 ≤ Γ ∗ μ dν = Γ ∗ ν dμ ≤ p μ(G) ∈ R. Hence, by the part (I) of the proof, there exists an open set B with C(B) < ε/2 such that (Γ ∗ ν)|G\B is continuous on G \ B. Finally, the open set O = A ∪ B has L-capacity less than ε and satisfies the assertion of the lemma. Indeed, (Γ ∗ μ)|F is continuous on F = G \ O since Γ ∗ μ ≡ Γ ∗ ν on G \ A ⊇ F , and since (Γ ∗ ν)|G\B is continuous on G \ B ⊇ F . (III) μ ∈ M with Γ ∗ μ ≡ ∞. For j ∈ N, we denote Cj = {x ∈ G| j − 1 ≤ d(x) < j } ∞ and μj = μ|Cj . It then follows that Γ ∗ μ = j =1 Γ ∗ μj . Since supp(μj ) ⊆ Cj G, we can apply the part (II) of the proof. Thus, let Oj be an open set with C(Oj ) < ε/2j and such that (Γ ∗ μj )|Fj is continuous on Fj = G \ Oj . Let us prove that the set O = ∞ j =1 Oj satisfies the assertion of the lemma. Indeed, C(O) < ε and (for F = G \ O) (Γ ∗ μ)|F

is continuous on F .

To see this, we fix x0 ∈ F , n0 ∈ N such that x0 ∈ Cn0 and an open neighborhood W of x0 with W { n0 − 2 < d(x) < n0 }. Then

532

11 L-capacity, L-polar Sets and Applications

Γ ∗μ=

n 0 −2

Γ ∗ μj + Γ ∗ μn0 −1 + Γ ∗ μn0 +

j =1

∞

Γ ∗ μj .

j =n0 +1

Now, for j = 1, . . . , n0 −2, it holds Γ ∗μj ∈ H({d(x) > n0 −2}), since supp(μj ) ⊆ {d(x) ≤ n0 − 2}. Moreover, for any j ≥ n0 + 1, we have supp(μj ) ⊆ {d(x) ≥ n0 }, whence Γ ∗ μj ∈ H({d(x) < n0 }). As a consequence, the function ∞ j =n0 +1 Γ ∗ μj is L-harmonic near x0 since it is majorized by Γ ∗ μ ≡ ∞. Finally, (Γ ∗ μj )|F is continuous at x0 for all j ∈ N, since F ⊆ Fj and Γ ∗ μj ≡ (Γ ∗ μj )|Fj on F . This ends the proof. With Lemma 11.9.11 at hand and a Riesz-representation argument, we can easily prove the quasi-continuity of arbitrary L-superharmonic functions. Proof (of Theorem 11.9.1). We consider a sequence {Ωj }j ∈N of open sets exhausting Ω with Ω j Ω, and we let μj = (μu )|Ωj . Then μj ∈ M0 . Thus, by Lemma 11.9.11, for given ε > 0, there exists an open set Oj such that and (Γ ∗ μj )|Fj is continuous on Fj = G \ Oj . Then O = Ω ∩ j Oj is an open subset of Ω whose L-capacity does not exceed ε. Finally, u|F is continuous on F = Ω \ O. Indeed, if x0 ∈ F , then x0 ∈ Ωj0 for a suitable j0 ∈ N. Now, by Riesz representation Theorem 9.4.4 on page 442, there exists h ∈ H(Ωj0 ) such that u = h + Γ ∗ μj0 on Ωj0 . Since F ⊆ Fj0 , this completes the proof. C(Oj ) < ε/2j

Bibliographical Notes. The topics developed in this chapter within the classical theory of Laplace’s operator can be found in all the main monographs devoted to potential theory; see the references in the Bibliographical Notes of Chapter 8. We also mention the following references. The paper by N.S. Trudinger and X.-J. Wang [TW02b] (and references therein) where several results of potential theory are established for a class of quasilinear subelliptic operators including sub-Laplacians; see also the series of papers by the same authors [TW97,TW99,TW02a] where fully non-linear Hessian operators are treated; see D. Labutin [Lab02] for the study of subharmonic functions related to this last class of operators; see the axiomatic non-linear potential theory contained in the monograph by J. Heinonen, T. Kilpelainen and O. Martio [HKM93] and in I.G. Markina and S.K. Vodopyanov [MV97a,MV97b], in a more general subelliptic context. For recent results on capacity, see [AH96,CDG96,Cart45,DG98,Hei95b,Ne88, KR87]. Some of the topics presented in this chapter also appear in [BC05,BC04].

11.10 Exercises of Chapter 11

533

11.10 Exercises of Chapter 11 Ex. 1) Prove that C({x ∈ G : Γ (x) ≥ 1/r}) = r. (Hint: Let K := {x ∈ G : Γ (x) ≥ 1/r}. Compute Γ ∗ μ(0) = Γ =1/r Γ (y) dμ(y), where μ is the L-capacitary distribution of K and recall that μ is supported in ∂K.) Ex. 2) Prove that for any compact set K, we have C(δλ (K)) = λQ−2 C(K),

C(α ◦ K) = C(K)

for every λ > 0 and every α ∈ G. (Hint: Use Theorem 11.5.6.) Ex. 3) Prove Ex. 11.2.2, page 491. Ex. 4) Show that C(K) = sup K K

1/(2−Q) Γ (x, y) dμ(x) dμ(y) μ ∈ M, μ(K) ≥ 1

if K ⊂ G is compact. Ex. 5) Re-derive Theorem 11.5.11 (page 507) as a consequence of Proposition 11.8.4. Ex. 6) Show that the hypothesis “u bounded from above” cannot be removed in the extended maximum principle of Theorem 11.2.7. (Hint: Take u = Γ in Ω = Bd (0, 1) \ {0} and E = {0}.) + Ex. 7) Let Z ⊂ G be L-polar. Let u ∈ S (G). Show that u(x) if x ∈ Z, RuZ = 0 otherwise. +

Derive RuZ ≡ 0. (Hint: Let y ∈ / Z and v ∈ S (G) be such that v(y) ∈ R and v|Z ≡ ∞. Then v/n ∈ ΦZu for every n ∈ N, so that. . . .) Ex. 8) (Reciprocity law.) Prove that: If Ω is an open subset of G with L-Green functionGΩ and μ and ν are (positive) measures on Ω, then Ω GΩ ∗ μ dν = Ω GΩ ∗ ν dμ. (Hint: Use Fubini–Tonelli’s theorem and the symmetry of the GΩ .) Ex. 9) Following the notation in Definition 11.9.2, prove that: • μ, ν is well defined; • it holds μ, ν = ν, μ (use the reciprocity law); • if μ ∈ M0 and Γ ∗ μ is bounded then μ ∈ E + ; • if μ ∈ E + and λ ∈ M satisfies Γ ∗ λ ≤ Γ ∗ μ, then λ ∈ E + and λ ≤ μ . Ex. 10) Following (11.39) (page 529) and using Theorem 9.3.10 (page 438), prove that: • the generic element Tx,r (z) of T in (11.39) is a non-negative and continuous function of z ∈ G; • it vanishes outside Bd (x, r), it equals (r/2)2−Q − r 2−Q on Bd (x, r/2) and it equals Γ (z−1 ◦ x) − r 2−Q in Bd (x, r) \ Bd (x, r/2);

534

11 L-capacity, L-polar Sets and Applications

•

show that, for every α ∈ G, the function z → Tx,r (α ◦ z) still belongs to T : precisely, it holds Tx,r (α ◦ z) = Tα −1 ◦x,r (z). Ex. 11) Following Definition 11.9.6 (page 529), prove that if μ1 , μ2 ∈ M satisfy f dμ = f dμ for every f in a total set T , then μ1 = μ2 . 1 2 G G Ex. 12) Let u : Bd (0, 1) → R be an L-harmonic function. Let us denote by μ the L-harmonic measure of Bd (0, 1) at x0 = 0. Then the function λ → |u(δλ (x))|p dμ(x), 1 ≤ p < ∞, ∂Bd (0,1)

is monotone increasing for 0 < r < 1. (Hint: x → |u(x)|p is Lsubharmonic. Then use the Poisson–Jensen formula.) Ex. 13) Let μ, ν be Radon measures in G and suppose that Γ ∗ μ ≤ Γ ∗ ν. Then μ(G) ≤ ν(G). Hint: It suffices to prove that μ(B(0, R)) ≤ ν(G) for every R > 0. Let K := B(0, R). As usual, μK denotes the L-capacitary distribution for K. Recall that VK = Γ ∗ μK = 1 on the interior of K, i.e. on B(0, R). Then we have μ(B(0, R)) = 1dμ = Γ ∗ μK dμ ≤ Γ ∗ μK dμ B(0,R) B(0,R) G = Γ ∗ μ dμK ≤ Γ ∗ ν dμK = Γ ∗ μK dν G G G = VK dν ≤ 1 dν = ν(G). G

G

Indeed, recall that VK ≤ 1 on G (see (11.16a)). Ex. 14) Let ν, σn be Radon measures on G (for every n ∈ N) with the following properties: (i) There exists a compact set K such that supp(σn ) ⊆ K, ∀ n ∈ N; (ii) Γ ∗ σn is decreasing; (iii) on G \ K, it holds Γ ∗ σn −→ Γ ∗ ν. Then, ν(G) = limn→∞ σn (G). Hint: Let A := B(0, r) with r 1 such that K ⊂ B(0, r). Since Γ ∗ σn ∈ H(G \ K) and {Γ ∗ σn }n is decreasing and non-negative, it holds Γ ∗ ν ∈ H(G \ K), whence supp(ν) ⊆ K. Let μA denote the Lcapacitary distribution for A. Note that Γ ∗ νA = 1 on B(0, r) ⊃ K and supp(μA ) ⊆ ∂A ⊂ G \ K. Thus Fubini’s theorem and monotone (decreasing) convergence imply 1 dσn = Γ ∗ μA dσn = Γ ∗ μA dσn = Γ ∗ σn dμA σn (G) = K G K G = Γ ∗ σn dμA → Γ ∗ ν dμA = Γ ∗ ν dμA supp(μA ) supp(μA ) G = Γ ∗ μA dν = Γ ∗ μA dν = 1 dν = ν(G). G

K

K

11.10 Exercises of Chapter 11

535

+

Ex. 15) Let u ∈ S (G). Let, for n ∈ N, Kn ⊆ Kn+1 be compact subsets of G. Suppose Ω := n∈N Kn is open. Then RuKn = RuΩ . lim

n→∞

(Hint: Provide details for the following arguments. { RuKn }n is a decreasing + + sequence in S (G). Set v := limn→∞ RuKn . Then v ∈ S (G) and v ≤ u on G. Moreover, v ≤ RuKn ≤ RuΩ

()

on G.

u Being RuKn = RuKn up to an L-polar subset of ∂Kn , say Pn , we get RKn (x) = / Pn u(x) for every x ∈ Kn \ Pn . Let P := n∈N Pn . If x ∈ Ω \ P , then x ∈ for every n ∈ N, and there exists n0 ∈ N such that x ∈ Kn0 ⊆ Kn for every n ≥ n0 . Hence RuKn (x) = u(x) ∀ n ≥ n0 . Letting n → ∞, we derive v(x) = u(x) for every x ∈ Ω \ P . Hence (recall u , so that v ≥ Ru . By (), this that P is L-polar) v = u on Ω. Thus v ∈ ΦΩ Ω u gives v = RΩ .) Ex. 16) In this exercise, we state the main properties of the exterior L-capacity C ∗ defined in Definition 11.5.14. For the relevant proofs, one can follow verbatim the classical capacitability theory as presented, e.g. in [Helm69]. (1) Starting from the strong sub-additivity property in (11.20) of Proposition 11.5.10, one first shows that if {Ki }1≤i≤m and {Ci }1≤i≤m are finite families of compact sets in G such that Ci ⊂ Ki (i ≤ m), then m m m

C(Ki ) − C(Ci ) . Ki − C Ci ≤ C i=1

i=1

i=1

(2) The second step is to derive from the above that if {Ui }1≤i≤m and {Vi }1≤i≤m are finite families of open sets in G such that Vi ⊂ Ui (i ≤ m), then m m m m Ui + C∗ (Vi ) ≤ C∗ (Ui ) + C∗ Vi . C∗ i=1

i=1

i=1

i=1

(3) From (1) and (2) it is possible to derive that if {Ei }1≤i≤m and {Fi }1≤i≤m are arbitrary finite families of subsets of G such that Fi ⊂ Ei (i ≤ m), then m m m m ∗ ∗ ∗ C Ei + C (Fi ) ≤ C Fi + C ∗ (Ei ). i=1

i=1

i=1

C∗

i=1

is an exterior capacity. We say (4) From (3) it is possible to derive that that a non-negative real-valued function φ ∗ (·) defined on the subsets of G is an exterior capacity if:

536

11 L-capacity, L-polar Sets and Applications

(i) E ⊂ F ⇒ φ ∗ (E) ≤ φ ∗ (F ); (ii) if {Ej }j ⊂ G is such that Ej ⊆ Ej +1 , then φ ∗ ( j Ej ) = limj →∞ φ ∗ (Ej ); (iii) if {Kj }j G is such that Kj +1 ⊆ Kj , then φ ∗ ( j Kj ) = limj →∞ φ ∗ (Kj ). (5) From (4) we infer that C ∗ is numerably sub-additive. Indeed, if {Ej }j ⊆ j }j , where E j := G is a sequence of arbitrary sets, we apply (ii) to {E j i=1 Ei , so that j ) ≤ j = lim C ∗ (E C∗ Ej = C ∗ C ∗ (Ej ), E j

j

j

j

for (apply (3) with empty Fi ’s) j ) = C C (E ∗

∗

j i=1

Ei

≤

j i=1

C ∗ (Ei ) ≤

∞

C ∗ (Ei ).

i=1

(6) As a consequence of (4) and the L-capacitability of open sets and compact sets, we infer that if Ω = j Kj with Ω open and Kj ⊆ Kj +1 compact sets, then C(Ω) = lim C(Kj ). j →∞

12 L-thinness and L-fine Topology

In this chapter, we deal with the notion of fine topology generated by the superharmonic functions related to a sub-Laplacian L. We mainly investigate the relationship between L-thinness and L-regularity of boundary points for the Dirichlet problem. As usual, throughout the chapter G = (RN , ◦, δλ ) is a (homogeneous) Carnot group and L is a sub-Laplacian on G. Moreover, Γ = d 2−Q is the fundamental solution for L.

12.1 The L-fine Topology: A More Intrinsic Tool Since L-superharmonic functions play a central rôle in the potential theory for L, it seems more natural to introduce a new topology—finer than the Euclidean one— with respect to which any L-superharmonic function is continuous. In the sequel, if a function u takes on a ∞-value at a point x0 , we say that u is continuous (in the extended sense) at x0 , provided limx→x0 u(x) = ∞. Moreover, if X is a topological space, A ⊆ X is any subset of X and x ∈ X, we say that x is a limit point of A if, for every open set U containing x, there exists a ∈ A ∩ U such that a = x. We denote by Der(A) (the derived set of A) the set of the limit points of A. Definition 12.1.1 (The L-fine topology). The L-fine topology on G is the smallest topology on RN with respect to which all L-superharmonic functions are continuous (in the extended sense). Throughout the chapter, all topological concepts (continuous, open, neighborhood, etc.) will be prefixed by “L-fine” when related to the L-fine topology. Instead, they will not be prefixed or prefixed by “Euclidean” or “metric”, when related to the usual Euclidean topology of RN . A few simple remarks are in order. Remark 12.1.2. (1) Since L-superharmonic functions are l.s.c., a basis for the L-fine topology is given by all finite intersections of the sets

538

12 L-thinness and L-fine Topology

{x ∈ Ω : u(x) < β }

(12.1)

as Ω ranges over all (Euclidean-)open subsets of RN , u is any function in S(Ω), and β ranges over ]−∞, ∞]. (2) The L-fine topology is properly larger than the Euclidean one (see Example 11.2.4, page 492). (3) In particular, if limx→x0 u(x) = λ in the Euclidean sense, then the same is true in the L-fine sense too, whereas the converse may be false. Vice versa, if {xn }n∈N is a sequence in G converging to x0 ∈ G in the L-fine sense, then xn → x0 in the Euclidean sense too. (4) Hence, the collection of L-fine limit points of a set A is contained (possibly properly) in the set of usual limit points of A, DerL-fine (A) ⊆ DerEucl. A. For example, if x0 ∈ G\{0}, the set of the L-fine limit points of A = {(1+ n1 ) x0 }n∈N is empty whereas x0 is a limit point of A (see also Proposition 12.1.3 below). Proposition 12.1.3. Let Z ⊂ G be an L-polar set. Then the set of the L-fine limit points of Z is empty. In particular, the points of Z are L-finely isolated points of Z. Proof. By contradiction, let x be an L-fine limit point of Z. Then x is also a (Euclidean) limit point of Z. By Example 11.2.4 (page 492), there exists u ∈ S(G) such that u(x) < lim u(y) =: r < ∞. Z y→x

Set ε := 12 (r − u(x)). Then there exists a neighborhood U of x such that u(y) > r − ε for every y ∈ (U ∩ Z) \ {x}. Since u is L-fine continuous at x (being Lsuperharmonic), there is an L-fine neighborhood U fine of x such that u(y) < u(x)+ε for every y ∈ U fine . But r − ε = 12 (r + u(x)) = u(x) + ε, so that (U ∩ Z) ∩ U fine = (U ∩ U fine ) ∩ Z is contained in {x}. This means that U ∩ U fine is an L-fine neighborhood of x having in common with Z at most x, whence x is not an L-fine limit point of Z. This is a contradiction.

12.2 L-thinness at a Point Consider a bounded open set Ω ⊂ G and a point x ∈ ∂Ω. We have seen in Proposition 7.1.5 (page 385) that if G \ Ω contains a suitable non-characteristic ball, then x is regular for the Dirichlet problem related to L. Hence, for L-irregular boundary points of Ω, the complement of Ω must be quite thin (in a suitable sense, which also takes into account the “degeneracy” of the operator L). The notion of L-thinness will give a more precise sense to this. Definition 12.2.1 (L-thinness of a set at a point). A set E ⊆ G is called L-thin at x ∈ G if x is not an L-fine limit point of E.

12.2 L-thinness at a Point

539

In other words, E is L-thin at x

⇔

x∈ / DerL-fine (E)

⇔

{x} ∩ DerL-fine (E) = ∅.

A few simple remarks are in order. Remark 12.2.2. Obviously, E is not L-thin at x iff x is an L-fine limit point of E, i.e. E is not L-thin at x

⇔

x ∈ DerL-fine (E).

Hence, if E is not L-thin at x, then x is a (Euclidean) limit point of E (indeed, DerL-fine (E) ⊆ DerEucl. E). Equivalently, a sufficient condition that a set E is L-thin at x is that x is not a metric limit point of E. From Proposition 12.1.3 we immediately get the following result. Corollary 12.2.3. A L-polar set is L-thin at every point. The following result will be important in the sequel. Theorem 12.2.4 (Characterization of L-thinness at a point. I). A set E ⊆ G is L-thin at a (metric limit) point x of E if and only if there exists an L-superharmonic function u on a (metric) neighborhood of x such that u(x) < lim inf u(y). E y→x

(12.2)

Proof. To avoid heavy notation, it is not restrictive to suppose that x ∈ / E. (⇒): Suppose first that E is L-thin at x. Then there exists an L-fine neighborhood U fine of x such that E∩U fine = ∅. By taking finite intersections of sets as in (12.1), as a basis for L-fine neighborhoods, we infer the existence of β1 , . . . , βn ∈ ]−∞, ∞] and u1 , . . . , un ∈ S(W ), W being an open neighborhood of x, such that U fine = n fine (i.e. u (x) < β ), by choosing ε = i i=1 {y ∈ W : ui (y) < βi }. Since x ∈ U n i fine we can replace U by the set {y ∈ W : ui (y) < ui (x) + ε}. min{βi − ui (x)}, i=1 Now let u := ni=1 ui . We claim that u satisfies (12.2). By the lower semi-continuity of the ui ’s at x, there exists a neighborhood U ⊂ W of x such that ui (y) > ui (x) − ε/n

for every y ∈ U and i = 1, . . . , n.

(12.3)

Now take any y ∈ U ∩ E. Then, since E ∩ U fine = ∅, there exists i0 ∈ {1, . . . , n} such that (12.4) ui0 (y) ≥ ui0 (x) + ε. Collecting together (12.3) and (12.4), we get u(y) =

ui (y) + ui0 (x) >

i=i0

In short, u(y) > u(x) +

i=i0

ε n

ui (x) −

n−1 ε ε + ui0 (x) + ε = u(x) + . n n

for every y ∈ U ∩ E which at once implies (12.2).

540

12 L-thinness and L-fine Topology

(⇐): Suppose now there exists an L-superharmonic function u on a neighborhood Ω of x such that u(x) < lim infE y→x u(y) =: r. We can assume that r < ∞ by replacing, if needed, u by min{u, u(x) + 1}. Set ε = 12 (r − u(x)). Consider U fine := {y ∈ Ω : u(y) < u(x) + ε}, which is an L-fine neighborhood of x. From lim infE y→x u(y) = r we infer that there exists a neighborhood U of x such that infE∩V u > r − ε. Now we argue verbatim as in the last part of the proof of Proposition 12.1.3. Since the value of an L-superharmonic function at a point coincide with its lim inf there, (12.2) gives another equivalent definition of L-thinness: E is L-thin at x if and only if there exists an L-superharmonic function u on a neighborhood of x such that

lim inf u(y) < lim inf u(y). y→x

E y→x

(12.5)

This last condition is the analogue of L-thinness at infinity given in (10.2), page 474. Another remarkable result (making L-thinness easier to handle with) is the following one. Theorem 12.2.5 (Characterization of L-thinness at a point. II). A set E ⊆ G is Lthin at a point x ∈ G (limit point of E) if and only if there exists an L-superharmonic function w on G, which is the L-potential of a compactly supported Radon measure, such that w(x) < lim w(y) = ∞. E y→x

(12.6)

Proof. The “if” part follows from Theorem 12.2.4. To prove the converse, we assume that E is L-thin at x (and, without affecting the generality, we may suppose x ∈ / E). It is enough1 to prove the existence of an open neighborhood Ω of x and u ∈ S(Ω) such that (12.6) holds with w replaced by u. To this end, let u be as in Theorem 12.2.4. We can assume that the right-hand side of (12.2) is finite (otherwise there is nothing to prove). Let B = Bd (x, ε) with ε > 0 small enough. By arguing as in the (footnote of the) previous paragraph, we can suppose that u = Γ ∗ ν, where ν is a Radon measure supported in B. For n large enough, we have 1 Indeed, in this case we can take any open set O with x ∈ O ⊂ O Ω and apply Riesz

representation Theorem 9.4.4 (page 442) to get u(y) = (Γ ∗ ν)(y) + h(y)

∀ y ∈ O,

where ν = μu |O and h is a suitable function in H(O). Hence, we have lim u(y) = lim (Γ ∗ ν)(y) + h(x),

E →x

E →x

whereas u(x) = (Γ ∗ ν)(x) + h(x).

Consequently, (12.6) holds if and only if (Γ ∗ ν)(x) < limE →x (Γ ∗ ν)(y), and the choice w := Γ ∗ ν fulfills the requirement of the assertion of the theorem.

12.2 L-thinness at a Point

Γ (x, y) dν(y) + Bd (x,1/n)

541

Γ (x, y) dν(y) = u(x) ∈ R, B\Bd (x,1/n)

and the second integral in the left-hand side increases2 to u(x) as n ↑ ∞. As a consequence, setting νn := ν|Bd (x,1/n) , wehave limn→∞ (Γ ∗ νn )(x) = 0. This ensures the existence of εn ↓ 0 such that n un (x) < ∞, where un := Γ ∗ νεn . Then the function v := n un is finite at x and belongs to S(B). Let δ := lim infE y→x (u(y) − u(x)) > 0. By applying once again the Riesz representation theorem, we get u = un + hn , where hn ∈ H(Bd (x, εn )). This gives lim inf (un (y) − un (x)) ≥ lim inf (un (y) + hn (y)) + lim inf (−hn (y) − un (x))

E y→x

E y→x

E y→x

= lim inf u(y) − hn (x) − un (x) = lim inf (u(y) − u(x)) = δ E y→x

E y→x

for every n ∈ N, so that (by summing up) p p lim inf un (y) − un (x) ≥ p δ. E y→x

n=1

p

n=1

p

This gives lim infE y→x n=1 un (y) ≥ n=1 un (x) + p δ ≥ p δ. Since v = p u ≥ u , we derive (letting p → ∞) lim infE y→x v(y) = ∞. n n n=1 n From Theorem 12.2.5 one immediately obtains the following assertion. Corollary 12.2.6. Let Z, E ⊆ G, y ∈ G. Suppose Z is L-polar. Then E \Z is L-thin at y if and only if E is. Proof. Suppose first that E is L-thin at y. Then, by Theorem 12.2.4, there exists an L-superharmonic function u on a neighborhood of y such that u(y) < lim infE z→y u(z). This gives u(y) < lim inf u(z) ≤ lim inf u(z), E z→y

E\Z z→y

so that E \ Z is L-thin at y, again by Theorem 12.2.4. Vice versa, suppose that E \ Z is L-thin at y and (thanks to Theorem 12.2.5) consider an L-superharmonic function u on G such that u(y) < limE\Z z→y u(z) = + ∞. Since Z \ {y} is L-polar, there exists w ∈ S (G) such that w ≡ ∞ on Z \ {y} and w(y) ∈ R. Consequently, we have lim (u(z) + w(z)) = ∞ > u(y) + w(y).

E z→y

This proves that E is L-thin at y by making use of Theorem 12.2.5. 2 This follows from ν({x}) = 0. Otherwise, we would have

∞ > u(x) ≥

Bd (x,1/n)

Γ (x, y) dν(y) → ∞ · ν({x}) = ∞.

542

12 L-thinness and L-fine Topology

12.3 L-thinness and L-regularity The aim of this section is to prove that the L-regularity of a boundary point can be characterized in terms of L-thinness (see Theorem 12.3.6 below). To this end, we provide some preliminary results. Some have an interest in their own and are collected in the following subsection. 12.3.1 Functions Peaking at a Point We need a definition and some preliminary results. Definition 12.3.1 (Function peaking at a point). If u is an extended real-valued function defined on a neighborhood of x, we say that u peaks at x if u(x) > supy ∈V / u(y) for every small neighborhood V of x. +

Lemma 12.3.2. For every fixed x ∈ G, there exists w ∈ S (G) ∩ C(G) such that w has a strict absolute maximum at x (in particular, w peaks at x). Proof. Let r > 0. Then the function w(z) := Mr (Γ (z−1 ◦ ·))(x) has the required properties (see Theorem 9.3.10, page 438; see also Fig. 12.1).

Fig. 12.1. The function w of Lemma 12.3.2

The relevance of functions peaking at a point within the theory of L-thinness becomes apparent from the following result. +

Lemma 12.3.3 (Characterization of L-thinness. III). Let w ∈ S (G) be a function which peaks at x. Then a set E is L-thin at x if and only if Rw E (x) w(x). Proof. Since E is L-thin at x if and only if E \ {x} is L-thin at x (see, e.g. Corolw lary 12.2.6) and since Rw E = RE\{x} (see Proposition 11.8.3, page 525), we can suppose that x ∈ / E. As a consequence, from Theorem 11.6.5 (page 513) we derive w Rw E (x) = RE (x).

12.3 L-thinness and L-regularity

543

(⇐): Let us suppose Rw E (x) < w(x) and prove that E is L-thin at x. We can suppose that x is a limit point of E (otherwise, nothing is left to prove). By the + definition of L-réduite function, there exists v ∈ S (G) such that v ≥ w on E and v(x) < w(x). This gives lim inf v(y) ≥ lim inf w(y) ≥ lim inf w(y) ≥ w(x) > v(x),

E y→x

y→x

E y→x

whence E is L-thin at x by Theorem 12.2.4. (⇒): Let now E be L-thin at x. Suppose first that x is not a limit point of E, and let V be a neighborhood of x such that V ∩ E = ∅. Then w w(x) > sup w ≥ sup w ≥ Rw E (x) = RE (x). G\V

E

The third inequality follows from the fact that the constant function supE w belongs to ΦEw (see the notation at the beginning of Section 11.5, page 500). We are left with + the case when x is a limit point of E. By Theorem 12.2.5, there exists v ∈ S (G) v := min{v, v(x) + 1}, we such that v(x) < limE y→x v(y) = ∞. Replacing v by can suppose that v(x) < limE y→x v(y) < ∞ and v is bounded from above. Let α and β be such that (12.7) v(x) < α < β < lim v(y). E y→x

For every λ > 0, we set wλ (·) := w(x) + λ (v(·) − α). First, we claim that there exists a neighborhood U of x and 0 < λ 1 such that wλ > 0 on U for any 0 < λ ≤ λ. Indeed, from w(x) > 0 (w peaks at x) and the lower semi-continuity of w we get lim infy→x w(y) ≥ w(x) > 0. Hence, there exists a neighborhood U of x and ε > 0 such that infU w ≥ ε. We choose 0 < λ 1 such that λ |v(y) − α| ≤ ε/2 (recall that v is bounded) for every y ∈ U . This gives, for every y ∈ U and every 0 < λ ≤ λ, wλ (y) = w(y) + λ(v(y) − α) ≥ ε − λ |v(y) − α| ≥ ε/2. The claim is proved. According to (12.7), there exists a neighborhood U ⊆ U of x such that v(y) > α for every y ∈ E ∩ U . Then we have wλ (y) > w(x) ≥ w(y)

∀ y ∈ E ∩ U.

(12.8)

Moreover, w(x) − w(y) ≥ γ > 0 for every y ∈ / U (again by the peaking property of w). Since v is bounded, there exists a small positive λ0 < λ such that λ0 |v(y) − α| ≤ γ ≤ w(x) − w(y) ∀ y ∈ / U. This proves that, for y ∈ / U , we have λ0 |v(y) − α| ≥ −λ0 |v(y) − α| ≥ −(w(x) − w(y)) = w(y) − w(x),

544

12 L-thinness and L-fine Topology

so that wλ0 (y) = w(x) + λ0 (v(y) − α) ≥ w(y)

∀y ∈ / U.

This last inequality, together with (see also (12.8)) the inequality wλ0 (y) ≥ w(y)

∀ y ∈ E ∩ U,

gives wλ0 (y) ≥ w(y) for every y ∈ E. Moreover, from wλ0 (y) ≥ 0 on U (see the claim above) and wλ0 ≥ w ≥ 0 on E we derive that wλ0 (y) ≥ 0. This proves that + w ∈ S (G) is such that wλ0 ≥ w on E, whence, by the definition of L-réduite, Rw E (x) ≤ wλ0 (x) = w(x) + λ0 (v(x) − α) < w(x). This completes the proof. Theorem 12.3.4 (L-thinness and L-polarity. I). Let E ⊆ G be any set. Then the subset of E where E is L-thin is an L-polar set. In other words, the set of the L-finely isolated points of E is an L-polar set. +

Proof. Let E be L-thin at x0 ∈ E. Let w ∈ S (G) ∩ C(G) be such that w peaks at w Rw x0 (see Lemma 12.3.2). By Lemma 12.3.3, we have E (x0 ) w(x0 ) = RE (x0 ) (for w x0 ∈ E). Then x0 belongs to P := {x ∈ G : Rw E (x0 ) < RE (x0 )} which is a L-polar subset of E ∩ ∂E (see Theorem 11.6.5, page 513). From the arbitrariness of x0 , the assertion follows. Corollary 12.3.5 (L-thinness and L-polarity. II). A set Z ⊂ G is L-polar if and only if Z is L-thin at any of its points. Proof. If Z is L-polar, then Z is L-thin at every point of G (see Proposition 12.1.3). The converse assertion follows immediately from Theorem 12.3.4. 12.3.2 L-thinness and L-regularity We are now in a position to state and prove the main results of the section. Theorem 12.3.6 (Non-L-thinness and L-barriers). Let F ⊆ G be a closed set. Let x0 ∈ ∂F . Then F is not L-thin at x0 if and only if there exists a (L-barrier) function b with the following properties: b ∈ S(G \ F ), b ≥ 0 on G \ F , b > 0 on W := Bd (x0 , 1) \ F and limW y→x0 b(y) = 0. From Theorem 12.3.6 and Bouligand’s Theorem 6.10.4 (page 371) we immediately derive the following L-regularity criterion. Theorem 12.3.7 (L-regularity criterion in terms of L-thinness). Let Ω ⊂ G be a bounded and connected open set. Let x0 ∈ ∂Ω. Then x0 is an L-regular point for Ω if and only if G \ Ω is not L-thin at x0 . Equivalently, x0 is an L-irregular point if and only if G \ Ω is L-thin at x0 .

12.3 L-thinness and L-regularity

545

Fig. 12.2. The barrier function of Theorem 12.3.6

Proof (of Theorem 12.3.6). Let F ⊆ G be closed, and let x0 ∈ ∂F (⊆ F ). (⇒): Suppose F is not L-thin at x0 . Let w(z) := M1 (Γ (z−1 ◦ ·))(x0 ). Then (see Theorem 9.3.10, page 438) w ∈ S(G) ∩ C(G), w peaks at x0 and w > 0 on G. Consider the L-balayage Rw F of w relative to F . Then Rw F (x0 ) = w(x0 ). Indeed, since F is not L-thin at x0 , by Lemma 12.3.3 (and the general properties of w the L-balayage) we have Rw F (x0 ) ≥ w(x0 ) ≥ RF (x0 ). Hence, w w Rw lim inf F (y) ≥ lim inf RF (y) ≥ RF (x0 ) = w(x0 ).

G\F y→x0

y→x

(12.9)

Let us now consider the function b := w − Rw F. The function b is non-negative3 on G and b ∈ S(G \ F ). This last fact follows Rw from w ∈ S(G), F ∈ H(G \ F ) and F = F . We now claim that b > 0 on W := Bd (x0 , 1) \ F . Indeed, it is enough to prove that b > 0 on every connected component C of W . Suppose, to the contrary, that b vanishes at some point of C. By Rw the minimum principle,4 this gives b ≡ 0 on C, whence w ≡ F on C, so that Lw = 0 on C,

(12.10)

for Rw F is L-harmonic on G \ F ⊃ W ⊇ C. Now, a direct computation (see (9.10b) in Theorem 9.3.10, page 438) shows that, for every z ∈ C, we have5 2−Q L(d 2 (x0 , z)) 2 Q Γ (2Q−2)/(2−Q) (x0−1 ◦ z) · |∇L Γ |2 (x0−1 ◦ z). = 2−Q

Lw(z) =

3 This follows from the general fact w ≥ Rw F if w ∈ S(G). 4 Note that b ∈ S(W ) since W is open, W ⊆ G \ F and b ∈ S(G \ F ). 5 Recall that d = Γ 1/(2−Q) , the general formula L(α(u)) = α (u) |∇ u|2 + α (u) Lu L

(where u : R → R is regular enough) and LΓ = 0 outside the origin.

546

12 L-thinness and L-fine Topology

The far right-hand term cannot vanish on the open set C (otherwise ∇L Γ ≡ 0 on C, i.e. Γ ≡ 0 on C, see Proposition 1.5.6, page 69) in contradiction with (12.10). Moreover, we have Rw 0 ≤ lim sup b(y) = lim sup b(y) = lim sup(w(y) − F (y)) W y→x0

F y→x / 0

F y→x / 0

≤ w(x0 ) − lim inf Rw F (y) ≤ 0. F y→x / 0

Here, we exploited the continuity of w and (12.9). This finally proves that lim supW y→x0 b(y) = 0. (⇐): Let b be as in the assertion of Theorem 12.3.6 (see also Fig. 12.2). Let us suppose by contradiction that F is not L-thin at x0 ∈ ∂F . Then, by making use of Theorem 12.2.5, it is not difficult6 to prove the existence of a bounded function v ∈ S(G) and a neighborhood U ⊆ Bd (x0 , 1) of x0 such that v(x0 ) = 1 and v ≤ −1 on U ∩ (F \ {x0 }). Let B1 := Bd (x0 , ρ) with 0 < ρ 1 such that B1 ⊂ U . Then v ≤ −1 on F ∩ ∂B1 (⊆ U ∩ F \ {x0 }). Since F ∩ ∂B1 is compact, this fact, together with the upper semi-continuity of v, of F ∩ ∂B1 such that v ≤ −1/2 implies the existence of an open neighborhood W 7 on W . Hence, for every λ > 0 ∩ (∂B1 \ F ). v − λ b ≤ 0 on W

(12.11)

Being b > 0 on ∂B1 \ F , we have inf b > 0,

∂B1 \W

(12.12)

is a compact set which does not intersect a neighborhood of F and b for ∂B1 \ W is l.s.c. outside F . Hence, the boundedness of v and (12.12) ensure that there exists λ0 1 such that . (12.13) v − λ0 b ≤ 0 on ∂B1 \ W and every ε > 0, one has (recalling that v − λ0 b is Thus, for every y ∈ ∂B1 \ W u.s.c. and using (12.13))

lim sup v(z) − λ0 b(z) − ε Γ (x0−1 ◦ z) ≤ 0. (12.14) B1 \F z→y

The same holds for y = x0 (since Γ (0) = ∞) and for every y ∈ B1 ∩ (∂F \ {x0 }), for this last set is contained in U ∩ (F \ {x0 }) where v ≤ −1. Now, being v − λ0 b − ε Γ (x0−1 ◦ ·) ∈ S(B1 \ F ), from the maximum principle one gets (also collecting (12.11), (12.14)) 6 See Ex. 5 at the end of the present Chapter. 7 Indeed, v ≤ −1/2 on W , b > 0 on W = Bd (x0 , 1) \ F and Bd (x0 , 1) ⊇ U ⊃ B1 ⊃ ∂B1 .

12.4 Wiener’s Criterion for Sub-Laplacians

547

v − λ0 b − ε Γ (x0−1 ◦ ·) ≤ 0 on B1 \ F . Letting ε → 0+ , we derive v ≤ λ0 b on B1 \ F , so that lim sup v(z) ≤ lim sup λ0 b(z) = 0

B1 \F z→x0

B1 \F z→x0

by the definition of the barrier function b. Since lim sup v(z) ≤ −1

F z→x0

(for v ≤ −1 on U ∩ (F \ {x0 })), we finally get lim sup v(z) ≤ 0. z→x0

But this is in contradiction to lim supz→x0 v(z) = v(x0 ) (a consequence of the fact that v is L-subharmonic in x0 ) and v(x0 ) = 1 (by construction). This contradiction proves the theorem.

12.4 Wiener’s Criterion for Sub-Laplacians The aim of this section is to prove a test for L-thinness similar to the well-known Wiener criterion for the classical Laplace operator (in this case, see, e.g. [AG01, Section 7.7]). For a generalization to more general operators, see also [NS87]. As usual, throughout the section, G = (RN , ◦, δλ ) is a Carnot group and L is a sub-Laplacian on G. Moreover, Γ = d 2−Q is the fundamental solution for L. We shall also denote by β ≥ 1 the constant appearing in the following inequalities: d(a ◦ b) ≤ β d(a) + d(b) ∀ a, b ∈ G, 1 d(a ◦ b) ≥ |d(a) − d(b)| ∀ a, b ∈ G. β

(12.15a) (12.15b)

These are (subtle) improvements of the pseudo-triangle inequality for d. (12.15a) is proved in Proposition 5.14.1 on page 306. Then, by choosing a = x ◦ y and b = y −1 in (12.15a) (and using the symmetry of d), we derive d(x) − d(y) ≤ β d(x ◦ y); then, replacing x and y in the latter with, respectively, y −1 and x −1 (and using twice the symmetry of d) we get d(y) − d(x) = d(y −1 ) − d(x −1 ) ≤ β d(y −1 ◦ x −1 ) = β d((x ◦ y)−1 ) = d(x ◦ y). This proves (12.15b). 12.4.1 A Technical Lemma Throughout the section, we use the following notation: given a point y ∈ G and a constant α > 1, for every n ∈ N we set

Cn := x ∈ G : α n ≤ Γ (y −1 ◦ x) ≤ α n+1 . (12.16)

548

12 L-thinness and L-fine Topology

In other words, let α := we have

1 1 Q−2 , α

Cn = Bd y, α n \ Bd y, α n+1 .

We begin with a crucial lemma concerning with suitable estimates of the fundamental solution Γ for L. We remark that the proof of Lemma 12.4.1 below is more delicate in our context than in the classical case (see [AG01, Lemma 7.7.1]). We shall henceforth suppose that α > β Q−2 ,

(12.17)

where β is the constant appearing in (12.15a) (note that in the Euclidean case, β = 1 and (12.17) reduces to α > 1). Lemma 12.4.1. Let α > 1 (be as in (12.17)). Then there exists c = c(α) > 0 such that, for every y ∈ G, for every Radon measure μ on Bd (y, 12 ), for every n ∈ N, n ≥ 2 and every x ∈ Cn it holds −1 Γ (x ◦ z) dμ(z) ≤ c Γ (y −1 ◦ z) dμ(z). (12.18) G\(Cn−1 ∪Cn ∪Cn+1 )

G

Proof. Let y ∈ G and α > 1 be fixed throughout. Let also x ∈ Cn , i.e. α n ≤ d 2−Q (y −1 ◦ x) ≤ α n+1 .

(12.19)

We have

G \ (Cn−1 ∪ Cn ∪ Cn+1 ) = z ∈ G : Γ (y −1 ◦ z) > α n+2 ∪

∪ z ∈ G : Γ (y −1 ◦ z) < α n−1 =: I ∪ II. We distinguish two cases: (i): z ∈ I, i.e. (ii): z ∈ II, i.e.

d 2−Q (y −1 ◦ z) > α n+2 , d 2−Q (y −1 ◦ z) < α n−1 .

(12.20a) (12.20b)

(i): Suppose first that z ∈ I . Then (see (12.15b), (12.19), (12.20a)) d(x −1 ◦ z) ≥

1 n+2 1 1

d(x −1 ◦ y) − d(y −1 ◦ z) ≥ α 2−Q (α Q−2 − 1). β β

Equivalently (exploiting once again (12.20a) in the last inequality), Γ (x

−1

◦ z) ≤

β 1

α Q−2 − 1

Q−2

α n+2 =: c(α) α n+2 < c(α) Γ (y −1 ◦ z).

12.4 Wiener’s Criterion for Sub-Laplacians

549

We have thus proved (indeed, we only used the second inequality in (12.19)) x ∈ Cn , z ∈ I

⇒

Γ (x −1 ◦ z) ≤ c(α). Γ (y −1 ◦ z)

(12.21)

(ii): Suppose now that z ∈ II. In this case, we also need an estimate similar to that in the far right-hand of (12.21), namely x ∈ Cn , z ∈ II

⇒

Γ (x −1 ◦ z) ≤ c(α). Γ (y −1 ◦ z)

(12.22)

Since this estimate will intervene to estimate the integral in the left-hand side of (12.18), and the measure μ appearing there is supported in Bd (y, 12 ), we can suppose that (12.23) d(y −1 ◦ z) ≤ 1/2. Now, (12.22) follows if we, equivalently, prove that d(y −1 ◦ z) ≤ c(α) d(x −1 ◦ z)

(12.24)

under the following8 hypotheses on z (see (12.20b), (12.23)) n−1

α 2−Q < d(y −1 ◦ z) ≤ 1/2.

(12.25)

Suitable bounds for d(x −1 ◦ z) are given by the following inequalities: n

d(x −1 ◦ z) ≤ β d(x −1 ◦ y) + d(y −1 ◦ z) ≤ β α 2−Q + d(y −1 ◦ z), n

d(x −1 ◦ z) ≥ d(y −1 ◦ x ◦ x −1 ◦ z) − β d(y −1 ◦ x) ≥ d(y −1 ◦ z) − β α 2−Q . The first estimate has been obtained by applying (12.15a) with a = x −1 ◦ y, b = y −1 ◦ z, jointly with the first inequality in (12.19); the second estimate has been obtained by applying (12.15a) with a = y −1 ◦ x, b = x −1 ◦ z, jointly with the first inequality in (12.19). Thus, n

n

d(y −1 ◦ z) − β α 2−Q ≤ d(x −1 ◦ z) ≤ d(y −1 ◦ z) + β α 2−Q . n−1

(12.26)

n

Set λ := d(y −1 ◦ z), μ := d(x −1 ◦ z), A := α 2−Q and B := β α 2−Q . According to the hypotheses (12.25), (12.26) and the needed assertion (12.24), we have to prove A < λ ≤ 1/2 λ−B ≤μ≤λ+B

⇒

λ ≤ c. μ

(12.27)

Now, (12.27) follows from a simple exercise of optimization (see Ex. 10 at the end of the present Chapter9 ) with c = A/(A − B), i.e. n−1

8 We are tacitly supposing that n is large enough, so that α 2−Q < 1/2, otherwise there is

nothing to prove. 9 Note that the hypothesis (12.17) on α ensures that A ∈ ]0, 1/2[ and B ∈ ]0, A[.

550

12 L-thinness and L-fine Topology n−1

α 2−Q

c= α

n−1 2−Q

−βα

n 2−Q

1 = c(α). 1 + β α 1/(2−Q)

=

(Note that c depends only on B/A and does not depend on n.) Finally, (12.18) follows straightforwardly from (12.21) and (12.22). The lemma is completely proved. 12.4.2 Wiener’s Criterion for L Let α > 1 be as in Lemma 12.4.1 at the beginning of the section. We also maintain the notation Cn given by (12.16). We are ready to state and prove our Wiener’s test for sub-Laplacians. Theorem 12.4.2 (Wiener’s criterion for L). Let C ∗ denote the exterior L-capacity relative to G. Let also E ⊆ G and y ∈ G. Then the following statements are equivalent: (i) E is L-thin at y,n ∗ (ii) It holds ∞ n=1 α C (E ∩ Cn ) < ∞, R1 (y) < ∞, (iii) It holds ∞ ∞n=1∗ E∩Cn (iv) It holds α C ({x ∈ E : Γ (y −1 ◦ x) ≥ t}) dt < ∞. Proof. First, we remark that it is not restrictive to suppose that y is a limit point of E, otherwise (for suitable large n and t) E ∩ Cn and E ∩ Bd (y, t 1/(2−Q) ) would be contained in {y} so that (i) through (iv) all hold (recall that a set is L-thin in the complement of its limit points). The proof is split into four steps: we prove one by one the equivalences (ii) ⇔ (iii), (i) ⇔ (iii), (ii) ⇔ (iv). R1E∩Cn is an L-potential (see Theo(ii) ⇔ (iii): Since E ∩ Cn is bounded, R1E∩Cn is L-harmonic rem 11.8.1), say Γ ∗ μn (where μn is supported in Cn for in the complement of E ∩ Cn ). Let us set Γ ∗ μn (x). u(x) := n≥1

From Theorem 11.8.1 (page 519) we also infer C ∗ (E ∩ Cn ) = μn (G). Let us assume (ii). Then (iii) holds too, for we have u(y) = Γ ∗ μn (y) = Γ (z−1 ◦ y) dμn (z) n≥1

≤

n≥1

n≥1 Cn

α n+1 μn (G) =

α n+1 C ∗ (E ∩ Cn ) < ∞.

n≥1

Vice versa, let us assume (iii). Then (ii) holds too, for we have

(12.28)

12.4 Wiener’s Criterion for Sub-Laplacians

α n+1 C ∗ (E ∩ Cn ) =

n≥1

≤

α n+1 μn (Cn )

n≥1

Γ (z−1 ◦ y) dμn (z) =

n≥1 Cn

551

Γ ∗ μn (y) < ∞.

(12.29)

n≥1

(Proving the inequalities “≤” in (12.28) and (12.29) we used the fact that, by definition, α n ≤ Γ (z−1 ◦ y) ≤ α n+1 on Cn .) (i) ⇔ (iii): First, suppose that (iii) holds. Then10 there exists a sequence bn R1E∩Cn (y) < ∞. Set diverging to ∞ such that n bn R1E∩Cn (x). v(x) := bn n

By Theorem 11.6.5 (page 513), there exists an L-polar set Pn ⊆ E ∩ Cn such that ∀ x ∈ (E ∩ Cn ) \ Pn . (12.30) R1E∩Cn (x) = 1 Set A = n An , P = n Pn . From (12.30) we infer v(x) = n bn = ∞ for every x ∈ (E ∩ A) \ P . As a consequence (A being a neighborhood of y) E \ P is L-thin at y. This ensures that E is also L-thin at y (see Corollary 12.2.6). + Now, suppose that (i) holds. Then, by Theorem 12.2.5, there exists w ∈ S (G) such that w(y) < lim w(x) = ∞. E x→y

(12.31)

1 Set E0 := ∞ k=k0 E ∩ C2k where k0 is chosen so that C2k0 ⊂ Bd (y, 2 ). Since E0 is Rw bounded, by Theorem 11.8.1 (page 519), there exists a measure ν0 such that E0 = Γ ∗ ν0 . By Theorem 11.6.5 (page 513), there exists an L-polar set F0 ⊆ E0 ∩ ∂E0 w such that Rw E0 = RE0 on G \ F0 . This ensures that Γ ∗ ν0 = w on E0 \ F0 , whence (from (12.31) and E0 \ F0 ⊆ E) we derive lim(E0 \F0 ) x→y Γ ∗ ν0 (x) = ∞, i.e. Γ (x −1 ◦ z) dν0 (z) = ∞. (12.32) lim (E0 \F0 ) x→y

Furthermore, it holds Γ (y −1 ◦ z) dν0 (z) = Γ ∗ ν0 (y) = Rw E0 (y) ≤ w(y) < ∞.

(12.33)

As a consequence, by applying Lemma 12.4.1 we get Γ (x −1 ◦ z) dν0 (z) ≤ c w(y) < ∞ ∀ x ∈ Cn , n ≥ 2. (12.34) G\(Cn−1 ∪Cn ∪Cn+1 )

10 Here, we use the following simple analysis lemma: If a < ∞ with a ≥ 0, then n n n there exists a sequence {bn } such that bn ↑ ∞ and n an bn < ∞. Indeed, setting An = −k and define b = k n j ≥n aj , we have An ↓ 0. Choose {nk }k such that Ank ≤ 4

whenever n = nk , nk + 1, . . . , nk+1 − 1.

12 L-thinness and L-fine Topology

552

Now, (12.32) and (12.34) may agree only if (for a suitable diverging sequence {βn }) Γ (x −1 ◦ z) dν0 (z) ≥ βn ∀ x ∈ C2n ∩ (E0 \ F0 ), n ≥ 1. C2n

In particular, also setting μn := (ν0 )|C2n , this gives (Γ ∗ μn )(x) ≥ 1

∀ x ∈ (E ∩ C2n ) \ F0 , n 1. +

By the definition of L-balayage (being Γ ∗ μn ∈ S (G)), this proves R1(E∩C2n )\F0 , Γ ∗ μn ≥ i.e. thanks to Proposition 11.8.3 (page 525), Γ ∗ (ν0 |C2n ) ≥ R1E∩C2n ,

n 1.

By arguing analogously with E1 := E ∩ C2k+1 ,

(12.35)

Rw E1 = Γ ∗ ν1 ,

k≥k0

we have R1E∩C2n+1 , Γ ∗ (ν1 |C2n+1 ) ≥

n 1.

(12.36)

From (12.33), (12.35) and (12.36) we get (iii). Indeed, one has R1Cn (y) = + n even

n1

≤

n odd

Γ ∗ (ν0 |C2n )(y) +

n1

Γ ∗ (ν1 |C2n+1 )(y)

n1

≤ (Γ ∗ ν0 )(y) + (Γ ∗ ν1 )(y) ≤ 2 w(y) < ∞. (ii) ⇔ (iv): First, we show the implication “⇒”, ∞ ∞

C ∗ x ∈ E : Γ (y −1 ◦ x) ≥ t dt =

α n+1

n n=1 α

α

≤

∞

C ∗ x ∈ E : Γ (y −1 ◦ x) ≥ α n

n=1

[· · ·]

α n+1

dt αn

m −1 ◦ x) ≤ α m+1 } (from E ∩ { Γ (y −1 ◦ x) ≥ α n } ⊆ ∞ m=n E ∩ {α ≤ Γ (y ∗ and the sub-additivity of the exterior L-capacity C ) ∞ ∞ m ∞ C ∗ (E ∩ Cm )(α n+1 − α n ) = C ∗ (E ∩ Cm ) (α n+1 − α n ) ≤ =

n=1 m=n ∞ ∗

∞

m=1

m=1

C (E ∩ Cm ) (α m+1 − α) ≤

m=1

n=1

C ∗ (E ∩ Cm ) α m+1 < ∞.

12.5 Exercises of Chapter 12

553

Finally, we show the implication “⇐”, ∞

C ∗ (E ∩ Cn ) α n ≤

n=2

∞

α n C ∗ x ∈ E : Γ (y −1 ◦ x) ≥ α n

n=2

αn α (use the monotonicity of C ∗ and α n = α−1 α n−1 dt) n ∞

α α ≤ C ∗ x ∈ E : Γ (y −1 ◦ x) ≥ t dt n−1 α−1 n=2 α ∞

α = C ∗ x ∈ E : Γ (y −1 ◦ x) ≥ t dt < ∞. α−1 α

This completes the proof. Collecting together the L-regularity criterion of Theorem 12.3.7 and Wiener’s criterion for L of Theorem 12.4.2, we derive the following remarkable result (again, α > 1 is a constant as in Lemma 12.4.1): Theorem 12.4.3 (Wiener’s regularity test for L). Let C ∗ denote the exterior Lcapacity relative to G. Let Ω be an open and connected subset of G and y ∈ ∂Ω. Then the following statements are equivalent: (i) y is an L-regular point for Ω; (ii) G \ Ω is not L-thin at y; (iii) It holds ∞ α n C ∗ (Cn \ Ω) = ∞; n=1 ∞ 1 (iv) It holds n=1 RCn \Ω (y) = ∞; ∞ (v) It holds α C ∗ ({x ∈ / Ω : Γ (y −1 ◦ x) ≥ t}) dt = ∞. Here, Cn is as in (12.16) at the beginning of the section. Bibliographical Notes. For the topics presented in this chapter, we were inspired by the exposition of the same subjects in [Helm69, Chapter 10] and [AG01, Chapter 7]. The Wiener test in the context of the nilpotent Lie groups was proved by H. Hueber [Hu85] and then generalized for Hörmander vector fields by P. Negrini and V. Scornazzani [NS87]. For further results on the Wiener test, see, e.g. [HH87,Ne88] and on the fine topology see, e.g. [BR67,CC72].

12.5 Exercises of Chapter 12 Ex. 1) Prove that a set E is L-thin at x ∈ G if and only if E \ {x} is L-thin at x (prove this by the very definition of L-thinness).

554

12 L-thinness and L-fine Topology

Ex. 2) Prove that any non-empty L-fine open set contains infinitely many distinct points. (Hint: If u is L-superharmonic, then u(x) = lim infy→x u(y).) Ex. 3) Prove that the L-fine topology is not locally compact. (Hint: Let F be the L-fine closure of an L-fine neighborhood of a point. Then, by Exercise 1, F contains an infinite set Z = {xn }n∈N . By Proposition 12.1.3, no subsequence of Z can converge.) Ex. 4) A set is L-finely compact if and only if it is finite. (Hint: See the hint for Exercise 2.) Ex. 5) Prove that E is L-thin at x if and only if there exists a bounded function v ∈ S(G) and a neighborhood U of x such that v(x) = 1 and v ≤ −1 + on U ∩ (E \ {x}). (Hint: By Theorem 12.2.5, let u ∈ S (G) be such that u(x) < limE y→x u(y) = ∞. Replace u by w = min{u, u(x) + 1} and take U := {y : u(y) > u(x) + 1/2} and v := 1 + 4(u(x) − u).) Ex. 6) (The exterior L-cone regularity condition). In the following statement, we say that C is a (truncated) L-cone if δλ (x) ∈ C

∀ x ∈ C, ∀ λ ∈]0, 1[,

where {δλ }λ>0 is the family of dilations of the Carnot group G = (RN , ◦, δλ ). As usual, L is a sub-Laplacian on L. Let Ω ⊆ RN be a bounded open set, and let y ∈ ∂Ω. Assume Ω has at y the property of the exterior L-cone, i.e. there exists a bounded open (truncated) L-cone C such that RN \ Ω ⊇ y ◦ C. Then y is an L-regular point for Ω. (Hint: We follow the notation of Theorem 12.4.3. Use the results of Exercise 2 at the end of Chapter 11 (page 533) to show that C ∗ (Cn \ Ω) ≥ c0 (1/α)n . This follows from the fact that Cn \ Ω ⊇ δr n {ξ ∈ C : r ≤ d(ξ ) ≤ 1}, with r = α 1/(2−Q) . Thus, n α n C ∗ (Cn \ Ω) ≥ n co α n α −n = ∞, and the L-regularity of y follows from Wiener’s test in Theorem 12.4.3.) Ex. 7) In the case of the classical Laplace operator, give a proof of the exterior L-ball regularity condition in Ex. 9 at the end of Chapter 7 by making use of Wiener’s criterion and geometrical arguments (instead of Bouligand’s theorem). Do the same in the case of the Heisenberg–Weyl group on R3 . Ex. 8) Let Ω ⊆ RN be a bounded open set, and let y ∈ ∂Ω. Assume there exists R > 0 and γ ∈ ]0, 1[ such that |Bd (y, r) \ Ω | ≥ γ r Q

∀ r ∈ ]0, R[.

Then y is an L-regular point for Ω. Here | · | stands for the Lebesgue measure.

12.5 Exercises of Chapter 12

555

Ex. 9) Let Ω ⊆ RN be a bounded open set, and let y ∈ ∂Ω. For every r > 0, let us set R1Ω (y) , Ωr (y) = Bd (y, r) \ Ω. Vr := r

Then y is not an L-regular point for Ω if and only if Vr (y) → 0 as r → 0. Ex. 10) Let A, B be fixed constants with 0 < B < A < 1/2. Prove that A ≤ λ ≤ 1/2 λ−B ≤μ≤λ+B

⇒

λ A ≤ . μ A−B

This proves (12.27) in the proof of Lemma 12.4.1. Ex. 11) Prove that if A is L-thin at x then, for every B ⊆ A, B is L-thin at x. Vice versa, if A is not L-thin at x then, for every B ⊇ A, B is not L-thin at x. Derive the following facts: be open, connected and bounded subsets of G. Suppose also that Let Ω, Ω If x0 is L-regular for Ω and Ω ⊆ Ω, then x0 is L-regular x0 ∈ ∂Ω ∩ ∂ Ω. ⊇ Ω, then x0 is L Vice versa, if x0 is L-irregular for Ω and Ω for Ω. irregular for Ω. Derive this both from Theorem 12.3.7 and from Bouligand’s Theorem 6.10.4 (page 371). Ex. 12) Let 0 < r1 < r2 < ∞. Let x0 ∈ G be fixed. Set Bi := Bd (x0 , ri ) (for i = 1, 2) and C := B2 \ B1 . Prove that the annulus C is an L-regular set by constructing explicit L-barrier functions (see Definition 6.10.3, page 371). (Hint: w1 := r 2−Q − d 2−Q (x0−1 ◦ ·) is an L-barrier function for the points in ∂B1 , whereas w2 := −r 2−Q + d 2−Q (x0−1 ◦ ·) is an L-barrier function for the points in ∂B2 .) Note that, by Theorem 12.3.7, the L-regularity of a point y ∈ ∂C implies that (actually, is equivalent to) G \ C = B1 ∪ (G \ B2 ) is not L-thin at y. In turn, this is equivalent to

y ∈ DerL-fine B1 ∪ (G \ B2 ) . Since Der(A ∪ B) = Der(A) ∪ Der(B) in any topological space (prove this), then the above argument shows that ∂C ⊆ DerL-fine (B1 ) ∪ DerL-fine (G \ B2 ). Recalling that DerL-fine (A) ⊆ DerEucl. (A), this easily proves that DerL-fine (B1 ) = B1 , DerL-fine (G \ B2 ) = G \ B2 . Is it true that we also have DerL-fine (B1 ) = B1 and DerL-fine (G \ B2 ) = G \ B2 ?

13 d-Hausdorff Measure and L-capacity

The aim of this chapter is to provide some estimate of the “smallness” of L-polar sets in terms of the Hausdorff measure naturally related to the quasi-distance defined by an L-gauge d on G. To this end, we first need the relevant definitions of d-Hausdorff measure and d-Hausdorff dimension, besides several preliminary results. One of the main results is contained in Theorem 13.2.5. Finally, in Section 13.3, we recall some results contained in the remarkable paper [BRSC03], in order to emphasize the new phenomena which can occur when the Hausdorff dimension is considered in the sub-elliptic context of Carnot groups (for instance, the Heisenberg–Weyl group H1 ). As usual, throughout the section G = (RN , ◦, δλ ) is a homogeneous Carnot group and L is a sub-Laplacian on G. Moreover, Γ is the fundamental solution for L and d is any L-gauge function (i.e. a positive constant times Γ 1/(2−Q) ). Q denotes, as usual, the homogeneous dimension of G. We recall that d is a quasi-distance on G satisfying the pseudo-triangle inequality d(x −1 ◦ y) ≤ c (d(x) + d(y))

for all x, y ∈ G.

(13.1)

The constant c≥1 will always denote that appearing in (13.1). As usual, the d-balls related to d are denoted by Bd (x, r).

13.1 d-Hausdorff Measure and Dimension Definition 13.1.1 (The d-Hausdorff measure). An increasing function φ : (0, ∞) → (0, ∞] vanishing as t → 0+ will be referred to as measure function. Given any set E ⊆ G and any ρ > 0, we set (ρ) φ(rk ) : E ⊆ Bd (xk , rk ), rk < ρ ∀ k ∈ N , (13.2) Mφ (E) := inf k

k

558

13 d-Hausdorff Measure and L-capacity

where the infimum is taken over all possible coverings E by a (finite or) countable family of d-balls {Bd (xk , rk )}k∈N such that the rays rk are strictly less than ρ (any such covering will be referred to as ρ-covering of E). (ρ) Since the map ]0, ∞] ρ → Mφ is decreasing, it is well posed, (ρ)

(ρ)

mφ (E) := lim Mφ (E) = sup Mφ (E). ρ→0+

(13.3)

ρ>0

In the sequel, mφ (E) will be referred to as the d-Hausdorff φ-measure. When φ(t) = (ρ) (ρ) t α ( for α > 0), we simply write M(α) instead of Mt α and m(α) instead of mt α . In the sequel, m(α) (E) will be referred to as the d-Hausdorff (α)-measure of the set E. (ρ)

Remark 13.1.2. With the above notation, it can be proved that Mφ and mφ are outer measures on G. Properly restricted to their related measurable sets, they are Borel regular measures. We leave the proof as an exercise. We begin with some lemmas. Lemma 13.1.3 (The d-Hausdorff dimension). For every bounded set E ⊂ G, there exists a number α(E) ∈ [0, Q] (where Q is the homogeneous dimension of G) such that m(α) (E) = ∞ for all α < α(E),

m(α) (E) = 0 for all α > α(E). (13.4)

We thus have α(E) = inf{α > 0 : m(α) (E) = 0}.

(13.5)

We call α(E) the d-Hausdorff dimension of E. Proof. Let α > Q be fixed and take any ε ∈ ]0, 1[. Let Ω be any bounded open set containing E. Then there exists a countable collection of disjointd-balls Bk := rk rk Bd (x k , 4 ) (with 2 < ε for every k) and a set N such that Ω = N ∪ k Bk and such that k Bd (xk , c rk ) ⊇ Ω. This gives (for a suitable structural constant cQ ) (ε) M(α) (E)

rk Q α α−Q Q Q α−Q ≤ (c rk ) ≤ ε (c rk ) = (4c) ε 4 k k k = cQ ε α−Q meas(Bk ) ≤ cQ ε α−Q meas(Ω) → 0 as ε → 0+ . k

This proves that m(α) (E) = 0 for every α > Q. If we define α(E) as in (13.5), we thus get α(E) ∈ [0, Q]. Let now α > α(E) and ε > 0. Hence, there exist a suitable β ∈ ]α(E), α[ and β an ε-covering {Bd (xk , rk )}k of E such that k rk < 1. Consequently, α−β β (ε) M(α) (E) ≤ rkα = rk rk ≤ ε α−β → 0 as ε → 0+ . k

k

13.1 d-Hausdorff Measure and Dimension

559

This proves the second part of (13.4). Let now α ∈ ]0, α(E)[. By the definition of α(E), there exists β ∈ ]α, α(E)[ with m(β) (E) > 0. If {Bd (xk , rk )}k is an ε-covering α−β β (ε) of E, then k rkα = k rk rk ≥ ε α−β M(β) (E), whence (ε)

(ε)

M(α) (E) ≥ ε α−β M(β) (E) → ∞

as ε → 0+ .

This proves the first part of (13.4), completing the proof of the lemma. Example 13.1.4. The d-Hausdorff dimension of G is Q. Indeed, there exists a positive constant c such that m(Q) (E) = c H N (E)

for every E ⊆ G,

where H N stands for the usual (Euclidean) Hausdorff N-dimensional measure on RN ≡ G. The proof is left as an exercise. Remark 13.1.5. It is easy to see that m(α) is invariant under left-translation and homogeneous of degree α with respect to the dilation of G. In other words, for every x ∈ G and every λ > 0, we have m(α) (x ◦ E) = m(α) (E),

m(α) (δλ (E)) = λα m(α) (E),

where E is any subset of G. The proof is left as an exercise. We next aim to prove Theorem 13.1.7 below. In order to do this, we need a lemma having an interest in its own. Lemma 13.1.6. Suppose K G is not an L-polar set. Then there exists a Radon measure μ supported in K with μ(K) > 0 such that Γ ∗ μ ∈ C(G, R). Proof. By Proposition 11.5.11 (page 507), since K is not L-polar, we have C(K) = μK (K) > 0. Since Γ ∗ μK ≤ 1, by Lemma 11.3.3 (page 495) there exists C K such that μK (K \ C) < μK (K)/2

and Γ ∗ (μK |C ) ∈ C(G, R).

We claim that μ := (μK )|C fulfills the assertion of the lemma. Indeed, μK (C) = μK (K) − μK (K \ C) > μK (K)/2 > 0. This ends the proof. We are now ready to give the proof of the following result. Theorem 13.1.7 (d-Hausdorff measure and L-polarity. I). Let E ⊂ G be a bounded L-capacitable set satisfying m(Q−2) (E) < ∞ (as usual, Q is the homogeneous dimension of G). Then E is L-polar.

560

13 d-Hausdorff Measure and L-capacity

For a partial converse of Theorem 13.1.7, see also Theorem 13.2.5. Proof. We argue by contradiction. Let E be a non-L-polar capacitable set, i.e. C(E) > 0. This also gives C∗ (E) > 0, i.e. there exists K E with C(K) > 0. By the sub-additivity of the L-capacity, there exists an open d-ball B1 of radius 1 such that C(K ∩ B1 ) > 0. Then, it is not restrictive to suppose that K is contained in such a d-ball B1 . From C(K) > 0 and Lemma 13.1.6 we derive the existence of a Radon measure μ supported in K such that μ(K) > 0 and u := Γ ∗ μ ∈ C(G, R). For any 0 < ρ < 1 and x ∈ G, we set

d(y −1 ◦ x) −1 dμ(y). uρ (x) := Γ (y ◦ x) max 0, 1 − 2ρ G For any fixed x0 ∈ G, we have uρ (x0 ) ≤ lim inf uρ (x) ≤ lim sup uρ (x) x→x0

x→x0

≤ lim sup u(x) + lim sup(uρ (x) − u(x)) x→x0

x→x0

= u(x0 ) − lim inf(u(x) − uρ (x)) x→x0

d(y −1 ◦ x) dμ(y) Γ (y −1 ◦ x) 1 − max 0, 1 − = u(x0 ) − lim inf x→x0 G 2ρ

≤ u(x0 ) − lim inf[· · ·] G x→x0

d(y −1 ◦ x0 ) = u(x0 ) − dμ(y) Γ (y −1 ◦ x0 ) 1 − max 0, 1 − 2ρ G = uρ (x0 ). In the first and the last inequalities, we used Fatou’s lemma; in the first equality, we used the continuity of u. This proves that uρ is continuous on G. Being Γ ∗ μ finite-valued, we have μ({x}) = 0 for every x ∈ G. As a consequence, it holds (by dominated convergence)

lim uρ (x) = lim [· · ·] dμ ρ→0+

ρ→0+ G\{x}

=

G\{x}

Γ (y

−1

d(y −1 ◦ x) ◦ x) lim max 0, 1 − dμ(y) = 0. 2ρ ρ→0+

The convergence is also monotone, for uρ (x) ≤ uρ (x) whenever 0 < ρ < ρ. By Dini’s theorem, the convergence is uniform on compact sets. Consequently, there exists a sequence ρn ↓ 0 such that uρn < 2−n on B1 for every n ∈ N. Let us introduce the function φ : [0, ∞[ → R, φ(t) :=

∞ 1 2−Q t χ[0,ρn ] (t). 2 n=1

13.2 d-Hausdorff Measure and L-capacity

561

Since χ[0,ρn ] (d(y −1 ◦ x)) = χBd (x,ρn ) (y) (and by monotone convergence), we have

φ(d(y −1 ◦ x)) dμ(y) =

K

∞ 1 n=1

≤ =

2

G

∞ 1 n=1 ∞

2

G

Γ (y −1 ◦ x) χBd (x,ρn ) (y) dμ(y) Γ (y

−1

d(y −1 ◦ x) dμ(y) ◦ x) max 0, 2 − ρn

uρn (x).

n=1

Hence,

φ(d(y −1 ◦ x)) dμ(y) ≤ 1

for every x ∈ B1 .

K

Let now n be large enough so that ρn < sup{d(a −1 ◦ b) : a ∈ K, b ∈ G \ B1 }.

(13.6)

Let us consider a ρn -covering of K, say {Bd (xk , rk )}k , consisting of d-balls all intersecting K. By (13.6), it is easily seen that xk ∈ B1 for every k ∈ N. This gives

φ(d(y −1 ◦ xk )) dμ(y) ≥ [· · ·] 1≥ K Bd (xk ,rk )

(φ is decreasing) ≥ φ(rk ) dμ(y) = φ(rk ) μ(Bd (xk , rk )) Bd (xk ,rk )

n ≥ (rk )2−Q μ(Bd (xk , rk )). 2

(13.7)

The last inequality follows from rk < ρn and the very definition of φ. From (13.7) we derive the second inequality in the following chain of inequalities: μ(K) ≤

μ(Bd (xk , rk )) ≤

k

2 (rk )Q−2 . n k

This gives (from E ⊇ K and again rk < ρn ) n (ρn ) (ρn ) M(Q−2) (E) ≥ M(Q−2) (K) ≥ (rk )Q−2 ≥ μ(K) → ∞ as n → ∞. 2 k

This gives m(Q−2) (E) = ∞, contradicting the assertion of the theorem.

13.2 d-Hausdorff Measure and L-capacity The aim of this section is to prove Theorem 13.2.5 (page 568). In order to do it, we need some lemmas. The first one is a simple result from the real analysis.

13 d-Hausdorff Measure and L-capacity

562

Lemma 13.2.1. Let μ be a Radon measure on RN . Let g : (0, R] → [0, ∞) be a continuous decreasing function. Finally, let f : RN → [0, ∞) be a μ-measurable function. Set

n(t) :=

dμ(y).

(13.8)

f (y)
Suppose n(t) is finite-valued on (0, ∞). Then g is integrable in the generalized Riemann–Stieltjes sense on (0, R] with respect to n(t) if and only if g ◦ f is μintegrable on the set {y ∈ RN : 0 < f (y) < R}. Moreover,

g(f (y)) dμ(y) = 0
R

g(t) dn(t)

(13.9)

0

whenever one of the integrals is finite. Proof. Since t → n(t) is decreasing, it is of bounded variation on every compact sub-interval of (0, R]. Hence g(t) (which is continuous) is dn(t)-integrable on every compact sub-interval of (0, R]. Suppose first that g is integrable in the generalized Riemann–Stieltjes sense on (0, R] with respect to n(t). If ε > 0, there exists δ = δ(ε) > 0 such that R

R ε g(t) dn(t) − g(t) dn(t) < . 2 0 δ We can assume δ(ε) < ε. Moreover, there exists a partition of [δ, R], say δ = t0 < t1 < · · · < tp = R, such that

R

g(t) dn(t) −

δ

p−1 k=0

ε g(tk ) (n(tk+1 ) − n(tk )) < . 2

(13.10)

We remark that n(tk+1 ) − n(tk ) = tk ≤f (y) g(tk+1 ), whence p−1 k=0

g(tk )

tk ≤f (y)
dμ(y) ≥

p−1

k=0 tk ≤f (y)
g(f (y)) dμ(y)

=

g(f (y)) dμ(y). δ(ε)≤f (y)
As a consequence, it holds

R

R

g(t) dn(t) > 0

g(t) dn(t) −

δ(ε)

≥

p−1 ε ≥ g(tk ) (n(tk+1 ) − n(tk )) − ε 2 k=0

g(f (y)) dμ(y) − ε. δ(ε)≤f (y)
13.2 d-Hausdorff Measure and L-capacity

Making use of (13.10), we have

g(f (y)) dμ(y) −

0

δ(ε)≤f (y)
R

g(t) dn(t) < ε.

Being g ≥ 0 and δ(ε) < ε, from Beppo Levi’s theorem we have

g(f (y)) dμ(y) g(f (y)) dμ(y)

563

(13.11)

if ε → 0+ .

0
δ(ε)≤f (y)
Hence (13.11) proves the assertion. Let us suppose that g(f (y)) is μ-integrable on {y ∈ RN : 0 < f (y) < R}. Let δ ∈ ]0, R[ be fixed and ε > 0. Since g is continuous on [δ, R], there exists σ = σ (δ, ε) such that | g(x ) − g(x )| < ε/n(R) for every x , x ∈ [δ, R] with |x − x | < σ . Let us now fix any arbitrary partition δ = t0 < t1 < · · · < tp = R of [δ, R] such that tk+1 − tk < σ , and for k = 0, . . . , p − 1, let us choose τk ∈ [tk , tk+1 ]. We have p−1

g(τk )(n(tk+1 ) − n(tk ))

k=0 p−1 p−1 (n(tk+1 ) − n(tk )) + g(tk ) (n(tk+1 ) − n(tk )) ≥ −ε/n(R) · k=0 k=0

g(f (y)) dμ(y). ≥ −ε + δ≤f (y)
Analogously, p−1

g(τk )(n(tk+1 ) − n(tk )) ≤ ε +

g(f (y)) dμ(y). δ≤f (y)
k=0

This proves that

g(f (y)) dμ(y) −

δ≤f (y)
p−1

g(τk ) (n(tk+1 ) − n(tk )) < ε

k=0

for every partition {tk }k of [δ, R] with tk+1 − tk < σ and any τk ∈ [tk , tk+1 ]. In other words, g(t) is dn(t)-integrable on [δ, R] and

R g(t) dn(t) = g(f (y)) dμ(y). δ

δ≤f (y)
Being δ arbitrary and taking into account the hypothesis

lim g(f (y)) dμ(y) = g(f (y)) dμ(y), δ→0+ δ≤f (y)
0
we derive that g is integrable in the generalized sense of Riemann–Stieltjes on (0, R] w.r.t. n(t) and (13.9) holds.

13 d-Hausdorff Measure and L-capacity

564

If we take f (y) := d(y −1 ◦ z) and g(t) := t 2−Q in (13.9), we obtain

0
d 2−Q (y −1 ◦ z) dμ(y) =

1

t 2−Q dmz (t),

(13.12)

0

where mz (t) = μ(Bd (z, t)). Lemma 13.2.2. Let μ be any compactly supported Radon measure on G. Let also φ be a strictly positive measure function such that

1

I (φ, Q) := 0

φ(t) dt < ∞. t Q−1

(13.13)

Then there exists a constant C > 0 (precisely, it holds C = (Q − 2) I (φ, Q)) such that μ(Bd (z, r)) φ(r) 0
(Γ ∗ μ)(z) ≤ μ(G) + C sup

∀ z ∈ G.

(13.14)

Proof. We split the proof in three steps. (I): z ∈ G is such that (Γ ∗ μ)(z) < ∞. This case ensures that μ({z}) = 0. Let us set mz (r) := μ(Bd (z, r)), r > 0. First of all, we claim that lim ε 2−Q mz (ε) = 0.

(13.15)

ε→0+

Indeed, from μ({z}) = 0 we have mz (0+ ) = 0, so that

ε

2−Q 2−Q + 2−Q mz (ε) = ε (mz (ε) − mz (0 )) = ε dmz (t) ≤ ε 0

where the far right-hand side vanishes as ε → 0, for making use of (13.12), we have

1

t

2−Q

dmz (t) =

0

0
ε

t 2−Q dmz (t),

0

1 0

t 2−Q dmz (t) < ∞. Indeed,

d 2−Q (y −1 ◦ z) dμ(y) ≤ (Γ ∗ μ)(z),

where the far right-hand side is finite, by our assumption. Then

d 2−Q (y −1 ◦ z) dμ(y) (Γ ∗ μ)(z) = + + −1 −1 {y=0} 0
2−Q −1 ≤ 0+ d (y ◦ z) dμ(y) + μ(G \ Bd (z, 1))

(see (13.12)) = 0

0
1

t 2−Q dmz (t) + μ(G \ Bd (z, 1))

13.2 d-Hausdorff Measure and L-capacity

565

1 mz (1) − ε 2−Q mz (ε) − (2 − Q)t 1−Q mz (t) dt ε→0+ ε + μ(G \ Bd (z, 1))

(by parts) = lim

(see (13.15)) = mz (1) + (Q − 2)

1

0

1

t 1−Q mz (t) dt + μ(G \ Bd (z, 1))

μ(Bd (z, t)) φ(t) dt φ(t) 0 μ(Bd (z, t)) , ≤ μ(G) + (Q − 2) I (φ, Q) sup φ(t) 0
= μ(G) + (Q − 2)

t 1−Q

where I (φ, Q) is as in (13.13). This proves (13.14). (II): z ∈ G is such that (Γ ∗ μ)(z) = ∞ and μ({z}) = 0. Let R 1 be such that supp(μ) ⊂ Bd (0, R). We have (also using μ({z}) = 0)

R 2−Q −1 ∞ = (Γ ∗ μ)(z) = d (y ◦ z) dμ(y) = t 2−Q dmz (t). 0
0

Since μ is a Radon measure, this also ensures that

1 t 2−Q dmz (t) = ∞.

(13.16)

0

(13.14) is proved in case (II), if we show that μ(Bd (z, r)) = ∞. φ(r) 0
Suppose by contradiction that there exists M > 0 such that mz (r) = μ(Bd (z, r)) ≤ M φ(r)

for all r ∈ ]0, 1[.

(13.17)

This gives (see also (13.16) and (13.14))

1

1 ∞= t 2−Q dmz (t) = lim t 2−Q dmz (t) ε→0+ ε 0

1 = lim mz (1) − ε 2−Q mz (ε) + (Q − 2) t 1−Q mz (t) dt ε→0+

(see (13.17)) ≤ mz (1) + (Q − 2) M

ε 1

t 1−Q φ(t) dt < ∞,

0

a contradiction. (III): z ∈ G is such that (Γ ∗ μ)(z) = ∞ and μ({z}) > 0. From φ(0+ ) = 0 we derive μ(Bd (z, r)) μ(Bd (z, 1/n)) μ({z}) sup ≥ ≥ −→ ∞ φ(r) φ(1/n) φ(1/n) 0

566

13 d-Hausdorff Measure and L-capacity

We have the following general covering lemma. Lemma 13.2.3 (Covering). Let D = {Bd (xα , rα )}α∈I be any collection of d-balls with supα∈I rα < ∞. Let E ⊂ G be a bounded set covered by D. Then there exists a disjoint at most countable sub-collection {Bd (xαk , rαk )}k∈N of D such that E⊆ Bd (xαk , 3c2 rαk ). k∈N

Here, c is the constant appearing in the pseudo-triangle inequality (13.1). Proof. We ignore the d-balls non-intersecting E, and we still denote by I the resulting sub-family of indices. Let also set Bα := Bd (xα , rα ). Let α1 ∈ I be such that rα1 ≥ 12 supα∈I rα . Let α2 ∈ I (if it exists, otherwise the process ends) be such that Bα2 ∩ Bα1 = ∅,

rα2 ≥

1 sup rα : Bα ∩ Bα1 = ∅ . 2

Proceeding by induction, chosen α1 , . . . , αk , we let αk+1 ∈ I (if it exists, otherwise the process ends) be such that Bαk+1 ∩ Bαj = ∅ for every j = 1, . . . , k, rαk+1 ≥

1 sup{rα : Bα ∩ Bαj = ∅ for every j = 1, . . . , k}. 2

(13.18)

Set A := {αk : k ∈ N}. We split the proof in two steps. / A. Since the selection process (I): The selection process (13.18) is finite. Let α ∈ is finite, we have Bα ∩ Bαj = ∅ for some j . Consequently, a simple application of the pseudo-triangle inequality (13.1) proves that Bα ⊆ Bd (xαj , 3 c2 rαj ), and the proof is complete. (II): The selection process (13.18) is infinite. Since E ∩ Bα = ∅ for every α ∈ I, the Bαk ’s are mutually disjoint and E is bounded, we claim that in this case rαk −→ 0 as k → ∞.

(13.19)

Suppose by contradiction that (13.19) is false. This means that there exist ε0 > 0 and a sequence {kj }j in N such that rαkj > ε0 for every j ∈ N. Consequently, we have ∞=

j ∈N

meas(Bd (xαkj , ε0 )) ≤

j ∈N

meas(Bαkj ) = meas Bαkj . j ∈N

This is impossible since k Bαk is bounded. This last fact is proved as follows. Since E is bounded, there exists H > 0 such that E ⊂ Bd (0, H ); if ξ ∈ Bαk ∩ E, we have d(z) ≤ c(d(z, ξ ) + d(ξ )) ≤ c2 d(z, xk ) + c2 d(xk , ξ ) + c d(ξ ) ≤ 2c2 rαk + c H ≤ 2c2 sup rα + c H < ∞ α

for every k and every z ∈ Bαk . The claimed (13.19) is proved.

13.2 d-Hausdorff Measure and L-capacity

567

If I = A, the proof is complete. Otherwise, let α ∈ / A. If rαk+1 ≥ 12 rα for every k ∈ N, then (thanks to (13.19)) the selection process is finite, which is not the case. Hence, there exists k (necessarily k = 0, for rα1 ≥ 12 rα ) such that rαk+1 < 12 rα . Let k0 be the least positive integer such that 1 rα . 2

(13.20)

Bα ∩ Bαj = ∅

(13.21)

rαk0 +1 < Then there exists j ∈ {1, . . . , k0 } such that

(indeed, if it were Bα ∩ Bαj = ∅ for every j = 1, . . . , k0 , we would have rαk0 +1 ≥ 1 2 rα by the very selection process (13.18), contradicting (13.20)). Then (13.21), together with rαj ≥ 12 rα (recall that k0 is the least index satisfying (13.20)), proves Bα ⊆ Bd (xαj , 3c2 rαj ) by a simple application of the pseudo-triangle inequality (13.1). This ends the proof. Lemma 13.2.4. Let E ⊂ G be a bounded L-polar set. Then, for every measure function φ such that

1 t 1−Q φ(t) dt < ∞, (13.22) 0

we have mφ (E) = 0. Proof. From Corollary 11.2.5 (page 492), being E bounded, there exists μ ∈ M0 such that Γ ∗ μ ≡ ∞ on E. If φ is as in the assertion of the present lemma, we set

φ1 (t) := min φ(3 c2 t), φ(1/2) . Here c≥1 is the constant appearing in the pseudo-triangle inequality (13.1). It is 1 easily seen that 0 t 1−Q φ1 (t) dt is finite, thanks to (13.22). Fix a > 0 such that μ(G) < φ(1/2). a

(13.23)

From Lemma 13.2.2 applied to the measure function φ1 we get μ(Bd (x, r)) φ1 (r) 0
(Γ ∗ μ)(x) ≤ μ(G) + C(Q, φ1 ) sup

for any x ∈ G. If, in particular, we take x ∈ E (so that Γ ∗ μ(x) = ∞), being μ(G) < ∞, we derive the existence of r(x) ∈]0, 1[ such that μ(Bd (x, r(x))) > a. φ1 (r(x))

(13.24)

568

13 d-Hausdorff Measure and L-capacity

By the covering Lemma 13.2.3 applied to the family

Bx := Bd (x, r(x)) : x ∈ E , there exists a sequence xk ∈ E such that the Bxk ’s are mutually disjoint and E⊆ Bd (xk , 3c2 r(xk )). (13.25) k

This gives (using (13.24) in the first inequality below) 1 1 1 φ1 (r(xk )) < μ(Bxk ) = μ Bxk ≤ μ(G). a a a k

k

(13.26)

k

Taking into account the choice (13.23) of a, from (13.26) it follows that no summand φ1 (r(xk )) coincides with φ(1/2). By the very definition of φ1 , this gives φ1 (r(xk )) = φ(3 c2 r(xk ))

∀ k ∈ N.

(13.27)

This yields (3c2 )

Mφ

(E) ≤

φ(3c2 r(xk )) =

k

φ1 (r(xk )) ≤ μ(G)/a.

(13.28)

k

In the first inequality we used (13.25), in the second we exploited (13.27), in the third we took into account (13.26). Since we can take any a 1 satisfying (13.23), we can let a tend to ∞ in (13.28) to derive that (3c2 ) Mφ (E) = 0. (ρ)

From the fact that φ is increasing and positive it easily1 follows that Mφ (E) = 0 for any ρ > 0, whence mφ (E) = 0. The proof is complete. As a consequence of Lemma 13.2.4, we have a partial converse of Theorem 13.1.7. Theorem 13.2.5 (d-Hausdorff measure and L-polarity. II). Let E ⊂ G be a bounded L-polar set. Then we have m(α) (E) = 0 for every α > Q − 2.

(13.29)

Hence, the d-Hausdorff dimension of a bounded L-polar set is ≤ Q − 2. Proof. This follows from Lemma 13.2.4, by taking φ(t) = t α with α > Q − 2 and the very definition of d-Hausdorff dimension (see Lemma 13.1.3). Collecting together Theorems 13.1.7 and 13.2.5, we have the following result. 1 See Ex. 4 at the end of the chapter.

13.3 New Phenomena Concerning the d-Hausdorff Dimension

569

Remark 13.2.6. Let E be any bounded and L-capacitable subset of G, and let dimH (E) be the d-Hausdorff dimension of E. We have: (i) (ii) (iii) (iv)

If dimH (E) < Q − 2, then E is L-polar, If dimH (E) > Q − 2, then E is not L-polar, If dimH (E) = Q − 2 and m(Q−2) (E) ∈ [0, ∞[, then E is L-polar, If dimH (E) = Q − 2 and m(Q−2) (E) = ∞, we cannot conclude anything about the L-polarity of E.

13.3 New Phenomena Concerning the d-Hausdorff Dimension: The Case of the Heisenberg Group The aim of this section is to provide a brief overview (without any proof) of some remarkable recent results on the Hausdorff dimension in the Heisenberg group, by Z.M. Balogh, M. Rickly and F. Serra Cassano [BRSC03]. Let H1 be the Heisenberg–Weyl group on R3 . We denote its points by z = (x, y, t). The relevant dilation is δλ (z) = (λx, λy, λ2 t), and the composition law is given by z ◦ z = x + x , y + y , t + t + 2(x y − y x) . The homogeneous norm d(z) = ((x 2 + y 2 )2 + t 2 )1/4 is an L-gauge, where L is the canonical sub-Laplacian on H1 . We denote by e the usual Euclidean norm on R3 , i.e. e(x, y, t) = (x 2 + y 2 + t 2 )1/2 . As it is expected, the Hausdorff measure and the Hausdorff dimension of a subset of H1 ≡ R3 w.r.t. d and e may be different. For example, the d-Hausdorff dimension of H1 is 4 (see Example 13.1.4), whereas its e-Hausdorff dimension is 3. Furthermore, the d-Hausdorff dimension of a regular surface is 3, and the d-Hausdorff dimension of a regular curve can be 1 or 2 (see [Gro96,Str92]). β In the sequel of this section, we use the notation Heα and Hd to denote, respectively, m(α) when the relevant gauge is the Euclidean norm e and m(β) when the relevant gauge is the above d. We fix some notation. We define the following functions: ⎧ f (α) := min{2α, α + 1}, ⎪ ⎪ ⎨ ϕ(β) := max{ 12 β, β − 1}, f, ϕ, g, γ : [0, ∞) → [0, ∞), where ⎪ ⎪ ⎩ g(β) := min{β, 1 + β/2}, γ (α) := max{α, 2α − 2}. (Note that f = ϕ −1 and g = γ −1 .) The following theorem holds. Theorem 13.3.1 (Dimension jump theorem, [BRSC03]). For every real numbers α, β > 0, we have f (α)

Hd

g(β) He

γ (α)

Heα Hd

β Hd

,

Heϕ(β) .

570

13 d-Hausdorff Measure and L-capacity

(Here, μ λ means that the measure μ is absolutely continuous with respect to the measure λ.) For example, the above theorem (and the very definition of absolutely continuous measures) give f (α)

Heα (E) = 0 ⇒ Hd

Heα (E) > 0 ⇒ β Hd (E) = 0 β Hd (E)

(E) = 0, γ (α)

Hd

(E) > 0, g(β)

⇒ He

(E) = 0,

> 0 ⇒ Heϕ(β) > 0.

Together with Theorem 13.3.1, Balogh, Rickly and Serra Cassano proved the following theorem. Theorem 13.3.2 (Sharpness of the dimension jump, [BRSC03]). The following assertions hold: (i) Given α ∈ (0, 3], there exists a compact set Eα ⊂ H1 such that f (α)

Heα (Eα ) < ∞ and Hd

(Eα ) > 0

or, equivalently, given β ∈ (0, 4], there exists a compact set Eβ ⊂ H1 such that β

Heϕ(β) (Eβ ) < ∞ and Hd (Eβ ) > 0; (ii)’ For β ∈ (0, 2) ∪ {4}, there exists a compact set Eβ ⊂ H1 such that g(β)

β

Hd (Eβ ) < ∞ and He

(Eβ ) > 0

or, equivalently, given α ∈ (0, 2) ∪ {3}, there exists a compact set Eα ⊂ H1 such that γ (α)

Hd

(Eα ) < ∞ and Heα (Eα ) > 0;

(ii)” For β ∈ [2, 4) and ε ∈ (0, 1), there exists a compact set Eβ,ε ⊂ H1 such that β

g(β)−ε

Hd (Eβ,ε ) = 0 and He

(Eβ,ε ) > 0.

We explicitly describe some of the sets Eα ’s in the previous theorem (see [BRSC03] for the details). (i) Let 0 < s < 1. We consider the Cantor2 set Cs ⊂ [0, 1] of (Euclidean) Hausdorff dimension s. 2 We recall the construction of C . Let s ∈ (0, 1) be fixed and set σ := 2−1/s . Let k ∈ N∪{0} s s in the following way: and j ∈ {1, . . . , 2k }. We define Ik,j s = [0, 1], I0,1 s = [0, σ ], I s = [1 − σ, 1], I1,1 1,2

13.3 New Phenomena Concerning the d-Hausdorff Dimension

571

(a) If α ∈ (0, 1], we let Eα = {0} × {0} × Cα (here C1 := [0, 1]). Obviously, f (α) Heα (Eα ) is positive and finite and the same is true of Hd2α (Eα ) = Hd (Eα ) for, roughly speaking, the distance induced by the gauge d on the t-axis coincides with the 12 -power of the Euclidean distance. Note that the vertical segment E1 = {0} × {0} × [0, 1] has d-Hausdorff dimension 2, but e-Hausdorff dimension 1. Since it also has finite Hd2 -measure, by Theorem 13.1.7 we infer that E1 is L-polar, where L is the canonical sub-Laplacian of H1 (recall that here Q − 2 = 2). (b) If α ∈ (1, 2), the set Eα = Cα−1 × {0} × [0, 1] (a Cantor set of vertical segments) has the requisites in Theorem 13.3.2-(i). (c) If α = 2, it suffices to take E2 = {x 2 + y 2 + t 2 = 1} (see also [Pan82]). Note that E2 has positive and finite 2-dimensional Euclidean Hausdorff measure and positive 3-dimensional d-Hausdorff measure. (d) If α ∈ (2, 3), then the choice ∂Bd (0, r) Eα = r∈Cα−2

(a Cantor set of d-spheres) has the requisites in Theorem 13.3.2-(i). (e) If α = 3, take E3 = Bd (0, 1). Note that E3 has positive and finite 3-dimensional Euclidean Hausdorff measure and the same is true of its 4-dimensional d-Hausdorff measure. (ii) If β ∈ (0, 1], the set Eβ = Cβ × {0} × {0} (here C1 stands for [0, 1]) has positive β β and finite Hd - and He -measures. If β = 4, it suffices to take E4 = Bd (0, 1), as already discussed. The case β ∈ [2, 4) is more involved, and the reader is referred directly to [BRSC03]. Bibliographical Notes. In the case of the classical Laplace operator, the relationship between the Hausdorff measure and capacity can be found, e.g. in [AG01, Section 5.9] (whose exposition we followed here) and [HK76, Section 5.4]. See [BRSC03] for recent results on the Hausdorff measure in the context of the Heisenberg group. See also [Gro96,Str92]. s = [0, σ 2 ], I s = [σ − σ 2 , σ ], I s = [1 − σ, 1 − σ + σ 2 ], I s = [1 − σ 2 , 1], I2,1 2,2 2,3 1,2 .. . s = [a s , bs ] for j = 1, . . . , 2k , then if we are given Ik,j k,j k,j s s s k+1 ], Ik+1,2j −1 = [ak,j , ak,j + σ

s s − σ k+1 , bs ]. Ik+1,2j = [bk,j k,j

Now, we define Cs as Cs :=

∞ 2k k=0 j =1

Ik,j =:

∞

Cks .

k=0

s ⊂ Cks , so that Cs = limk→∞ Cks with the obvious meaning. Moreover, Note that Ck+1 s Ck is the union of 2k disjoint closed intervals of length σ k . It can be proved that Cs is a compact subset of [0, 1] with no interior points and such that 0 < Hes (Cs ) < ∞.

572

13 d-Hausdorff Measure and L-capacity

13.4 Exercises of Chapter 13 Ex. 1) Consider the Heisenberg–Weyl group (H1 , ◦) on R3 . Let z0 = (x0 , y0 , t0 ) ∈ H1 be fixed, with t0 = 0. Consider the following “segment of δλ -ray” R := {δλ (z0 ) : λ ∈ [a, b]}, where a < b are fixed positive real numbers. Finally, let ΔH1 be the canonical sub-Laplacian on H1 . Prove the following facts: • If Γ is the fundamental solution for ΔH1 , the function

u(z) := a

b

Γ (δλ (z0−1 ) ◦ z) dλ

belongs to S(H1 ) (w.r.t. ΔH1 ) and u(0) < ∞; • u ≡ ∞ on R; • Derive that R is a bounded ΔH1 -polar set. If d is a ΔH1 -gauge, what is the d-Hausdorff dimension of R? (Use the remarks in Section 13.3.) b Hint: u(0) = Γ (z0 ) a λ−2 dλ; u ∈ S(H1 ) by Theorem 8.2.20, page 410; by the explicit form of ◦ and Γ , if λ0 ∈ [a, b] is fixed, it holds

u(δλ0 (z0 )) = [· · ·] =

≈

λ0 +ε λ0 −ε

b

a

1 dλ {(λ0 − λ)4 (x02 + y02 )2 + (λ20 − λ2 )2 t02 }1/2

1 dλ = ∞. |λ − λ0 | 2λ0 |t0 |

Ex. 2) Let Ω ⊂ G be any bounded open set, and let ε > 0 be any positive real number. Prove that there exists a countable collection of disjoint d-balls rk Bk := Bd xk , 4 with

rk 2

< ε for every k and a set N such that ⎧ ⎨ Ω = N ∪ k Bk , ⎩

k

Bd (xk , c rk ) ⊇ Ω.

Ex. 3) Let μ be any compactly supported Radon measure on G. Suppose z ∈ G is such that (Γ ∗ μ)(z) < ∞ and prove that μ({z}) = 0. (Compare to Exercise 22, Chapter 9.) Ex. 4) Let E ⊂ G be any set and φ be a measure function. Suppose there exists ρ0 > 0 such that (ρ ) Mφ 0 (E) = 0. (ρ)

Prove that Mφ (E) = 0 for every ρ > 0, whence mφ (E) = 0.

13.4 Exercises of Chapter 13 (ρ)

(Hint: Since the map ]0, ∞] ρ → Mφ (prove this!), we have

(ρ) Mφ (E)

is non-negative and decreasing

= 0 for every ρ ≥ ρ0 . Let now ρ ∈]0, ρ0 [ be (ρ0 )

fixed. By the very definition of Mφ

()

573

(E) = 0, we have

∀ ε > 0 ∃ {Bd (xk (ε), rk (ε))}k∈N

⎧ r (ε) < ρ , ⎨ k 0 E ⊆ k Bd (xk (ε), rk (ε)), : ⎩ k φ(rk (ε)) < ε.

Since φ is positive on ]0, ∞[, in () we can take any ε ∈]0, φ(ρ)[. Hence, the third condition in the right-hand side of () gives φ(rk (ε)) ≤ φ(rk (ε)) < ε < φ(ρ) ∀ k ∈ N. k (ρ)

Since φ is increasing, this gives rk (ε) < ρ, so that Mφ (E) = 0.) Ex. 5) Prove the assertions of Example 13.1.4 and Remark 13.1.5. (Hint: The following facts may help. For every x ∈ G and r> 0, we have r Q = c H N (Bd (x, r)), where c = H N (Bd (0, 1)). If E ⊆ k Bd (xk , rk ), then x◦E ⊆ Bd (x ◦ xk , rk ), δλ (E) ⊆ Bd (δλ (xk ), λrk ), k

k

(ρ)

(ρ)

(λ ρ)

(ρ)

whence H(α) (x ◦ E) = H(α) (E) and H(α) (δλ (E)) = λα H(α) (E).) Ex. 6) Prove the assertion in Remark 13.1.2. More precisely, we have the following results. For any E ⊆ G, set diamd (E) := sup{d(x −1 ◦ y) : x, y ∈ E},

diamd (∅) := 0.

If φ is a measure function as in Definition 13.1.1, we let (ρ) Hφ (E) := inf φ(diamd (Ek )) : E ⊆ Ek , diamd (Ek ) < ρ ∀ k , k

(ρ) Mφ (E)

:= inf

φ(rk ) : E ⊆

k (ρ)

k

Bd (xk , rk ), rk < ρ ∀ k .

k (ρ)

• Prove that Hφ , Mφ , hφ , mφ are outer measures on G, where (ρ)

hφ (E) := sup Hφ (E), ρ>0

(ρ)

mφ (E) := sup Mφ (E). ρ>0

• Let A, B be arbitrary subsets of G. Prove that hφ (A ∪ B) = hφ (A) + hφ (B), distd (A, B) > 0 ⇒ mφ (A ∪ B) = mφ (A) + mφ (B). Here, we have set

(13.30)

574

13 d-Hausdorff Measure and L-capacity

distd (A, B) := inf{d(a −1 ◦ b) : a ∈ A, b ∈ B}. Note that distd (A, B) > 0 iff the usual Euclidean distance between A and B is positive.3 Hence, derive from (13.30) and from very general results of measure theory, that every (open and then every) Borel set in G is both hφ and mφ measurable. Hence, if restricted to their related measurable sets, hφ and mφ are Borel measures. Now, we take φ(t) = t α with α > 0, and write h(α) and m(α) instead of ht α and mt α . • Prove that 2−α m(α) (E) ≤ h(α) (E) ≤ (2c)α m(α) (E) for every E ⊆ G,

(13.31)

where c is the constant in the pseudo-triangle inequality for d. The following facts might help: r ≤ diam Bd (x, r) ≤ 2c r; if e ∈ E, then E ⊆ Bd (e, diam(E)) ⊂ Bd (e, 2diam(E)); ε/(2c)

Htεα (E) ≤ (2c)α Ht α

(E);

α ε Mt2ε α (E) ≤ 2 Ht α (E).

• Derive from (13.31) that, for every bounded set E ⊂ G, there exists a number α(E) ∈ [0, Q] (Q is the homogeneous dimension of G) such that h(α) (E) = m(α) (E) = ∞ for all α < α(E), h(α) (E) = m(α) (E) = 0 for all α > α(E). Thus, we have α(E) = inf{α > 0 : h(α) (E) = 0} = inf{α > 0 : m(α) (E) = 0}.

3 Indeed, if a ∈ A, b ∈ B, then observe that a −1 ◦ b → 0 in the Euclidean metric iff j j j j d(aj−1 ◦ bj ) → 0.

Part III

Further Topics on Carnot Groups

14 Some Remarks on Free Lie Algebras

The aim of this chapter is to investigate a few properties of the free nilpotent Lie algebras. After recalling the definition of the free Lie algebra fm,r with m generators and nilpotent of step r, we provide an algorithm (due to M. Hall) to construct a basis for fm,r , jointly with several examples. Furthermore, we describe another useful algorithm concerning with free Lie algebras in an analytic context, an algorithm due to M. Grayson and R. Grossman. In Section 14.2, we furnish a canonical way to construct free Carnot groups (i.e. Carnot groups whose Lie algebras is free and nilpotent) by means of the Campbell– Hausdorff formula. As we shall see in Chapter 16, free Carnot groups are the appropriate generalizations of the usual Euclidean group, when we are concerned with the “equivalence” of all sub-Laplacians.

14.1 Free Lie Algebras and Free Lie Groups We first recall the definition of free Lie algebra with m generators and nilpotent of step r (see, e.g. [VSC92, p. 45]; see also [Var84, p. 174]). Definition 14.1.1 (The free Lie algebra fm,r ). Let m ≥ 2 and r ≥ 1 be fixed integers. We say that fm,r is the free Lie algebra with m generators x1 , . . . , xm and nilpotent of step r if: (i) fm,r is a Lie algebra generated by its elements x1 , . . . , xm , i.e. fm,r = Lie{x1 , . . . , xm }; (ii) fm,r is nilpotent of step r; (iii) for every Lie algebra n nilpotent of step r and for every map ϕ from the set ϕ from {x1 , . . . , xm } to n, there exists a (unique) homomorphism of Lie algebras fm,r to n which extends ϕ.

578

14 Some Remarks on Free Lie Algebras

Remark 14.1.2. It is easy to see that, fixed m and r, the Lie algebra fm,r is unique up to isomorphism: it is enough to compare two algebras f, f satisfying the above definition and recognize that the morphism as in (iii) which takes the generators of f into the generators of f is the inverse of the morphism taking the generators of f into those of f. Moreover, if fm,r is isomorphic to fm ,r , then it necessarily holds m = m and r = r . We explicitly remark that it is not trivial to prove the existence of such an algebra fm,r . We refer the reader to [Var84] for this topic. Definition 14.1.3 (Free Carnot group). We say that a Carnot group G is a free Carnot group if its Lie algebra g is isomorphic to fm,r for some m and r. The following lemma concerning with free Lie algebras fm,r will be useful in the sequel. Lemma 14.1.4. Let fm,r be the free Lie algebra with m generators x1 , . . . , xm , nilpotent of step r. Let ϕ : span{x1 , . . . , xm } −→ span{x1 , . . . , xm } be a linear bijective map. Then there exists one (and only one) isomorphism of Lie algebras ϕ : fm,r −→ fm,r extending ϕ. Proof. We consider the maps ϕ, ϕ −1 : span{x1 , . . . , xm } −→ span{x1 , . . . , xm }. By the very definition of fm,r , there exist two homomorphisms of Lie algebras ϕ, −1 −1 (ϕ ) defined on fm,r into itself which extend ϕ and ϕ , respectively. We consider the map −1 ) ◦ ψ := (ϕ ϕ. The proof is complete if we show that ψ is the identity map. To this end, we note that, for every i = 1, . . . , m, one has −1 )(ϕ(x )) = ϕ −1 (ϕ(x )) = x , ψ(xi ) = (ϕ i i i since ϕ(xi ) ∈ span{x1 , . . . , xm }. By the definition of fm,r , there exists a unique homomorphism of the Lie algebra fm,r into itself which extends ψ|span{x1 ,...,xm } . Since both ψ and the identity map satisfy this requirement, the assertion immediately follows.

14.1 Free Lie Algebras and Free Lie Groups

579

We next give a model for fm,r . It is known that, setting H (m, r) := dim(fm,r ), fm,r may be represented by upper triangular matrices of order H (m, r) (see [Var84]). There exists another remarkable representation of fm,r via vector fields, useful in the applications and easy to be dealt with, due to M. Grayson and R. Grossman (see [GG90]), which we now briefly recall. This representation makes use of the so-called Hall basis for fm,r . Definition 14.1.5 (The Hall basis for fm,r ). Let fm,r be the free Lie algebra with m generators x1 , . . . , xm , nilpotent of step r. We define the elements of the Hall basis for fm,r in the following way: • x1 , . . . , xm are the first m elements of the Hall basis and they are referred to as the standard monomials of height 1; • We now let 2 ≤ n ≤ r, and we define the standard monomials of height n: suppose we have already defined the standard monomials of height 1, . . . , n − 1, and suppose we have ordered them in such a way that u precedes v (we write u < v) if the height of u is strictly lower than that of v; then the Lie bracket [u, v] is a standard monomial of height n if the sum of the heights of u and v equals n and if, moreover, the following conditions are satisfied: 1. u and v are standard monomials such that u > v; 2. if u = [x, y] is the form of the standard monomial u, then v ≥ y. The collection of all the standard monomials of height 1, . . . , r is referred to as the Hall basis for fm,r . The following result holds. Theorem 14.1.6 (Hall, [Hal50]). The Hall basis for fm,r is a basis (in the sense of vector fields) for fm,r . Proof. The assertion follows from the arguments in the paper by M. Hall [Hal50, Theorem 3.1], in which a general basis for free Lie rings is constructed exactly as in Definition 14.1.5 (but with no restriction on r, i.e. r = ∞). Since fm,r is nilpotent of step r, it is enough to consider the elements of the basis constructed in [Hal50] which are brackets of height r at most. Example 14.1.7. We next give some examples of Hall bases: • The Hall basis for f3,4 is given by: height 1: height 2:

x1 , x2 , x3 , [x2 , x1 ], [x3 , x1 ], [x3 , x2 ],

⎧ ⎨ [[x2 , x1 ], x1 ], [[x3 , x1 ], x1 ], height 3: [[x2 , x1 ], x2 ], [[x3 , x1 ], x2 ], [[x3 , x2 ], x2 ], ⎩ [[x2 , x1 ], x3 ], [[x3 , x1 ], x3 ], [[x3 , x2 ], x3 ],

580

14 Some Remarks on Free Lie Algebras

⎧ [[[x , x ]x ]x ], [[[x , x ]x ]x ], 2 1 1 1 3 1 1 1 ⎪ ⎪ ⎪ [[[x , x ]x ]x ], [[[x , x ]x ]x ], ⎪ ⎪ 2 1 1 2 3 1 1 2 ⎪ ⎪ ⎪ ⎪ [[[x2 , x1 ]x2 ]x2 ], [[[x3 , x1 ]x2 ]x2 ], [[[x3 , x2 ]x2 ]x2 ], ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ [[[x2 , x1 ]x1 ]x3 ], [[[x3 , x1 ]x1 ]x3 ], height 4: [[[x2 , x1 ]x2 ]x3 ], [[[x3 , x1 ]x2 ]x3 ], [[[x3 , x2 ]x2 ]x3 ], ⎪ ⎪ ⎪ ⎪ [[[x2 , x1 ]x3 ]x3 ], [[[x3 , x1 ]x3 ]x3 ], [[[x3 , x2 ]x3 ]x3 ], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [[x3 , x1 ], [x2 , x1 ]], [[x3 , x2 ], [x2 , x1 ]], ⎪ ⎪ ⎪ ⎪ ⎩ [[x , x ], [x , x ]]. 3

2

3

1

• The Hall basis for f4,2 is given by: height 1:

x1 , x2 , x3 , x4 , ⎧ ⎪ ⎨ [x2 , x1 ], [x3 , x1 ], [x4 , x1 ], [x3 , x2 ], [x4 , x2 ], height 2: ⎪ ⎩ [x4 , x3 ].

• The Hall basis for fm,2 is given by: height 1:

x1 , x2 , x3 , . . . , xm , ⎧ [x2 , x1 ], [x3 , x1 ], · · · [xm , x1 ], ⎪ ⎪ ⎪ ⎪ ⎨ [x3 , x2 ], · · · [xm , x2 ], height 2: .. .. ⎪ ⎪ . . ⎪ ⎪ ⎩ [xm , xm−1 ].

Remark 14.1.8. We explicitly remark that H (m, 2) = dim(fm,2 ) =

m

j=

j =1

m(m + 1) . 2

In particular, this implies, the following result. Remark 14.1.9. The Heisenberg group HN is a free Carnot group if and only if N = 1. Indeed, since the Lie algebra hN of HN has step 2 and 2N generators, a necessary condition for HN to be free is that 2N + 1 = dimhN = H (2N, 2) = 2N (2N + 1)/2, i.e. that N = 1. On the other hand, it is immediate to recognize that h1 is isomorphic to f2,2 . As we shall see more closely in Chapter 17, this shows that, even if the vector fields related to the Heisenberg group HN are naturally defined on R2N +1 , the free (lifted) counterpart of such vector fields acts on R2N (2N +1)/2 .

14.1 Free Lie Algebras and Free Lie Groups

581

We finally turn our attention to the cited model for fm,r by Grayson–Grossman, which we now describe. We order the elements of the Hall basis for fm,r according1 to Definition 14.1.5. We set H := dimfm,r , and we denote E1 , . . . , E H the elements of the Hall basis with such a fixed ordering. We fix an element Ei . According to the construction of the Hall basis in Definition 14.1.5, it certainly has the following form (if i > m) Ei = [Ej1 , Ek1 ],

with j1 > k1 .

Taking Ek1 fixed (whatever its height is) and arguing in the same way for Ej1 , one has Ei = [[Ej2 , Ek2 ]; Ek1 ], with j2 > k2 . After finitely many steps, we have obtained the expression Ei = [Ej1 , Ek1 ] = [[Ej2 , Ek2 ]; Ek1 ] = [[[Ej3 , Ek3 ], Ek2 ]; Ek1 ] = [[[[Ej4 , Ek4 ], Ek3 ], Ek2 ]; Ek1 ] = ··· = [[· · · [[Ejn , Ekn ], Ekn−1 ], . . . , Ek2 ]; Ek1 ], with 1 ≤ kn < jn ≤ m

and kl+1 ≤ kl

for 1 ≤ l ≤ n − 1.

This maximal expression for Ei involves n commutations (or, equivalently, there are n indices of the type k’s), and then we set, by definition, d(i) := n with the convention d(1) = d(2) = · · · = d(m) := 0. 1 A somewhat “canonical” way to order the elements of the Hall basis is the following one:

1. x1 , x2 , . . . , xm is a fixed ordering of the standard monomials of height 1. 2. If 2 ≤ n ≤ r, we now show how to canonically order the standard monomials of height n. Suppose the standard monomials of heights 1, . . . , n − 1 have already been ordered. Let u and v be two standard monomials of height n in the form u = [x1 , y1 ],

v = [x2 , y2 ].

We then follow the rules: • if y1 = y2 , then u precedes v if and only if y1 precedes y2 ; • if y1 = y2 , then u precedes v if and only if x1 precedes x2 .

582

14 Some Remarks on Free Lie Algebras

This process associates in a natural way a multi-index I (i) = (a1 , a2 , . . . , aH ) to Ei , where as := cardinality of the set {t | kt = s}. In other words, for every s = 1, . . . , H , as is the number of times the element Es appears within Ekn , Ekn−1 , . . . , Ek2 ; Ek1 . By definition, we set I (1) = I (2) = · · · = I (m) := (0, . . . , 0). Moreover, we say that Ei is a direct descendant of all the elements of the type Ejl Ej1 , Ej2 , . . . , Ejn−1 , Ejn , and we write jl ≺ i. We remark that ≺ is a partial ordering, and that if a ≺ b, then every entry of I (a) is ≤ of the corresponding entry of I (b). For every pair a, b satisfying a ≺ b, we define the monomial a≺b

Pa,b :=

⇒

(−1)d(b)−d(a) I (b)−I (a) x . (I (b) − I (a))!

For example, if Ei = [[[[[[[E2 , E1 ], E1 ], E1 ], E2 ], E4 ], E4 ]; E7 ] Ek = [[[E2 , E1 ], E1 ]; E1 ],

and

then d(i) = 7, d(k) = 3, d(2) = 0 and I (i) = (3, 1, 0, 2, 0, 0, 1, 0, . . . , 0), I (k) = (3, 0, 0, 0, 0, 0, 0, 0, . . . , 0), whereas

x13 x2 x42 x7 x2 x42 x7 , Pk,i = . 3! 2! 2! The following remarkable result holds (see [GG90, Theorem 2.1]). P2,i = −

Theorem 14.1.10 (Grayson–Grossman, [GG90]). Let r ≥ 1, m ≥ 2, and let H be the dimension of fm,r . Then the vector fields ∂ , ∂x1 ∂ ∂ E2 := + P2,j , ∂x2 ∂xj E1 :=

j 2

.. . ∂ ∂ Em := + Pm,j ∂xm ∂xj j m

with polynomial coefficients on

RH

have the following properties:

(14.1)

14.1 Free Lie Algebras and Free Lie Groups

583

(1) E1 , . . . , Em are homogeneous of degree one with respect to the dilation canonically induced on RH by the stratification RH ≡ fm,r = V1 ⊕ · · · ⊕ Vr , where Vi is the vector space spanned by the standard monomials of the Hall basis with height i; (2) The elements of the Hall basis generated by the vector fields E1 , . . . , Em satisfy Ei (0) =

∂ , ∂xi

i = 1, . . . , H ;

(3) The Lie algebra of vector fields on RH generated by E1 , . . . , Em is isomorphic to fm,r . Remark 14.1.11. We explicitly remark that the vector fields in (14.1) satisfy conditions (H0)–(H1)–(H2) in Section 4.2, page 191. Then they naturally define, as described in Section 4.2 (see, in particular, Theorem 4.2.10), a Carnot group G which is free (of step r and with m generators), according to Definition 14.1.3: indeed, by (3) in the above Theorem 14.1.10, we immediately infer that the Lie algebra of G above Theorem 14.1.10, the canonical is isomorphic to fm,r . Moreover, by (2) in the 2 sub-Laplacian ΔG is simply given by ΔG = m j =1 Ej . Example 14.1.12. To end this section, we give an example of a model for f4,2 . Let E1 , E2 , E3 , E4 be the generators of f4,2 . We set E5 := [E2 , E1 ], E8 := [E3 , E2 ],

E6 := [E3 , E1 ], E9 := [E4 , E2 ],

E7 := [E4 , E1 ], E10 := [E4 , E3 ].

With the above notation, one has d(1) = d(2) = d(3) = d(4) = 0, d(5) = d(6) = d(7) = d(8) = d(9) = d(10) = 1, I (1) = I (2) = I (3) = I (4) = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0), I (5) = I (6) = I (7) = (1, 0, 0, 0, 0, 0, 0, 0, 0, 0), I (8) = I (9) = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0), I (10) = (0, 0, 1, 0, 0, 0, 0, 0, 0, 0), 2 ≺ 5, 3 ≺ 6, 4 ≺ 7, P2,5 = P3,6 = P4,7 = −x1 , P3,8 = P4,9 = −x2 , P4,10 = −x3 .

3 ≺ 8,

4 ≺ 9,

4 ≺ 10,

Then, according to Theorem 14.1.10, a model for f4,2 is given by the Lie algebra of vector fields generated by the following vector fields on R10 :

584

14 Some Remarks on Free Lie Algebras

E1 :=

∂ , ∂x1 ∂ ∂ − x1 , ∂x2 ∂x5 ∂ ∂ ∂ − x1 − x2 , E3 := ∂x3 ∂x6 ∂x8 ∂ ∂ ∂ ∂ − x1 − x2 − x3 E4 := ∂x4 ∂x7 ∂x9 ∂x10

E2 :=

with the commutator identities E5 = [E2 , E1 ] =

∂ , ∂x5

E6 = [E3 , E1 ] =

∂ , ∂x6

∂ , ∂x7 ∂ , E8 = [E3 , E2 ] = ∂x8 ∂ , E9 = [E4 , E2 ] = ∂x9 ∂ = [E4 , E3 ] = . ∂x10 E7 = [E4 , E1 ] =

E10

As observed in Remark 14.1.11, the following second order differential operator on R10 E12 + E22 + E32 + E42 = (∂x1 )2 + (∂x2 − x1 ∂x5 )2 + (∂x3 − x1 ∂x6 − x2 ∂x8 )2 + (∂x4 − x1 ∂x7 − x2 ∂x9 − x3 ∂x10 )2 is then the canonical sub-Laplacian related to a suitable free homogeneous Carnot group on R10 , with 4 generators and nilpotent of step 2. Finally, we explicitly remark that the above is not the only possible model for f4,2 (see, e.g. Example 14.2.5 in Section 14.2).

14.2 A Canonical Way to Construct Free Carnot Groups 14.2.1 The Campbell–Hausdorff Composition First of all, we recall some references on the so-called Campbell–Hausdorff formula. Roughly speaking, if X and Y are two non-commuting indeterminates, the (Baker–)Campbell–(Dynkin–)Hausdorff formula states that the formal expression

14.2 A Canonical Way to Construct Free Carnot Groups

585

“ log(exp(X) exp(Y )) ” can be expressed in terms of an infinite sum of iterated commutators of X and Y . This statement can be made precise in many contexts such as for formal power series, for matrix algebras, for general normed Banach algebras, for finite-dimensional Lie groups, for solutions of differential equations, etc. (See Bibliographical Notes at the end of the chapter.) Since we are mainly interested in the setting of Lie groups and Lie algebras, we briefly recall how the Campbell–Hausdorff formula naturally arises in this context (see also Section 2.2, page 121, where we discussed this topic; for the reader’s convenience, the main results we need here are reproduced). Let (n, [·,·]) be an abstract nilpotent Lie algebra. For X, Y ∈ n, we set2 X Y :=

(−1)n+1 n≥1

n

= X+Y +

pi +qi ≥1 1≤i≤n

(ad X)p1 (ad Y)q1 · · · (ad X)pn (ad Y)qn −1 Y ( nj=1 (pj + qj )) p1 ! q1 ! · · · pn ! qn !

1 [X, Y ] 2

1 1 [X, [X, Y ]] − [Y, [X, Y ]] 12 12 1 1 − [Y, [X, [X, Y ]]] − [X, [Y, [X, Y ]]] 48 48 + {brackets of height ≥ 5}.

+

(14.2)

Since n is nilpotent, (14.2) is a finite sum and determines a binary operation on n, which is defined in a universal way by a sum of Lie monomials with rational coefficients. Now, let (F, ∗) be a Lie group with the Lie algebra f. Suppose the exponential map Exp : f → F has an inverse function Log globally defined. Then the operation

f × f (X, Y ) → Log Exp (X) ∗ Exp (Y ) ∈ f is well posed on f. With this last composition law, f is obviously a Lie group isomorphic to (F, ∗) via Exp . The following result states that this last composition law is (under suitable hypotheses on F) precisely the universal one defined in (14.2), i.e. the following Campbell–Hausdorff formula holds:

Log Exp (X) ∗ Exp (Y ) = X Y for all X, Y ∈ f. (14.3) 2 Here, we use the notation (ad A)B = [A, B]. Moreover, if q = 0, the term in the sum n

(14.2) is, by convention, (ad X)p1 (ad Y)q1 · · · (ad X)pn−1 (ad Y)qn−1 (ad X)pn −1 X. Clearly, if qn > 1, or qn = 0 and pn > 1, the term is zero.

586

14 Some Remarks on Free Lie Algebras

Theorem 14.2.1 (Corwin and Greenleaf, [CG90], Theorem 1.2.1). Let (F, ∗) be a connected and simply connected Lie group. Suppose that the Lie algebra f of F is nilpotent. Let be the operation defined in (14.2). Then defines a Lie group structure on f and Exp : (f, ) → (F, ∗) is a Lie group isomorphism. In particular, if Log is the inverse function of Exp , then Log : (F, ∗) → (f, ) is a Lie group isomorphism and (14.3) holds. We then recall the third fundamental theorem of Lie, which is another deep result in the theory of Lie groups. Theorem 14.2.2 (Varadarajan, [Var84], Theorem 3.15.1). Let f be a finite-dimensional Lie algebra. Then there exists a connected and simply connected Lie group whose Lie algebra is isomorphic to f. From the above two theorems we obtain the following result (for its proof, see Corollary 2.2.15, page 130). Theorem 14.2.3. If f is a finite-dimensional nilpotent Lie algebra, then the operation introduced in (14.2) defines a Lie group structure on f. We call the Campbell– Hausdorff operation on f. 14.2.2 A Canonical Way to Construct Free Carnot Groups We now apply the results of the previous section in order to show a standard way to construct Carnot groups on RN . This construction makes also use of fm,r , the free nilpotent Lie algebra of step r with m generators, which we treated in Section 14.1. Let r ≥ 1 and m ≥ 2 be given, and set H := dim(fm,r ). Let E1 , . . . , Em be a system of generators for fm,r . Suppose that {Ei }i≤H is an enumeration of the Hall basis for fm,r which preserves the natural ordering ≤ given in Definition 14.1.5. We define a family of dilations {Dλ }λ>0 on fm,r by H H Dλ xi E i = λαi xi Ei , i=1

i=1

where αi is the height of Ei . Finally, we identify fm,r with RH in the natural way by introducing the linear isomorphism π : RH → fm,r

such that x →

H

xi E i .

i=1

It is now not difficult to prove the following result (we explicitly observe that, by Theorem 14.2.3, (fm,r , ) is a Lie group). Theorem 14.2.4 (Free homogeneous Carnot groups). With the above notation, for every x, y ∈ RH and λ > 0, we set

x ◦ y := π −1 π(x) π(y) , δλ (x) := π −1 Dλ (π(x)) . Then G := (RH , ◦, δλ ) is a free homogeneous Carnot group.

14.2 A Canonical Way to Construct Free Carnot Groups

587

Example 14.2.5. We give a simple descriptive example of this construction for a free homogeneous Carnot group G = (R10 , ◦, δλ ) of step 2 and with 4 generators. Suppose f4,2 is generated by E1 , . . . , E4 . The Hall basis E1 , . . . , E10 for f4,2 is given by E1 , E2 , E3 , E4 ;

[E2 , E1 ], [E3 , E1 ], [E4 , E1 ], [E3 , E2 ], [E4 , E2 ], [E4 , E3 ].

The group of dilations on G is given by δλ (x) = (λx1 , . . . , λx4 , λ2 x5 , . . . , λ2 x10 ).

(14.4)

The Campbell–Hausdorff formula on a Lie algebra nilpotent of step 2 is given by 1 X Y = X + Y + [X, Y ]. 2 The group law x ◦ y is simply obtained by explicitly writing

xi E i yi E i i≤10

i≤10

with respect to the Hall basis, making use of bilinearity, skew-symmetry and the Jacobi identity xi E i yi E i i≤10

i≤10

1 xi yj [Ei , Ej ] = · · · 2 i≤10 i≤10 i,j ≤10

1 = (xi + yi )Ei + x5 + y5 + (x2 y1 − x1 y2 ) E5 + · · · 2 i≤4

1 + x10 + y10 + (x4 y3 − x3 y4 ) E10 . 2

=

xi E i +

This yields, for every x, y ∈ G, ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ x◦y =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

yi E i +

x 1 + y1 x2 + y2 x3 + y3 x4 + y4 x5 + y5 + 12 (x2 y1 − x1 y2 ) x6 + y6 + 12 (x3 y1 − x1 y3 ) x7 + y7 + 12 (x4 y1 − x1 y4 ) x8 + y8 + 12 (x3 y2 − x2 y3 ) x9 + y9 + 12 (x4 y2 − x2 y4 )

x10 + y10 + 12 (x4 y3 − x3 y4 )

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

588

14 Some Remarks on Free Lie Algebras

With the above composition law ◦ and the dilation δλ defined in (14.4), G = (R10 , ◦, δλ ) is a free homogeneous Carnot group of step 2 and with 4 generators. Now, by making use of (1.34) (page 19), it is very simple to find the first 4 elements Z1 , . . . , Z4 of the Jacobian basis of the algebra of G: given ϕ ∈ C ∞ (R10 ), we have (Z1 ϕ)(x) = (∂/∂y1 )|y=0 ϕ(x ◦ y) 1 = ∂x1 ϕ(x) + x2 ∂x5 ϕ(x) + 2 (Z2 ϕ)(x) = (∂/∂y2 )|y=0 ϕ(x ◦ y) 1 = ∂x2 ϕ(x) − x1 ∂x5 ϕ(x) + 2 (Z3 ϕ)(x) = (∂/∂y3 )|y=0 ϕ(x ◦ y) 1 = ∂x3 ϕ(x) − x1 ∂x6 ϕ(x) − 2 (Z4 ϕ)(x) = (∂/∂y4 )|y=0 ϕ(x ◦ y) 1 = ∂x4 ϕ(x) − x1 ∂x7 ϕ(x) − 2

1 1 x3 ∂x6 ϕ(x) + x4 ∂x7 ϕ(x), 2 2 1 1 x3 ∂x8 ϕ(x) + x4 ∂x9 ϕ(x), 2 2 1 1 x2 ∂x8 ϕ(x) + x4 ∂x10 ϕ(x), 2 2 1 1 x2 ∂x9 ϕ(x) − x3 ∂x10 ϕ(x). 2 2

As a consequence, the following second order differential operator on R10 Z12 + Z22 + Z32 + Z42 2

1 1 1 = ∂x1 + x2 ∂x5 + x3 ∂x6 + x4 ∂x7 2 2 2 2 1 1 1 + ∂x2 − x1 ∂x5 + x3 ∂x8 + x4 ∂x9 2 2 2 2

1 1 1 + ∂x3 − x1 ∂x6 − x2 ∂x8 + x4 ∂x10 2 2 2 2

1 1 1 + ∂x4 − x1 ∂x7 − x2 ∂x9 − x3 ∂x10 2 2 2 is the canonical sub-Laplacian related to a free homogeneous Carnot group on R10 , with 4 generators and nilpotent of step 2 (for another example, see Example 14.1.12). Bibliographical Notes. The motivation for this brief investigation on free Lie algebras is primarily due to the Rothschild and Stein lifting theorem, according to which any sum of squares of Hörmander vector fields can be “approximated as close as we want” (in a suitable sense; see [RS76]) by a sub-Laplacian on a (larger) manifold, which is a stratified group whose Lie algebra is free.

14.3 Exercises of Chapter 14

589

14.3 Exercises of Chapter 14 Ex. 1) Prove that fm,r is isomorphic to fm ,r if and only if m = m and r = r . Ex. 2) Write down the Hall bases for f2,3 , f2,4 and f4,3 . Ex. 3) This exercise concerns with the canonical way of constructing Carnot groups via the Campbell–Hausdorff formula described in Section 14.2. Suppose it is given an algebra generated by the two elements X1 , X2 with the following commutator relations (indeed, it should be noted that these relations are consistent with the properties of the bracket operation: bilinearity, skewsymmetry and the Jacobi identity; see Definition 2.1.39): X1 , X2 , [X1 , X2 ] and [X1 , [X1 , X2 ]] are linearly independent, whereas there exists α ∈ R such that [X2 , [X1 , X2 ]] = α [X1 , [X1 , X2 ]] and all commutators of height ≥ 4 vanish. Consider the Campbell–Hausdorff operation for an algebra nilpotent of step 3 XY =X+Y +

1 1 1 [X, Y ] + [X, [X, Y ]] − [Y, [X, Y ]]. 2 12 12

Write down explicitly the Campbell–Hausdorff operation

x1 X1 + x2 X2 + x3 [X1 , X2 ] + x4 [X1 , [X1 , X2 ]]

y1 X1 + y2 X2 + y3 [X1 , X2 ] + y4 [X1 , [X1 , X2 ]] and write the result with respect to the basis X1 , X2 , [X1 , X2 ] and [X1 , [X1 , X2 ]]. Denote the 4-tuple of the coordinates thus obtained as (x1 , x2 , x3 , x4 ) ◦ (y1 , y2 , y3 , y4 ). Then, arguing as in Theorem 14.2.4, ◦ defines on R4 a homogeneous Carnot group. Verify that ◦ is given by ⎞ ⎛ x1 + y1 ⎟ ⎜ x2 + y2 ⎟ ⎜ ⎟ ⎜ x3 + y3 + 1 (x1 y2 − x2 y1 ) 2 ⎟. ⎜ ⎟ ⎜ ⎠ ⎝ x4 + y4 + 12 (x1 y3 − x3 y1 ) + α2 (x2 y3 − x3 y2 ) +

1 12 (x1

− y1 ) (x1 y2 − x2 y1 ) +

α 12 (x2

− y2 ) (x1 y2 − x2 y1 )

Ex. 4) We follow the ideas in Ex. 1 to re-derive the Heisenberg–Weyl group on R3 : the Campbell–Hausdorff operation for a Lie algebra h of step 2 has the form 1 X Y = X + Y + [X, Y ]. 2 We consider an algebra of step two with the basis Z1 , Z2 and Z3 := [Z1 , Z2 ]. Then we have

ξ1 Z1 + ξ2 Z2 + ξ3 Z3 η1 Z1 + η2 Z2 + η3 Z3

= (ξ1 Z1 + ξ2 Z2 + ξ3 Z3 + η1 Z1 + η2 Z2 + η3 Z3

590

14 Some Remarks on Free Lie Algebras

+

1 ξ1 Z1 + ξ2 Z2 + ξ3 Z3 , η1 Z1 + η2 Z2 + η3 Z3 2

[Z1 , Z2 ] = Z3 , [Z2 , Z1 ] = −Z3 and commutators of height > 2 vanish

1 1 = (ξ1 + η1 )Z1 + (ξ2 + η2 )Z2 + ξ3 + η3 + ξ1 η2 − η1 ξ2 Z3 . 2 2

By the natural identification h ξ1 Z1 + ξ2 Z2 + ξ3 Z3 ←→ (ξ1 , ξ2 , ξ3 ) ∈ R3 , we have re-obtained the Heisenberg–Weyl group H1 on R3 whose composition law is

1 1 ξ ◦ η = ξ 1 + η1 , ξ2 + η2 , ξ3 + η3 + ξ 1 η2 − η1 ξ 2 . 2 2 Carry out a similar exercise with an algebra of step two whose basis is X1 , . . . , X N ;

Ex. 5) Ex. 6)

Ex. 7)

Ex. 8)

Y1 , . . . , YN ;

[X1 , Y1 ]

with the following commutator identities: a) [Xj , Yj ] = [X1 , Y1 ] for all j = 1, . . . , N , b) [Xi , Xj ] = 0 = [Yi , Yj ] for all i, j = 1, . . . , N , c) [Xi , Yj ] = 0 for all i, j = 1, . . . , N such that i = j . Recognize the Lie group in (4.16), page 201. Give a detailed proof of Theorem 14.2.4. By means of Example 14.1.7 and the ideas in Section 14.2, find a Carnot group whose Lie algebra is isomorphic to f4,2 . Finally, consider the general case of fm,2 . a) Use Grayson–Grossman’s Theorem 14.1.10 to find vector fields generating f5,2 . b) Show that they satisfy conditions (H0)–(H1)–(H2) in Section 4.2 (page 191) and construct the relevant homogeneous Carnot group as described in that section. c) Construct a Carnot group whose Lie algebra is isomorphic to f5,2 following the ideas in Section 14.2. Compare to point (b). Let m, r ∈ N (m ≥ 2) and consider fm,r . Set H (m, r) = dim(fm,r ). Finally, let f be any finite-dimensional (real) Lie algebra and set H := dim(f). What of the following conditions are sufficient in order to have fm,r and f isomorphic as Lie algebras? a) H (m, r) = H . (Hint: Consider f2,2 and f3,1 or, equivalently, the Lie algebra of the Heisenberg–Weyl group H1 on R3 and the Lie algebra of the Euclidean group (R3 , +).) b) H (m, r) = H and fm,r and f are nilpotent of step r. (Hint: Consider f3,2 and the Lie algebra f, nilpotent of step 2, whose basis is {X, Y, A, B, C, [X, Y ]} such that A, B, C commute with each other and with X and Y ; roughly speaking, f is the Lie algebra of the sum of H1 and (R3 , +) (see Section 4.1.5, page 190).)

14.3 Exercises of Chapter 14

591

c) fm,r and f are both nilpotent of step r and Lie-generated by m of their elements. (Hint: Consider f3,2 and the Lie algebra f, nilpotent of step 2, whose basis is {X, Y, A, [X, Y ]} such that A commutes with X and Y ; roughly speaking, f is the Lie algebra of the sum of H1 and (R1 , +).) d) H (m, r) = H and fm,r and f are Lie-generated by m of their elements. (Hint: Consider f3,2 and the Lie algebra f, nilpotent of step 3, whose basis is X1 , X2 , A, X3 := [X1 , X2 ], X4 := [X1 , [X1 , X2 ]], X5 := [X1 , [X1 , [X1 , X2 ]]] , such that A commutes with Xi (i = 1, . . . , 5), [X1 , X5 ] = 0, [Xi , Xj ] = 0 for every 1 < i < j ≤ 5. Roughly speaking, f is obtained from the filiform Lie algebra h in Example 4.3.5 (page 208) when i = 5, by adding an extra generator A commuting with every other generator.) e) H (m, r) = H and fm,r and f are nilpotent of step r and both are Liegenerated by m of their elements. Hint: Condition (e) is actually sufficient. Indeed, suppose f = Lie{F1 , . . . , Fm } (here F1 , . . . , Fm ∈ f) is nilpotent of step r. Let also x1 , . . . , xm denote the generators for fm,r , as in Definition 14.1.1. According to this very definition (replacing n with f, what is possible since f is nilpotent of step r), if we consider the map ϕ sending xi in Fi (for every i = 1, . . . , m), there exists a homomorphism of Lie algebras ϕ from fm,r to f which extends ϕ. We are done if we show that ϕ is a vector space isomorphism. Since fm,r and f have the same dimension, it suffices to prove that ϕ is onto. This follows by the argument below f = Lie{F1 , . . . , Fm } = Lie{ϕ(x1 ), . . . , ϕ(xm )} = Lie{ ϕ (x1 ), . . . , ϕ (xm )} = ϕ (Lie{x1 , . . . , xm }) = ϕ (fm,r ). Provide the proofs of the above equalities.

15 More on the Campbell–Hausdorff Formula

The main aim of this chapter is to sketch a proof of Lemma 4.2.4 in Section 4.2 (page 194). This lemma played a crucial rôle in the construction of Carnot groups. We could refer to this lemma as a sort of Campbell–Hausdorff formula for stratified vector fields. Our proof here is articulated in the following way. In Section 15.1, we show in details how to derive Lemma 4.2.4 from a general version of the Campbell–Hausdorff formula for formal power series. Then, in Sections 15.2 and 15.3, we provide two proofs of this formula for formal power series: the former is due to D. Djokovi´c [Djo75], the latter is due to M. Eichler [Eic68]. The first proof makes use of results from the theory of formal power series; the second one rests on some properties of Lie polynomials. For the sake of brevity, we shall leave to the reader the details of the proofs of these properties, which are actually very intuitive. As a result, the proofs of the Campbell–Hausdorff formula are surprisingly simple and short, as in the spirit of the cited authors. Finally, Section 15.4 provides the version of the Campbell–Hausdorff formula mostly cited in analytic contexts. Once again, it is unavoidable to make use of the formula for formal power series. Furthermore, the main task is to show how to control and give precise estimates of the remainder terms in the relevant series expansions.

15.1 A Proof of the Campbell–Hausdorff Formula for Stratified Vector Fields We recall the notation of Section 4.2: X1 , . . . , Xm is a given set of smooth vector fields on RN satisfying hypotheses (H0)–(H1)–(H2) of Section 4.2, i.e. the following conditions are fulfilled: (H0) X1 , . . . , Xm are linearly independent and δλ -homogeneous of degree one with respect to a suitable family of dilations {δλ }λ>0 of the following type δλ : RN → RN ,

δλ (x) = δλ (x (1) , . . . , x (r) ) := (λx (1) , . . . , λr x (r) ),

594

15 More on the Campbell–Hausdorff Formula

where r ≥ 1 is an integer, x (i) ∈ RNi for i = 1, . . . , r, N1 = m and N1 + · · · + Nr = N. (H1) We set W (k) = span{XJ | J ∈ {1, . . . , m}k }, where X(j1 ,...,jk ) = [Xj1 , . . . [Xjk−1 , Xjk ] . . .] if J = (j1 , . . . , jk ). Then, dim(W (k) ) = dim{XI (0) : X ∈ W (k) } for every k = 1, . . . , r. (H2) dim(Lie{X1 , . . . , Xm }I (0)) = N. Let a = Lie{X1 , . . . , Xm }. (k)

(k)

For every k = 1, . . . , r, Z1 , . . . , ZNk will be a fixed basis for W (k) . We know that (1)

(1)

(r)

(r)

{Z1 , . . . , ZN } := {Z1 , . . . , ZN1 , . . . , Z1 , . . . ZNr } is a basis for a. For every ξ = (ξ1 , . . . , ξN ) = (ξ (1) , . . . , ξ (r) ) we set ξ ·Z =

N

ξj Zj =

j =1

Nk r

(k)

(k)

ξj Zj .

k=1 j =1

We know that1 the map (x, t) → exp(t ξ · Z)(x) is well defined for every x ∈ RN and t ∈ R. Moreover, Exp : RN −→ RN ,

Exp (ξ ) := exp(ξ · Z)(0)

is a global diffeomorphism.2 Its inverse function is denoted by Log . In Section 4.2, we defined a composition law on RN by means of x, y ∈ RN ,

x ◦ y := exp(Log (y) · Z)(x).

Our aim there was to show that G := (RN , ◦, δλ ) is a Carnot group whose Lie algebra g coincides with a. The main task of the proof definitely was to show that ◦ is associative. To this end, we needed the cited Lemma 4.2.4, which stated that For every X, Y ∈ a, there exists a unique V ∈ a such that exp(Y ) exp(X)(x) = exp(V )(x) (15.1) for every x ∈ RN . In this section, we at last prove (15.1) as a consequence of the following theorem. 1 We recall that, given a smooth vector field X on RN , exp(tX)(x) := γ (t, x) denotes the

solution γ (·) = γ (·, x) to the ordinary differential system γ˙ = XI (γ ), γ (0) = x.

2 We explicitly remark that we are using the notation Exp to denote a map on RN instead of

a map on an algebra of vector fields.

15.1 The Campbell–Hausdorff Formula for Stratified Fields

595

Theorem 15.1.1 (The Campbell–Hausdorff formula for stratified vector fields). Let X, Y ∈ a be fixed. Let V be the differential operator defined by the formal expansion r j r X k2 Y k1 (−1)j +1 j k2 ! k1 ! j =1 k1 +k2 =1 n y x y x n n 1 1 = V + summands of the type Y X · · · Y X with (yi + xi ) > r . i=1

(15.2) Then V turns out to be a vector field belonging to Lie{X, Y } (hence belonging to a) such that (15.3) exp(Y ) exp(X)(x) = exp(V )(x) ∀ x ∈ RN . We explicitly remark that (15.3) proves (15.1), i.e. it proves Lemma 4.2.4. For example, a direct computation shows that 1 1 1 V = X Y := X + Y + [X, Y ] + [X, [X, Y ]] − [Y, [X, Y ]] 2 12 12 1 1 [X, [Y, [X, Y ]]] + · · · . − [Y, [X, [X, Y ]]] − 48 48 We recognize here the Campbell–Hausdorff composition law X Y defined in (14.2), page 585. Before proceeding with the proof of Theorem 15.1.1, we give an example of how the formal expansion in (15.2) behaves. For instance, taking r = 3, we have 3 (−1)j +1 j =1

j

3 k1 +k2 =1

X k1 Y k2 k1 ! k2 !

j

3 X1 Y 0 X0 Y 2 X1 Y 1 X2 Y 0 (−1)j +1 X 0 Y 1 + + + + = j 0! 1! 1! 0! 0! 2! 1! 1! 2! 0! j =1 X1 Y 2 X2 Y 1 X3 Y 0 j X0 Y 3 + + + + 0! 3! 1! 2! 2! 1! 3! 0!

3 Y2 X2 Y3 XY 2 X2 Y X3 j (−1)j +1 Y +X+ + XY + + + + + = j 2 2 6 2 2 6 j =1

1 X2 Y3 XY 2 X2 Y X3 Y2 + XY + + + + + = Y +X+ 2 2 6 2 2 6

Y2 X2 Y3 XY 2 X2 Y X3 2 1 Y +X+ + XY + + + + + − 2 2 2 6 2 2 6

Y2 X2 Y3 XY 2 X2 Y X3 3 1 Y +X+ + XY + + + + + + 3 2 2 6 2 2 6

596

15 More on the Campbell–Hausdorff Formula

X2 Y3 XY 2 X2 Y X3 Y2 + XY + + + + + =Y +X+ 2 2 6 2 2 6

1 2 Y X2 Y3 XY 2 − Y + YX + + Y XY + + XY + X 2 + + X2 Y 2 2 2 2 X3 Y3 Y 2X X3 X2 Y + + + + XY 2 + XY X + + 2 2 2 2 2 1 3 + (Y + Y 2 X + Y XY + Y X 2 + XY 2 + XY X + X 2 Y + X 3 ) 3 + {summands of height ≥ 4}

1 1 1 1 2 1 2 1 =X+Y +X − +Y − + XY 1 − + YX − 2 2 2 2 2 2

1 1 1 1 1 1 1 1 + Y3 + X3 − − + − − + 6 4 4 3 6 4 4 3

1 1 1 1 − − + + X2 Y 2 2 4 3

1 1 1 1 1 1 1 2 2 1 2 + XY − − + + YX − + +Y X − + 4 3 2 4 2 3 4 3

1 1 1 1 + XY X − + + Y XY − + 2 3 2 3 + {summands of height ≥ 4} 1 1 2 1 2 1 1 X Y+ Y X+ XY 2 = X + Y + XY − Y X + 2 2 12 12 12 1 1 1 Y X 2 − Y XY − XY X + 12 6 6 + {summands of height ≥ 4} 1 1 1 = X + Y + [X, Y ] + [X, [X, Y ]] − [Y, [X, Y ]] 2 12 12 + {summands of height ≥ 4}. Proof (of Theorem 15.1.1). Throughout the proof, X, Y ∈ a are fixed. We begin by noticing that (see the arguments in Remark 1.1.3, page 9) the map R × RN (t, x) → exp(tX)(x) has polynomial component functions. Moreover, we have n (yi + xi ) > r

⇒

Y yn X xn · · · Y y1 X x1 I ≡ 0.

(15.4)

i=1

This follows by recalling that any field in a is a sum of vector fields δλ -homogeneous of degree at least 1 and by observing that the component functions of the identity map I are δλ -homogeneous monomials of degree at most r. Since exp(tX)(x) is an analytic function of t, (1.7) (page 7) gives exp(X) ≡

r 1 k X I. k! k=0

15.1 The Campbell–Hausdorff Formula for Stratified Fields

597

We henceforth fix x ∈ RN and set Φ(t1 , t2 ) := exp(t1 Y ) exp(t2 X)(x) ,

t1 , t2 ∈ R.

Clearly, any component function of Φ is a polynomial in t1 , t2 . From (1.15) (page 10), for every smooth vector field Z on RN , f ∈ C ∞ (RN , RN ), k ∈ N and every z ∈ RN , we have

k d f (exp(tZ)(z)) = (Z k f )(z). (15.5) k dt t=0 This gives, for every k1 , k2 ∈ N,

k1 +k2 ∂ Φ(t1 , t2 ) k1 k2 ∂t1 ∂t2 (t1 ,t2 )=(0,0)

k1

k2 ∂ ∂ = I exp(t1 Y ) exp(t2 X)(x) k2 k1 ∂t2 t2 =0 ∂t1 t1 =0

k2 ∂ = (Y k1 I ) exp(t2 X)(x) k2 ∂t2 t2 =0 = (X k2 Y k1 I )(x).

(15.6)

Indeed, in the second equality we used (15.5) with f = I , k = k1 , Z = Y , z = exp(t2 X)(x), whereas in the third equality we used (15.5) with f = Y k1 I , k = k2 , Z = X, z = x. We now use Taylor’s formula for Φ. Since Φ is a polynomial, exploiting (15.4), we derive r

exp(Y ) exp(X)(x) = Φ(1, 1) =

k1 +k2 =0

1 (X k2 Y k1 I )(x). k1 ! k2 !

(15.7)

We now introduce the following higher order differential operator W (t, X, Y ) :=

r (−1)j +1

j

j =1

r k1 +k2 =1

t k1 +k2 k2 k1 X Y k1 ! k2 !

j .

We formally expand W (t, X, Y ), and we order it as a polynomial in t setting W (t, X, Y ) =

r k=1

2

t Zk (X, Y ) + k

r

t k Zk (X, Y )

k=r+1

=: V (t, X, Y ) + R(t, X, Y ). We explicitly remark that the differential operator V appearing in (15.2) in the statement of the theorem is simply given by V (t, X, Y ) with t = 1. It is easy to recognize that, by (15.4), any power of R(t, X, Y ) annihilates the identity map, i.e.

598

15 More on the Campbell–Hausdorff Formula

(R(t, X, Y ))k I ≡ 0

for every k ≥ 0.

(15.8)

We claim that Zk (X, Y ) ∈ Lie{X, Y }

for every k ≥ r.

(15.9)

Before proving the claimed (15.9), we first show that (15.9) implies the assertion of the theorem. Indeed, if we set V := V (1, X, Y ), then from the definition of V (t, X, Y ) and (15.9) we derive that V ∈ Lie{X, Y }. Finally, we have to prove that (15.3) is satisfied with the above choice of V . Indeed, we have exp(V )(x) =

r 1 k V I (x) k! k=0

r k 1 V (1, X, Y ) + R(1, X, Y ) I (x) = k! k=0

r k 1 W (1, X, Y ) I (x) = k! k=0 r r r 1 (−1)j +1 = k! j k=0

= I (x) +

j =1 r

k1 +k2 =1

k1 +k2 =1

1 X k2 Y k1 k1 ! k2 !

j k I (x)

1 X k2 Y k1 I (x) = exp(Y ) exp(X)(x) . k1 ! k2 !

We now motivate every equality appearing above: 1) 2) 3) 4)

the first equality follows from the fact that V ∈ a and from (1.7); the second one follows from (15.8) and simple homogeneity arguments; the third one is the very definition of W (1, X, Y ); the fourth equality is a consequence of the formal power series expansion of the identity 1 + x = exp(log(1 + x)), jointly with (15.4): more precisely, the fourth equality is a consequence of the identity k r r r2 1 (−1)j +1 j H =I +H + cj H j k! j k=0

j =1

and of (15.4); 5) the last equality follows from (15.7).

j =r+1

∀H ∈ a

15.2 The Campbell–Hausdorff Formula for Formal Power Series–1

599

Finally, we are left with the proof of the claimed (15.9). A proof of it can be derived from a general result by Djokovi´c [Djo75], which we present in Section 15.2. Indeed, in [Djo75] an assertion analogous to (15.9) is derived in the more general context3 of the formal power series in two non-commuting indeterminates X and Y . This completes the proof of the theorem. Remark 15.1.2. An odd fact, arising from the proof

of Theorem 15.1.1, has to be remarked: from (15.7) and the fact that exp(X) = rk=0 k!1 X k I (see (1.7)) we derived r r r X k2 Y k1 1 I I (x) = X k2 Y k1 I (x). k1 ! k2 ! k2 ! k1 ! k1 =0

k2 =0

k1 +k2 =0

The order of composition of X and Y seems reversed, on the face of it. Thus, formal computations such as “

Y k1 X k2 k1

k1 !

k2

k2 !

=

k1 +k2

1 Y k1 X k2 k1 ! k2 !

”

are erroneous in our context of composition of exponential maps. This is another reason why we proved in detail our Campbell–Hausdorff-type Theorem 15.1.1 rather than reducing it, with no further justification, to formal power series arguments.

15.2 A Proof of the Campbell–Hausdorff Formula for Formal Power Series The aim of this section is to present the arguments (with few more details) in [Djo75] for a proof of the Campbell–Hausdorff formula for formal power series. Here X and Y will denote general non-commuting indeterminates in a multiplicative algebra over the field Q of the rational numbers (for instance, X and Y may be vector fields on RN , as in the previous section). We shall perform computations in the algebra of formal power series in X and Y . We first fix a very natural4 notation: eX :=

Xk k≥0

k!

(15.10)

3 At this point, a simple remark has to be given, in order to justify the fact that we can

replace general calculus between two non-commuting indeterminates X and Y with our vector fields X, Y ∈ a. To this end, we explicitly remark that the identities between formal power series used in [Djo75] in order to prove (15.9) readily reduce, in our context, to identities between finite sums of vector fields: it will be enough to make use of arguments such as (15.4). 4 As Remark 15.1.2 has shown, though this is a very natural definition, when X is a vector field on RN , eX should not be confused with the exponential exp(X) of a vector field as the integral curve to the system γ˙ = X(γ ).

600

15 More on the Campbell–Hausdorff Formula

denotes the formal expression of the exponential of X. The Campbell–Hausdorff formula simply states that there exists Z which is a formal infinite sum of summands in Lie{X, Y } such that eY · eX = eZ . More precisely, see Theorem 15.2.2 below. In the next result and for the future references, we need the definition of Liepolynomial. Definition 15.2.1 (Lie polynomial). If K is any field, we denote by K[X, Y ] the vector space (over K) of the polynomials in two non-commuting indeterminates X, Y , i.e. K[X, Y ] is K-spanned by the formal expressions X α1 Y β1 · · · X αi Y βi (called, a monomial of degree α1 + β1 + · · · + αi + βi ), where i ∈ N and α1 , β1 , · · · , αi , βi ∈ N ∪ {0}. The degree of a polynomial is the maximum degree of its monomials. Let X and Y be two non-commuting indeterminates in an associative algebra. Set [X, Y ] = XY − Y X (which we call a Lie-bracket). Given k ∈ N, k ≥ 2, and Z1 , . . . , Zk ∈ {X, Y }, we say that the Lie-bracket [Z1 [Z2 [Z3 [· · · [Zk−1 , Zk ] · · ·]]]], is a Lie(-bracket) monomial of length k in X and Y . We also say that X and Y are Lie-monomials of length 1. A linear combination of Lie monomials of length k in X and Y is called a homogeneous Lie-polynomial of length k. Any finite sum of Lie-bracket monomials is called a Lie-polynomial. Analogous definitions are given in the setting of Lie algebras. Obviously, K[X, Y ] can be equipped with an associative algebra structure in an obvious way.5 We are now ready to state and prove the following theorem. Theorem 15.2.2 (The Campbell–Hausdorff formula for formal power series). For every positive integer k, there exists a homogeneous Lie-bracket-polynomial Zk of length k (with rational coefficients) in two non-commuting indeterminates such that, using the notation Zk (·, ·) to express the dependence of Zk on the two indeterminates, we have: if Z(X, Y ) =

∞

Zk (X, Y )

then eY · eX = eZ

(15.11)

k=1

for every couple of non-commuting indeterminates X, Y . For example, Z1 (X, Y ) = X + Y, 1 Z2 (X, Y ) = [Y, X], 2 5 For example, XY 2 · Y 3 X 4 = XY 5 X 4 and XY 2 · X 4 = XY 2 X 4 , XY 2 X 5 · XY 3 X 4 Y = XY 2 X 6 Y 3 X 4 Y , etc.

15.2 The Campbell–Hausdorff Formula for Formal Power Series–1

1 1 [Y, [Y, X]] − [X, [Y, X]], 12 12 1 1 [Y, [X, [Y, X]]]. Z4 (X, Y ) = − [X, [Y, [Y, X]]] − 48 48

601

Z3 (X, Y ) =

(15.12)

Proof. For every t ∈ R, we have e t Y · et X =

t k1 +k2 (t Y )k2 (t X)k1 · = Y k2 X k1 k2 ! k1 ! k1 !k2 !

k2 ≥0

=I+

k1 ≥0

k1 ,k2 ≥0

t k1 +k2

k1 +k2 ≥1

k1 !k2 !

Y k2 X k1 =: I + HX,Y (t).

If HX,Y (t) is as above, we set ZX,Y (t) := ln(I + HX,Y (t)) =

(−1)j +1 j ≥1

j

k1 +k2 ≥1

t k1 +k2 k2 k1 Y X k1 !k2 !

j . (15.13a)

From the formal power series expansion of the identity eln(1+x) = 1 + x, it then follows that (15.13b) eZX,Y (t) = I + HX,Y (t) = et Y · et X . Now, (15.11) is formally fulfilled with the natural choice Z(X, Y ) := ZX,Y (1).

(15.13c)

The theorem is then proved if we show that ZX,Y (1) can be put in the form ∞ Z(X, Y ) = k=1 Zk (X, Y ) with the Zk ’s as in the assertion of the theorem. To this end, it suffices to prove the following power series expansion w.r.t. t: ZX,Y (t) =

∞

t k · Zk (X, Y ),

(15.13d)

k=1

where Zk (X, Y ) ∈ Lie{X, Y }

for every k ∈ N

(15.13e)

and Zk (X, Y ) is homogeneous of length k. To begin with, we observe that, differentiating the far right-hand side of (15.13b) w.r.t. t (and writing henceforth Z(t) instead of ZX,Y (t)), we have ∂t eZ(t) = ∂t et Y · et X = ∂t et Y · et X + et Y · ∂t et X = Y · et Y · et X + et Y · et X · X = Y · eZ(t) + eZ(t) · X. Hence, we obtain ∂t eZ(t) · e−Z(t) = eZ(t) · X · e−Z(t) + Y.

(15.13f)

602

15 More on the Campbell–Hausdorff Formula

As a general fact, it holds eZ · X · e−Z =

Zm Zk Zm · X · Zk ·X· = . (−1)k (−1)k m! k! m! k!

m≥0

k≥0

m,k≥0

Then, setting m + k =: n, the expression for eZ · X · e−Z becomes6

n 1 n Z n−k · X · Z k . eZ · X · e−Z = (−1)k k n! n≥0

(15.13g)

k=0

Throughout the sequel, we use the following classical notation for the left and right multiplication: LZ (X) := Z · X, RZ (X) := X · Z, (Ad Z)(X) := LZ − RZ (X) = Z · X − X · Z. Obviously, LZ and RZ commute.7 As a consequence, we can apply Newton’s binomial expansion to get

n n n n n−k k L Ad Z (X) = LZ − RZ (X) = (−1)k · RZ (X) k Z k=0

n n Z n−k · X · Z k . = (−1)k (15.13h) k k=0

Taking into account (15.13h), the expression (15.13g) turns out to be eZ · X · e−Z =

1 n Ad Z (X) =: eAd Z (X). n! n≥0

This gives (see (15.13f)) ∂t eZ(t) · e−Z(t) = Y + eAd Z(t) (X).

(15.13i)

On the other hand, a direct computation yields 1 (−1)k Z(t) −Z(t) m k ∂t e ∂t (Z(t) ) · Z(t) ·e = m! k! m≥1 k≥0

n 1 n ∂t (Z(t)m ) Z(t)n−m =: ( ). (k =: n − m) = (−1)n−m n! m n≥1

6 Here we wrote

m=1

n 1 1 = . m! k! k n!

7 We obviously have L (R (X)) = Z · X · Z = R (L (X)). Z Z Z Z

15.2 The Campbell–Hausdorff Formula for Formal Power Series–1

603

Thus, since one has ∂t (Z(t) ) = m

m−1

Z(t)k · (∂t Z(t)) · Z(t)m−k−1 ,

k=0

we derive that

n m−1 1 n Z(t)k · (∂t Z(t)) · Z(t)n−k−1 (−1)n−m m n! n≥1 m=1 k=0 n

n−1 1 n = (−1)n−m Z(t)k · (∂t Z(t)) · Z(t)n−k−1 m n!

( ) =

n≥1

k=0

m=k+1

(we set r := n − m and use properties of the binomial coefficients)

n−1 n−k−1 1 r n (−1) = Z(t)k · (∂t Z(t)) · Z(t)n−k−1 =: (

). r n! n≥1

k=0

r=0

An inductive argument shows that

m n n−1 = (−1)m , (−1)r r m

0 ≤ m ≤ n − 1.

r=0

Consequently, we infer

n−1

1 n−k−1 n − 1 (−1) Z(t)k · (∂t Z(t)) · Z(t)n−k−1 (

) = k n! n≥1

1 = n! n≥1

(see (15.13h))

k=0

n−1

j

(−1)

j =0

n−1 j

Z(t)n−1−j · (∂t Z(t)) · Z(t)j

1 (Ad Z(t))n−1 ∂t Z(t) . = n! n≥1

Finally, comparing to (15.13i), we have Y + eAd Z(t) (X) =

1 (Ad Z(t))n−1 ∂t Z(t) , n! n≥1

i.e. by the definition of the exponential of Ad Z(t), Y+

1 1 (Ad Z(t))n (X) = (Ad Z(t))n−1 ∂t Z(t) . n! n! n≥0

(15.13j)

n≥1

Now, the power series n≥1 x n−1 /n! furnishes the formal expansion of ϕ(x) := (ex − 1)/x. Then, since ϕ(0) = 1 = 0, there exists a reciprocal series expansion

604

15 More on the Campbell–Hausdorff Formula

of n≥1 x n−1 /n!, say g(x) := n≥0 an x n . More precisely, the following formal identity holds

an x n · x n−1 /n! = 1. n≥0

n≥1

As a consequence, multiplying both sides of (15.13j) times g(Ad Z(t)), we obtain 1 n (Ad Z(t)) (X) g Ad Z(t) Y + n! n≥0

1 = (Ad Z(t))n−1 ∂t Z(t) an (Ad Z(t))n · n! n≥0

n≥1

= ∂t Z(t). This proves that Z(t) satisfies the differential equation

1 ∂t Z(t) = g(Ad Z(t)) Y + (Ad Z(t))n (X) . n!

(15.13k)

n≥0

By reordering the sum in (15.13a) which defines Z(t), we obtain Z(t) =

(−1)j +1 j ≥1

If we set bk :=

∞

j

k=1

k1 +k2 =k

k1 +k2 =k

j 1 k2 k1 Y X tk . k1 ! k2 !

1 Y k2 X k1 , k1 ! k2 !

k ≥ 1,

by means of the formula for the j -th power of a power series, we derive

∞ (−1)j +1 k t Z(t) = j j ≥1

=

∞

k=1

tk

j =1

k=1

=

∞ k=1

=:

∞ k=1

∞ (−1)j +1

t

k

j

∞ (−1)j +1 j =1

j

t k · Zk (X, Y ).

bn1 · · · bnj

n1 +···+nj =k n1 ,...,nj ≥1

bn1 · · · bnj

n1 +···+nj =k n1 ,...,nj ≥1

(r1 +s1 )+···+(rj +sj )=k r1 +s1 ≥1,...,rj +sj ≥1

Y r1 X s1 · · · Y rj X sj r1 !s1 ! · · · rj !sj !

15.3 The Campbell–Hausdorff Formula for Formal Power Series–2

605

Following this notation, the left-hand side in (15.13k) (i.e. the derivative of Z(t)) equals ∞ t k · (k + 1) Zk+1 (X, Y ). k=0

Following again the notation Z(t) = k≥1 t k Zk , recalling that g(x) = n≥0 an x n , and by using the formula for the power of a series in t, the right-hand side of (15.13k) equals X+Y +

∞

∞ 1 t n!

Ad Zm1 · · · Ad Zmn (X) m1 +···+mn =k k=1 n=1 ∞ ∞ + tk an Ad Zm1 · · · Ad Zmn X + Y m1 +···+mn =k k=1 n=1 ∞ ∞ 1 + tl Ad Zh1 · · · Ad Zhj (X) . j! h1 +···+hj =l l=1 j =1 k

It has to be noticed that the coefficient of t k in this last expression is a linear combination of terms of the type Ad Zm1 · · · Ad Zmn (X) and Ad Zm1 · · · Ad Zmn (Y ) with m1 + · · · + mn = k. In particular, by equating the coefficients of t k from both sides of (15.13k), one has Z1 (X, Y ) = X + Y, whereas Z2 (X, Y ) is a linear combination of commutators of height 2 of X + Y with X (and Y ); moreover Z3 (X, Y ) is a linear combination of commutators of height 3 of Z2 (X, Y ) (or X + Y ) and X (or Y ) and so on. An inductive argument now proves that Zk (X, Y ) is a linear combination of commutators of height k of X and Y . In particular, Zk (X, Y ) ∈ Lie{X, Y } for every k ∈ N. This actually demonstrates (15.13e), and the proof is complete.

15.3 Another Proof of the Campbell–Hausdorff Formula for Formal Power Series In this section, we provide another proof of the Campbell–Hausdorff formula for formal power series, as given by M. Eichler in [Eic68]. This proof essentially rests on some properties of Lie polynomials. (In due course of the proof, we shall leave to the reader the details for these properties, which are indeed very intuitive.) In the rest of the section, we consider the associative algebra of formal power series in two non-commuting indeterminates A, B over the field of the rational numbers Q. In particular, we shall use the definition of the exponential as in (15.10) and the notion of Lie-polynomial (see Definition 15.2.1). We are now ready to give another proof of Theorem 15.2.2.

606

15 More on the Campbell–Hausdorff Formula

Proof (of Theorem 15.2.2). We aim to prove that eA · eB = e

∞

n=1 Fn (A,B)

,

(15.14)

where, for every n ∈ N, Fn (A, B) is a Lie-polynomial in A, B which is homogeneous of length n. First of all, we remark that it certainly exist unique homogeneous polynomials Fn ’s in A, B satisfying (15.14) (note that we are not yet asking for Lie-polynomials). Indeed, there exists a unique formal power series Z in A, B such that eA · eB = eZ . It suffices to take

A B n m A /n! B /m! Z := log(e · e ) = log 1 + n+m≥1

or, more precisely, Z=

(−1)j +1 j ≥1

j

j An /n! B m /m! .

n+m≥1

Then we reorder the summands defining Z in such a (unique) way that they are homogeneous polynomials in A, B. Our main task is to prove that the resulting Fn ’s are Lie-polynomials in A, B. We find F1 and F2 . F1 results from the choice (j = 1, n = 1, m = 0) or from (j = 1, n = 0, m = 1) thus giving F1 (A, B) = A + B. F2 results from the choice (j, n, m) ∈ {(1, 1, 1), (1, 2, 0), (1, 0, 2), (2, 1, 0), (2, 0, 1)}, so that B2 1 A2 + − (A + B)2 2 2 2 A2 B2 1 2 = AB + + − (A + AB + BA + B 2 ) 2 2 2 1 1 1 = AB − BA = [A, B]. 2 2 2

F2 (A, B) = AB +

Hence, F1 and F2 are homogeneous Lie-polynomials of the desired degree. We now argue by induction. Let n ≥ 3 and suppose that F1 , . . . , Fn−1 are Lie-polynomials of degree 1, . . . , n−1, respectively. We are done if we show that Fn is a Lie-polynomial (its degree is obviously n for it is the sum of polynomials of degree n). The crucial device is to take three non-commuting indeterminates A, B, C and to use associativity to equal the expansions of

15.3 The Campbell–Hausdorff Formula for Formal Power Series–2

(eA · eB ) · eC = eA · (eB · eC ).

607

(15.15)

By (using twice) (15.14), the left-hand side of (15.15) is ∞

∞

∞ e j =1 Fj (A,B) · eC = e i=1 Fi j =1 Fj (A,B),C . The reader should appreciate that we are here using the universality of the Fj ’s as functions of two “arbitrary” non-commuting indeterminates.8 Analogously, the righthand side of (15.15) is

∞

∞ Fi A, ∞ Fj (B,C) Fj (B,C) A i=1 j =1 j =1 e ·e . =e We thus have W :=

∞

Fi

∞

i=1

Fj (A, B), C

j =1

=

∞ i=1

∞ Fj (B, C) . A,

Fi

(15.16)

j =1

It is easy to verify that whenever Fi and Fj are Lie-polynomials in two indeterminates, then Fi (Fj (A, B), C) and Fi (A, Fj (B, C)) are Lie-polynomials in A, B, C. Moreover, it is also not difficult to verify that the homogeneous polynomials which are the summands into which a Lie-polynomial splits up are homogeneous Liepolynomials. We now compare homogeneous terms of degree n (in A, B, C) on both sides of (15.16). Such terms appear in summands of the type Fi (Fj (A, B), C),

Fi (A, Fj (B, C))

if and only if i = 1, j = n; i = n, j = 1; or i + j = n (in the case i > 1, j > 1). Thus, with the remarks of the preceding paragraph at hand, by our inductive assumption, we see that all homogeneous terms of degree n in both expressions for W are Lie-polynomials, except at most for Fn (A, B) + Fn (A + B, C) in the left-hand side of (15.16) and except for Fn (A, B + C) + Fn (B, C) in its right-hand side. For the sake of brevity, we introduce the equivalence relation ∼ between polynomials p1 , p2 in A, B, C by saying that p1 ∼ p2 whenever p1 − p2 is a Liepolynomial. The assertion made at the end of the preceding paragraph thus rewrites as (15.17) Fn (A, B) + Fn (A + B, C) ∼ Fn (A, B + C) + Fn (B, C). 8 After all, the arbitrary nature of the non-commuting indeterminates lies in the very concept

of “indeterminates”.

608

15 More on the Campbell–Hausdorff Formula

We now make some trivial remarks on the ∼ relation. If α, β ∈ Q, then (since α A and β A commute) we have eα A · eβ A = eα A+β A = eF1 (α A,β A) . Consequently, for every n ≥ 2, Fn (α A, β A) = 0, i.e. Fn (α A, β A) ∼ 0.

(15.18)

In particular, this holds for α = 1 and β = 0. Now, we show that (15.17) and (15.18) imply Fn (A, B) ∼ 0, which is our goal. First, the choice C := −B in (15.17) yields Fn (A, B) ∼ −Fn (A + B, −B) + Fn (A, 0) + Fn (B, −B) ∼ −Fn (A + B, −B), i.e.

Fn (A, B) ∼ −Fn (A + B, −B).

(15.19)

Analogously, the choice A := −B in (15.17) yields Fn (B, C) ∼ Fn (−B, B) + Fn (0, C) − Fn (−B, B + C) ∼ −Fn (−B, B + C), i.e. by renaming B → A, C → B, Fn (A, B) ∼ −Fn (−A, A + B).

(15.20)

Applying (15.20), (15.19) and again (15.20), we get Fn (A, B) ∼ −Fn (−A, A + B) ∼ − − Fn (−A + A + B, −A − B) = Fn (B, −A − B) ∼ −Fn (−B, B − A − B) = −Fn (−B, −A) = −(−1)n Fn (B, A). The last equality is a consequence of the length-homogeneity n of Fn as a polynomial. This gives Fn (A, B) ∼ (−1)n+1 Fn (B, A).

(15.21)

Now, we insert C = − 12 B in (15.17). This gives

1 1 1 Fn (A, B) ∼ −Fn A + B, − B + Fn A, B − B + Fn B, − B 2 2 2

1 1 ∼ −Fn A + B, − B + Fn A, B , 2 2 i.e.

1 1 Fn (A, B) ∼ Fn A, B − Fn A + B, − B . 2 2

(15.22)

15.3 The Campbell–Hausdorff Formula for Formal Power Series–2

609

Analogously, taking A := − 12 B in (15.17) and renaming B → A, C → B, we get

1 1 Fn (A, B) ∼ Fn A, B − Fn − A, A + B . (15.23) 2 2 An application of (15.22) to both summands in the right-hand side of (15.23) yields

1 1 1 1 A, B − Fn A + B, − B Fn (A, B) ∼ [· · ·] = Fn 2 2 2 2

1 1 1 1 1 1 A + B, − A − B . − Fn − A, A + B + Fn 2 2 2 2 2 2 Now, we apply (15.20) to the third summand in the far right-hand of the above expression and (15.19) to the second and fourth summands. We thus get

1 1 1 1 1 Fn (A, B) ∼ Fn A, B + Fn A + B, B 2 2 2 2 2

1 1 1 1 1 A, B − Fn B, A + B + Fn 2 2 2 2 2 = 21−n Fn (A, B) + 2−n Fn (A + B, B) − 2−n Fn (B, A + B) ∼ 21−n Fn (A, B) + 2−n (1 + (−1)n ) Fn (A + B, B). In the equality we used the degree-homogeneity n of Fn , and we used (15.21) for the last ∼ sign. This gives (1 − 21−n ) Fn (A, B) ∼ 2−n (1 + (−1)n ) Fn (A + B, B),

(15.24)

which proves that Fn (A, B) ∼ 0 whenever n = 1 in odd. As for the case when n is even, we argue as follows. We write A − B instead of A in (15.24) (1 − 21−n ) Fn (A − B, B) ∼ 2−n (1 + (−1)n ) Fn (A, B), and we use (15.19) in the above left-hand side getting (after multiplication times (1 − 21−n )−1 ) Fn (A, −B) ∼ −(1 − 21−n )−1 2−n (1 + (−1)n ) Fn (A, B).

(15.25)

We write −B instead of B in (15.25), and we finally apply (15.25) once again obtaining Fn (A, B) ∼ (1 − 21−n )−2 2−2n (1 + (−1)n )2 Fn (A, B). When n is even and n = 2, this is possible only if Fn (A, B) ∼ 0. This ends the proof.

610

15 More on the Campbell–Hausdorff Formula

15.4 The Campbell–Hausdorff Formula for General Smooth Vector Fields At this point of the exposition, the Campbell–Hausdorff formula has been exhaustively proved in the abstract setting of formal power series and in the more concrete setting of the vector fields in the algebra of a Carnot group. As a matter of fact, it is easy to recognize that our proof of Theorem 15.1.1 can be applied for any couple of vector fields X, Y which are δλ -homogeneous of positive degree with respect to any dilation δλ : RN → RN of the form δλ (x1 , . . . , xN ) = (λα1 x1 , . . . , λαN xN ) with positive αi ’s. The reason for this fact is that the cited δλ -homogeneity ensures that exp(X), exp(Y ) are indeed given by finite sums and there exists a large m ∈ N such that X m I, Y m I ≡ 0. As a consequence, all the computations with formal power series in two indeterminates make precise sense, and the given proof of the Campbell–Hausdorff formula for formal power series can be straightforwardly adapted. We have so far provided the proof of the Campbell–Hausdorff formula for many interesting cases in literature, but we are now concerned in giving the most useful version of such an important formula. Theorem 15.4.1 below is the version which analysts mostly refer to. For example, it is, almost verbatim, the version needed in the very recent paper [Mor00, Proposition 2.3] or a restatement of Proposition 4.3 in the pioneering paper [NSW85] by Nagel, Stein and Wainger or, furthermore, an analogue of the statement needed for the celebrated lifting by Rothschild and Stein (see [RS76, Section 10]). The aim of this section is to provide its proof. As usual, given a smooth vector field X on an open set Ω ⊆ RN and fixed x ∈ Ω, we denote by t → γ (t) = exp(tX)(x) the solution to the first order differential system in RN γ˙ = XI (γ ), γ (0) = x.

(∗)

Equivalently (see for example, Ex. 7 at the end of Chapter 1), if D(X, x) is the maximal domain of existence of the solution of (∗), then the solution of the system ν˙ (s) = (tX)I ν(s) , ν(0) = x exists at s = 1 for every t ∈ D(X, x), and it holds ν(1) = γ (t). Theorem 15.4.1 (The Campbell–Hausdorff formula for smooth vector fields). Let X and Y be smooth vector fields on the open set Ω ⊆ RN . Then the following formal equality holds:

15.4 The Campbell–Hausdorff Formula for Smooth Vector Fields

611

st exp(sY )◦exp(tX) = exp sY +tX + [X, Y ]+ s k t j Ck,j (X, Y ) . (15.26) 2 k+j >2

Here Ck,j (X, Y ) denotes a finite linear combination of commutators of X and Y . Any summand in Ck,j (X, Y ) contains k times the field X and j times the field Y . The precise meaning of (15.26) is the following: For any fixed compact set K ⊂ Ω and any given couple of integers k0 , j0 , there exists a real number r0 > 0 (depending on K, k0 , j0 , X, Y ) such that if |s|, |t| < r0 , then we have (see also Figure 15.1)

st exp(sY ) exp(tX)(x) = exp sY + tX + [X, Y ] 2 + s k t j Ck,j (X, Y ) (x) 1 ≤ k ≤ k0 , 1 ≤j ≤j0 k+j >2

+ O(s k0 +1 ) + O(t j0 +1 )

(15.27)

for every x ∈ K, where the above O’s mean |O(r m )| ≤ c r m for every |r| ≤ r0 and uniformly in x ∈ K (the constant c depends on r0 , K, k0 , j0 , X, Y ). Finally, the Ck,j ’s are “universal” Lie-polynomials in two non-commuting indeterminates, as given in Theorem 15.2.2. For example, 1 [X, Y ], 2 1 1 [X, [X, Y ]], C2,1 (X, Y ) = − [Y, [X, Y ]], C1,2 (X, Y ) = 12 12 1 1 C2,2 (X, Y ) = − [Y, [X, [X, Y ]]] − [X, [Y, [X, Y ]]], 48 48 C1,3 (X, Y ) = C3,1 (X, Y ) = 0.

C1,1 (X, Y ) =

Proof. Consider the notation in the statement of the Campbell–Hausdorff formula for formal power series (Theorem 15.2.2). Then replace X and Y in that statement with, respectively, sY and tX, where X, Y are still two non-commuting indeterminates and s, t are real numbers. Within the context of formal power series, formulas from (15.11) to (15.13d) then yield ZsY,tX n (tX)j (sY)k · = , (15.28) j! k! n! j ≥0

∞

k≥0

n≥0

where ZsY,tX = n=1 Zn (sY, tX), and Zn (·, ·) is a “universal” homogeneous Liebracket-polynomial of length n in two indeterminates. Now, for every fixed k, j ∈ N ∪ {0}, we group together the terms containing s k t j in the Zn (sY, tX)’s. Denote the resulting group by Ck,j (X, Y). Thus, ZsY,tX rewrites as

612

15 More on the Campbell–Hausdorff Formula

Fig. 15.1. The Campbell–Hausdorff flow

ZsY,tX = sY + tX +

st s k t j Ck,j (X, Y). [X, Y] + 2

(15.29)

k+j >2

For the future references, we see that (15.28) and (15.29) group together to give (tX)j (sY)k · j! k! j ≥0 k≥0 n 1

sY + tX + = s k t j Ck,j (X, Y) . n! n≥0

(15.30)

k+j ≥2

The Ck,j (X, Y )’s in the assertion of the present theorem are defined by making use of the above “universal” Ck,j ’s. Note that, obviously, any summand in Ck,j (X, Y ) contains k times the field X and j times the field Y . Throughout the proof, we fix a compact set K ⊂ Ω and a couple of integers k0 , j0 . By general results on differential systems of ODE’s (see, e.g. [Har82]), there exists a positive r0 = r0 (X, Y, K) such that the set exp(sY ) exp(tX)(x) x ∈ K, |s|, |t| ≤ r0 makes sense and is a compact subset of Ω. Analogously, using general results on the dependence on the initial data for ODE’s, it is possible to prove the existence of r0 > 0 (we may assume it is the same as above) such that the integral curve of the vector field

15.4 The Campbell–Hausdorff Formula for Smooth Vector Fields

V (s, t) := sY + tX +

st [X, Y ] + 2

s k t j Ck,j (X, Y )

613

(15.31)

1≤k≤k0 , 1≤j ≤j0 k+j >2

exists at unit time, whenever x ∈ K and |s|, |t| ≤ r0 . (The Ck,j ’s have been defined in the first part of the proof.) Thus, the functions u(s, t) := exp(sY ) exp(tX)(x) , v(s, t) := exp(V (s, t))(x), are well-posed on Q0 := (−r0 , r0 ) × (−r0 , r0 ) for every fixed x ∈ K. Our aim is to prove that the maps (s, t) → u(s, t), v(s, t) have the same Taylor expansion at (s, t) = (0, 0), up to order k0 w.r.t. s and up to order j0 w.r.t. t. To begin with, we exhibit the expansion of u. Recalling that, for every function f ∈ C ∞ (Ω, RN ), every smooth vector field Z on Ω, every z ∈ Ω and every n ∈ N ∪ {0}, it holds

n d (f (exp(rZ)(z))) = (Z n f )(exp(rZ)(z)), (15.32) dr we infer

j k ∂ k+j ∂ ∂ u(s, t) = exp(sY ) exp(tX)(x) k j j k ∂s ∂t ∂t ∂s (use (15.32) with r = s, n = k, f = I , Z = Y , z = exp(tX)(x))

j k ∂ = Y I exp(sY ) exp(tX)(x) j ∂t (use (15.32) with r = t, n = j , f = Y k I (exp(sY )(·)), Z = X, z = x) = X j Y k I exp(sY )(·) (exp(tX)(x)).

Thus, we have proved that, for every j, k ∈ N ∪ {0},

k+j ∂ u(s, t) = X j Y k I exp(sY )(·) (exp(tX)(x)). k j ∂s ∂t

(15.33)

Now, when s = 0 = t, the map exp(sY )(·) is the identity map, so that the above computation reduces to (15.6), i.e.

k+j ∂ u(0, 0) = (X j Y k I )(x). ∂ sk ∂ t j As a consequence, the Taylor polynomial of u(s, t) at (0, 0) up to order k0 w.r.t. s and up to order j0 w.r.t. t is given by sk t j I (x) + sY I (x) + tXI (x) + (X j Y k I )(x) k!j ! =

j0 j t j =0

j!

X

j

1≤k≤k0 1≤j ≤j0

k0 k s k=0

k!

Y

k

I (x).

614

15 More on the Campbell–Hausdorff Formula

We now estimate the remainder. From the Taylor formula with the Lagrangeremainder up to height j0 + k0 applied to the i-th component of u, say ui , for every (s, t) ∈ Q0 we infer the existence of (σi , τi ) ∈ Q0 such that ui (s, t) =

0≤k≤k0 0≤j ≤j0

sk t j (X j Y k Ii )(x) + k!j !

+

k+j =k0 +j0 +1

k>k0 vel j >j0 k+j ≤k0 +j0

sk t j (X j Y k Ii )(x) k!j !

sk t j j k X Y Ii exp(σi Y )(·) (exp(τi X)(x)). k!j !

Clearly, the second sum in the above right-hand side is given by O(s k0 +1 ) + O(t j0 +1 ), uniformly in x ∈ K (due to the smoothness of X, Y and the compactness of K). The same estimate is true of the third sum, for (see (15.33)) |X j (Y k Ii (exp(σi Y )(·)))(exp(τi X)(x))| ≤ M(Q0 , K, X, Y ) uniformly in x ∈ K. All these facts prove that j k 0 0 tj j sk k u(s, t) = X Y I (x) + O(s k0 +1 ) + O(t j0 +1 ). j! k! j =0

(15.34)

k=0

We now turn to the expansion of v(s, t). By the very definition v(s, t) = exp(V (s, t))(x), we have v(s, t) = γs,t (1), where r → γs,t (r) is the solution to γ˙s,t (r) = V (s, t)I (γs,t (r)), γs,t (0) = x. A simple computation (see the note to Ex. 5) shows that

n d γs,t (r) = V (s, t)n I (γs,t (r)), dr so that, by Taylor’s expansion with an integral remainder, we have v(s, t) = γs,t (1) =

N +1

n 1 N d d 1 1 γs,t (0) + (1 − r)N γs,t (r) dr n! d r N! 0 dr n=0

=

N n=0

1 1 V (s, t)n I (x) + n! N!

1

(1 − r)N V (s, t)N +1 I (γs,t (r)) dr.

0

Let us remark that, for r ∈ [0, 1], (s, t) ∈ Q0 and x ∈ K, all the points γs,t (r) belong to a fixed compact set, say K0 . Hence, if we choose N = k0 + j0 , we have (see also (15.31))

15.4 The Campbell–Hausdorff Formula for Smooth Vector Fields

V (s, t)k0 +j0 +1 I (γs,t (r))

= sY + tX +

615

k0 +j0 +1 s t Ck,j (X, Y ) I (γs,t (r)) k j

1≤k≤k0 , 1≤j ≤j0 k+j ≥2

≤ c0 (s k0 +1 + t j0 +1 ) × max sup |H I (ξ )| : H ∈ R[X, Y ] is a monomial with degree ξ ∈KO

≤ (k0 + j0 )(k0 + j0 + 1)

= O(s k0 +1 ) + O(t j0 +1 ) uniformly in x ∈ K. Here we have denoted by R[X, Y ] the set of the polynomials in the (non-commuting) indeterminates X, Y (see Definition 15.2.1). This shows that v(s, t) =

k 0 +j0 n=0

1 V (s, t)n I (x) + O(s k0 +1 ) + O(t j0 +1 ), n!

(15.35)

uniformly in x ∈ K. Collecting together (15.34), (15.35) and the definition (15.31) of V (s, t), we see that the asserted (15.27) will follow if we prove that k j 0 0 tj j sk k X Y I (x) j! k! j =0

=

k=0

k 0 +j0 n=0

1 sY + tX + n!

n s t Ck,j (X, Y ) I (x)

k j

1≤k≤k0 , 1≤j ≤j0

k+j ≥2

+ O(s

k0 +1

) + O(t

j0 +1

(15.36)

),

again uniformly in x ∈ K. Now, let us return to identity (15.30), an identity in the context of formal power series in two non-commuting indeterminates X, Y. By the very definition of formal power series, we derive from (15.30) an identity between terms involving X up to j0 times and Y up to k0 times, namely j0 k0 (tX)j (sY)k · j! k! j =0

≡

k=0

k 0 +j0 n=0

1 sY + tX + n!

n s t Ck,j (X, Y) k j

1≤k≤k0 , 1≤j ≤j0 k+j ≥2

{modulo terms with X more than j0 times or Y more than k0 times}. (15.37)

616

15 More on the Campbell–Hausdorff Formula

In particular, (15.37) holds true if X, Y are our smooth vector fields in Ω, as differential operators in the (non-commutative) algebra T (Ω, RN ) o f the smooth vector fields on Ω. As a consequence, (15.36) will follow from (15.37) if we prove the following fact. After carrying out the n-th power in the second line of (15.36), those terms where X appears more than j0 times or where Y appears more than k0 times, when applied to I and evaluated at x, all together give O(s k0 +1 ) + O(t j0 +1 ), uniformly in x ∈ K. This is straightforwardly seen. Indeed, in the second line of (15.36) we have only a finite number of summands (depending on j0 , k0 ) and any summand ((tX)α1 (sY )β1 · · · (tX)αi (sY )βi )I (x) with α1 + · · · + αi > j0 or β1 + · · · + βi > k0 is bounded by t α1 +···+αi s β1 +···+βi sup X α1 Y β1 · · · X αi Y βi I (x) ≤

x∈K

c0 t j0 +1 c0 s k0 +1

if α1 + · · · + αi > j0 , if β1 + · · · + βi > k0 .

Here c0 depends only on Q0 , k0 , j0 , K and on max sup |H I (x)| : H ∈ R[X, Y ] monomial with degree ≤ (k0 + j0 )2 . x∈K

This completes the proof. Bibliographical Notes. Classical references on the Campbell–Hausdorff formula are Bourbaki [Bou89], M. Hausner and J.T. Scwartz [HS68], G. Hochschild [Hoc68], N. Jacobson [Jac62], V.S. Varadarajan [Var84]. Some applications of this tool to analysis are discussed, for example, in [Hor67, RS76,VSC92], see also the paper [GG90], already mentioned in Section 14.1. We would also like to cite [DST91,Egg93,Oki95,Ote91,Str87,Tho82] for other remarkable applications, referring the reader to the references therein for further details.

15.5 Exercises of Chapter 15 Ex. 1) Given a smooth vector field X on RN and a point x ∈ RN , we recall that by exp(X)(x) we mean the point reached at unit time (if this point exists) by the integral curve of X starting at x. In (1.7) of Chapter 1 (page 7), we found the exponential-type expansion

15.5 Exercises of Chapter 15

exp(X)(x) = I (x) + XI (x) +

1 2 X I (x) + · · · . 2!

617

(15.38)

Suppose that there exists a dilation δλ on RN of the form δλ (x) = δλ (x1 , . . . , xN ) = (λσ1 x1 , . . . , λσN xN ) with 0 < σ1 ≤ · · · ≤ σN such that X is δλ -homogeneous of positive degree. Prove that any integral curve of X exists on the whole R (that is to say that X is complete) and there exists n ∈ N such that X n I ≡ 0 (here X n denotes the n-th power of X as a differential operator). Deduce that formula (15.38) makes a precise sense, for the sum is finite. Ex. 2) By means of the above exercise, find the integral curves of the vector fields on R2 given by ∂x , x ∂y and (x − 12 ) ∂y + ∂x both by solving a suitable system of ODE’s and by the above formula (15.38). Then check the Campbell– Hausdorff formula (15.3), which, in this case, gives

1 exp(∂x ) exp(x ∂y )(x, y) = exp x ∂y + ∂x + [x ∂y , ∂x ] (x, y). 2 The reader is once again invited to appreciate the interchange of the order of the two vector fields from one side to the other of the formula

1 exp(X) exp(Y )(x, y) = exp(Y X)(x, y) = exp Y +X + [Y, X] (x, y). 2 Ex. 3) Taking into account the actual expressions (15.12) (page 601) of the first four terms in the Campbell–Hausdorff formula, verify directly for n = 1, 2, 3, 4 the alternating skew-symmetry relation Zn (A, B) = (−1)n+1 Zn (B, A). Compare to (15.21) (page 608). Ex. 4) Find the Campbell–Hausdorff composition up to step five. Ex. 5) Prove Theorem 15.4.1 when k0 = 1 = j0 , with direct arguments and without the aid of the Campbell–Hausdorff formula. The assertion to be proved is the following one. Let X, Y be smooth vector fields on the open set Ω ⊆ RN , and let K ⊂ Ω be any fixed compact set. Then there exists a real number r0 > 0 (depending on K, X, Y ) such that if |s|, |t| < r0 , then, for every x ∈ K, we have exp(sY ) exp(tX)(x)

st = exp sY + tX + [X, Y ] (x) + O(s 2 ) + O(t 2 ), (15.39) 2 where the above O’s mean |O(r 2 )| ≤ c r 2 for every |r| ≤ r0 and uniformly in x ∈ K.

618

15 More on the Campbell–Hausdorff Formula

Hint: The Taylor expansion w.r.t. (s, t) up to order 2 of the left-hand side of (15.39) is9 given by I (x) + s Y I (x) + t XI (x) +

st XY I (x) + O(s 2 ) + O(t 2 ). 2

Prove by a simple computation that this is also the expansion of

st exp sY + tX + [X, Y ] (x). 2 This can be derived by recalling that, by the Taylor formula with an integral remainder, we have

st exp sY + tX + [X, Y ] (x) 2

st = exp r sY + tX + [X, Y ] (x) 2 r=1

st = I (x) + sY + tX + [X, Y ] I (x) 2

2 st 1 sY + tX + [X, Y ] I (x) + 2 2 3

1 st 1 (1 − r)2 sY + tX + [X, Y ] I (γs,t,x (r)) dr, + 2 0 2 where r → γs,t,x (r) is the integral curve of the vector field sY + tX + st 2 [X, Y ] starting at x. Notice that the integral remainder in the above far right-hand side is O(s 2 ) + O(t 2 ), whereas the third line of the above expression equals, modulo O(s 2 ) + O(t 2 ), 9 See (15.6); see also Ex. 8 at the end of Chapter 1 where it is proved that if X is a smooth vector field on an open set Ω ⊆ RN , f ∈ C ∞ (Ω, RN ) and k ∈ N, it holds

d dt

k

f (γ (t)) = (X k f )(γ (t)),

whenever γ satisfies γ˙ (t) = XI (γ (t)). Indeed, when k = 1, we have d f (γ (t)) = (∇f )(γ (t)), γ˙ (t) = (∇f )(γ (t)), XI (γ (t)) = (Xf )(γ (t)). dt Moreover, arguing by induction and by using the inductive step and the case n = 1 with f replaced by X k f , we have k 1+k d d d d f (γ (t)) = f (γ (t)) = (X k f )(γ (t)) dt dt dt dt = X(X k f ) (γ (t)) = (X 1+k f )(γ (t)).

15.5 Exercises of Chapter 15

st st (XY I (x) − Y XI (x)) + Y XI (x) 2 2 st = I (x) + s Y I (x) + t XI (x) + XY I (x). 2

I (x) + sY I (x) + tXI (x) +

This ends the proof.

619

16 Families of Diffeomorphic Sub-Laplacians

2 Let G be a homogeneous Carnot group, and let ΔG = m i=1 Xi be the canonical sub-Laplacian on G. Suppose it is given a positive-definite symmetric matrix A = (ai,j )1≤i,j ≤m . We consider the second order operator modeled on the matrix A and the vector fields Xi ’s m LA := ai,j Xi Xj . (16.1) i,j =1

For example, in the classical case G = (RN , +), ΔA =

N

ai,j ∂xi ∂xj

i,j =1

is coefficient second order operator of elliptic type, whereas Δ = Na constant 2 is the usual Laplace operator. It is well known that a linear change of (∂ ) x i i=1 coordinates in RN transforms the above operator ΔA into Δ. Thus, the operator ΔA is “equivalent” to the operator Δ (in a new system of coordinates). Following the above naive idea, in order to study the operator LA in (16.1), it is natural to ask the following question. Does there exist a diffeomorphism T = TA : G → G such that, in the new coordinate system defined by T , the operator (16.1) is turned1 into ΔG ? Unfortunately, when the Xi ’s have not constant coefficients, classical changes of basis may fail to apply, and simple examples (provided in this chapter) show that T may not exist at all. Then, to face the above question, it seems natural to make an additional assumption on the group G. Indeed, we shall prove the existence of such a diffeomorphism T whenever G is a free Carnot group. Finally, in Section 16.4, we present an example of application to PDE’s of the topics developed in this chapter. 1 We say that T turns L into Δ if L (u ◦ T ) = (Δ u) ◦ T for every u ∈ C ∞ (G). A A G G

622

16 Families of Diffeomorphic Sub-Laplacians

16.1 An Isomorphism Turning the Operator the Canonical Sub-Laplacian ΔG

i,j

ai,j Xi Xj into

Let G be a homogeneous Carnot group, and let ΔG =

m

Xi2

i=1

be the canonical2 sub-Laplacian on G. Suppose it is given a positive-definite symmetric matrix A = (ai,j )1≤i,j ≤m . We then consider the following new second order differential operator modeled on the matrix A and the vector fields Xi ’s LA :=

m

ai,j Xi Xj .

(16.2)

i,j =1

This is exactly what is done in the classical case G = (RN , +) when, starting from the classical Laplace operator Δ :=

N ∂ 2 , ∂ xi i=1

one considers the following constant coefficient second order operator of elliptic type ΔA =

N i,j =1

ai,j

∂2 . ∂ xi ∂ xj

It is well known and easy to prove that a linear change of coordinates in RN (naturally related to the matrix A−1/2 ) transforms the above operator ΔA into Δ. Indeed, consider the new coordinates in RN given by y = T (x) := C x,

where C is a symmetric matrix such that C 2 = A−1 ,

i.e. C = B −1 is the inverse of the symmetric (square root of A) matrix B such that B 2 = A. With this choice, for every u ∈ C ∞ (RN ), u = u(y), we have 2 We recall that the Jacobian basis of g (=algebra of G) is the basis of vector fields in g Then, the canonagreeing at the origin with the coordinate partial derivatives of RN ≡ G. 2 ical sub-Laplacian on G is the second order differential operator ΔG = m i=1 Xi , where X1 , . . . , Xm are the first m vector fields of the Jacobian basis for g. Here m = N1 is the dimension of the first layer of the stratification of g or, equivalently, m = N1 is the dimension of the layer of coordinates where the dilation group δλ of G

δλ (x) = δλ (x (1) , x (2) , . . . , x (r) ) = (λx (1) , λ2 x (2) , . . . , λr x (r) ), acts like the multiplication times λ.

x (i) ∈ RNi ,

16.1 An Isomorphism Turning

i,j ai,j Xi Xj into ΔG

623

N N ΔA u(T (x)) = ai,j ∂xi ch,j (∂yh u)(T (x)) = = =

i,j =1 N

h,k=1 N

h=1 N

ai,j ch,j ck,i (∂y2k ,yh u)(T (x))

i,j =1

(t C · A · C)k,h (∂y2k ,yh u)(T (x))

h,k=1 N

(∂y2h ,yh u)(T (x)) = (Δu)(T (x)),

h=1

since C is symmetric and C · A · C = B −1 · B 2 · B −1 = IN , the identity matrix of order N . Thus, the operator ΔA is “equivalent” to the operator Δ (in a new system of coordinates). Following the above naive idea, in order to study the operator LA in (16.2), it is natural to ask the following question. Does there exist a diffeomorphism T = TA : G → G such that, in the new coordinate system defined by T , the operator (16.2) is turned into ΔG ? Here, we said that T turns LA into ΔG if LA (u ◦ T ) = (ΔG u) ◦ T C ∞ (G).

for every u ∈ following assertion.

A further justification for this natural question is given by the

Proposition 16.1.1 ({LA : A} and {L}). Any sub-Laplacian on G is of the form (16.2) for a suitable symmetric positive definite matrix A. Vice versa, any operator LA in that form is a sub-Laplacian m Yk2

LA =

with Yk =

m

1/2 ) k,j j =1 (A

Xj .

k=1

(16.3)

2 Proof. First, suppose that L = m k=1 Yk is a sub-Laplacian on G. By the definition given in Section 1.5 (page 62), this means that {Y1 , . . . , Ym } and {X1 , . . . , Xm } are both bases for the first layer of the stratification of the algebra of G. In particular, there exists a non-singular m × m matrix D = (dk,j )k,j such that Yk =

m

dk,j Xj

for every k = 1, . . . , m.

j =1

Then we have m m m m 2 Yk = dk,i Xi dk,j Xj k=1

=

k=1 m

i=1m

i,j =1

k=1

j =1 dk,i dk,j Xi Xj =:

m i,j =1

ai,j Xi Xj = LA ,

624

16 Families of Diffeomorphic Sub-Laplacians

where we have set A = (ai,j )i,j :=

m

= tD · D

dk,i dk,j

k=1

i,j

(clearly, A is a symmetric positive-definite matrix). Vice versa, if A is a symmetric positive-definite matrix and LA is the operator (16.2), denoting by B = (bi,j )1≤i,j ≤m the symmetric positive-definite matrix such that A = B 2 (i.e. B = A1/2 ), one has LA =

m

ai,j Xi Xj =

i,j =1

=

m k=1

m m

bi,k bk,j Xi Xj

i,j =1 k=1

m

bk,i Xi

m

bk,j Xj

j =1

i=1

=

m

Yk2 ,

k=1

where Yk := m j =1 bk,j Xj for every k = 1, . . . , m. This proves that LA is a subLaplacian. Proposition 16.1.1 shows that, looking for a change of coordinates T turning LA into ΔG , it is natural to look for a T turning Yk =

m (A1/2 )k,j Xj j =1

into Xk , for k = 1, . . . , m. In the simple case when Xi = ∂/∂xi , we showed at the beginning of the section that the problem always has a solution via a linear change of coordinates in RN . However, when the Xi ’s have not constant coefficients, classical changes of basis may fail to apply. Moreover, and this is the most striking fact, simple examples (see Remark 16.2.2 and Remark 16.2.3 in the next section) show that T may not exist in the general case! Some further assumptions either on the coefficient matrix A or on the structure of the group G must be made. To face this question, rather than restricting the form of the matrix A, it seems more natural to make an additional assumption on the group G. Indeed, we shall prove the existence of such a diffeomorphism T , whenever G is a free Carnot group (we recall below the relevant definition; see also Chapter 14, page 577). The free group setting turns out to be the most natural one in order to generalize the classical Euclidean case. Though, we stress that T is not necessarily a linear application, in the general case (see Remark 16.2.1 in the next section). We now recall the definition of fm,r , the free nilpotent Lie algebra of step r with m (≥ 2) generators x1 , . . . , xm . By definition, fm,r is the unique (up to isomorphism) nilpotent Lie algebra of step r generated by m of its elements x1 , . . . , xm such that, for every nilpotent Lie algebra n of step r and for every map ϕ from {x1 , . . . , xm } to n, there exists a (unique) Lie algebra morphism

ϕ from fm,r to n extending ϕ. (See Chapter 14 for more details.) We say that the Carnot group G is a free Carnot

16.1 An Isomorphism Turning

i,j ai,j Xi Xj into ΔG

625

group if its Lie algebra g is isomorphic to fm,r , for some m and r. RN equipped with the ordinary Abelian structure is an example of free Carnot group. The Heisenberg group H1 is also a free Carnot group, while Hn is not free for any n ≥ 2, as can be seen by a dimensional argument. We are now in the position to prove the main result of this chapter. Theorem 16.1.2 (Equivalence of sub-Laplacians on free groups). Let G be a free homogeneous Carnot group, and let A be a given positive-definite symmetric matrix. Let X1 , . . . , Xm denote the first vector fields in the Jacobian basis related to G, i.e. ΔG =

m

Xk2

k=1

is the canonical sub-Laplacian on G. Finally, let the Yk ’s be defined by Yk =

m

(A1/2 )k,j Xj ,

k = 1, . . . , m.

(16.4)

j =1

Consider the related (non-canonical) sub-Laplacian LA =

m

Yk2 =

k=1

m

ai,j Xi Xj .

i,j =1

Then there exists a Lie group automorphism TA of G such that Yk (u ◦ TA ) = (Xk u) ◦ TA , LA (u ◦ TA ) = (ΔG u) ◦ TA

k = 1, . . . , m,

(16.5a) (16.5b)

for every smooth function u : G → R. Moreover, TA has polynomial component functions and commutes with the dilations of G. Proof. First of all, we remark that (16.5b) immediate consequence of (16.5a) m is an 2 2 since LA = m k=1 Yk whereas ΔG = k=1 Xk . On the other hand, by the very definition (16.4) of Yk , (16.5a) reads m 1/2 (A )k,j Xj (u ◦ TA ) = (Xk u) ◦ TA , k = 1, . . . , m, j =1

which, in turn, is trivially equivalent to Xk (u ◦ TA ) =

m (A−1/2 )k,j (Xj u) ◦ TA ,

k = 1, . . . , m.

(16.6)

j =1

Moreover, if we already knew that TA (denoted henceforth simply by T ) is a Lie group morphism, then it would be sufficient to prove that (16.6) holds at the origin. Indeed, suppose that

626

16 Families of Diffeomorphic Sub-Laplacians

Xk (u ◦ T )(0) =

m bk,j (Xj u)(0)

∀ u ∈ C ∞ (RN ), k = 1, . . . , m,

(16.7)

j =1

where we have set for brevity B = (bk,j )k,j = A−1/2 . We fix y ∈ G, v ∈ C ∞ (RN ) and apply (16.7) to u(x) = v(T (y) ◦ x). Since T is a Lie group morphism and Xi is left-invariant, we have Xk v ◦ T (y) = Xk v T (y ◦ ·) (0) = Xk v T (y) ◦ T (·) (0) =

m m bk,j Xj v(T (y) ◦ ·) (0) = bk,j (Xj v)(T (y)), j =1

j =1

i.e. (16.6) holds. We now turn to show (16.7). We first observe that, by the chain rule and since B is symmetric, (16.7) is equivalent to JT (0) (XI )(0) = (XI )(0) B.

(16.8) Here we have denoted by I the identity map on G, by XI = X1 I · · · Xm I the (N × m)-matrix whose i-th column is Xi I and by JT the Jacobian matrix of T . We thus aim to construct a Lie group automorphism T of G satisfying (16.8). This will complete the proof of the first statement of the theorem. Consider the linear map defined as follows ϕ : span{X1 , . . . , Xm } → span{X1 , . . . , Xm },

Xi →

m

bi,j Xj .

j =1

Since X1 , . . . , Xm are linearly independent, ϕ is well posed. Moreover, B being a non-singular matrix, ϕ is a bijective linear map. Since G is a free Carnot group and its Lie algebra g is nilpotent of step r and generated by {X1 , . . . , Xm }, clearly g is isomorphic to fm,r . Then, by Lemma 14.1.4 (page 578), there exists a unique Lie algebra automorphism

ϕ : g → g extending ϕ. For the simplicity of notation, we also denote

ϕ by ϕ. We are now in the position to define T . We set Log

ϕ

Exp

T : G −→ g −→ g −→ G,

T := Exp ◦ ϕ ◦ Log ,

(16.9)

where Exp denotes the exponential map and Log its inverse function. Clearly, T is bijective, T −1 = Exp ◦ ϕ −1 ◦ Log , whence both T and T −1 are C ∞ maps. Moreover, T is a Lie group automorphism of G: indeed, when g is equipped with the Campbell–Hausdorff3 group law , then Exp , Log and any Lie algebra morphism of g are Lie group morphisms. Indeed, the 3 See Section 14.2, page 584, for the definition and the properties of .

16.1 An Isomorphism Turning

i,j ai,j Xi Xj into ΔG

627

fact that Exp and Log are Lie-algebra morphisms between (g, ) and (G, ◦) is a consequence of Theorem 14.2.1 (page 586). The fact that any Lie-algebra morphism of (g, [·, ·]) into itself is actually a Lie-group morphism of (g, ) into itself is easily seen.4 Let us prove the matrix equality (16.8). We recall that, by definition, Im (XI )(0) = 0 where Im is the identity matrix of order m and 0 is the null matrix of order (N −m)× m. If g is equipped with the well-known Jacobian basis, we have (see, for instance, the arguments in Remark 1.3.30, page 50) JLog (0) = (JExp (0))−1 = IN , whence JT (0) = JExp (0) Jϕ (0) JLog (0) = Jϕ (0). Thus, in order to prove (16.8), it is enough to prove that the (N × m)-matrix of the first m-columns of Jϕ (0) is equal to Im B B= , 0 0 which straightforwardly follows from the definition of ϕ and from the fact that X1 , . . . , Xm are the first m vectors of the Jacobian basis. We now turn to the proof of the last statement of the theorem. We recall that Exp and Log have polynomial components (see Theorem 1.3.28, 49). Hence T has polynomial component functions since ϕ is linear. Finally, we prove that T commutes with δλ , the dilations on G. Recalling that δλ has the form δλ (x) = δλ (x (1) , . . . , x (r) ) = (λx (1) , . . . , λr x (r) ),

x (i) ∈ RNi ,

let us accordingly denote by (1)

(1)

(r)

(r)

Z1 , . . . , ZN1 , . . . , Z1 , . . . , ZNr the Jacobian basis of g, and let δλ also denote the map

4 Indeed, let ϕ : g → g be a Lie-algebra morphism, i.e. ϕ([X, Y ]) = [ϕ(X), ϕ(Y )] for every

X, Y ∈ g. Then, comparing to the definition (14.2) of the operation, we have 1 1 ϕ(X Y ) = ϕ X + Y + [X, Y ] + [X, [X, Y ]] + · · · 2 12 = ϕ(X) + ϕ(Y ) + = ϕ(X) ϕ(Y ).

1 1 [ϕ(X), ϕ(Y )] + [ϕ(X), [ϕ(X), ϕ(Y )]] + · · · 2 12

This amounts to say that ϕ is a Lie-group morphism of (g, ) into itself.

628

16 Families of Diffeomorphic Sub-Laplacians

g

Ni Ni r r (i) (i) (i) (i) ξj Zj → λi ξj Zj ∈ g. i=1 j =1

i=1 j =1

With this notation, both Exp and Log commute with δλ (see (1.71), page 49, in Theorem 1.3.28). Thus, we only have to show ϕ ◦ δλ = δλ ◦ ϕ. (k) Zj

(k)

Now, is a δλ -homogeneous vector field of degree k, hence in particular Zj can be expressed as a homogeneous Lie polynomial of degree k in the generators (k) X1 , . . . , Xm . By the definition of the Lie algebra morphism ϕ, ϕ(Zj ) is a homogeneous Lie polynomial of degree k in X1 , . . . , Xm as well. Hence, there exist scalars (k) ci,j such that (k) ϕ(Zj )

Nk (k) (k) = ci,j Zi . i=1

Consequently, for every k = 1, . . . , r and j = 1, . . . , Nk , we have (k)

(k)

(k)

(ϕ ◦ δλ )(Zj ) = ϕ(λk Zj ) = λk ϕ(Zj ) = λk

Nk

(k)

i=1

=

Nk

(k) (k) ci,j λk Zj

i=1

=

Nk

(k) (k) ci,j δλ (Zj )

i=1

(k)

ci,j Zj

= δλ

Nk

(k) (k) ci,j Zj

i=1

(k)

= (δλ ◦ ϕ)(Zj ). This completes the proof. We explicitly remark that, in the proof of Theorem 16.1.2, we have proved that the diffeomorphism T has the following form Log

ϕ

Exp

T : G −→ g −→ g −→ G, T := Exp ◦ ϕ ◦ Log , where ϕ is a linear map which, w.r.t. the Jacobian basis, is related to a block-diagonal matrix of the type ⎛ (1) ⎞ C 0 ··· 0 (2) ⎜ 0 (k) ··· 0 ⎟ C ⎜ . , where C (k) = ci,j 1≤i,j ≤N . .. .. ⎟ .. ⎝ .. ⎠ k . . . 0

0

· · · C (r)

16.2 Examples and Counter-examples In this section, we give some examples and counter-examples concerning with the topics of the previous section. We follow all the notation of Section 16.1.

16.2 Examples and Counter-examples

629

Remark 16.2.1. The diffeomorphism TA constructed in Theorem 16.1.2 may fail to be linear. Proof. We consider the free homogeneous Carnot group G = (R3 , ◦, δλ ), where x ◦ y = (x1 + y1 , x2 + y2 , x3 + y3 + x1 y2 ) and δλ (x) = (λ x1 , λ x2 , λ2 x3 ). The Jacobian basis for g is given by X1 = ∂1 , X2 = ∂2 + x1 ∂3 , X3 = [X1 , X2 ] = ∂3 . Moreover, we have 1 Exp (ξ1 X1 + ξ2 X2 + ξ3 X3 ) = ξ1 , ξ2 , ξ3 + ξ1 ξ2 , 2 1 Log (x) = x1 X1 + x2 X2 + x3 − x1 x2 X3 . 2 Let now B = (bi,j )i,j ≤2 be an assigned symmetric matrix. Since G is free, there exists a unique Lie algebra morphism ϕ from g into itself which maps the generators X1 , X2 respectively in b1,1 X1 + b1,2 X2 ,

b1,2 X1 + b2,2 X2 .

In Jacobian coordinates, ϕ is represented by the block-diagonal matrix b1,1 b1,2 0 . 0 b1,2 b2,2 0 0 det(B) We now set T = Exp ◦ ϕ ◦ Log . A direct computation gives ⎛

⎞ b1,1 x1 + b1,2 x2 ⎠. b1,2 x1 + b2,2 x2 T (x) = ⎝ 1 1 2 2 2 det(B) x3 + 2 b1,1 b1,2 x1 + 2 b1,2 b2,2 x2 + b1,2 x1 x2

In particular, if we choose b1,1 = 2, b1,2 = b2,1 = 1, b2,2 = 4, we obtain T (x) = (2 x1 + x2 , x1 + 4 x2 , 7 x3 + x12 + 2 x22 + x1 x2 ).

630

16 Families of Diffeomorphic Sub-Laplacians

With the notation in the proof of Theorem 16.1.2, we see that, choosing 17 6 + 49 − 49 −2 , A=B = 6 5 − 49 + 49 the diffeomorphism T = TA turns Y1 =

4 7

X1 −

1 7

and

X2

Y2 = − 17 X1 +

2 7

X2

respectively into X1 and X2 . As a consequence, TA turns the sub-Laplacian LA =

17 2 6 6 5 2 X − X1 X2 − X2 X1 + X 49 1 49 49 49 2

into the canonical sub-Laplacian ΔG = X12 + X22 . We remark that TA is not linear. We observe that the group G in the above remark is isomorphic to H1 . However, on H1 , TA is always linear. This follows from the fact that, in H1 , the map Exp is linear (with the obvious choices of basis in H1 and in its Lie algebra). Remark 16.2.2. If G is not free, a diffeomorphism TA satisfying (16.5a) of Theorem 16.1.2 may not exist. Proof. We consider the group G = H2 , the Heisenberg group on R5 . If the points of H2 are denoted by (a, b, c), with a, b ∈ R2 , c ∈ R, we have (a, b, c) ◦ (α, β, γ ) = (a + α, b + β, c + γ + 2b, α − 2a, β) and δλ (a, b, c) = (λ a, λ b, λ2 c), and the canonical sub-Laplacian on H2 is given by 2 (A2j + Bj2 ), j =1

where Aj = ∂aj + 2 bj ∂c ,

Bj = ∂bj − 2 aj ∂c ,

j = 1, 2.

(H2 , ◦, δλ ) is then a homogeneous Carnot group of step 2 with 4 generators. The dimension of the Lie algebra of H2 is 5, whereas dim(f4,2 ) = 10, hence H2 is not free. We observe that there does not exist any diffeomorphism on H2 turning the set of vector fields

16.2 Examples and Counter-examples

631

Y = {A1 + B2 , A2 , B1 , B2 } into the set X = {A1 , A2 , B1 , B2 }. Indeed, each vector field in X has exactly one non-vanishing commutator with any other vector field in X ; on the contrary, A1 + B2 ∈ Y has two non-vanishing commutators with the other vector fields in Y. According to Remark 16.2.2 (we follow the notation therein), we may expect that the two sub-Laplacians on H2 (A1 + B2 )2 + A22 + B12 + B22

and A21 + A22 + B12 + B22

cannot be turned into each other, thus furnishing an example of two sub-Laplacians on the same group which are not “equivalent”. However, since (16.5a) is only sufficient but not necessary to deduce (16.5b), further comments must be made: these are given in the next theorem. Theorem 16.2.3. If G is not free, a diffeomorphism TA satisfying (16.5b) of Theorem 16.1.2 may not exist. Proof. The proof of this result makes use of some Liouville-type theorems for subLaplacians proved in Section 5.8, page 269. We shall exhibit a rather elaborated counter-example. First of all, if A = (ai,j )i,j ≤m is a fixed symmetric matrix and the Xi ’s are smooth vector fields, we provide a lemma characterizing the maps T turning the operator i,j ai,j Xi Xj into the operator i Xi2 . Lemma 16.2.4. Suppose A = (ai,j )i,j ≤m is a fixed symmetric matrix and Xi =

N

(i)

αk (x) ∂k ,

i = 1, . . . , m,

k=1

are smooth vector fields on RN . Then a map T ∈ C 2 (RN , RN ) satisfies m m 2 ai,j Xi Xj (u ◦ T ) = Xi u ◦ T ∀ u ∈ C ∞ (RN ) i,j =1

(16.10)

i=1

(i.e. T turns the operator i,j ai,j Xi Xj into the operator i Xi2 ) if and only if the following system of quasi-linear PDE’s is satisfied ⎧ m m ⎪ ⎪ (i) ⎪ a X X T = (Xi αk ) ◦ T ∀ k = 1, . . . , N , ⎪ i,j i j k ⎪ ⎪ ⎪ i=1 ⎨ i,j =1 (16.11) ⎪ m m ⎪ ⎪ ⎪ (i) (i) ⎪ ai,j Xi Tl Xj Tk = (αk αl ) ◦ T ∀ k, l = 1, . . . , N . ⎪ ⎪ ⎩ i,j =1

i=1

632

16 Families of Diffeomorphic Sub-Laplacians

Proof. The left-hand side of (16.10) is equal to N

ai,j Xi Tl Xj Tk · (∂k,l u) ◦ T +

i,j

k,l=1

k=1

With the position Xi = N

N

N

(i) k=1 αk (x) ∂k ,

m (i) (i) (αk αl ) ◦ T

k,l=1

ai,j Xi Xj Tk · (∂k u) ◦ T .

i,j

the right-hand side of (16.10) equals

· (∂k,l u) ◦ T +

i=1

N k=1

m (i) (Xi αk ) ◦ T

· (∂k u) ◦ T .

i=1

Now, if we put u(x) = xk in (16.10) and equate the left-hand and right-hand sides of (16.10) expressed above, we obtain m

ai,j Xi Xj Tk =

i,j =1

m (i) (Xi αk ) ◦ T

∀ k = 1, . . . , N.

(16.12)

i=1

Moreover, if we put u(x) = xk xl in (16.10), and (16.12) holds, we get (recall that ai,j = aj,i ) m i,j =1

ai,j Xi Tl Xj Tk =

m (i) (i) (αk αl ) ◦ T

∀ k, l = 1, . . . , N.

(16.13)

i=1

This proves that (16.10) is equivalent to (16.11). We explicitly remark that, if we set JTX := (Xj Ti )1≤i≤N, 1≤j ≤m

(the “X-Jacobian” matrix of the map T )

and if we consider the N × m matrix X whose j -th column is given by the N-tuple of the coefficient functions of Xj I , i.e. (j ) X := (X1 I · · · Xm I ) = αi 1≤i≤N, 1≤j ≤m ,

(16.14)

then the second equation in (16.11) turns out to be equivalent to JTX · A · t JTX = X · t X ◦ T .

(16.15)

We now return to the proof of Theorem 16.2.3. We consider the special case of the Heisenberg group H2 (whose points are denoted by (x1 , . . . , x5 )), with the natural choice of the fields X1 , . . . , X4 , i.e. X1 := ∂1 + 2 x3 ∂5 , X3 := ∂3 − 2 x1 ∂5 ,

X2 := ∂2 + 2 x4 ∂5 , X4 := ∂4 − 2 x2 ∂5 .

We aim to prove that there does not exist any map T ∈ C 2 (H2 , H2 ) satisfying (16.10), for a suitable matrix A to be specified in the sequel (henceforth, we write

16.2 Examples and Counter-examples

633

LA := m i,j =1 ai,j Xi Xj ). To this end, we suppose by contradiction that there exists such a T . Obviously, it is not restrictive to suppose that T (0) = 0 (since X1 , . . . , X4 are left-translation invariant). With the above choice of X1 , . . . , X4 (and following the notation in (16.14)), one has ⎛ ⎞ 1 0 0 0 2 x3 1 0 0 2 x4 ⎜ 0 ⎟ ⎜ ⎟ (16.16) X · tX = ⎜ 0 0 1 0 −2 x1 ⎟. ⎝ ⎠ 0 0 0 1 −2 x2 2 x3 2 x4 −2 x1 −2 x2 4(x12 + x22 + x32 + x42 ) Hence, the first equation in (16.11) becomes LA T k =

4

ai,j Xi Xj Tk = 0

∀ k = 1, . . . , 5.

(16.17)

i,j =1

The second equation in (16.11) gives, in particular, 4

ai,j Xi Th Xj Th = 1

∀ h = 1, . . . , 4,

(16.18a)

i,j =1 4

ai,j Xi T5 Xj T5 = 4 (T12 + T22 + T32 + T42 ).

(16.18b)

i,j =1

Moreover, from (16.17) and (16.18a) we easily obtain 4 ai,j Xi Tk Xj Tk = 2. LA (Tk )2 = 2 Tk LA Tk + 2

(16.19)

i,j =1

Suppose now that A = (ai,j )i,j ≤4 is a symmetric positive-definite matrix. Then, by Proposition 16.1.1, LA is a sub-Laplacian on H2 . In particular, we are in the position to apply the Liouville-type theorems for sub-Laplacians in Section 5.8 (page 269), Chapter 5. By the hypoellipticity of LA , from (16.17) it follows that Tk ∈ C ∞ (RN ) for every k = 1, . . . , 5. Let now k ∈ {1, . . . , 4} be fixed. We note that (Tk )2 is a non-negative function such that LA ((Tk )2 ) is a polynomial of H2 -degree5 zero (see (16.19)). As a consequence, by Liouville Theorem 5.8.4 (page 270), (Tk )2 is a polynomial function of H2 -degree at most 2 for every k = 1, . . . , 4. In particular, this proves that Tk is bounded from below by a polynomial function of H2 -degree at most 1. Hence, by Theorem 5.8.2, page 270 5 For the definition of G-degree, see Definition 1.3.3, page 33.

634

16 Families of Diffeomorphic Sub-Laplacians

(being LA Tk = 0), it follows that, for any k = 1, . . . , 4, Tk is a polynomial of H2 degree at most 1. Having supposed that T (0) = 0, there exists a constant matrix B = (bi,j )i,j ≤4 such that ⎛ ⎞ ⎛ ⎞ x1 T1 ⎜ x2 ⎟ ⎜ T2 ⎟ (16.20) ⎝ ⎠ = B · ⎝ ⎠. T3 x3 T4 x4 From (16.17) and (16.18b) we obtain 4 LA (T5 )2 = 2 T5 LA T5 + 2 ai,j Xi T5 Xj T5 = 8 (T12 + T22 + T32 + T42 ). i,j =1

From (16.20) we then infer that LA ((T5 )2 ) is a polynomial function of H2 -degree at most 2. Arguing as above, we derive that T5 is a polynomial function of H2 -degree at most 2. Then there exist c5 ∈ R, a 1×4 constant matrix D = (di )i≤4 and a symmetric 4 × 4 constant matrix C = (ci,j )i,j ≤4 such that (setting x = t (x1 , x2 , x3 , x4 )) T5 (x) = c5 x5 + D · x + t x · C · x. First, we show that the entries of D are zero. From (16.20) we obtain Xi Tl = ∂i Tl = bl,i for every l = 1, . . . , 4 and i = 1, . . . , 4. Hence, if we take 1 ≤ l ≤ 4 and 1 ≤ k ≤ 4 in the second equation of (16.11), then we derive (see also (16.16)) B · A · t B = I4

(16.21)

(In denoting henceforth the m × n identity matrix). In particular, we notice that B is non-singular. If in the second equation of (16.11) we take l ∈ {1, . . . , 4} and k = 5, we obtain ⎞ ⎛ ⎞ ⎛ 2 T3 X 1 T5 ⎜ X T ⎟ ⎜ 2 T4 ⎟ B ·A·⎝ 2 5⎠=⎝ ⎠. X 3 T5 −2 T1 X 4 T5 −2 T2 By evaluating both sides of this last identity at 0 (and recalling the form of T5 ), one has (Xi T5 )(0) = (∂i T5 )(0) = di , hence B · A · D = 0. Since A and B are non-singular, it follows D = 0. Consequently, T5 is a homogeneous polynomial of the form T5 (x) = c5 x5 + t x · C · x,

where t C = C,

x = t (x1 , x2 , x3 , x4 ). (16.22)

By writing (16.15) in a more compact form and making use of (16.20) and (16.22), one gets

16.2 Examples and Counter-examples

JTX =

B , 2 tx · tC

where we have set := C + c5 I C

635

and I :=

Thus, with a block notation, we infer B · A · tB X t X JT · A · JT = t · A · tB 2 x · tC

0 −I2

I2 0

.

· x 2B · A · C · A · C · x 4 tx · tC

.

Again from (16.20) and (16.16), we also recognize that 2 I· B · x I4 (X · t X) ◦ T = . 2 t x · t B · t I 4 t x · t B · t I· I· B · x Collecting together all the above facts, we have proved that (16.15) is equivalent to ⎧ t ⎪ ⎪ B · A · B = I4 , ⎪ ⎨ · x = 2 I· B · x, 2B · A · C t t ⎪2 x · C · A · t B = 2 t x · t B · t I, ⎪ ⎪ ⎩ t t · A · C · x = 4 t x · t B · t I· I· B · x. 4 x· C The third equation follows from the second one by transposing, whereas the fourth one follows from the first and the second equations. As a consequence, (16.15) turns out to be equivalent to B · A · t B = I4 , = 2 I· B, 2B · A · C or equivalently,

B · A · t B = I4 , = t B · I· B. C

(16.23)

We now explicitly observe that Iis skew-symmetric, then the same is true of t B · I·B. From the second equation in (16.23) we infer that the matrix ⎞ ⎛ c11 c12 c13 + c5 c14 c12 c22 c23 c24 + c5 ⎟ =⎜ C ⎠ ⎝ c13 − c5 c23 c33 c34 c14 c24 − c5 c34 c44 must be skew-symmetric. This is possible if and only if C = 0. Thus, (16.23) is equivalent to B · A · t B = I4 , (16.24) c5 I = t B · I· B. From the second equation of (16.24) it follows that c5 = 0 (thus, T is bijective!).

636

16 Families of Diffeomorphic Sub-Laplacians

Finally, we choose the symmetric positive-definite matrix A of the following form: A = t S · S, where S is a suitable non-singular matrix. From the first equation in (16.24) we then have (B · S) · t (B · S) = I4 , i.e. B = O · S −1 , with an orthogonal matrix O. In particular, the second equation of (16.24) becomes t

1 t S · I· S = O · I· O, c5

We choose

where O is an orthogonal matrix. ⎛

1 ⎜0 S=⎝ 0 0

0 1 0 0

0 0 1 0

Thus, (16.25) is equivalent to ⎛ ⎞ 0 0 1 0 1 t 0 0 1 ⎟ ⎜ 0 O · I· O, ⎝ ⎠= −1 0 0 −1 c5 0 −1 1 0

(16.25)

⎞ 1 0⎟ ⎠. 0 1

where O is an orthogonal matrix.

(16.26) In particular, the eigenvalues of these matrices should be equal. But the eigenvalues of c15 t O · I·O (being O orthogonal) are ± c15 i, whereas the eigenvalues of the matrix √ in the left-hand side of (16.26) are ±i 3±2 5 . This gives a contradiction, proving that, with the above choice of S and consequently of A, it does not exist any map T ∈ C 2 (H2 , H2 ) satisfying (16.10). This ends the proof of Theorem 16.2.3. Remark 16.2.5. In the proof of Theorem 16.2.3, we explicitly proved that the operator LA = i,j ai,j Xi Xj on H2 , where ⎛

⎞ ⎛ ⎞ 1 1 0 0 1 0⎟ ⎜0 1 0 0⎟ ⎠=⎝ ⎠, 0 0 0 1 0 1 1 0 0 2 2 cannot be turned into the canonical sub-Laplacian ΔH2 = i Xi by any C 2 -map T . However, by Proposition 16.1.1, we know that LA is (exactly as ΔH2 ) a subLaplacian on H2 . Since the symmetric square root B of A is ⎞ ⎛ 2 √ 0 0 √1 5 5 ⎜ 0 1 0 0 ⎟ ⎟ B=⎜ ⎝ 0 0 1 0 ⎠, √1 0 0 √3 1 ⎜0 t A= S·S =⎝ 0 1

0 1 0 0

0 0 1 0

⎞ ⎛ 0 1 0⎟ ⎜0 ⎠·⎝ 0 0 1 0

5

0 1 0 0

0 0 1 0

5

again from the result in Proposition 16.1.1 we know that LA is equal to the following sub-Laplacian

16.3 Canonical or Non-canonical?

LA =

m Yk2 ,

with Yk =

637

m (A1/2 )k,j Xj , j =1

k=1

i.e. 2 1 Y1 = √ X1 + √ X4 , 5 5

Y2 = X2 ,

Y3 = X3 ,

1 3 Y4 = √ X1 + √ X4 . 5 5

This proves that the following two sub-Laplacians on H2 cannot be turned into each other by any C 2 -map: ΔH2 = (∂1 + 2 x3 ∂5 )2 + (∂2 + 2 x4 ∂5 )2 + (∂3 − 2 x1 ∂5 )2 + (∂4 − 2 x2 ∂5 )2 , 2 2 1 4 2 LA = √ ∂1 + √ ∂4 + √ x3 − √ x2 ∂5 5 5 5 5 + (∂2 + 2 x4 ∂5 )2 + (∂3 − 2 x1 ∂5 )2 2 1 3 2 6 + √ ∂1 + √ ∂4 + √ x3 − √ x2 ∂5 5 5 5 5 2 2 2 2 = X1 + X2 + X3 + 2 X4 + X1 X4 + X4 X1 = (∂1 + 2 x3 ∂5 )2 + (∂2 + 2 x4 ∂5 )2 + (∂3 − 2 x1 ∂5 )2 + 2 (∂4 − 2 x2 ∂5 )2 + 2 (∂1 + 2 x3 ∂5 ) (∂4 − 2 x2 ∂5 ). However, another natural question still arises: is it possible to create a new Carnot group G on R5 , (different but) diffeomorphic to H2 , such that its canonical subLaplacian ΔG is turned into (i.e. equivalent to) the non-canonical LA ? We answer to this question in the next section.

16.3 Canonical or Non-canonical? In this section, we deal with the following problem. Let G = (RN , ◦, δλ ) be a homogeneous Carnot group with an assigned (not necessarily canonical) sub-Laplacian L. We know that there may not exist any C 2 -map on G itself turning L into the canonical sub-Laplacian ΔG of G (see, for instance, Theorem 16.2.3). We aim to show that there exists a different homogeneous Carnot group H = (RN , ∗, δλ ) (with the same underlying manifold RN and the same group of dilations δλ ) and a Lie-group isomorphism from G to H turning L into the canonical sub-Laplacian ΔH . Roughly speaking, even if we are given a non-canonical sub-Laplacian L, up to an isomorphism of Lie groups, we can consider L to be equivalent to a canonical sub-Laplacian, modulo a change of the Lie group. m 2 Let us denote by L = j =1 Yj a given sub-Laplacian on G and by ΔG = m 2 j =1 Xj the canonical sub-Laplacian on G. By definition, there exists a nonsingular matrix B = (bi,j )i,j ≤m

638

16 Families of Diffeomorphic Sub-Laplacians

such that Yj =

m

bi,j Xi

for every j = 1, . . . , m.

(16.27)

i=1

With reference to the particular form δλ (x) = δλ (x (1) , x (2) , . . . , x (r) ) = (λx (1) , λ2 x (1) , . . . , λr x (r) ),

x (i) ∈ RNi ,

of the dilations on G, we introduce a block-diagonal matrix ⎛ C (1) ⎜ 0 C=⎜ ⎝ .. . 0

0 C (2) .. . ···

··· .. . .. . 0

0 ⎞ .. . ⎟ ⎟, ⎠ 0 (r) C

(16.28)

where C (i) is a matrix of order Ni × Ni , for every i = 1, . . . , r, and we choose C (i) in such a way that C (1) = B −1

and C (2) , . . . , C (r) are non-singular.

(16.29)

In the sequel, we also denote by C the linear map C : RN → RN ,

x → y = C · x.

We now consider a new Lie group H = (RN , ∗), where ∗ is defined by H × H (x, y) → x ∗ y := C C −1 (x) ◦ C −1 (y) ∈ H.

(16.30)

We claim that: 1) C : (G, ◦) → (H, ∗) is a Lie group isomorphism; 2) H = (RN , ∗, δλ ) is a homogeneous Carnot group (this is not obvious, since we know that the homogeneity is not an invariant of equivalence classes of abstract Carnot groups); 3) C turns L on G into the canonical sub-Laplacian ΔH of H. Now, 1) is trivial (see also Example 2.1.48, page 112): for every ξ, η ∈ G we have, by (16.30), C(ξ ) ∗ C(η) = C C −1 (C(ξ )) ◦ C −1 (C(η)) = C(ξ ◦ η). We now prove 2). We must show that: 2’) the first m vector fields of the Jacobian basis of H are Lie-generators of the Lie algebra h of H; and 2”) δλ is a Lie group morphism of (H, ∗).

16.3 Canonical or Non-canonical?

639

First of all, we show that C turns Yj into the j -th vector field of the Jacobian basis of H.

(16.31)

We denote by H1 , . . . , HN the Jacobian basis of h, the Lie algebra of H. In order to prove (16.31), we have to show that C turns Yj into Hj for every j = 1, . . . , m. Now, from Example 2.1.48 (page 112) we know that the map C turns the vector

j on H such that (see (2.17a) in Examfield Yj into a ∗-left-invariant vector field Y ple 2.1.25, page 101)

j I (y) = JC (C −1 (y)) · (Yj I )(C −1 (y)). Y In particular, we have

j I (0) = JC (C −1 (0)) · (Yj I )(C −1 (0)) = C · (Yj I )(0) Y m bi,j Xi I (0) (see (16.27)) = C · i=1

= C · (b1,j , . . . , bm,j , 0, . . . , 0)T (1)

= (C (1) · bj , 0(2) , . . . , 0(r) ), (1)

where bj denotes the j -th column of B. By the choice (16.29) of C (1) , this proves

j I (0) is the j -th vector of the canonical basis of RN . Since the same is true of that Y Hj I (0) (by the definition of Jacobian basis) and the left-invariant vector fields are determined by their value at the origin, this proves (16.31).

j = dC(Yj ). Hence, the above From Theorem 2.1.50 (page 114) we know that Y computation proves that (16.32) dC(Yj ) = Hj . Since X1 , . . . , Xm is a system of Lie-generators for the Lie-algebra g of G and the Xj ’s are linear combinations of the Yj ’s, then every element of g is a linear combination of commutators of the type U = [U1 [U2 [U3 [· · · [Un−1 , Un ] · · ·]]]], with U1 , . . . , Un ∈ {Y1 , . . . , Ym }. Now, take any H ∈ h; since dC : g → h is a Lie-algebra isomorphism (see Theorem 2.1.50, page 114), then (dC)−1 (H ) ∈ g is a linear combination of brackets as the above U . Consequently, H is a linear combination of elements in h of the type (dC)(U ) = (dC)([U1 [U2 [U3 [· · · [Un−1 , Un ] · · ·]]]]) = [(dC)U1 [(dC)U2 [(dC)U3 [· · · [(dC)Un−1 , (dC)Un ] · · ·]]]] = [V1 [V2 [V3 [· · · [Vn−1 , Vn ] · · ·]]]],

640

16 Families of Diffeomorphic Sub-Laplacians

where V1 , . . . , Vn ∈ {(dC)Y1 , . . . , (dC)Ym } = {H1 , . . . , Hm } (see (16.32)). This proves (2’). In order to prove (2”), we first remark that δλ commutes with C, i.e. δλ (C(x)) = C(δλ (x))

∀ x ∈ RN , ∀ λ > 0.

(16.33)

Indeed, following our usual notation, we have ⎛ C (1)

0

··· .. . .. .

0 ⎞ ⎛ λ x (1) ⎞ ⎛ C (1) λ x (1) ⎞ .. ⎜ λ2 x (2) ⎟ ⎜ C (2) λ2 x (2) ⎟ . ⎟ ⎟·⎜ . ⎟=⎜ ⎟ .. ⎠ ⎝ .. ⎠ ⎝ ⎠ . 0 r (r) (r) r (r) (r) λ x C λ x C

⎜ 0 C (2) C(δλ (x)) = ⎜ . .. ⎝ . . . 0 ··· 0 ⎛ (1) (1) ⎞ C x ⎜ C (2) x (2) ⎟ ⎟ = δλ (C(x)). = δλ ⎜ .. ⎝ ⎠ . C (r) x (r)

Obviously, (16.33) implies that δλ commutes with C −1 too. Now, (2”) is equivalent to δλ (x ∗ y) = (δλ (x)) ∗ (δλ (y)) ∀ x, y ∈ H, ∀ λ > 0, which follows from the simple argument below (δλ (x)) ∗ (δλ (y)) = (see (16.30)) C C −1 (δλ (x)) ◦ C −1 (δλ (y)) = (see (16.33)) C (δλ (C −1 (x))) ◦ (δλ (C −1 (y))) (δλ is a Lie group morphism of (G, ◦)) = C δλ (C −1 (x)) ◦ (C −1 (y)) = (see (16.33)) δλ C (C −1 (x)) ◦ (C −1 (y)) = (see (16.30)) δλ (x ∗ y). Finally, (3) is a straightforward consequence of (16.31). In the next subsection, we shall see an explicit example of non-canonical subLaplacian turning into a canonical one. Remark 16.3.1. The above computations also prove the following fact. Let (RN , ◦, δλ ) be a homogeneous Carnot group. As usual, we denote the points of G by x = (x (1) , . . . , x (r) )

with x (i) ∈ RNi

and the dilation group by δλ (x) = (λx (1) , . . . , λr x (r) ). Let C (1) , . . . , C (r) be fixed non-singular matrices, with C (i) of dimension Ni × Ni for every i = 1, . . . , r. We denote by C the matrix as in (16.28). Then C defines a linear change of coordinates on RN . The expression of ◦ in the new coordinates defined by ξ = C(x)

16.3 Canonical or Non-canonical?

641

is precisely the composition ∗ defined in (16.30). Then, H = (RN , ∗, δλ ) is a homogeneous Carnot group isomorphic to G = (RN , ◦, δλ ), and C turns a left-invariant vector field X on (G, ◦) into the left-invariant vector field dC(X) on H = (RN , ∗); in particular, dC(X) coincides with C · XI (0) at the origin, i.e. dC(X) I (0) = C · XI (0). (16.34) This last fact can be proved as follows. From Example 2.1.48 (page 112) we have dC(X) I (y) = JC (C −1 (y)) · (XI )(C −1 (y)) = C · (XI )(C −1 (y)). When y = 0, we have (16.34). 16.3.1 Example of a Non-canonical Sub-Laplacian Turning into a Canonical One In Theorem 16.2.3, we proved that the following sub-Laplacian on H2 2 2 1 4 2 L = √ ∂1 + √ ∂4 + √ x3 − √ x2 ∂5 5 5 5 5 + (∂2 + 2 x4 ∂5 )2 + (∂3 − 2 x1 ∂5 )2 2 1 2 3 6 + √ ∂1 + √ ∂4 + √ x3 − √ x2 ∂5 5 5 5 5 cannot be turned into the canonical sub-Laplacian ΔH2 of H2 itself by any C 2 -map. Nonetheless, we have just showed a constructive method which allows us to turn the above L into the canonical sub-Laplacian on another homogeneous Carnot group. Indeed (following the notation in Section 16.3), we have G = (R5 , ◦, δλ ), where ⎞ ⎛ x 1 + y1 x2 + y2 ⎟ ⎜ ⎟ ⎜ x◦y =⎜ x3 + y3 ⎟ ⎠ ⎝ x4 + y4 x5 + y5 + 2(x3 y1 + x4 y2 − x1 y3 − x2 y4 ) and δλ (x1 , x2 , x3 , x4 , x5 ) = (λx1 , λx2 , λx3 , λx4 , λ2 x5 ). Moreover, L=

4 Yj2 ,

where Yj =

j =1

with

4

bi,j Xi

i=1

⎛

√2 5

⎜ 0 B = (bi,j )i,j ≤4 = ⎜ ⎝ 0

√1 5

0 1 0 0

0 0 1 0

√1 5

⎞

0 ⎟ ⎟. 0 ⎠

√3 5

642

16 Families of Diffeomorphic Sub-Laplacians

More explicitly, since X1 = ∂1 + 2 x3 ∂5 , X3 = ∂3 − 2 x1 ∂5 ,

X2 = ∂2 + 2 x4 ∂5 , X4 = ∂4 − 2 x2 ∂5 ,

the Yj ’s are given by 2 1 Y1 = √ X1 + √ X4 , 5 5

Y2 = X2 , ⎛

We have

Y3 = X3 , √3 5

⎞ 0 − √1 5 0 0 ⎟ ⎟. 1 0 ⎠ √2 0

0 1 0 0

⎜ 0 C (1) = B −1 = ⎜ ⎝ 0 − √1

5

1 3 Y4 = √ X1 + √ X4 . 5 5

5

Consequently, we define ⎛ C :=

C (1) 0

0 1

√3 5

⎜ 0 ⎜ =⎜ ⎜ 01 ⎝ −√

5

0

0 1 0 0 0

0 − √1 5 0 0 1 0 √2 0 5 0 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎠ 1

and the relevant linear map ⎞ x1 − √1 x4 5 x2 ⎟ ⎜ ⎟ ⎜ x→ y =Cx =⎜ x3 ⎟. ⎠ ⎝ √1 2 − x1 + √ x4 5 5 x5 ⎛

C : R5 → R5 ,

√3 5

Following (16.30), we equip R5 with a new Lie group law ∗ defined by y ∗ η := C C −1 (y) ◦ C −1 (η) = y 1 + η 1 , y 2 + η 2 , y 3 + η3 , y 4 + η4 , 2 y5 + η5 + √ y3 (2η1 + η4 ) + (y1 + 3y4 )η2 − (2y1 + y4 )η3 5 − y2 (η1 + 3η4 ) . We explicitly remark that this Lie group composition law can be written as in (3.6) (page 158) of Section 3.2, i.e. y ∗ η = (y1 + η1 , y2 + η2 , y3 + η3 , y4 + η4 , y5 + η5 + H (y1 , y2 , y3 , y4 ), (η1 , η2 , η3 , η4 )),

16.3 Canonical or Non-canonical?

where

⎛

0 1 ⎜ −1 H =√ ⎝ 2 5 0

1 0 0 3

643

⎞ −2 0 0 −3 ⎟ ⎠. 0 1 −1 0

Finally, we observe that the first four vector fields of the Jacobian basis of (R5 , ∗) are given by6 4 2 H1 = ∂y1 + √ y3 − √ y2 ∂y5 , 5 5 2 6 H2 = ∂y2 + √ y1 + √ y4 ∂y5 , 5 5 4 2 H3 = ∂y3 + − √ y1 − √ y4 ∂y5 , 5 5 2 6 H4 = ∂y4 + √ y3 − √ y2 ∂y5 . 5 5 We obviously recognize that the Hj ’s are the expression of the Yj ’s w.r.t. the new system of coordinates y = C x, i.e. (Hj f )(C(x)) = Yj (f ◦ C)(x)

∀ f ∈ C ∞ (RN , R), ∀ x ∈ RN .

As a consequence, the sub-Laplacian L on H2 is turned by C into the canonical sub-Laplacian of H∗ = (R5 , ∗) ΔH∗ =

4

Hj2

j =1

2 4 2 = ∂y1 + √ y3 − √ y2 ∂y5 5 5 2 2 6 + ∂y2 + √ y1 + √ y4 ∂y5 5 5 2 4 2 + ∂y3 + − √ y1 − √ y4 ∂y5 5 5 2 2 6 + ∂y4 + √ y3 − √ y2 ∂y5 . 5 5 6 Here, we use the formula

(Hj ϕ)(y) = (∂/∂ηj )|η=0 ϕ(y ◦ η).

644

16 Families of Diffeomorphic Sub-Laplacians

16.4 Further Reading: An Example of Application to PDE’s We close the chapter, by giving an example of application to PDE’s of the topics treated in Section 16.1. To this end, let us introduce the parabolic-type operator in non-divergence form ai,j (x) Xi Xj − ∂t , (16.35) i,j

being {Xi }i the generators of the Lie algebra of a Carnot group G and ai,j Hölder continuous functions. The fundamental solution for (16.35) can be constructed via the Levi-parametrix method7 , provided suitable uniform estimates of the fundamental solutions for the frozen operators are established (see [BLU03]). By the frozen operators, we mean the constant coefficient operators HA = LA − ∂t =

m

ai,j Xi Xj − ∂t ,

A = (ai,j )i,j ∈ MΛ ,

i,j =1

where MΛ is the set of m × m symmetric matrices A such that Λ−1 |ξ |2 ≤ Aξ, ξ ≤ Λ|ξ |2

∀ξ ∈ Rm

(Λ ≥ 1 being a fixed constant). The Levi-parametrix method for the operator (16.35) requires the following Gaussian uniform estimates Xi · · · Xi (∂t )q ΓA (x, t) − Xi · · · Xi (∂t )q ΓB (x, t) p p 1 1 2 dG (x) 1/r −(Q+p+2q)/2 ≤ cΛ,p,q A − B t exp − cΛ t (16.36) for every x ∈ G, t > 0, A, B ∈ MΛ , where we have denoted by ΓA the fundamental solution for HA and by dG a fixed symmetric homogeneous norm8 on G. A natural question in approaching (16.36) (whose proof can be found in [BLU02]) is to ask whether the sub-Laplacians LA ’s are all diffeomorphic to the canonical operator ΔG , via a change of variables. This naive idea stands at the basis of the Leviparametrix method in the classical case. As we showed in Section 16.1, in the general case of a Carnot group, this problem is not trivial. Theorem 16.1.2 allows us to obtain the fundamental solution ΓA for HA as the composition of TA with the fundamental solution ΓG for HG , where HG = ΔG − ∂t . 7 See, e.g. [Kal92] for a reference to the Levi-parametrix method in the classical case. 8 I.e. a continuous function d : RN → [0, ∞[, smooth away from the origin, such that G dG (δλ (x)) = λ dG (x), dG (x −1 ) = dG (x) and dG (x) = 0 iff x = 0; see Section 5.1,

page 229.

16.5 Exercises of Chapter 16

645

Indeed, if G is free, it turns out that ΓA (x, t; ξ, τ ) = | det JTA (x)| ΓG (TA (x), t; TA (ξ ), τ )

(16.37)

for every x, ξ ∈ RN and t, τ ∈ R. A crucial step in order to obtain the uniform estimates in (16.36) is then to establish ad hoc uniform estimates for TA (these can be found in [BU04b, Theorem 2.7]). In order to handle the case of an arbitrary Carnot group G, a possible approach

in such a way that is to lift G to a free Carnot group G ΔG is lifted to ΔG

(the lifting technique is treated in Chapter 17). Indeed, the following result holds (see Theorem 17.1.5).

on RH (with H ≥ N) such There exists a free homogeneous Carnot group G H N that, denoting by π : R → R the projection on the first N coordinates, we have

i (u ◦ π) = (Xi u) ◦ π ∀ u ∈ C ∞ (RN ), X m 2

2 where m

, rei=1 Xi and i=1 Xi are the canonical sub-Laplacians ΔG and ΔG spectively. This result, together with Theorem 16.1.2, allows to prove (16.36) in the general case. We briefly describe how (we refer to [BLU02] for the complete proofs). A family of sub-Laplacians {LA } on an arbitrary Carnot group G is lifted to a

The related family {Γ A } of fundamental

A } on a free G. family of sub-Laplacians {L solutions fulfills identity (16.37). By means of the uniform estimates of TA , this allows to derive (16.36) for {Γ A }. Finally, the fundamental solutions ΓA are explicitly represented by integrating Γ A over the added variables. As a consequence, estimate (16.36) can be obtained on G. Bibliographical Notes. For the details of the topics presented in Section 16.4, we refer the reader to [BLU02,BLU03]. Some of the topics presented in this chapter also appear in [BU04b].

16.5 Exercises of Chapter 16 Ex. 1) Prove that RN equipped with the ordinary Abelian structure is a free Carnot group. Then prove that the Heisenberg group H1 is also a free Carnot group. Ex. 2) Consider the Heisenberg group H1 . Let B be a 2 × 2 symmetric non-singular matrix. Arguing as in Theorem 16.1.2, write down explicitly a diffeomorphism T which turns the sub-Laplacian LB −2 into ΔH1 . Recognize that T so constructed turns out to be linear. Compare with Remark 16.2.1. Ex. 3) Consider the following sub-Laplacian on the Heisenberg group H2 L = X12 + X22 + X42 + (X2 + X3 )2 ,

646

16 Families of Diffeomorphic Sub-Laplacians

where 4j =1 Xj2 is the canonical sub-Laplacian of H2 . Following the arguments of Section 16.3, find a new group law ∗ on R5 such that H∗ := (R5 , ∗) is a homogeneous Carnot group isomorphic to H2 , via a Lie group isomorphism which turns L into the canonical sub-Laplacian ΔH∗ of H∗ . Write down explicitly ΔH∗ . Ex. 4) Let (G, ◦) be a free homogeneous Carnot group of step r, and let dG be a symmetric homogeneous norm on G. Given a fixed constant Λ ≥ 1, let MΛ denote the set of symmetric m × m constant matrices A such that Λ−1 |ξ |2 ≤ Aξ, ξ ≤ Λ|ξ |2

∀ ξ ∈ Rm .

In what follows we shall denote by cΛ any positive constant depending only on Λ and the structure of G. Finally, A will stand for the matrix norm max|ξ |=1 |Aξ |. Consider the diffeomorphism TA constructed in Theorem 16.1.2 and set, for x ∈ G, JA (x) = | det JTA (x)|. Prove that the following properties hold for any A, A1 , A2 ∈ MΛ and x ∈ G: a) JA is constant in x, b) (cΛ )−1 ≤ JA ≤ cΛ , c) |JA1 − JA2 | ≤ cΛ A1 − A2 , d) (cΛ )−1 dG (x) ≤ dG (TA (x)) ≤ cΛ dG (x), e) dG ((TA2 (x))−1 ◦ TA1 (x)) ≤ cΛ A1 − A2 1/r dG (x). Hint: First prove Ex. 5 below. Then recall that TA is defined in (16.9), where ϕ = ϕA is a linear map represented in Jacobian coordinates by a blockdiagonal matrix whose entries are polynomials in the entries of A−1/2 . Use the following facts det JExp = det JLog ≡ 1 to prove (a). Then use (b) of Ex. 5 to prove (b). Now, observing that JA is in the form JA = |Ψ (A−1/2 )|, being Ψ a polynomial function in the entries of A−1/2 , estimate |JA1 − JA2 | by the mean value theorem. Use then (b) and (c) of Exercise 5 to prove (c). Exploiting the fact that TA commutes with the dilations of G (see Theorem 16.1.2) and that dG is δλ -homogeneous of degree 1, reduce the proofs of (d) and (e) to the case x ∈ SG := {ξ ∈ G | dG (ξ ) = 1}. Then prove that the map MΛ ×SG (A, ξ ) → TA (ξ ) is continuous, observing that TA (ξ ) is a polynomial function both in ξ and in the entries of A−1/2 , and using (c) of Ex. 5. Observe also that dG (TA (ξ )) > 0 for all ξ ∈ SG , and then prove (d). Finally, apply (5.5) to dG and K := {TA (ξ )| A ∈ MΛ , ξ ∈ SG } in order to obtain (e).

16.5 Exercises of Chapter 16

647

Ex. 5) With the notation of Ex. 4, prove that: a) If A ∈ MΛ , then A−1 ∈ MΛ ; b) If A ∈ MΛ , then Λ−1 ≤ A, A−1 ≤ Λ and Λ−1/2 ≤ A1/2 , A−1/2 ≤ Λ1/2 ; c) For every A, B ∈ MΛ , we have A1/2 − B 1/2 , A−1 − B −1 , A−1/2 − B −1/2 ≤ cΛ A − B. Ex. 6) In the setting and with the notation of Ex. 4 (and using the very results of Ex. 4, besides Theorem 16.1.2), prove the following facts: a) For every A ∈ MΛ , we have ΓA (x) = JA · ΓG (TA (x)),

x ∈ RN ,

where ΓG and ΓA denote, respectively, the fundamental solutions for the canonical sub-Laplacian ΔG =

m

Xk2

k=1

and for the sub-Laplacian LA =

m

ai,j Xi Xj ;

i,j =1

b) The fundamental solution ΓA satisfy the following uniform estimates 2−Q c−1 ≤ ΓA (x) ≤ cΛ (dG (x))2−Q , Λ (dG (x))

x ∈ RN \ {0},

for every A ∈ MΛ (as usual, Q denotes the homogeneous dimension of G); c) The derivatives of the fundamental solution ΓA satisfy the following uniform estimates Xi · · · Xi ΓA (x) ≤ cΛ,p (dG (x))2−Q−p , x ∈ RN \ {0}, p 1 for every A ∈ MΛ and for every i1 , . . . , ip ∈ {1, . . . , m}.

17 Lifting of Carnot Groups

As we discussed in the Preface of this book, L.P. Rothschild and E.M. Stein [RS76] obtained sharp regularity results for the sum of squares of Hörmander vector fields m 2 by using analysis on nilpotent Lie groups. A crucial step in [RS76] is the X j =1 j 1 , . . . , X m (on a larger manifold) which “lift” construction of new vector fields X the Xj ’s and which can be locally approximated by left-invariant vector fields on a stratified group. The aim of this chapter is to study the lifting procedure in the special case when the Xj ’s generate the Lie algebra of a Carnot group G. In this case, we prove that G which preserves the homogeneous structure can be directly lifted to a free group G of G, besides being itself a homogeneous Carnot group. We also give an example of application of the lifting technique to PDE’s. Indeed, in Section 17.3 we write an explicit formula for the fundamental solutions for all the sub-Laplacians on Carnot groups of step two. This formula is given in terms only of the fundamental solution for the canonical sub-Laplacian on a fixed free Carnot group.

17.1 Lifting to Free Carnot Groups We first recall the basic notation for Carnot groups that we are going to use throughout the chapter. We denote, as usual, by G = (RN , ◦, δλ ) a homogeneous Carnot group. The dilations on G have the usual form δλ (x) = δλ (x (1) , . . . , x (r) ) = (λx (1) , . . . , λr x (r) ).

(17.1)

Here x (i) ∈ RNi for i = 1, . . . , r and N1 + · · · + Nr = N . We denote by g the Lie algebra of G. For i = 1, . . . , N , let Zi be the vector field in g that agrees at the origin with ∂/∂xi , i.e. the Zi ’s form the Jacobian basis for g. We know that the crucial hypothesis on G is that the Lie algebra generated by Z1 , . . . , ZN1 is the whole g. We also say that G is of step r and has m := N1 generators. Let us also explicitly recall that g is thus an N -dimensional nilpotent Lie

650

17 Lifting of Carnot Groups (i)

algebra of step r generated by Z1 , . . . , Zm . Moreover, if Zj ∈ g agrees at the origin (i)

(i)

with ∂/∂xj , then Zj is δλ -homogeneous of degree i (see Remark 1.4.5, page 58). As usual, we denote by Q = rj =1 j Nj the homogeneous dimension of G. The canonical sub-Laplacian on G is the second order differential operator ΔG =

m Zi2 . i=1

If Y1 , . . . , Ym is any basis for span{Z1 , . . . , Zm }, the second order differential operator m L= Yi2 i=1

is called a sub-Laplacian on G. We now recall the definition of fm,r , the free nilpotent Lie algebra of step r with m (≥ 2) generators x1 , . . . , xm (see also Chapter 14). By definition, fm,r is the unique (up to isomorphism) nilpotent Lie algebra of step r generated by m of its elements F1 , . . . , Fm such that, for every nilpotent Lie algebra n of step r and for every map ϕ from fm,r ϕ from {F1 , . . . , Fm } to n, there exists a (unique) Lie algebra morphism to n extending ϕ. We say that the Carnot group G is a free Carnot group if its Lie algebra g is isomorphic to fm,r for some m and r. Following the above notation, by the definition of fm,r , there exists a unique Lie algebra morphism Π from fm,r to g such that Π(Fi ) = Zi

for every i = 1, . . . , m.

Clearly, Π is surjective, whence dim fm,r = dim ker(Π) + dim g. We set H = dim fm,r . The following result will be relevant in the sequel. Proposition 17.1.1 (A lifting basis F ). There exists a basis F = {F1 , . . . , FH } for fm,r such that

Π(Fj ) = Zj Π(Fj ) = 0

for every j = 1, . . . , N, for every j = N + 1, . . . , H ,

(17.2)

and each Fj (j = 1, . . . , H ) is a homogeneous Lie polynomial in F1 , . . . , Fm . Proof. Let αj denote the δλ -homogeneity degree of Zj . In particular, α1 = · · · = αm = 1. For a multi-index I = (i1 , . . . , ik ) with i1 , . . . , ik ∈ {1, . . . , m}, we set |I | = k (the height of I ) and define

17.1 Lifting to Free Carnot Groups

651

ZI := [Zi1 , [Zi2 . . . [Zik−1 , Zik ] . . . ]]. Then, by simple homogeneity arguments, we have (see Proposition 1.3.9, page 36, and Proposition 1.1.7, page 12) (j ) (j ) cI ZI , cI ∈ R, for every j = 1, . . . , N , Zj = I ∈Ij

where Ij is a set of multi-indices all with height αj . If we analogously set FI := [Fi1 , [Fi2 . . . [Fik−1 , Fik ] . . . ]], the first N elements of the basis F can be chosen as (j ) Fj = c I FI , j = 1, . . . , N. I ∈Ij

Indeed Π(Fj ) = Zj and F1 , . . . , FN are linearly independent homogeneous Lie N +1 , . . . , F H be a basis for polynomials in the generators F1 , . . . , Fm . Let now F ker(Π). We can write (j ) j = qI FI , j = N + 1, . . . , H, F I ∈ Aj (j )

for a certain set of multi-indices Aj and scalars qI ’s. For every k = 1, . . . , r, we (k) set Aj := {I ∈ Aj : |I | = k }. We evidently have (1)

(r)

Aj = Aj ∪ · · · ∪ Aj . Then

j = F

(j )

q I FI + · · · +

(1)

I ∈ Aj

(j ) (1) + · · · + F (r) . qI FI =: F j j

(r)

I ∈ Aj

The system of vectors := {F (k) : j = N + 1, . . . , H, k = 1, . . . , r} F j has the following properties: 1) each of its vectors is a homogeneous Lie polynomial in the generators F1 , . . . , F m ; 2) the system spans ker(Π); (k) ∈ ker(Π). 3) for every j = N + 1, . . . , H and k = 1, . . . , r, we have F j Let us prove this last assertion: for j = N + 1, . . . , H , we have j ) = Π(F ) + · · · + Π(F ). 0 = Π(F j j (1)

(r)

652

17 Lifting of Carnot Groups

) is a δλ -homogeneous vector field of degree k, unless it vanWe notice that Π(F j (k) ) must ishes. Consequently, since the cited δλ -degrees are all different, each Π(F j vanish identically (see Proposition 1.3.9, page 36). a Finally, the proposition is proved by extracting from the system of vectors F basis FN +1 , . . . , FH for ker(Π). (k)

In the sequel, we shall consider some abstract finite-dimensional algebras; we hence introduce a useful notation, which will allow us to deal with ordinary Rn spaces. Let h be a finite-dimensional nilpotent real Lie algebra. It is possible to prove (see Theorem 14.2.3, page 586) that h is equipped with a Lie group structure by the so-called Campbell–Hausdorff composition law defined by (14.2), page 585. For X, Y ∈ h, the first few terms in the sum (14.2) (which is a finite sum since h is nilpotent) are given by 1 1 1 [X, Y ] + [X, [X, Y ]] − [Y, [X, Y ]] + · · · . 2 12 12 We explicitly remark that is defined in a universal way (independent of h) as a Lie polynomial in X and Y . We now fix a basis XY =X+Y +

E = (E1 , . . . , EN ) for h (N := dim h), and we identify h with RN via the map N

πE : h → RN ,

ξi Ei → (ξ1 , . . . , ξN ).

i=1

The group law is then turned into a group law E on RN in the natural way a E b := πE πE−1 (a) πE−1 (b) , a, b ∈ RN . In other words, if a, b ∈ RN , then a E b is the only element of RN such that

N

N N ai E i bi Ei =: (a E b)i Ei . i=1

i=1

i=1

(RN ,

1 The Lie groups E ) and (h, ) are clearly isomorphic via πE . We stress that the Lie group morphism πE is also a linear map. 1 It is evident that the particular form of the operation on RN does depend on the choice E of the basis E on h. However, if E1 and E2 are two different bases of h, obviously the two Lie groups (RN , ) and (RN , ) are canonically Lie-isomorphic via the natural linear

E1

change of basis from E1 to E2 :

E2

(h, )

(h, )

πE1

(RN , E1 )

πE2 change of basis

(RN , E2 ).

17.1 Lifting to Free Carnot Groups

653

With the above notation, we now consider the Lie groups (RN , Z ) and F ), where Z is the Jacobian basis of g, whereas F is the basis of fm,r introduced in Proposition 17.1.1. First of all, we remark that the map (RH ,

−1 π := πZ ◦ Π ◦ πF : (RH , F ) → (RN , Z )

is a surjective Lie group morphism coinciding with the usual projection of RH onto RN RH (ξ1 , . . . , ξN , ξN +1 , . . . , ξH ) → (ξ1 , . . . , ξN ) ∈ RN . The following diagram may be considered Π

(fm,r , )

(g, )

−1 πF

(RH , F )

πZ canonical projection

(RN , Z ).

Indeed, if ξ ∈ RH , we have −1 (πZ ◦ Π ◦ πF )(ξ ) = (πZ ◦ Π)

= πZ (see (17.2)) = πZ

H

ξi Fi

i=1 H

i=1 N

ξi Π(Fi )

ξi Zi

= (ξ1 , . . . , ξN ).

i=1

We have thus proved a first kind of lifting result (we notice that the Lie algebra of vector fields g may be replaced by any finite-dimensional nilpotent Lie algebra h generated by m of its elements). Proposition 17.1.2 (Lifting via a lifting basis F ). Let h be a finite-dimensional Lie algebra generated by m of its elements, nilpotent of step r. Let N := dim (h). Let E be a fixed arbitrary linear basis of h. Then there exists a basis F of fm,r such that the projection −→ (RN , E ), π: (RH , F ) (ξ1 , . . . , ξN , ξN +1 , . . . , ξH ) → (ξ1 , . . . , ξN ) is a surjective Lie group homomorphism. We now turn to the Lie algebras of the Lie groups (RN , Z ) and (RH , F ) considered above: let rN and rH denote these Lie algebras, respectively. Since π is a Lie group morphism, then its differential dπ : rH → rN

654

17 Lifting of Carnot Groups

is a Lie algebra morphism (see Theorem 2.1.50, page 114) with the following property ∀ E ∈ rH , ∀ ξ ∈ RH . (17.3) dπ(E) π(ξ ) = dπ(Eξ ) Roughly speaking, the differential of a Lie group morphism which coincides with a projection gives a lifting of vector fields. More precisely, we have the following lemma. Lemma 17.1.3. With the above notation, if E ∈ rH , then E is a lifting of dπ(E) in the following sense: if f ∈ C ∞ (RH ) depends only on ξ1 , . . . , ξN , i.e. there exists g ∈ C ∞ (RN ) such that f (ξ1 , . . . , ξH ) = g(ξ1 , . . . , ξN ) for every ξ ∈ RH , then we have Ef (ξ1 , . . . , ξH ) = dπ(E)g (ξ1 , . . . , ξN )

∀ ξ ∈ RH .

Proof. Let g ∈ C ∞ (RN ) and f ∈ C ∞ (RH ) be such that f = g ◦ π. We have to prove Eξ (f ) = dπ(E) π(ξ ) (g) for every ξ ∈ RH . From (17.3) we immediately obtain dπ(E) π(ξ ) (g) = dπ(Eξ ) (g) = Eξ (g ◦ π) = Eξ (f ). This ends the proof.

The rest of the lifting method consists in the transferring of this result to the group G, after a suitable definition of the larger group which projects onto G. We refer the reader to the following diagram of Lie group morphisms: Π

Φ

(fm,r , ) −→ πF π

(g, ) πZ

Exp

←→ (G, ◦)

(G × RH −N , •) ←→ (RH , F ) −→ (RN , Z ). We recall that (RN , Z ) is isomorphic to (g, ) via πZ . On the other hand, the exponential map Exp : (g, ) → (G, ◦) is a Lie group isomorphism (see Theorem 2.2.13, page 129). As a consequence, the map −1 Ψ := Exp ◦ πZ is a Lie group isomorphism from (RN , Z ) to (G, ◦). We then look for a suitable group structure on G × RH −N and a Lie group isomorphism Φ : (RH , F ) → (G × RH −N , •)

17.1 Lifting to Free Carnot Groups

such that

655

−1 ◦ π ◦ Φ −1 ϑ := Ψ ◦ π ◦ Φ −1 = Exp ◦ πZ

is the projection of G × RH −N onto G. To this end, we set Φ(ξ1 , . . . , ξH ) := (Ψ (ξ1 , . . . , ξN ), ξN +1 , . . . , ξH ). Clearly, Φ is an invertible map of class C ∞ and the same is true of its inverse map. We then define on G × RH −N the composition law “induced” by Φ (g1 , a1 ) • (g2 , a2 ) := Φ Φ −1 (g1 , a1 ) F Φ −1 (g2 , a2 ) , so that Φ becomes a Lie group isomorphism between the groups (RH , F ) and (G× RH −N , •). Finally, the Lie group morphism ϑ := Ψ ◦ π ◦ Φ −1 : (G × RH −N , •) → (G, ◦) is, by construction, the natural projection. Indeed, ϑ(x1 , . . . , xH ) = (Ψ ◦ π)(Ψ −1 (x1 , . . . , xN ), xN +1 , . . . , xH ) = Ψ (Ψ −1 (x1 , . . . , xN )) = (x1 , . . . , xN ). We denote by g rH −N the Lie algebra of (G × RH −N , •). The proof of the following result is simply a restatement of the proof of Lemma 17.1.3. Lemma 17.1.4. If W ∈ g rH −N , then W is a lifting of dϑ(W ) in the following sense: if f ∈ C ∞ (G × RH −N ) depends only on x1 , . . . , xN , i.e. there exists g ∈ C ∞ (G) such that f (x1 , . . . , xH ) = g(x1 , . . . , xN )

for every x ∈ G × RH −N ,

then we have Wf (x1 , . . . , xH ) = dϑ(W )g (x1 , . . . , xN )

∀ x ∈ G × RH −N .

Proof. Let g ∈ C ∞ (G) and f ∈ C ∞ (G × RH −N ) be such that f = g ◦ ϑ, i.e. f (x1 , . . . , xH ) = g(x1 , . . . , xN )

∀ x ∈ G × RH −N .

We have to prove that Wx (f ) = (dϑ(W ))ϑ(x) (g)

∀ x ∈ G × RH −N .

To this end, it suffices to notice that (dϑ(W ))ϑ(x) (g) = (dϑ(Wx ))(g) = Wx (g ◦ ϑ) = Wx (f ). This ends the proof.

656

17 Lifting of Carnot Groups

We claim that the first N vector fields of the Jacobian basis W1 , . . . , WH for g rH −N lift orderly the Jacobian basis Z1 , . . . , ZN for g. Indeed, by Lemma 17.1.4, it suffices to show that for every k = 1, . . . , N . dϑ(Wk ) = Zk To this end, we remark that, for all f ∈ C ∞ (G), we have dϑ(Wk ) 0 (f ) = (Wk )0 (f ◦ ϑ) = (∂xk f )(0) = Zk 0 f, which proves the assertion, since two left-invariant vector fields are equal if and only if they coincide at the origin. We are now in the position to prove the main result of this section. Theorem 17.1.5 (Lifting). Let G be a homogeneous Carnot group on RN of step r and m (= N1 ) generators. Then there exists a free homogeneous Carnot group G on RH (H = dim fm,r ) with the properties (i) and (ii) stated below. We fix the following notation: δλ (x) = δλ (x (1) , x (2) , . . . , x (r) ) = (λx (1) , λ2 x (2) , . . . , λr x (r) ) and

δλ ( x) = δλ ( x (1) , x (2) , . . . , x (r) ) = (λ x (1) , λ2 x (2) , . . . , λr x (r) )

respectively, with the usual notation denote the dilations on G and G, x (i) ∈ RNi ,

i = 1, . . . , r,

N1 + · · · + Nr = N,

i = 1, . . . , r,

1 + · · · + N r = H ; N

(i)

i = 1, . . . , r,

j = 1, . . . , Ni ,

and analogously x (i) ∈ RNi , the vector fields Zj ,

denote the Jacobian basis of the Lie algebra g of G, and analogously (i) , Z j

i = 1, . . . , r,

i , j = 1, . . . , N

Then: denote the Jacobian basis of the Lie algebra g of G. has step r and m generators, and its Lie algebra is isomorphic to fm,r . (i) G (ii) For a certain i0 ∈ {1, . . . , r}, we have i = Ni , N

i = 1, . . . , i0

i > Ni , and N

i = i0 + 1, . . . , r;

R Ni

→ RNi denotes the projection on the first Ni coordinates moreover, if (i) : H N and : R → R is defined by x (1) ), . . . , (r) ( x (r) )), ( x ) = ((1) ( then

(i) (u ◦ ) = Z (i) u ◦ Z j j

∀ u ∈ C ∞ (RN ),

i ≤ r, j ≤ Ni ,

lifts Z . Moreover, is a Lie group morphism. i.e. Z j j (i)

(i)

(17.4)

17.1 Lifting to Free Carnot Groups

657

Proof. Let (G × RH −N , •) be the previously defined Lie group on RH . We show can be constructed from G × RH −N by a permutation of the coordinates. that G Let F = (F1 , . . . , FH ) be the basis for fm,r as in Proposition 17.1.1. Then Fj is a homogeneous Lie polynomial of degree αj in the generators F1 , . . . , Fm . We stress that, by Proposition 17.1.1, αj is also the δλ -homogeneity degree of Zj for j = 1, . . . , N , i.e. the dilation δλ on RN can be written as δλ (x1 , . . . , xN ) = (λα1 x1 , . . . , λαN xN ). We also observe that only F1 , . . . , Fm have degree 1. We define the dilations on fm,r as follows H

H δλ ξi Fi := λαi ξi Fi . i=1

i=1

First, we prove that δλ is a Lie algebra automorphism of fm,r . Indeed, for all i, j ∈ {1, . . . , H }, one has δλ [Fi , Fj ] = λαi +αj [Fi , Fj ] = [λαi Fi , λαj Fj ] = [ δλ (Fi ), δλ (Fj )]. The first equality holds since [Fi , Fj ] is a homogeneous Lie polynomial in F1 , δλ is also a Lie group automorphism of (fm,r , ) by . . . , Fm of degree αi + αj . Then Remark 2.2.16, page 130. As a consequence, the map −1 δλ∗ := πF ◦ δλ ◦ πF

is a Lie group automorphism of (RH , F ). Analogously, the map δλ := Φ ◦ δλ∗ ◦ Φ −1 is a Lie group automorphism of (G × RH −N , •). If x ∈ G × RH −N , we have δλ (x) = ((Ψ ◦ δλ ◦ Ψ −1 )(x1 , . . . , xN ), λαN+1 xN +1 , . . . , λαH xH ) = (δλ (x1 , . . . , xN ), λαN+1 xN +1 , . . . , λαH xH ). Indeed, Ψ commutes with the dilations of G (see Theorem 1.3.28, page 49). We then reorder the coordinates of G × RH −N in the following way. Let (x, y) ∈ G × RH −N , where x = (x (1) , . . . , x (r) ) ∈ G and y = (yN +1 , . . . , yH ) ∈ RH −N . We can suppose that the coordinates of y are ordered in such a way that αN +1 ≤ · · · ≤ αH . Setting i0 := αN +1 − 1, we can write y = (y (i0 +1) , . . . , y (r) ), where to each coordinate ys of y (k) (i0 + 1 ≤ k ≤ r) corresponds a degree of homogeneity αs equal to k. We now set

658

17 Lifting of Carnot Groups

P : G × RH −N → RH , (x, y) → (x (1) ; . . . ; x (i0 ) ; x (i0 +1) , y (i0 +1) ; . . . ; x (r) , y (r) ). as RH with the group structure naturally induced by P from (G × We define G H −N , •). R With respect to the coordinates of G, δλ assumes the usual form (1) λ x ; . . . ; λi0 x (i0 ) ; λi0 +1 x (i0 +1) , λi0 +1 y (i0 +1) ; . . . ; λr x (r) , λr y (r) . We now set := ϑ ◦ P −1 . It is then easy to recognize that (17.4) follows from Lemma 17.1.4. is a homogeneous Carnot In order to complete the proof, we have to show that G group with Lie algebra g isomorphic to fm,r . To this purpose, let (Gm,r , ∗) be a free homogeneous Carnot group2 on RH whose Lie algebra is isomorphic to fm,r . It is not restrictive to consider fm,r to be the Lie algebra of Gm,r itself. If we denote by Exp ∗ the exponential map from (fm,r , ) to (Gm,r , ∗), then the map −1 Exp ∗ ◦ πF : (RH , F ) → (Gm,r , ∗)

is a Lie group isomorphism (see Theorem 2.2.13, page 129). As a consequence, by Theorem 2.1.50, page 114, the differential −1 d(Exp ∗ ◦ πF )

is a Lie algebra isomorphism from rH to fm,r . Let E1 , . . . , EH be the Jacobian basis for rH . We now prove that −1 d(Exp ∗ ◦ πF )(Ei ) = Fi ,

i = 1, . . . , m.

(17.5)

In particular, since F1 , . . . , Fm are generators for fm,r , this will prove that E1 , . . . , Em are generators for rH . If f ∈ C ∞ (Gm,r ), we have −1 −1 d (Exp ∗ ◦ πF )(Ei ) 0 (f ) = (Ei )0 f ◦ Exp ∗ ◦ πF H

d = (∂ξi |ξ =0 )f Exp ∗ ξj Fj = f Exp ∗ (t Fi ) dt t=0 j =1 d = f expFi (t) = (Fi )0 (f ). dt t=0 Since a left invariant vector field is determined by its value at the origin, this proves (17.5). An analogous argument shows that dΦ : rH → g rH −N 2 The existence of such a group can be easily derived by the third fundamental theorem of

Lie (see Theorem 2.2.14, page 130) and by Theorem 2.2.18, page 131.

17.2 An Example of Lifting

659

maps the first m fields of the Jacobian basis for rH into the first m fields of the Ja is obtained from G × RH −N by the Lie cobian basis for g rH −N . Finally, since G group isomorphism P (which permutes the coordinates, leaving the first m coordi is a homogeneous Carnot group. Moreover, nates unaltered), we can assert that G g is isomorphic to fm,r via the Lie algebra isomorphism −1 d(Exp ∗ ◦ πF ◦ Φ −1 ◦ P −1 ).

This ends the proof. In Section 17.2 below, we give an example of the lifting method just described. The reader is referred to this example for a straightforward comprehension of the topics presented so far.

17.2 An Example of Lifting We here give an example of a homogeneous Carnot group, and we lift it to a free homogeneous Carnot group. We shall follow the notation introduced in Section 17.1. We consider the homogeneous Carnot group G on R5 with the group law and dilation defined as follows: x ◦ y = (x1 + y1 , x2 + y2 , x3 + y3 , x4 + y4 , x5 + y5 + x1 y3 + x2 y4 ) and δλ (x) = (λ x1 , λ x2 , λ x3 , λ x4 , λ2 x5 ). The Jacobian basis Z for g is given by Z1 = ∂1 , Z2 = ∂2 , Z3 = ∂3 + x1 ∂5 , Z4 = ∂4 + x2 ∂5 , Z5 = ∂5 . The Campbell–Hausdorff formula on a Lie algebra nilpotent of step 2 is 1 X Y = X + Y + [X, Y ]. 2 Hence the group (R5 , Z ) has the composition law given by ξ Z η = ξ 1 + η1 , ξ2 + η 2 , ξ3 + η3 , ξ4 + η4 ,

1 ξ5 + η5 + (ξ1 η3 − ξ3 η1 + ξ2 η4 − ξ4 η2 ) . 2

With few modifications, we recognize that Z is the usual group law on H2 , the Heisenberg group on R5 . The Jacobian basis X for r5 is given by

660

17 Lifting of Carnot Groups

ξ3 ∂5 , 2 ξ1 X3 = ∂3 + ∂5 , 2

X1 = ∂1 −

ξ4 ∂5 , 2 ξ2 X4 = ∂4 + ∂5 , 2 X2 = ∂2 −

X5 = ∂5 .

We now turn to consider f4,2 . We have dim f4,2 = 10. Let Π : f4,2 → g be the Lie algebra morphism such that Π(Fi ) = Zi for every i = 1, . . . , 4. The following is a basis F for f4,2 as in Proposition 17.1.1: F1 , F2 , F3 , F4 , [F1 , F3 ]; [F1 , F2 ], [F1 , F4 ], [F2 , F3 ], [F3 , F4 ], [F2 , F4 ] − [F1 , F3 ]. One can easily recognize that the last 5 vectors in F form a basis for ker(Π). The Lie group (R10 , F ) has the composition law given by ξ F η = ξ 1 + η1 , ξ2 + η2 , ξ3 + η3 , ξ4 + η 4 , 1 ξ5 + η5 + (ξ1 η3 − ξ3 η1 + ξ2 η4 − ξ4 η2 ), 2 1 ξ6 + η6 + (ξ1 η2 − ξ2 η1 ), 2 1 1 ξ7 + η7 + (ξ1 η4 − ξ4 η1 ), ξ8 + η8 + (ξ2 η3 − ξ3 η2 ), 2 2

1 1 ξ9 + η9 + (ξ3 η4 − ξ4 η3 ), ξ10 + η10 + (ξ2 η4 − ξ4 η2 ) . 2 2 The first 5 vector fields of the Jacobian basis for r10 are then given by ξ3 ξ2 ξ4 ∂5 − ∂6 − ∂7 , 2 2 2 ξ4 ξ1 ξ3 ξ4 E2 = ∂2 − ∂5 + ∂6 − ∂8 − ∂10 , 2 2 2 2 ξ1 ξ2 ξ4 E3 = ∂3 + ∂5 + ∂8 − ∂9 , 2 2 2 ξ2 ξ1 ξ3 ξ2 E4 = ∂4 + ∂5 + ∂7 + ∂9 + ∂10 , 2 2 2 2 E5 = ∂5 .

E1 = ∂1 −

It is evident that Ei lifts dπ(Ei ) = Xi for every i = 1, . . . , 5, as stated in Lemma 17.1.3. By a straightforward calculation of the exponential map from g to G, we have −1 Ψ = Exp ◦ πZ : (R5 , Z ) → (G, ◦),

1 Ψ (ξ ) = ξ1 , ξ2 , ξ3 , ξ4 , ξ5 + (ξ1 ξ3 + ξ2 ξ4 ) . 2

In particular, the Lie group isomorphism Φ : (R10 , F ) → (G × R5 , •) is

17.3 An Example of Application to PDE’s

661

1 Φ(ξ ) = ξ1 , ξ2 , ξ3 , ξ4 , ξ5 + (ξ1 ξ3 + ξ2 ξ4 ); ξ6 , ξ7 , ξ8 , ξ9 , ξ10 . 2 As a consequence, the group law on G × R5 is given by x • y = x1 + y1 , x2 + y2 , x3 + y3 , x4 + y4 , 1 x5 + y5 + x1 y3 + x2 y4 , x6 + y6 + (x1 y2 − x2 y1 ), 2 1 1 x7 + y7 + (x1 y4 − x4 y1 ), x8 + y8 + (x2 y3 − x3 y2 ), 2 2

1 1 x9 + y9 + (x3 y4 − x4 y3 ), x10 + y10 + (x2 y4 − x4 y2 ) . 2 2 The first 5 vector fields of the Jacobian basis for g r5 are then given by x2 x4 ∂6 − ∂7 , 2 2 x1 x3 x4 ∂6 − ∂8 − ∂10 , W2 = ∂2 + 2 2 2 x2 x4 ∂8 − ∂9 , W3 = ∂3 + x1 ∂5 + 2 2 x1 x3 x2 W4 = ∂4 + x2 ∂5 + ∂7 + ∂9 + ∂10 , 2 2 2 W5 = ∂5 . W1 = ∂1 −

As stated in Lemma 17.1.4, Wi lifts dϑ(Wi ) = Zi for every i = 1, . . . , 5. Finally, in which lifts G has the same group this case, the free homogeneous Carnot group G 5 law as (G × R , •) since the permutation of the coordinates P can be chosen as the are given by identity map. We remark that the dilations of G δλ (x) = λ x1 , λ x2 , λ x3 , λ x4 , λ2 x5 , λ2 x6 , λ2 x7 , λ2 x8 , λ2 x9 , λ2 x10 .

17.3 An Example of Application to PDE’s In this section, we focus our attention on Carnot groups of step two. We recall a characterization of the composition law on such groups, and we explicitly exhibit their lifting. As a consequence, we derive a direct formula for the fundamental solutions for all the sub-Laplacians on groups of step two. This formula is given in terms only of the fundamental solution for the canonical sub-Laplacian on a fixed free Carnot group. The latter fundamental solution can be written in a somewhat explicit form by means of a result by Beals, Gaveau and Greiner [BGG96]. Let us start by recalling the result below which easily follows from Theorem 1.3.15, page 39 (see also Theorem 3.2.2, page 160).

662

17 Lifting of Carnot Groups

Remark 17.3.1. The N-dimensional homogeneous Carnot groups of step two and m generators are characterized by being (RN , ◦) with the following Lie group law (N = m + n, x ∈ Rm , t ∈ Rn )

xj + ξj , j = 1, . . . , m (x, t) ◦ (ξ, τ ) = , tj + τj + 12 x, B (j ) ξ , j = 1, . . . , n where the B (j ) ’s are m × m matrices whose skew-symmetric parts 1 (j ) (B − (B (j ) )T ) 2 are linearly independent. We fix G = (RN , ◦), a Carnot group of steptwo as above, i.e. we fix matrices m 2 as in Remark 17.3.1. Let L = j =1 Yj be a fixed sub-Laplacian on G. Then there exists a non-singular m × m matrix A = (ak,j )k,j such that B (1) , . . . , B (n)

Yj =

m

ak,j Zk ,

k=1

where Z1 , . . . , Zm denote the first m vector fields of the Jacobian basis of g. Our aim here is to write the fundamental solution for L in terms only of the matrices A, B (1) , . . . , B (n) and the fundamental solution for the canonical sub-Laplacian of the prototype free Carnot group (Fm,2 , ), which we introduced in Section 3.3, page 163. For the reader’s convenience, we recall the notation. We set N = m(m + 1)/2, and we denote the points of RN = Rm × Rm(m−1)/2 by (x, γ ), where x ∈ Rm and γ ∈ Rm(m−1)/2 , and we agree to write the coordinates of γ by γi,j , where (i, j ) varies in the set I = {(i, j ) | 1 ≤ j < i ≤ m}. Then the composition law on RN is given by

xh + xh , h = 1, . . . , m . (x, γ ) (x , γ ) = + 1 (x x − x x ), (i, j ) ∈ I γi,j + γi,j j i 2 i j It is easily proved that the Lie algebra of (Fm,2 , ) is (isomorphic to) fm,2 . Then Fm,2 is a free homogeneous Carnot group of step two on RN , with m generators. Since, by the assumption, the skew-symmetric parts of B (1) , . . . , B (n) are linearly independent and the matrix A is non-singular, then the matrices AT

B (r) − (B (r) )T A, 2

r = 1, . . . , n,

17.3 An Example of Application to PDE’s

663

are also linearly independent. Hence, there exist indices (i1 , j1 ), . . . , (in , jn ) ∈ I such that the following n × n matrix AT

B (r) − (B (r) )T A 2 is , js 1≤r,s≤n

is non-singular. We denote by K = (kr,s )r,s the inverse of the above matrix. We also define the subset of indices C = I \ {(i1 , j1 ), . . . , (in , jn )}. With the above notation, we shall also denote the points of Fm,2 by (x, t, β), where x ∈ Rm , t ∈ Rn , and β = (βh,k )(h,k)∈C ∈ Rm(m−1)/2−n . We are now in the position to state the main result of this section. Proposition 17.3.2. With the above notation, let ΓL and Γm,2 denote respectively the fundamental solutions of L and of the canonical sub-Laplacian ΔFm,2 . Then the following formula holds | det K| ΓL (x, t) = dβ | det A| β∈Rm(m−1)/2−n 1 × Γm,2 A−1 x, kr,s ts − x, B (s) x 4 s≤n

(s) − (B (s) )T T B A − βh,k A ,β . 2 h,k r≤n (h,k)∈C

The above result can be applied in order to manage, in a uniform way, families of sub-Laplacians (letting A vary) and families of step two Carnot groups with the same number of generators (letting B (1) , . . . , B (n) vary). Indeed, an example of application is given in [BLU02,BLU03] where uniform estimates for some families of sub-Laplacians are derived and, as a consequence, the fundamental solutions for non-divergence form operators with Hölder-continuous coefficients are constructed. Remark 17.3.3. We explicitly remark that a rather explicit formula is given by Beals, Gaveau and Greiner in [BGG96] for the fundamental solution of the canonical subLaplacian on any step two Carnot group G: V (ρ) f (x, t, ρ)(2−Q)/2 dρ, ΓΔG (x, t) = Rn

where f is the action associated to a complex Hamiltonian problem and V solves a transport equation. Collecting together this formula in the case of the free group Fm,2 and our Proposition 17.3.2, we obtain the following formula

664

17 Lifting of Carnot Groups

ΓL (x, t) =

| det K| dβ dρ | det A| β∈Rm(m−1)/2−n ρ∈Rm(m−1)/2

1/2 S(ρ) 1 S(ρ) coth S(ρ) A−1 x, A−1 x × det sinh S(ρ) 2 n 1 −ι ρir ,jr kr,s ts − x, B (s) x 4 s≤n r=1

−

(h,k)∈C

(2−m2 )/2 (s) − (B (s) )T T B + A βh,k A ρi,j βi,j . 2 h,k i,j ∈C

Here, S(ρ) is the matrix S(ρ) = ι i,j ∈I ρi,j S (i,j ) , where the matrices S (i,j ) have been introduced in Section 3.3, page 163, and ι is the imaginary unit. So, we also have an even more explicit representation | det K| dβ dρ ΓL (x, t) = | det A| β∈Rm(m−1)/2−n ρ∈Rm(m−1)/2 ∞

2p −1/2 (−1)p (i,j ) − ρi,j S × det (2p + 1)! p=0 i,j ∈I ∞

2p 1 (−1)p 2(2p) B2p (i,j ) −1 −1 − × ρi,j S A x, A x 2 (2p)! p=0 i,j ∈I n 1 −ι ρir ,jr kr,s ts − x, B (s) x 4 s≤n r=1 (s) − (B (s) )T T B A − βh,k A 2 h,k (h,k)∈C

(2−m−2n)/2 + ρi,j βi,j . i,j ∈C

As an example, if m = 3, Γ3,2 has been explicitly written in [BGG96, p. 322]. Hence, the above formula writes Gamma(7/2) | det K| 2|ρ| ΓL (x, t) = − dβ dρ 7/2 | det A| β∈R3−n sinh(2|ρ|) (2π) ρ∈R3 −1 2 A x, ρ A−1 x, ρ2 −1 2 × + |ρ| coth(2|ρ|) |A x| − 2|ρ|2 |ρ|2 n 1 −ι ρr kr,s ts − x, B (s) x 4 s≤n r=1

17.3 An Example of Application to PDE’s

665

B (s) − (B (s) )T A βh,k AT 2 h,k (h,k)∈C

−7/2 3 ρr βr−n . +

−

r=n+1

Proof (of Proposition 17.3.2). As the first step, we turn the arbitrary sub-Laplacian L on G into the canonical sub-Laplacian on a new group H. This can be done by the isomorphism of Lie groups H (ξ, τ ) → (Aξ, τ ) ∈ G which turns the composition ◦ on G into

ξj + ξj , j = 1, . . . , m, . (ξ, τ ) • (ξ , τ ) = τj + τj + 12 ξ, AT B (j ) Aξ , j = 1, . . . , n Then, it is easy to see that we have the relation ΓL (x, t) = | det A|−1 ΓΔH (A−1 x, t).

(17.6)

As the second step, we turn the composition law • into a composition law ∗ whose associated matrices (according to Remark 17.3.1) are skew-symmetric. This can be done by identifying H with its Lie algebra via the exponential map. Then, the isomorphism of Lie groups

1 P (x, t) → x, tj + x, AT B (j ) Ax ∈ H 4 turns the composition • on H into

xj + xj , j = 1, . . . , m, , (x, t) ∗ (x , t ) = (j ) (j ) T ) tj + tj + 12 x, AT ( B −(B )Ax , j = 1, . . . , n 2 where the associated matrices are skew-symmetric. It is easy to see that we have the relation 1 ΓΔH (ξ, τ ) = ΓΔP ξ, τj − ξ, AT B (j ) Aξ . (17.7) 4 As the third step, we lift P to a free Carnot group V, as in Theorem 17.1.5. The fundamental solution of the canonical sub-Laplacian on P can be obtained by integrating the fundamental solution of the canonical sub-Laplacian on V with respect to the added variables, i.e. ΓΔV (x, t, β) dβ. (17.8) ΓΔP (x, t) = β∈Rm(m−1)/2−n

We need the explicit form of the lifting for an arbitrary group of step two (with associated skew-symmetric matrices). This is given in the following lemma, which can be proved by retracing the proof of Theorem 17.1.5.

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17 Lifting of Carnot Groups

Lemma 17.3.4. Let (RN , ◦) be a Carnot group of step two and m generators as in Remark 17.3.1, where its associated matrices B (j ) ’s are linearly independent and skew-symmetric. Then (RN , ◦) is lifted to the group (RN × Rm(m−1)/2−n , ◦) (according to Theorem 17.1.5), where (using the notation x ∈ Rm , t ∈ Rm(m−1)/2−n ) ⎛ xj + xj , j = 1, . . . , m ⎜ (x, t, β) ◦ (x , t , β ) = ⎝ tj + tj + 12 x, B (j ) x , j = 1, . . . , n + 1 (x x − x x ), βi,j + βi,j j i 2 i j

Rn , β ∈ ⎞ ⎟ ⎠.

(i, j ) ∈ C

The next step is to explicitly write a Lie group isomorphism between V and Fm,2 . This can be obtained by the composition Exp ◦ ϕ ◦ Log =: T , where Exp is the exponential map on Fm,2 , Log is the logarithmic map on V, and ϕ is the unique Lie algebra morphism mapping the first m vector fields of the Jacobian basis related to V into the first m vector fields of the Jacobian basis related to Fm,2 . A laborious computation shows that the map T : (V, ∗) → (Fm,2 , ) is defined by T (x, t, β) := x,

n s=1

kr,s ts −

(h,k)∈C

βh,k

(s) − (B (s) )T B AT A 2 h,k

,β . r≤n

It is easy to see that we have the relation ΓΔV (x, t, β) = | det K| Γm,2 (T (x, t, β)).

(17.9)

Finally, collecting together equations (17.6) to (17.9), we obtain the desired formula in Proposition 17.3.2.

17.4 Folland’s Lifting of Homogeneous Vector Fields The aim of this section is to provide another lifting theorem for vector fields more general than those considered in Section 17.1. An example will clarify the new situation. Let us consider on R2 the vector fields X = ∂x and Y = x ∂y . As we saw in Example 1.2.14 (page 18), they are not left-invariant with respect to any group law in R2 , so that the lifting Theorem 17.1.5 cannot be applied. In this section, we shall provide a lifting result which can be applied to X, Y . It will give a constructive way

17.4 Folland’s Lifting of Homogeneous Vector Fields

667

to build new vector fields (generating a homogeneous Carnot group!) lifting X and Y = ∂z + x ∂y on R3 , whose points are denoted by (x, z, y)). = ∂x and Y (namely, X This lifting result (due to Folland, see [Fol77]) only relies on the homogeneity properties of X and Y (namely, they are homogeneous of degree one, w.r.t. the dilation δλ (x, y) = (λx, λ2 y) on R2 ). Before entering into the details of this new lifting result, it is useful to focus on the example above and to see how this result works. Example 17.4.1. Let us denote the points of R2 by (x1 , x2 ), and let us consider the following vector fields on R2 : X1 = ∂x1 and X2 = x1 ∂x2 . They are δλ -homogeneous of degree one w.r.t. the dilation δλ (x1 , x2 ) = (λx1 , λ2 x2 ) on R2 . As a consequence, the Lie algebra a generated by them in T (R2 ) (i.e. the set of smooth vector fields on R2 equipped with the usual bracket operation) is nilpotent (of step two), namely, since [X1 , X2 ] = ∂x2 and all other commutators vanish, we have a = Lie{X1 , X2 } = span{X1 , X2 , [X1 , X2 ]},

whence dim(a) = 3.

As we saw in Theorem 2.2.13 (page 129; see also the relevant Definition 2.2.11), the Campbell–Hausdorff operation (truncated of step two) XY =X+Y +

1 [X, Y ] 2

defines on a a Lie group structure (a, ). We first realize this group as a Lie group on R3 in the usual way. We fix the basis X = (X1 , X2 , [X1 , X2 ]) for a, and we identify a with R3 via the map φ : R3 → a,

(ξ1 , ξ2 , ξ3 ) → ξ1 X1 + ξ2 X2 + ξ3 [X1 , X2 ].

The group law is then turned into a group law3 ◦ on R3 in the natural way a ◦ b := φ −1 φ(a) φ(b) , a, b ∈ R3 . The Lie group G = (R3 , ◦) is obviously isomorphic to (a, ), and the Lie algebra g of G is isomorphic (as a Lie algebra) to the Lie algebra a (see Ex. 3 of Chapter 2, page 148). A direct computation shows that in this case we have

1 a ◦ b = a1 + b1 , a2 + b2 , a3 + b3 + (a1 b2 − a2 b1 ) . 2 Denoting the points of R3 by ξ = (ξ1 , ξ2 , ξ3 ), we have g = span{Z1 , Z2 , Z3 }, where Z1 = ∂ξ1 −

1 1 ξ2 ∂ξ3 , Z2 := ∂ξ2 + ξ1 ∂ξ3 , Z3 := [Z1 , Z2 ] = ∂ξ3 . 2 2

The Lie algebra isomorphism between g and a is the linear map α : g → a such that 3 In the previous sections, we denoted it by , and we also wrote φ −1 = π . X X

668

17 Lifting of Carnot Groups

α(Z1 ) = X1 ,

α(Z2 ) = X2 ,

α(Z3 ) = [X1 , X2 ].

Let us denote by Log the logarithmic map related to the Lie group (G, ◦). Moreover, we write I for the identity map on R2 . We then introduce the following map π : R3 −→ R2 , π(ξ ) := e

α(Log (ξ ))

α(Log (ξ )) k I (0, 0). I (0, 0) := k!

(17.10)

k≥0

We explicitly remark that the map π is well defined. Indeed, if ξ ∈ R3 ≡ G, then Log (ξ ) ∈ g, so that α(Log ξ ) ∈ a. As a consequence, by the properties of δλ k homogeneity of the vector fields in a, any power α(Log (ξ )) with k ≥ 3 annihilates the identity map I , so that the sum in (17.10) is finite. We compute π explicitly,

1 2 (0) π(ξ ) = I + (α(ξ1 Z1 + ξ2 Z2 + ξ3 Z3 ))I + (α(ξ1 Z1 + ξ2 Z2 + ξ3 Z3 )) I 2

x=0 1 = I + (ξ1 ∂x1 + ξ2 x1 ∂x2 + ξ3 ∂x2 )I + (ξ1 ∂x1 + ξ2 x1 ∂x2 + ξ3 ∂x2 )2 I (0) 2 x=0

1 = I + ξ1 ∂x1 + ξ3 ∂x2 + ξ1 ξ2 ∂x2 I (0) 2 x=0

1 = ξ1 , ξ3 + ξ1 ξ2 . 2 We now verify that Z1 and Z2 (the vector fields in g corresponding to X1 and X2 in a via α) lift X1 and X2 via π in the sense that Xi |π(ξ ) = dξ π(Zi )

for all ξ ∈ R3 and i = 1, 2.

Indeed, since the component functions of Xi |π(ξ ) are given by Xi I (π(ξ )) whereas those of dξ π(Zi ) are given by Jπ (ξ ) · Zi I (ξ ), we only have to check the matrix identity

1 0 1 0 0 1 0 = 1 · , 0 1 1 0 ξ1 2 ξ2 2 ξ1 1 − 12 ξ2 12 ξ1 which actually holds. The ideas presented so far appear in the paper [Fol77] by G.B. Folland. We now aim to add a new feature: we are interested in a “change of coordinates” on R3 turning Zi into a vector field on R3 lifting Xi . To this aim, we define the following map

1 T : R3 → R3 , T (ξ ) = (π(ξ ), ξ2 ) = ξ1 , ξ3 + ξ1 ξ2 , ξ2 . 2 We obtained T by completing π with the only coordinate function (i.e. ξ2 ) not appearing in the first order expansion of π(ξ ). We remark that T is a smooth diffeomorphism of R3 onto itself (with polynomial coordinates and the same holds

17.4 Folland’s Lifting of Homogeneous Vector Fields

669

for T −1 ). We perform a change of coordinates on R3 by introducing new variables x = (x1 , x2 , x3 ) as follows x = T (ξ ),

x, ξ ∈ R3 .

We express Z1 and Z2 (the vector fields in g corresponding to X1 and X2 in a via α) 2 . We claim that Z 1 and Z 2 lift X1 1 and Z in these new coordinates and obtain Z 3 and X2 . Indeed, for every x ∈ R and i = 1, 2, we have ! i I (x) = JT (T −1 (x)) · (Zi I )(T −1 (x)), i I (x), ∇x , where Z i = Z Z so that (after a simple computation) 1 = ∂x1 Z

2 = ∂x3 + x1 ∂x2 . and Z

2 lift X1 and X2 , respectively. 1 and Z It is now immediately seen that Z The aim of the rest of the section is to generalize the previous example to sets of vector fields satisfying suitable hypotheses. 17.4.1 The Hypotheses on the Vector Fields Assume that we are given on Rn (whose points will be denoted by x = (x1 , . . . , xn )) a set of m smooth vector fields X1 , . . . , Xm satisfying the following conditions: (F1) X1 , . . . , Xm are linearly independent and dλ -homogeneous of degree one with respect to a suitable family of dilations {dλ }λ>0 of the following type dλ : R n → R n ,

dλ (x) = (λσ1 x1 , . . . , λσn xn ),

where 1 = σ1 ≤ · · · ≤ σn . (F2) dim(Lie{X1 , . . . , Xm }I (0)) = n. The reader should notice that these conditions are resemblant to conditions (H0) and (H2) on page 191, but condition (H1) on that page has no analogue. For example, the vector fields in Example 17.4.1 fulfill hypotheses (F1) and (F2) but not (H1). We henceforth use the following notation: a denotes the Lie algebra generated by the Xj ’s (sub-algebra of T (Rn ), the set of the smooth vector fields on Rn equipped with the usual bracket of differential operators, not necessarily left-invariant on a Lie-group), i.e. (17.11) a := Lie{X1 , . . . , Xm }. The first task of the section is to prove the following theorem (see also [Fol77, Theorem 1]). Theorem 17.4.2 (Folland’s lifting, [Fol77]). Let X1 , . . . , Xm satisfy the above conditions (F1) and (F2). Then there exist a homogeneous Carnot group G on RN with m generators and nilpotent of step r (where N ≥ m is the dimension of a and r the step of nilpotence of a), a polynomial surjective map

670

17 Lifting of Carnot Groups

π : RN → Rn , and a system of Lie-generators Z1 , . . . , Zm for the algebra of G such that Zi lift Xi via π, i.e. dξ π(Zi ) = (Xi )π(ξ ) for every ξ ∈ RN . (17.12) More precisely, Zi is the i-th vector of the Jacobian basis for G. We give a constructive proof of Theorem 17.4.2 consisting of six steps. STEP 1 (The properties of a). We leave as an exercise the verification of the following results. If X1 , . . . , Xm satisfy the above conditions (F1) and (F2), then the following facts hold: (i) Each Xj has the following form Xj =

n

ai,j (x) ∂xi ,

i=1

where ai,j is a dλ -homogeneous polynomial of degree σi − 1 (so that ai,j depends only on the xk ’s such that σk ≤ σi − 1). (ii) a is nilpotent (of step at most σn , say r). As a consequence, since a is spanned by high-order brackets of the form [Xik · · · [Xi1 , Xi2 ] · · ·],

i1 , . . . , ik ∈ {1, . . . , m}, k ∈ N,

and only a finite number of them are not vanishing (due to the nilpotence of a), then a is finite-dimensional. We set N := dim(a).

(17.13)

(iii) The Lie algebra a is stratified, i.e. (see also Definition 2.2.3, page 122) [a1 , ai−1 ] = ai if 2 ≤ i ≤ r, a = a1 ⊕ a2 ⊕ · · · ⊕ ar with (17.14) [a1 , ar ] = {0}. Here a1 = span{X1 , . . . , Xm } and r is the step of nilpotence of a. We explicitly remark that any operator in ai is dλ -homogeneous of degree i,

(17.15)

for it is a linear combination of elements of the type X j1 X j2 · · · X js

with j1 + j2 + · · · + js = i

(and any Xj is dλ -homogeneous of degree 1 by hypothesis (F1)). The above decomposition also allows us to define a group of dilations {δλ }λ>0 on a in the following way: r

r Ai := λi Ai , where Ai ∈ ai for i = 1, . . . , r. δλ : a → a, δλ i=1

i=1

(17.16)

17.4 Folland’s Lifting of Homogeneous Vector Fields

671

STEP 2 (The choice of a basis for a). Recalling hypothesis (F1) and (17.13), we can complete X1 , . . . , Xm to a basis of a X = (X1 , . . . , Xm , Xm+1 , . . . , XN ) satisfying the following two conditions: (1) we have (by making use of hypothesis (F2)) " # span X1 I (0), . . . , Xm I (0), Xm+1 I (0), . . . , XN I (0) = Rn ,

(17.17)

(2) X is adapted to the stratification of a, i.e. we have (1) (2) (r) (1) (2) (r) X = X1 , . . . , X m , ; X1 , . . . , Xm ; · · · ; X1 , . . . , Xm r 1 2

(17.18)

(1)

where m1 = m, Xi

= Xi for every 1 ≤ i ≤ m and, with reference to (17.14),

" # (i) ai = span X1(i) , . . . , Xm i

for every i = 2, . . . , r.

STEP 3 (The group operations). It is well known that (see Theorem 2.2.13, page 129; see also the relevant Definition 2.2.11), the Campbell–Hausdorff operation defines on a a Lie group structure (a, ). We can realize this group as a Lie group on RN in the usual way. Fixed a basis X for a as in Step 2, we identify a with RN via the map N φ : RN → a, ξ = (ξ1 , . . . , ξN ) → ξj X j . j =1

The group law is then turned into a group law4 ∗ on RN in the natural way a ∗ b := φ −1 φ(a) φ(b) , a, b ∈ RN . The Lie group G := (RN , ∗) is obviously isomorphic to (a, ), for the map φ : (G, ∗) → (a, ) is a Lie group isomorphism (which is also linear, when a and G ≡ RN are equipped with their vector space structures). Analogously, we transfer the dilations {δλ }λ>0 on a defined in (17.16) to a group of dilations {Dλ }λ>0 on G by setting (17.19) Dλ : G → G, Dλ (ξ ) := φ −1 δλ (φ(ξ )) . Since δλ is an automorphism of the Lie group (a, ) (see Ex. 5 at the end of the Chapter), it immediately follows that Dλ is an automorphism of the Lie group (G, ∗). We claim that G = (RN , ∗, Dλ ) is a homogeneous Carnot group. Let us prove this fact. With reference to the choice of the basis X for a in (17.18), for any ξ ∈ RN , we write ξ = (ξ (1) , . . . , ξ (r) ),

ξ (i) ∈ Rmi ,

i = 1, . . . , r.

4 In the previous sections, we denoted it by , and we also wrote φ −1 = π . X X

672

17 Lifting of Carnot Groups

First, we aim to show that, with this notation, Dλ (ξ (1) , ξ (2) , . . . , ξ (r) ) = (λξ (1) , λ2 ξ (2) , . . . , λr ξ (r) ). We have Dλ (ξ ) = φ

−1

N

−1 ξj X j δλ (φ(ξ )) = φ δλ

=φ

−1

δλ

j =1

mi r

ξj(i) Xj(i)

=φ

−1

i=1 j =1

= (λξ (1)

(17.20)

(1)

2 (2)

,λ ξ

mi r

ξj(i) λi

Xj(i)

i=1 j =1 r (r)

,...,λ ξ

).

(1)

Denote by Y1 , . . . , Ym the first m vector fields of the Jacobian basis of g. In order to prove that G is a homogeneous Carnot group, we have to show (1)

Lie{Y1 , . . . , Ym(1) } = g.

(17.21)

(i)

Let us prove this fact. Let Yj denote the vector field in g coinciding at 0 with ∂ξ (i) |0 j

(i.e. Yj(i) is the vector field in the Jacobian basis of g relative to the coordinate ξj(i) ). (i)

Then, Yj is Dλ -homogeneous of degree i: indeed, it holds (i)

(i)

(i)

Yj |ξ (u ◦ Dλ ) = d0 τξ (Yj |0 )(u ◦ Dλ ) = (Yj |0 )(u ◦ Dλ ◦ τξ ) (i)

= (Yj |0 )(u ◦ τDλ (ξ ) ◦ Dλ ) = (∂x (i) )(u ◦ τDλ (ξ ) )(λx (1) , . . . , λi x (i) , . . . , λr x (r) ) j

0

= λ (∂y (i) )(u ◦ τDλ (ξ ) )(y (1) , . . . , y (i) , . . . , y (r) ) j 0 (i) (i) i = λ (Yj |0 )(u ◦ τDλ (ξ ) ) = λi Yj D (ξ ) u. i

λ

Hence, g is split into the direct sum g = ⊕ · · · ⊕ H (r) , where H (i) is the set of the left-invariant vector fields, Dλ -homogeneous of degree i, i.e. H (i) = (i) (i) span{H1 , . . . , Hmi }. In order to prove (17.21), it is sufficient to find another set {Z1 , . . . , Zm } of independent vector fields in g which are Dλ -homogeneous of degree 1 and such that Lie{Z1 , . . . , Zm } = g. This is done in the next step. STEP 4 (Relations between g and a). Consider the following commutative (see Theorem 2.1.59, page 119) diagram: H (1)

(G, ∗)

φ

Exp G

g

(a, ) Exp a

dφ

Lie(a).

17.4 Folland’s Lifting of Homogeneous Vector Fields

673

It is proved in Ex. 4 at the end of the chapter that Exp a is linear and it is a Lie algebra isomorphism, when a is equipped with its former structure of Lie algebra (and Lie(a) with its obvious one). Hence (recall that dφ is a Lie algebra isomorphism, see Theorem 2.1.50, page 114) α : g → a,

α := φ ◦ Exp G = Exp a ◦ dφ

(17.22)

is a Lie algebra isomorphism. The reader should notice that the elements of a, which are vector fields on Rn , have now been put into a bijective correspondence with vector fields on RN (the elements of g). We consider the vector fields Z1 , . . . , Zm in g corresponding to X1 , . . . , Xm , respectively, i.e. Zi ∈ g :

α(Zi ) = Xi

for every i = 1, . . . , m.

(17.23)

We claim that Zi is the i-th vector of the Jacobian basis for g. Indeed, Zi |0 u = (α −1 (Xi ))|0 u = (dφ −1 (Log a (Xi )))0 u = (d0 φ −1 ((Log a (Xi ))0 ))u d −1 = (Log a (Xi ))0 (u ◦ φ ) = (u ◦ φ −1 )(tXi ) dt t=0 d ∂ = u(tei ) = (u(ξ )). dt t=0 ∂ξi ξ =0 For the fifth equality, we used the results of Ex. 6 at the end of the Chapter (recall that (tX) (sX) = (t + s)X for every X ∈ a and s, t ∈ R). Finally, we notice that (17.23), the fact that α is a Lie algebra isomorphism, and Lie{X1 , . . . , Xm } = a imply Lie{Z1 , . . . , Zm } = g. This completes the proof of Step 3. The aim of the following step is to find a “lifting” map π : RN → Rn relating the Zi ’s to the Xi ’s as in (17.12). STEP 5 (The lifting map π). Let us denote by Log the logarithmic map related to the Lie group (G, ◦). Moreover, we write I for the identity map on Rn . We then introduce the following map π : RN −→ Rn , π(ξ ) := e

α(Log (ξ ))

α(Log (ξ )) k I (0) := I (0). k!

(17.24)

k≥0

We explicitly remark that the map π is well defined. Indeed, if ξ ∈ RN ≡ G, then Log (ξ ) ∈ g, so that α(Log ξ ) ∈ a. As a consequence, by the properties of δλ homogeneity of the vector fields in a (see hypothesis (F1)), any power (α(Log (ξ )))k with k ≥ n+1 annihilates the identity map I of Rn , so that the sum in (17.24) is finite. By the very definition of α in (17.22), we have α(Log (ξ )) = φ(ξ ) = N j =1 ξj Xj , so that N k j =1 ξj Xj I (0) for all ξ ∈ RN . (17.25) π(ξ ) := k! k≥0

STEP 6 (The properties of π ). The map π has the following properties:

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17 Lifting of Carnot Groups

1. (H OMOGENEITY PROPERTY .) We have π(Dλ (ξ )) = dλ (π(ξ ))

∀ ξ ∈ RN , λ > 0.

(17.26)

2. (S URJECTIVE PROPERTY .) The map π : RN → Rn is onto. 3. (E XPONENTIAL PROPERTY .) For every X ∈ a, we have π(φ −1 (X)) = exp(X)(0),

(17.27)

where the right-hand side is meant as a formal exponential.5 4. (L IFTING PROPERTY .) It holds dξ π(Zi ) = (Xi )π(ξ )

for every ξ ∈ RN .

(17.28)

Let us begin with the homogeneity property. If we shortly write Dλ (ξ ) = (λβ1 ξ1 , . . . , λβN ξN ),

(17.29)

then from (17.25) we have π(Dλ (ξ )) :=

N

j =1 ξj

λβj Xj

k I (0).

k!

k≥0

Now, let us fix i0 ∈ {1, . . . , n}. The claimed (17.26) will follow if we show that N k N k βj j =1 ξj λ Xj j =1 ξj Xj σi0 Ii0 (0) = λ Ii0 (0). (17.30) k! k! k≥0

k≥0

Let us prove (17.30). From (17.15), the choice (17.18) of the basis X , the very definition (17.16) of δλ , (17.20) and the notation in (17.29) it follows that Xj is dλ homogeneous of degree βj . Moreover, any non-vanishing summand in the left-hand side of (17.30) has the following form (for some c ∈ R, i1 , . . . , is ∈ {1, . . . , N }) c λβi1 +···+βis ξi1 · · · ξis Xi1 · · · Xis Ii0 (0)

with βi1 + · · · + βis = σi0 ,

because the function Ii0 is dλ -homogeneous of degree σi0 . This proves (17.30). Let us turn to the surjective property. From (17.25) it is evident that π(ξ ) :=

N

ξj Xj I (0) + O(|ξ |2 ),

as ξ → 0.

j =1 5 Id est, the formal exponential of a high-order differential operator on Rn

exp(X)(0) :=

Xk I (0), k!

k≥0

which is well defined for the sum is finite (recall the dλ -homogeneity properties of the operators in a).

17.4 Folland’s Lifting of Homogeneous Vector Fields

This gives

675

Jπ (0) = X1 I (0) · · · XN I (0) ,

so that, recalling (17.17),

rank Jπ (0) = n.

By the rank theorem from elementary calculus, this proves the existence of an open neighborhood W ⊆ Rn of π(0) = 0 such that π : RN → W is surjective. We next demonstrate that π is also onto Rn . Indeed, if x ∈ Rn , there exists λ = λ(x) > 0 such that dλ (x) ∈ W . Let ξ ∈ RN be such that π(ξ ) = dλ (x). Then, thanks to (17.26), π(D1/λ (ξ )) = d1/λ (π(ξ )) = d1/λ (dλ (x)) = x. This shows that π is surjective. Obviously, π has polynomial entries, for the sum in (17.25) is finite. Let us prove (17.27). This is just the definition (17.24) of π jointly with (α ◦ Log ◦ φ −1 )(X) = (φ ◦ φ)(X) = X. Finally, Theorem 17.4.2 will be completely proved if we show that (17.28) holds (the lifting property of π). Let us fix ξ ∈ RN . Since in (17.12) we are testing the equality of two first order differential operators on Rn , (17.12) follows if we demonstrate that for every s = 1, . . . , n, (17.31) dξ π(Zi )Is = (Xi )π(ξ ) Is where Is (x) = xs is the s-th coordinate projection of Rn . Recalling that Zi is the i-th vector of the Jacobian basis for g, we have d dξ π(Zi )Is = Zi |ξ (Is ◦ π) = (Is ◦ π)(ξ ∗ (tei )) dt t=0 d = (Is ◦ π) φ −1 φ(ξ ) φ(tei ) dt t=0 N

d −1 ξj Xj (tXi ) = (Is ◦ π ◦ φ ) dt t=0 j =1 N

d = Is ◦ exp ξj Xj (tXi ) (0) dt t=0 j =1 N

d ξj Xj (0) g = Is ◦ exp(tXi ) exp dt t=0

= (Xi Is ) exp

N

j =1

ξj Xj (0) = (Xi Is )(π(ξ )),

j =1

which gives (17.31). Here we used the following facts: The first equality is the definition of the differential; the second equality follows from the left-invariance of Zi ; the third equality is the definition of ∗; the fourth equality is the definition of φ; the

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17 Lifting of Carnot Groups

fifth equality follows from (17.27); the sixth equality follows from the Campbell– Hausdorff formula (applied to the vector fields of a which have dλ -homogeneity properties; see (15.3) page 595); the seventh equality is trivial; the last equality follows again from (17.27). This ends the proof. Bibliographical Notes. Different proofs of the lifting procedure have been provided. Besides the pioneering paper by L.P. Rothschild and E.M. Stein [RS76], we refer the reader to L. Hörmander and A. Melin [HM78], G.B. Folland [Fol77], R.W. Goodman [Goo78]. In [Fol77], the case described in Section 17.4 is considered. Some of the topics presented in this chapter also appear in [BU05a].

17.5 Exercises of Chapter 17 Ex. 1) Let α ∈ R be fixed. Lift the Carnot group on R4 with the composition law ⎛ ⎞ x1 + y1 , ⎜ ⎟ x2 + y2 , ⎜ ⎟ 1 ⎜ ⎟. + y + (x y − x y ), x 3 3 2 1 2 1 2 ⎜ ⎟ 1 α ⎝ ⎠ x4 + y4 + 2 (x1 y3 − x3 y1 ) + 2 (x2 y3 − x3 y2 ) 1 α + 12 (x1 − y1 ) (x1 y2 − x2 y1 ) + 12 (x2 − y2 ) (x1 y2 − x2 y1 ) Verify that, following the notation and definitions in Section 17.1, the following facts hold: a) The Jacobian basis Z for g is

1 1 1 Z1 = ∂1 − x2 ∂3 − x3 + x2 (x1 + α x2 ) ∂4 , 2 2 12

1 1 1 Z2 = ∂2 + x1 ∂3 + − α x3 + x1 (x1 + α x2 ) ∂4 , 2 2 12

1 1 x1 + α x2 ∂4 , Z3 = ∂3 + 2 2 Z4 = ∂4 . b) The Campbell–Hausdorff law related to Z on R4 coincides with ◦. Hence, the Jacobian basis X for r4 is given by

1 1 1 ξ3 + ξ2 (ξ1 + α ξ2 ) ∂4 , X1 = ∂1 − ξ2 ∂3 − 2 2 12

1 1 1 X2 = ∂2 + ξ1 ∂3 + − α ξ3 + ξ1 (ξ1 + α ξ2 ) ∂4 , 2 2 12

17.5 Exercises of Chapter 17

X3 = ∂3 +

677

1 1 ξ1 + α ξ2 ∂4 , 2 2

X4 = ∂4 . c) Consider f2,3 . We have dim f2,3 = 5. Let Π : f2,3 → g be the Lie algebra morphism such that Π(Fi ) = Zi for every i = 1, 2. The Hall basis for f2,3 is given by F1 , F2 , F3 = [F2 , F1 ], F5 = [[F2 , F1 ], F2 ].

F4 = [[F2 , F1 ], F1 ],

The following is a basis F for f2,3 , as in Proposition 17.1.1, F1 ,

F2 ,

−F3 ,

F4 ;

F5 − α F4 .

d) The Lie group (R5 , F ) is such that ξ F η equals ⎞ ⎛ ξ1 + η1 , ⎟ ⎜ ξ2 + η2 , ⎟ ⎜ 1 ⎟ ⎜ ξ3 + η3 + 2 (ξ1 η2 − ξ2 η1 ), ⎟. ⎜ 1 α ⎟ ⎜ ξ4 + η4 + 2 (ξ1 η3 − ξ3 η1 ) + 2 (ξ2 η3 − ξ3 η2 ) ⎟ ⎜ 1 α ⎝ + 12 (ξ1 − η1 ) (ξ1 η2 − ξ2 η1 ) + 12 (ξ2 − η2 ) (ξ1 η2 − ξ2 η1 ), ⎠ 1 (ξ2 − η2 ) (ξ1 η2 − ξ2 η1 ) ξ5 + η5 + 12 (ξ2 η3 − ξ3 η2 ) + 12 Verify that the first 4 vector fields of the Jacobian basis for r5 lift Xi , for every i = 1, . . . , 4, as stated in Lemma 17.1.3. e) Verify that −1 : (R4 , Z ) −→ (G, ◦) ψ = Exp ◦ πZ satisfies ψ(ξ1 , ξ2 , ξ3 , ξ4 ) = (ξ1 , ξ2 , ξ3 , ξ4 ) ∈ G. In particular, the Lie group isomorphism Φ : (R5 , F ) → (G × R, •) is defined by Φ(ξ1 , ξ2 , ξ3 , ξ4 , ξ5 ) = (ξ1 , ξ2 , ξ3 , ξ4 ; ξ5 ) ∈ G × R. As a consequence, for the group law • on G × R, we have that x • y equals ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞ x 1 + y1 , ⎟ x2 + y2 , ⎟ ⎟ x3 + y3 + 12 (x1 y2 − x2 y1 ), ⎟. ⎟ x4 + y4 + 12 (x1 y3 − x3 y1 ) + α2 (x2 y3 − x3 y2 ) ⎟ 1 α (x1 − y1 ) (x1 y2 − x2 y1 ) + 12 (x2 − y2 ) (x1 y2 − x2 y1 ), ⎠ + 12 1 (x2 − y2 ) (x1 y2 − x2 y1 ) x5 + y5 + 12 (x2 y3 − x3 y2 ) + 12

Find the Jacobian basis W1 , . . . , W5 for g r1 . Verify that, in particular, it holds

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17 Lifting of Carnot Groups

1 1 1 1 x3 + x2 (x1 + α x2 ) ∂4 − x22 ∂5 , W1 = ∂1 − x2 ∂3 − 2 2 12 12

1 1 1 W2 = ∂2 + x1 ∂3 + − α x3 + x1 (x1 + α x2 ) ∂4 2 2 12

1 1 + − x 3 + x 1 x 2 ∂5 . 2 12 Verify that, as stated in Lemma 17.1.4, Wi lifts dϑ(Wi ) = Zi , for every i = 1, . . . , 4. Ex. 2) Give a detailed proof of Lemma 17.3.4, page 666. Ex. 3) Prove properties (i)–(ii)–(iii) in Step 1, page 670. Ex. 4) Prove the following result. Proposition 17.5.1. Let a be a (finite-dimensional) nilpotent Lie algebra. Consider the Lie group (a, ), where is the Campbell–Hausdorff operation6 on a. Denote by Lie(a) the Lie algebra of the Lie group (a, ) and by Exp a : Lie(a) → a the relevant exponential map. Then, Exp a is linear and it is a Lie algebra isomorphism, when a is equipped with its former structure of Lie algebra. (Hint: By the third fundamental theorem of Lie (see Theorem 2.2.14, page 130) there exists a connected, simply connected Lie group (G, ·) such that g := Lie(G) is isomorphic to a as Lie algebras. Let ϕ : a → g be the relevant Lie algebra isomorphism. Then ϕ : (a, ) → (g, ) is a Lie group isomorphism (see Remark 2.2.16, page 130). Consider the following commutative diagram (a, )

ϕ

(g, )

Exp G

(G, ·)

Exp a

Lie(a)

Exp G A dϕ

Lie(g)

d Exp G

g.

We have Exp a = ϕ −1 ◦ (Exp G )−1 ◦ Exp G ◦ dExp G ◦ dϕ = ϕ −1 ◦ dExp G ◦ dϕ, which is evidently a Lie algebra isomorphism.) Ex. 5) Let a be a (finite-dimensional) nilpotent Lie algebra. Consider the Lie group (a, ), where is the Campbell–Hausdorff operation on a. Suppose a is stratified, i.e. it admits a decomposition of the form [a1 , ai−1 ] = ai if 2 ≤ i ≤ r, a = a1 ⊕ a2 ⊕ · · · ⊕ ar with [a1 , ar ] = {0}. Define a group of dilations {δλ }λ>0 on a in the following way: 6 See Theorem 2.2.13, page 129; see also the relevant Definition 2.2.11.

17.5 Exercises of Chapter 17

δλ : a → a,

δλ

r

Ai

:=

i=1

r

λi Ai ,

679

where Ai ∈ ai for i = 1, . . . , r.

i=1

Prove that δλ is an automorphism of the Lie group (a, ). (Hint: First show that if X ∈ ai and Y ∈ aj ,then [X, Y ] ∈ ai+j (where ai := {0} for i > r). Then prove that δλ [X, Y ] = [δλ (X), δλ (Y )] for every X, Y ∈ a. End by using Remark 2.2.16.) Ex. 6) Let V be a real (finite-dimensional) vector space equipped with a Lie group structure by the operation ∗. Denote by v the relevant Lie algebra, by Exp and Log the relevant exponential and logarithmic maps. Furthermore, suppose that ()

(tv) ∗ (sv) = (t + s)v for every v ∈ V and s, t ∈ R.

Let us introduce the following notation: for every v ∈ V , Ξ (v) denotes the tangent vector at 0 ∈ V such that d f (tv) ∀ V ∈ C ∞ (V , R). Ξ (v)f = dt t=0 Prove the following facts: a) Show that Ξ (v) is indeed an element of T0 (V ); b) Show that T0 (V ) = {Ξ (v) : v ∈ V }, whence the natural identification between T0 (V ) and V ; c) Denote by α : v → T0 (V ) the usual identification between v and T0 (V ), i.e. α(X) = X0 for every X ∈ v. Prove that the flow of X ∈ v is t → t v, where v = Ξ −1 (X0 ). d) Derive that Exp (X) = Ξ −1 (X0 ), i.e. Exp ◦ α −1 ◦ Ξ (v) = v for every v ∈ V , or analogously, α ◦ Log = Ξ , that is, for every v ∈ V , it holds d (Log v)0 f = f (tv) for every f ∈ C ∞ (V ). dt t=0 (Hint: The main task is to prove (c). We must show that γ : R → V , γ (t) = tv solves the Cauchy problem γ˙ (t) = Xγ (t) , γ (0) = 0. Note that Xγ (t) = d0 τγ (t) (X0 ) = d0 τγ (t) (Ξ (v)). We thus have to prove that, for every f ∈ C ∞ (V ), ? d d f (t v) = d0 τγ (t) (Ξ (v))f = [· · ·] = f ((tv) ∗ (sv)), dt t ds s=0 which follows from ().)

680

17 Lifting of Carnot Groups

Ex. 7) Prove that the vector fields on R3 (whose points are denoted by (x, y, t)) defined by Y := ∂y X := ∂x + y 2 ∂t , are dλ -homogeneous of degree 1, where dλ : R3 −→ R3 ,

dλ (x, y, t) := (λx, λy, λ3 t),

and satisfy hypotheses (F1)–(F2) of page 669 (these are the fields considered in Section 4.4.2, page 212, as an example of fields not satisfying hypothesis (H1) of page 191). Lift them by the method illustrated in Section 17.4.

18 Groups of Heisenberg Type

In this chapter, we treat the so-called Heisenberg-type groups (also referred to as H-type groups), a significant class of Carnot groups of step two, generalizing the classical Heisenberg–Weyl groups. In Section 18.2, we give a direct characterization of Heisenberg-type groups, via a suitable choice of a coordinate system, with respect to which the canonical subLaplacian has a simple and suggestive representation. We compare the definition of (abstract) H-type group given in the present chapter to our previous definition of prototype H-type group (see Definition 3.6.1): roughly speaking, any abstract H-type group is naturally isomorphic to a prototype H-type group. In Section 18.3, we prove in a direct way the existence of a fundamental solution for some sub-Laplacians on Heisenberg-type groups, following the proof by A. Kaplan in [Kap80]. In Sections 18.4 and 18.5 we define an inversion and a Kelvintransform on H-type groups, generalizing those from the classical theory of Laplace’s operator. Unfortunately, this Kelvin transform has remarkable properties only on a sub-class of the H-type groups: the so-called Iwasawa-type groups (the relevant definition is given in Section 18.4). Indeed given a H-type group H, the following result holds: The H-Kelvin transform of a ΔH -harmonic function is ΔH -harmonic if and only if H is of Iwasawa-type.

18.1 Heisenberg-type Groups Let us begin with the definition of Heisenberg-type algebra and Heisenberg-type group. Definition 18.1.1 (H-type algebra, H-type group). A Heisenberg-type algebra (H-type algebra, in short) is a finite-dimensional real Lie algebra g which can be endowed with an inner product , such that [z⊥ , z⊥ ] = z, where z is the center of g and moreover, for every fixed z ∈ z, the map

682

18 Groups of Heisenberg Type

Jz : z⊥ → z⊥ defined by1

∀ w ∈ z⊥

Jz (v), w = z, [v, w]

(18.1)

is an orthogonal map whenever z, z = 1. A Heisenberg-type group (H-type group, in short) is a connected and simply connected Lie group whose Lie algebra is an H-type algebra. We recall that the center of g is, by definition, z = {z ∈ g | [z, g] = 0 ∀ g ∈ g}. In the following we shall always suppose that z is not the null subspace. We shall often use the notation b := z⊥ . Let us consider some examples. Example 18.1.2 (The Heisenberg–Weyl group is an H-type group). The classical Heisenberg–Weyl group HN is an H-type group. As usual, we shall denote by (x, y, t) the points of HN ≡ R2N +1 , x, y ∈ RN , t ∈ R. Consider the vector fields Xj := ∂xj + 2yj ∂t ,

Yj := ∂yj − 2xj ∂t ,

j = 1, . . . , N.

Then we have g = Lie{Xj , Yj | j = 1, . . . , N },

z = span{∂t }.

Let us now fix on g the coordinate system associated to the basis {X1 , . . . , XN , Y1 , . . . , YN , −4 ∂t }, and let us consider on g the standard inner product generated by this system. In other words, we identify g with R2N +1 in the following way g

N

(aj Xj + bj Yj ) − 4c ∂t ←→ (a, b, c) ∈ R2N +1 ,

j =1

and we set

N (a, b, c), (a , b , c ) := (aj aj + bj bj ) + c c . j =1

In this coordinate system, the Lie bracket has the following expression N [(a, b, c), (a , b , c )] = 0, 0, (aj bj − bj aj ) . j =1 1 J is well-posed, see (18.3) below. z

18.1 Heisenberg-type Groups

Then we have

683

b := z⊥ = span{Xj , Yj | j = 1, . . . , N },

which gives [b, b] = z. Let now z ∈ z be such that |z| = 1, i.e. z = ∓4 ∂t ≡ (0, 0, ±1). With the notation of Definition 18.1.1, we have N N (aj Xj + bj Yj ) = ± (−bj Xj + aj Yj ), Jz j =1

j =1

and, clearly, Jz is an orthogonal endomorphism of b.

Example 18.1.3. It can be proved (and it will be proved in Section 18.2) that the following group law on R6 defines an H-type group: 1 1 (2) (2) (2) (2) x ◦ y := x (1) + y (1) , x1 + y1 + P1 x (1) , y (1) , x2 + y2 + P2 x (1) , y (1) , 2 2 (2)

(2)

(2)

(2)

where x = (x (1) , x1 , x2 ) ∈ R6 , x (1) ∈ R4 , x1 , x2 ∈ R and ⎛ ⎞ ⎛ 0 1 1 0 0 −1 √ √ 2 ⎜ −1 0 0 −1 ⎟ 2⎜ 1 0 P1 := ⎝ ⎠ , P2 := ⎝ −1 0 0 1 −1 0 2 2 0 1 −1 0 0 1

⎞ 1 0 0 −1 ⎟ ⎠. 0 −1 1 0

Example 18.1.4. The Lie group obtained as the direct product of the group (R, +) with the Heisenberg group (H1 , ◦) is not an H-type group. More precisely, if we consider the algebra g generated by the vector fields on R4 ∂x1 ,

∂x2 + 2x3 ∂x4 ,

∂x3 − 2x2 ∂x4 ,

we observe that the center z of g is given by span{∂x1 , ∂x4 }, but ∂x1 cannot be obtained as linear combinations of brackets in g. Thus the hypothesis [b, b] = z in Definition 18.1.1 cannot be satisfied by any b ⊆ g. Remark 18.1.5. If a Lie algebra g has a center z such that dim(g) − dim(z) is odd, then g is not an H-type algebra. This fact will be proved in Section 18.2 (see also Remark 18.1.6 below). Remark 18.1.6. We have the following general result (see A. Kaplan [Kap80, Corollary 1]). Let N1 , N2 ∈ N \ {0}. Then there exists an H-type algebra of dimension N1 + N2 whose center has dimension N2 (thus N1 is the dimension of the orthogonal completion of the center) if and only if N2 < ρ(N1 ), where ρ is the so-called Hurwitz–Radon function, i.e. ρ : N → N,

ρ(n) := 8p + q,

where n = (odd) · 24p+q , 0 ≤ q ≤ 3.

We observe that ρ(N1 ) = 0 if N1 is odd. Thus there cannot be H-type algebras where the dimension of the orthogonal completion of the center is odd.

684

18 Groups of Heisenberg Type

We explicitly remark that the relation Jz (v), v = z, [v, v ]

∀ v ∈ b

(18.2)

in Definition 18.1.1 actually defines an endomorphism of b. Indeed, for fixed v ∈ b and z ∈ z, the map Ψ : b → R,

v → Ψ (v ) := z, [v, v ]

is linear. Hence, there exists exactly one w ∈ b (depending only on v and z) such that Ψ (v ) = w, v for every v ∈ b. Then we set Jz (v) := w. We finally show that, for fixed z ∈ z, Jz (·) is linear: if u, v ∈ b, α, β ∈ R, we have Jz (αu + βv), v = z, [αu + βv, v ] = αz, [u, v ] + βz, [v, v ] = αJz (u), v + βJz (v), v = αJz (u) + βJz (v), v ∀ v ∈ b. Then we have Jz (αu + βv) − αJz (u) − βJz (v) ∈ b⊥ ∩ b = {0}. Moreover, for fixed v ∈ b, the map J(·) (v) : z → b, z → Jz (v), is a linear map. Indeed, if z, z ∈ z and α, β ∈ R, we have J(αz+βz ) (v), v = αz + βz , [v, v ] = αz, [v, v ] + βz , [v, v ] = αJz (v), v + βJz (v), v = αJz (v) + βJz (v), v ∀ v ∈ b. Thus J(αz+βz ) (v) = αJz (v) + βJz (v) as above. This shows that the map J(·) : z → End(b),

z → Jz ,

defined by (18.2) is well-posed and linear. (18.3)

Let us make some other simple but useful remarks on H-type algebras. Remark 18.1.7. Let g be an H-type algebra. With the above notation, we have: 1) is a nilpotent Lie algebra of step two; 2) g = b ⊕ z, [b, b] = z, [b, z] = {0}; hence, if G is an H-type group, then G is a step two Carnot group; 3) the notion of H-type algebra (respectively, of H-type group) is invariant under Lie algebra (respectively, Lie group) isomorphisms. Proof. 1) Since b is the orthogonal completion of z = {0} with respect to , , we have g = b ⊕ z. Let now g1 , g2 , g3 ∈ g. We have gi = vi + zi , with vi ∈ b and zi ∈ z, for every i = 1, 2, 3. Since z is the center of g, every zi commutes with any element of g. As a consequence, [g1 , [g2 , g3 ]] = [v1 + z1 , [v2 + z2 , v3 + z3 ]] = [v1 , [v2 , v3 ]] = 0,

18.1 Heisenberg-type Groups

685

since we have [v2 , v3 ] ∈ [b, b] = z. 2) It directly follows from the above argument. 3) Let g be an H-type algebra, let g∗ be a Lie algebra isomorphic to g and let ϕ : g∗ → g be a Lie algebra isomorphism. Since ϕ is a vector space isomorphism, the bilinear form g1∗ , g2∗ ∗ := ϕ(g1∗ ), ϕ(g2∗ ),

g1∗ , g2∗ ∈ g∗ ,

is an inner product in g∗ . Let z∗ := ϕ −1 (z). It is immediate to recognize that z∗ is the center of g∗ . Moreover, setting b∗ := (z∗ )⊥ (orthogonal completion of z∗ with respect to , ∗ ), we have b∗ = ϕ −1 (z⊥ ) = ϕ −1 (b). It follows that [b∗ , b∗ ] = [ϕ −1 (b), ϕ −1 (b)] = ϕ −1 ([b, b]) = ϕ −1 (z) = z∗ . For z∗ ∈ z∗ fixed, we consider the linear map Jz∗∗ : b∗ → b∗ defined as follows: if v ∗ ∈ b, Jz∗∗ (v ∗ ) is defined by the identity Jz∗∗ (v ∗ ), (v )∗ ∗ = z∗ , [v ∗ , (v )∗ ]∗

∀ (v )∗ ∈ b∗ .

It is immediate to verify that Jz∗∗ = ϕ −1 ◦ Jϕ(z∗ ) ◦ ϕ. Let now z∗ ∈ z∗ be such that z∗ , z∗ ∗ = 1. Then ϕ(z∗ ), ϕ(z∗ ) = 1. Thus Jz∗∗ is an orthogonal map. Proposition 18.1.8. Let g be an H-type algebra. With the above notation, Jz (v), v = 0, Jz (v), v = −v, Jz (v ), |Jz (v)| = |z| · |v|, Jz (v), Jz (v) = z, z · |v|2 , [v, Jz (v)] = |v|2 · z

(18.4a) (18.4b) (18.4c) (18.4d) (18.4e)

for every z, z ∈ z and for every v, v ∈ b. Proof. Let z, z ∈ z and v, v ∈ b be fixed. From (18.2) we have Jz (v), v = z, [v, v] = 0, which gives (18.4a). It follows that 0 = Jz (v + v ), v + v = Jz (v), v + Jz (v), v + Jz (v ), v + Jz (v ), v = Jz (v), v + Jz (v ), v, which gives (18.4b). If z = 0, we have Jz ≡ 0 and (18.4c) trivially follows. If z = 0, from (18.3) it follows that Jz = |z| Jz/|z| . In this case, by the definition of H-type algebra, we have that Jz/|z| is an orthogonal map. As a consequence,

686

18 Groups of Heisenberg Type

|Jz (v)| = |z| · |Jz/|z| (v)| = |z| · |v|. This proves (18.4c). Using (18.3), it is immediate to verify that we have Jz+z (v), Jz+z (v) = Jz (v), Jz (v) + Jz (v), Jz (v) + 2 Jz (v), Jz (v). Thus, from (18.4c) it follows that 1 (Jz+z (v), Jz+z (v) − Jz (v), Jz (v) − Jz (v), Jz (v)) 2 1 = (|z + z |2 · |v|2 − |z|2 · |v|2 − |z |2 · |v|2 ) = z, z · |v|2 , 2

Jz (v), Jz (v) =

which gives (18.4d). Finally, we have z , [v, Jz (v)] = Jz (v), Jz (v) = z , z · |v|2 = z , |v|2 · z. The first equality follows from (18.2), the second equality follows from (18.4d). As a consequence, we have z , [v, Jz (v)] − |v|2 · z = 0

∀ z ∈ z.

Since |v|2 · z ∈ z and [v, Jz (v)] ∈ [b, b] = z, we immediately get (18.4e).

18.2 A Direct Characterization of H-type Groups The main aim of this section is to prove that (under a suitable system of coordinates) the group law of any H-type group has a somewhat explicit form (see (18.6) below). Roughly speaking, following our definitions from Section 3.6 (page 169), any abstract H-type group is naturally isomorphic to a prototype H-type group. Moreover, we furnish several useful details on H-type groups. By means of the natural identification of a Carnot group with its Lie algebra (see Section 2.2 on page 121 for details), we already proved that (see Theorem 3.2.2, page 160, for the precise statement) any Carnot group of step two is canonically isomorphic to RN equipped with the following group law (N = m + n, x (1) ∈ Rm , x (2) ∈ Rn ) (1) (1) xj + yj , j = 1, . . . , m (1) (2) (1) (2) , (x , x ) ◦ (y , y ) = (2) (2) xj + yj + 12 B (j ) x (1) , y (1) , j = 1, . . . , n (18.5) where the B (j ) ’s are m × m linearly independent skew-symmetric matrices. Let now G be a general H-type group according to Definition 18.1.1. We set m := dim(z⊥ ) and n := dim(z). Since G has step two and since the stratification of the Lie algebra g is evidently z⊥ ⊕ z (see Remark 18.1.7), in the sequel we shall fix on G a system of coordinates

18.2 A Direct Characterization of H-type Groups

687

(x, t), and we shall suppose that the dilations on G are δλ (x, t) = (λx, λ2 t) and that the group law has the form xj + ξj , j = 1, . . . , m (x, t) ◦ (ξ, τ ) = , tj + τj + 12 U (j ) x, ξ , j = 1, . . . , n (18.6) x, ξ ∈ Rm , t, τ ∈ Rn , for suitable skew-symmetric matrices U (j ) ’s. Our main aim is to give a necessary and sufficient condition on the U (j ) ’s such that the composition law (18.6) defines on Rm+n an H-type group. A complete answer is given by the following result. Theorem 18.2.1 (Characterization). G is an H-type group if and only if G is (isomorphic to) Rm+n with the group law (18.6) and the matrices U (1) , . . . , U (n) have the following properties: 1) U (j ) is an m × m skew-symmetric and orthogonal matrix for every j ≤ n; 2) U (i) U (j ) + U (j ) U (i) = 0 for every i, j ∈ {1, . . . , n} with i = j . Remark 18.2.2. Rm+n equipped with the group law (18.6) (the matrices U (j ) ’s being as in Theorem 18.2.1) will be referred to as a prototype H-type group. Theorem 18.2.1 states that any H-type group is naturally isomorphic to a prototype H-type group. indepenRemark 18.2.3. Conditions 1) and 2) imply that U (1) , . . . , U (n) are linearly dent. Namely, as it will appear from the proof, if 1) and 2) hold, then ns=1 zs U (s) is the product of |z| times an orthogonal matrix. For example, the following three matrices ⎛ ⎞ ⎛ 0 −1 0 0 0 0 ⎜1 0 0 0 ⎟ ⎜ 0 0 (1) (2) U =⎝ U =⎝ ⎠, 0 0 0 −1 −1 0 0 0 1 0 0 1 ⎛ ⎞ 0 0 0 1 0 1 0⎟ ⎜ 0 U (3) = ⎝ ⎠ 0 −1 0 0 −1 0 0 0

⎞ 1 0 0 −1 ⎟ ⎠, 0 0 0 0

satisfy conditions 1)–2) and, together with (18.6), define on R7 an H-type group whose center has dimension 3. Proof (of Theorem 18.2.1). First of all, we prove the “only if” part of the theorem. Let G be an H-type group: we make use of all the notation in Definition 18.1.1. Let B1 , . . . , Bm and Z1 , . . . , Zn be orthonormal bases for b := z⊥ and z, respectively. (s) The hypothesis [b, b] = z ensures that there exist scalars Ui,j such that [Bi , Bj ] =

n s=1

(s)

Uj,i Zs

∀ i, j ∈ {1, . . . , m}.

(18.7)

688

18 Groups of Heisenberg Type

With the notation in (18.7), we define the following square matrices (s) U (s) := Ui,j i,j ≤m ,

s = 1, . . . , n.

(18.8)

Since [Bi , Bj ] = −[Bj , Bi ], and recalling that Z1 , . . . , Zn is a basis of z, U (s) is skew-symmetric. We now recall that the group (G, ◦) is canonically isomorphic (via the exponential map) to its Lie algebra g (endowed with the Campbell–Hausdorff operation (X, Y ) → X + Y + 12 [X, Y ]). As a consequence, by means of the fixed basis B1 , . . . , Bm , Z1 , . . . , Zn on g, we can identify G to g ≡ Rm+n with the group are as in (18.8). law (18.6), where matrices U (j ) ’s the m We fix v = i=1 vi Bi and z = ni=1 zi Zi . We look for x1 , . . . , xm ∈ R such m that J z (v) = i=1 xi Bi satisfies condition (18.1) of Definition 18.1.1 for every m w = i=1 wi Bi . Precisely, m m m x j wj = xi Bi , wi Bi = Jz (v), w = z, [v, w] j =1

= = =

i=1 n

i=1

zi Zi ,

i=1 n

zi Zi ,

i,j =1

n

m

zi Zi ,

wj

j =1

=

m

i=1

m

m

vi Bi ,

wj

j =1

v i wj

n

(s) Uj,i Zs

s=1

m

(r) vi Uj,i zs

r,s=1 i=1 n m

wj Bj

vi wj [Bi , Bj ]

i,j =1 n

m j =1

i=1

i=1

=

m

Zr , Zs

(s) vi Uj,i zs

(18.9)

.

s=1 i=1

Since w is arbitrary, this gives xj =

(s)

vi Uj,i zs ,

s≤n i≤m

whence Jz : b → b,

m j =1

vj Bj →

m j =1

(s) vi Uj,i zs

Bj .

s≤n i≤m

As a consequence, w.r.t. the orthonormal basis B1 , . . . , Bm for b, the endomorphism Jz is represented by the following matrix

18.2 A Direct Characterization of H-type Groups

⎛

(s)

··· .. . ···

U1,1 ⎜ . zs ⎝ .. (s) s=1 Um,1

n

(s) ⎞ U1,m n .. ⎟ = zs U (s) . . ⎠ (s) s=1 Um,m

689

(18.10)

By Definition 18.1.1, this matrix has to be orthogonal, whenever ns=1 (zs )2 = 1. In particular, this implies that every U (s) is orthogonal (whence property 1) of the assertion is proved). We explicitly remark that since U (s) is both skew-symmetric and orthogonal, U (s) has no real eigenvalues, whence m is necessarily even. Moreover, −U (s) U (s) = Im (Im denotes the unit matrix of order m). If ns=1 (zs )2 = 1, the matrix ns=1 zs U (s) is orthogonal if and only if Im =

n

zs U

(s)

·

s=1

=−

r≤n

= Im −

n s=1

zr2 U (r) U (r)

−

T zs U

(s)

=−

zr zs U (r) U (s)

r,s≤n

zr zs U

(r)

U (s)

r,s≤n, r=s

zr zs U (r) U (s) .

r,s≤n, r=s

We have here used the fact that (U (r) )T = −U (r) and −(U (r) )2 = U (r) · (−U (r) ) = U (r) · (U (r) )T = Im , since U (r) is skew-symmetric and orthogonal. Therefore, we have n

zr zs U (r) U (s) = 0

∀ z1 , . . . , zn :

r,s=1, r=s

n

zs2 = 1.

(18.11)

s=1

√ √ If in (18.11) we take z = (0, . . . , 1/ 2, . . . , 1/ 2, . . . , 0), we obtain U (i) U (j ) + U (j ) U (i) = 0

for every i, j ∈ {1, . . . , n} with i = j ,

(18.12)

which is property 2) of the assertion. Since we also have zr zs U (r) U (s) = zr zs (U (r) U (s) + U (s) U (r) ), r,s≤n, r=s

r,s≤n, r<s

(18.11) turns out to be equivalent to (18.12). We now prove the “if” part of the theorem. Let U (1) , . . . , U (n) be matrices having properties 1)–2) of the assertion. Suppose Rm+n is endowed with the composition law (18.6). It is immediately verified that ◦ defines a Lie group, nilpotent of step at most two, in which the identity is the origin and the inverse of (x, t) is (−x, −t). Moreover, δλ (x, t) = (λx, λ2 t) is a group of automorphisms. An easy computation

690

18 Groups of Heisenberg Type

shows that the vector field in the algebra g of G = (Rm+n , ◦) that agrees at the origin with ∂/∂xj (j = 1, . . . , m) is given by m n 1 (s) Uj,i xi (∂/∂ts ), (18.13) Xj = (∂/∂xj ) + 2 s=1

i=1

and that g is spanned by X1 , . . . , X m ,

∂/∂t1 , . . . , ∂/∂tn .

From (18.13) and the skew-symmetry of U (s) we obtain [Xi , Xj ] =

n

(s) Uj,i (∂/∂ts )

s=1

for every i, j ∈ {1, . . . , m}. Now, since U (1) , . . . , U (n) are linearly independent (see (1) (n) Remark 18.2.3), the dimension of the vector space spanned by (Uj,i , . . . , Uj,i ) as i, j ∈ {1, . . . , m} equals n. As a consequence, G is a homogeneous Carnot group. For every s = 1, . . . , n, we set Zs = ∂/∂ts . We claim that z, the center of g, is spanned by Z1 , . . . , Zn . Indeed, we suppose by contradiction that (for suitable scalars αi ’s) m i=1 αi Xi ∈ z, i.e. m n m (s) αi Xi , Xj = αi Uj,i Zs for every j ∈ {1, . . . , m}. 0= i=1

s=1 i=1

Since the Zs ’s are linearly independent, this means that m

(s)

αi Uj,i = 0

i=1

for every s ≤ n and every j ≤ m, i.e. α = (α1 , . . . , αm ) belongs to the kernel of the transpose matrix of U (s) for every s ≤ n. Since every U (s) is orthogonal, this is possible only if α = 0, which proves the claim. Finally, let , be the standard inner product on g w.r.t. the basis X1 , . . . , Xm , Z1 , . . . , Zn . For what has been proved above, we have z⊥ = span{X1 , . . . , Xm } and [z⊥ , z⊥ ] = span{Z1 , . . . , Zn } = z. By means of a computation analogous to (18.9), it is easy to recognize that properties 1) and 2) of the assertion ensure that, with the above choice of , , G is an H-type group according to Definition 18.1.1. Indeed, for a fixed z = nj=1 zj Zj , setting

18.2 A Direct Characterization of H-type Groups

v :=

m

w :=

vi Xi ,

i=1

m

691

wi X i ,

i=1

we have

z, [v, w] = = =

n

zi Zi ,

i=1 n

zi Zi ,

i,j =1

n

m

zi Zi ,

m

wj

j =1

(Zr , Zs = δr,s ) =

m

i=1

m

vi Xi ,

wj

j =1

m

wj X j

vi wj [Xi , Xj ]

v i wj

i,j =1

n

(s) Uj,i Zs

s=1

n m

(s) vi Uj,i zr

r,s=1 i=1 n m

j =1

i=1

i=1

=

m

(r) vi Uj,i zr

Zr , Zs

.

r=1 i=1

As a consequence, by definition of , , setting Jz (v) :=

m

xj X j

j =1

with xj :=

n m

(r)

vi Uj,i zr ,

r=1 i=1

we have m

Jz (v), w = z, [v, w]

for every w = i=1 wi Xi . Repeating exactly the same computations previously done, we see that if U (1) , . . . , U (n) are orthogonal matrices satisfying U (r) · U (s) + U (s) · U (r) = 0 for every r, s ∈ {1, . . . , n} with r = s, then the map m

vj Xj → Jz (v)

j =1

defines an orthogonal endomorphism on span{X1 , . . . , Xm } for every choice of z = n n 2 j =1 zj Zj such that j =1 |zj | = 1. This completes the proof. From the explicitness of the operation (18.6) we can derive some more properties of H-type groups. We first give the explicit form of the canonical sub-Laplacian.

692

18 Groups of Heisenberg Type

Proposition 18.2.4. With the notation in the proof of Theorem 18.2.1, the canonical sub-Laplacian on the H-type group G is given by 1 2 ∂ |x| Δt + U (s) x, ∇x , 4 ∂ts n

ΔG = Δx +

(18.14)

s=1

where the U (s) are as in Theorem 18.2.1. Here we used the notation m n ∂ 2 ∂ 2 ∂ ∂ . , Δt = , ∇x = ,..., Δx = ∂xj ∂ts ∂x1 ∂xm j =1

s=1

Moreover, on functions u(x, t) = u(|x|, t), ΔG has the form 2 1 ∂ m−1 ∂ 1 ΔG = Δx + |x|2 Δt = + + r 2 Δt , r = |x| = 0. (18.15) 4 ∂r r ∂r 4 m 2 Proof. Since the canonical sub-Laplacian is ΔG = j =1 Xj , where Xj is given by (18.13), one has m 2 m n m n ∂ 2 1 (s) ∂ 1 (s) xi Uj,i + Uj,j ΔG = Δx + 4 ∂ts 2 ∂ts j =1 s=1

m 1 + 4

=

j =1 s=1

i=1

n

(r)

j,h,k=1 r,s=1, r=s m n 2 (s) ∂ + xi Uj,i ∂xj ∂ts i,j =1 s=1 n 1 (s) 2 ∂ 2 Δx + |U x| 4 ∂ts s=1 n 1 (r) (s)

+

2

U

(s)

xh xk Uj,h Uj,k

x, U

x

r,s=1, r<s

+

∂2 ∂tr ∂ts

m n

(s)

xi Uj,i

i,j =1 s=1

∂2 ∂xj ∂ts

∂2 ∂tr ∂ts

m n 2 1 (s) ∂ = Δx + |x|2 Δt + xi Uj,i 4 ∂xj ∂ts i,j =1 s=1

+

1 2

n

U (r) U (s) x, x

r,s=1, r<s

∂2 ∂tr ∂ts

m n 2 1 (s) ∂ = Δx + |x|2 Δt + xi Uj,i . 4 ∂xj ∂ts i,j =1 s=1 (s)

Here we used the following facts: Uj,j = 0 since U (s) is skew-symmetric; |U (s) x| = |x| since U (s) is orthogonal; from (18.12) we have

18.2 A Direct Characterization of H-type Groups

693

U (r) x, U (s) x = −U (s) U (r) x, x = U (r) U (s) x, x for every r = s. Again from (18.12) it follows that U (r) U (s) is skew-symmetric (for r = s) since (U (r) U (s) )T = (−U (s) )(−U (r) ) = U (s) U (r) = −U (r) U (s) , whence U (r) U (s) x, x = 0 for every r = s, since for every skew-symmetric matrix A we have A x, x = 0. This proves the first part of Proposition 18.2.4. We now prove the second part. The third differential summand in the right-hand side of (18.14) vanishes on functions u(|x|, t). Indeed, we have m n 2 (s) ∂ xi Uj,i u(|x|, t) ∂xj ∂ts i,j =1 s=1 m n ∂ 2u 1 (s) (|x|, t) = 0, xi xj Uj,i = r ∂r∂ts i,j =1

s=1

since for every s ≤ n, symmetric.

m

(s) i,j =1 xi xj Uj,i

= U (s) x, x = 0, being U (s) skew-

Remark 18.2.5. Starting from (18.13), another direct computation shows m n 1 2 (s) ∂ u ∂ u 2 2 2 |∇G u| = |∇x u| + |x| · |∇t u| + xi Uj,i 4 ∂xj ∂ts = |∇x u|2 +

1 2 |x| · |∇t u|2 + 4

s=1 n s=1

i,j =1

(s) ∂u U x, ∇x u . ∂ts

Remark 18.2.6. Let (G, ◦) be an H-type group. Suppose that the center of the algebra of G has dimension 1. Then G is isomorphic to a Heisenberg group Hk (here k = 1 2 (dim(G) − 1)). Indeed, by Theorem 18.2.1 (and by the hypothesis n = 1) it is not restrictive to suppose that G is Rm+1 equipped with the group law 1 (1) (x, t) ◦ (ξ, τ ) = x + ξ, t + τ + U x, ξ . 2 Let M be an m × m non-singular matrix and consider the following bijection : G → Rm+1 , M

(x, t) → (M x, t).

is Rm+1 equipped with the composition Clearly, if G (M x, t) ∗ (M ξ, τ ) := M((x, t) ◦ (ξ, τ )),

694

18 Groups of Heisenberg Type

∗) is a Lie group isomorphism. It is then sufficient to show : (G, ◦) → (G, then M ∗) is isomorphic to a Heisenberg group. In order to that there exists M such that (G, prove it, we notice that (M −1 ξ, τ) ( x , t) ∗ ( ξ, τ) = M x , t) ◦ (M −1 1 (1) −1 −1 −1 −1 =M M τ + U M x + M ξ, t + x, M ξ 2 1 = x + ξ , t + τ + M U (1) M −1 x, ξ 2 It is known that (see, e.g. [HJ85, Corollary 2.5.14]) every skewif ( x , t), ( ξ, τ ) ∈ G. symmetric orthogonal matrix is congruent to a block diagonal matrix of the type 0 −1 0 −1 J = diag ,..., . 1 0 1 0 Hence, if P is a non-singular matrix such that P T U (1) P = J and if we take M = P −1 , Remark 18.2.6 is proved. Remark 18.2.7. We end this section with the following natural question. Let Rm+n be equipped with a homogeneous Carnot group structure G = (Rm+n , δλ , ◦) by the usual dilations δλ (x, t) = (λx, λ2 t) and the usual operation xj + ξj , j = 1, . . . , m (x, t) ◦ (ξ, τ ) = , tj + τj + 12 B (j ) x, ξ , j = 1, . . . , n x, ξ ∈ Rm , t, τ ∈ Rn , (18.16) for suitable skew-symmetric linearly independent matrices B (j ) . We aim to answer to the following question. When is G an H-type group? Obviously, we cannot say that this happens if and only if 2 the B (j ) ’s satisfy conditions 1) and 2) of Theorem 18.2.1, for this is only the characterization of prototype H-type groups, but not the characterization of all H-type groups. (Consider the Heisenberg– Weyl group as a counterexample.) The proof of Theorem 18.2.1 helps us finding the answer to this question. Indeed, let G be an H-type group. Suppose B = {B1 , . . . , Bm , Z1 , . . . , Zn } is any fixed basis for the algebra g of G, which is an orthonormal basis w.r.t. an inner product endowing g with a structure of H-type algebra (see Definition 18.1.1, page 681). Moreover, identify G to g (via the exponential map) and then identify g to Rm+n (via coordinates w.r.t. B). Denote by H this last group, which is obviously isomorphic to G. We demonstrated in due course of the proof of Theorem 18.2.1 that H is a prototype H-type group. 2 The “if” part is true, the “only if” is not.

18.3 The Fundamental Solution on H-type Groups

695

Roughly speaking, if G is an H-type group, then a suitable choice of a new basis of g turns G into a prototype H-type group. As we already studied the action of a change of basis in the algebra of a homogeneous Carnot group in Remark 2.2.20 (precisely, see page 153), we can infer that3 there exist two non-singular matrices U (of order m × m) and V (of order n × n) such that the matrices n (i) T (j ) wi,j B U := U · · U, j =1

where V −1 = (wi,j )i,j ≤n , fulfill the requirements 1)–2) of Theorem 18.2.1. We thus have proved the following result. Corollary 18.2.8. The homogeneous Carnot group G = (Rm+n , δλ , ◦) with ◦ as in (18.16) is an H-type group if and only if there exist two non-singular matrices U (of order m × m) and V (of order n × n) such that the matrices n (i) T (j ) U := U · wi,j B · U, j =1

where V −1 = (wi,j )i,j ≤n fulfill the requirements 1)–2) of Theorem 18.2.1. In particular, if the second layer of the stratification of G is one-dimensional (i.e. if n = 1), then G is an H-type group if and only if there exists w ∈ R such that the matrices w B (j ) ’s are simultaneously congruent to a set of matrices U (j ) ’s as in 1)–2) of Theorem 18.2.1. Unfortunately, this result does not seem very operative in order to answer to the proposed question.

18.3 The Fundamental Solution for Sub-Laplacians on H-type Groups Throughout this section, (G, ◦) will denote a fixed H-type group with the Lie algebra g. Moreover, we shall denote by , the given inner product on g as in Definition 18.1.1. It is not restrictive to suppose that, if z is the center of g, b is the orthogonal completion of z and N1 := dim(b), N2 := dim(z), then G is a homogeneous Carnot group on RN1 +N2 with the dilations δλ (x) = δλ (x (1) , x (2) ) = (λx (1) , λ2 x (2) ),

x (1) ∈ RN1 , x (2) ∈ RN2 .

We finally set N := N1 +N2 , and we observe that Q = N1 +2N2 is the homogeneous dimension of G. Under our hypotheses, we have N2 ≥ 1 and thus Q ≥ 4. 3 This fact is proved in details at the cited page 153.

696

18 Groups of Heisenberg Type

We fix an orthonormal basis X1 , . . . , XN1 of b and an orthonormal basis Z1 , . . . , ZN2 of z. Then X1 , . . . , XN1 , Z1 , . . . , ZN2 is an orthonormal basis of g, and we have v=

N1 v, Xj Xj ,

N1 v, Xj 2

|v|2 =

j =1 N2 z= z, Zj Zj ,

N2 |z| = z, Zj 2 2

j =1

|Xj |2 = N1 ,

j =1

(18.17a)

∀ z ∈ z,

(18.17b)

j =1

|v + z|2 = |v|2 + |z|2 N1

∀ v ∈ b,

j =1

N2

∀ v ∈ b,

∀ z ∈ z,

|Zj |2 = N2 .

(18.17c) (18.17d)

j =1

The aim of this section is to find the fundamental solution for the sub-Laplacian L :=

N1

Xj2 .

j =1

To this end, we start by introducing the following functions on G: v : G → b,

N1

v(x) :=

Log (x), Xj Xj ,

j =1

z : G → z,

N2 Log (x), Zj Zj . z(x) := j =1

From the definition it immediately follows that v and z are characterized by the following property: ∀ x ∈ G,

x = Exp (v(x) + z(x)),

v(x) ∈ b,

z(x) ∈ z.

(18.18)

We want to prove the following result, due to A. Kaplan [Kap80]. Theorem 18.3.1 (A. Kaplan [Kap80]). Let X1 , . . . , XN1 be an orthonormal basis of b, and let L be the sub-Laplacian L=

N1

Xj2 .

j =1

Then, with the above notation, there exists a positive constant c such that the function (2−Q)/4 Φ(x) := c · |v(x)|4 + 16|z(x)|2 is the fundamental solution for L.

18.3 The Fundamental Solution on H-type Groups

697

Remark 18.3.2. We stress that Theorem 18.3.1 gives a fundamental solution for L = N1 2 j =1 Xj only when X1 , . . . , XN1 is an orthonormal basis of b and not for any subLaplacian L on the H-type group G. We observe that d(x) := (|v(x)|4 + 16|z(x)|2 )1/4 is a symmetric homogeneous norm on G. Indeed, from the definition of v(x) and z(x) it immediately follows d ∈ C ∞ (G \ {0}) ∩ C(G) and v(δλ (x)) =

N1

N1 λLog (x), Xj Xj = λv(x),

Log (δλ (x)), Xj Xj =

j =1

z(δλ (x)) =

N2

j =1 N2 Log (δλ (x)), Zj Zj = λ2 Log (x), Zj Zj = λ2 z(x).

j =1

j =1

Therefore, d(δλ (x)) = λd(x) for every λ > 0. Arguing analogously, we prove that v(x −1 ) = −v(x) and z(x −1 ) = −z(x), which give d(x −1 ) = d(x). Remark 18.3.3. When G is a prototype H-type group as in Theorem 18.2.1, from Theorem 18.3.1 we get the following more explicit formula for the fundamental solution Φ of the canonical sub-Laplacian ΔG in (18.14). Namely, with the notation of Theorem 18.2.1, the fundamental solution of the operator 1 2 ∂ |x| Δt + U (s) x, ∇x , 4 ∂ts n

ΔG = Δx +

s=1

is given by Φ(x) := c · (|x|4 + 16|t|2 )(2−Q)/4 , for some positive constant c. Proof m (of2 Remark 18.3.3). Let X1 , . . . , Xm be defined as in (18.13), so that ΔG = j =1 Xj , and let Zi = ∂ti , i = 1, . . . , n. Then {X1 , . . . , Xm } is an orthonormal basis of b and {X1 , . . . , Xm , Z1 , . . . , Zn } is an orthonormal basis of g (see the proof of Theorem 18.2.1). By means of Theorem 18.3.1, we are only left to prove that |v(x, t)| = |x|,

|z(x, t)| = |t|.

(18.19)

To this end, we need to investigate the exponential map Exp : g → G. For a fixed (a, b) ∈ Rm+n , we have (by the definition of Exp ) m n aj X j + bi Zi = γ (1), Exp j =1

where γ solves

i=1

698

18 Groups of Heisenberg Type

γ˙ (s) =

m

j =1 aj Xj (γ (s)) +

n

i=1 bi Zi (γ (s)),

γ (0) = 0. Setting γ (s) = (x(s), t (s)), we get ⎧ x˙ (s) = a , j ⎨ j t˙i (s) = bi + 12 U (i) x(s), a, ⎩ x(0) = 0, t (0) = 0, so that xj (s) = s aj and s t˙i (s) = bi + U (i) a, a = bi , 2 since U (i) is skew-symmetric. Therefore, ti (s) = s bi , and we finally get m n aj X j + bi Zi = γ (1) = (x(1), t (1)) = (a, b). Exp j =1

i=1

As a consequence, we get Log : G → g,

(x, t) →

m

xj X j +

n

j =1

and then v(x, t) =

m

xj X j ,

j =1

z(x, t) =

ti Zi

i=1 n

ti Zi

i=1

which finally give (18.19). Remark 18.3.4. We refer to Example 5.4.7 on page 250 for another, more direct proof of Remark 18.3.3 above. Proof (of Theorem 18.3.1). We introduce a family of regular functions Φε approximating Φ, and we compute L(Φε ). If ε > 0 and x ∈ G, we set (2−Q)/4 . Φε (x) := c · (|v(x)|2 + ε 2 )2 + 16|z(x)|2 It is clear (being Exp and Log analytic functions) that Φε is an analytic function for every fixed ε > 0, and Φ is analytic on G\{0}. From the definition of the exponential map we have L(Φε )(x) =

N1 j =1

Xj2 (Φε )(x) =

N1 2 d Φε (x ◦ Exp (tXj )). dt t=0 j =1

18.3 The Fundamental Solution on H-type Groups

699

Let us fix ε > 0 and x ∈ G. In order to compute L(Φε )(x), we need to compute the first and the second derivatives of 2 j = 1, . . . , N1 . φj (t) := |v(x ◦ Exp (tXj ))|2 + ε 2 + 16|z(x ◦ Exp (tXj ))|2 , We observe that φj (0) = (|v(x)|2 + ε 2 )2 + 16|z(x)|2 does not depend on j ; then we set φ(0) := φj (0). In the above notation, setting also k := (Q − 2)/4, we then have L(Φε )(x) =

N1 2 N1 d d −k −kφj (t)−k−1 φj (t) c (φ (t)) = c j dt dt t=0 t=0 j =1

j =1

= c k(k + 1)

N1

φj (0)−k−2 (φj (0))2 − c k

j =1

= c k φ(0)

φj (0)−k−1 φj (0)

j =1

−k−2

N1

(k + 1)

N1

(φj (0))2

j =1

− φ(0)

N1

φj (0)

.

j =1

We now fix j = 1, . . . , N1 and t ∈ R. We look for simple expressions for v(x ◦ Exp (tXj )),

z(x ◦ Exp (tXj )).

We recall that, since G is a step two nilpotent group, the following Campbell– Hausdorff formula holds: 1 ∀ A, B ∈ g. Exp (A) ◦ Exp (B) = Exp A + B + [A, B] 2 Thus we have Exp v(x ◦ Exp (tXj )) + z(x ◦ Exp (tXj )) (by (18.18)) = x ◦ Exp (tXj ) = Exp (Log (x)) ◦ Exp (tXj ) (by Campbell–Hausdorff) t = Exp Log (x) + tXj + [Log (x), Xj ] (by (18.18)) 2 t = Exp v(x) + z(x) + tXj + [v(x) + z(x), Xj ] (z(x) ∈ z, ∀ x ∈ G) 2 t = Exp v(x) + tXj + z(x) + [v(x), Xj ] . 2 Comparing the first term of this equality to the last one and observing that v(x) + tXj ∈ b, from (18.18) we get

t z(x) + [v(x), Xj ] ∈ z, 2

700

18 Groups of Heisenberg Type

v(x◦Exp (tXj )) = v(x)+tXj ,

t z(x◦Exp (tXj )) = z(x)+ [v(x), Xj ]. (18.20a) 2

Hence we have 2 t φj (t) = |v(x) + tXj |2 + ε 2 + 16|z(x) + [v(x), Xj ]|2 2 2 = |v(x)|2 + t 2 + 2 t v(x), Xj + ε 2 t2 2 2 + 16 |z(x)| + |[v(x), Xj ]| + t z(x), [v(x), Xj ] . 4 This gives

φj (t) = 2 |v(x)|2 + t 2 + 2 t v(x), Xj + ε 2 · (2t + 2 v(x), Xj ) t + 16 |[v(x), Xj ]|2 + z(x), [v(x), Xj ] ; 2

φj (0) = 8v(x), Xj 2 + 4(|v(x)|2 + ε 2 ) + 8|[v(x), Xj ]|2 ; φj (0) = 4(|v(x)|2 + ε 2 ) · v(x), Xj + 16 z(x), [v(x), Xj ] = 4(|v(x)|2 + ε 2 ) · v(x), Xj + 16 Jz(x) (v(x)), Xj = 4(|v(x)|2 + ε 2 ) · v(x) + 4 Jz(x) (v(x)), Xj . In particular, we obtain N1 N1 (φj (0))2 = 16 (|v(x)|2 + ε 2 ) · v(x) + 4 Jz(x) (v(x)), Xj 2 j =1

j =1

(by (18.17a))

= 16 |(|v(x)|2 + ε 2 ) · v(x) + 4 Jz(x) (v(x))|2

(by (18.4a))

= 16 (|v(x)|2 + ε 2 )2 · |v(x)|2 + 162 |Jz(x) (v(x))|2

(by (18.4c))

= 16 |v(x)|2 {(|v(x)|2 + ε 2 )2 + 16 |z(x)|2 } = 16 |v(x)|2 φ(0);

N1

φj (0) = 8

j =1

N1

v(x), Xj 2 + 4 N1 (|v(x)|2 + ε 2 ) + 8

j =1

(by (18.17a))

N1

|[v(x), Xj ]|2

j =1

= 8 |v(x)|2 + 4 N1 (|v(x)|2 + ε 2 ) + 8

N1

|[v(x), Xj ]|2 .

j =1

Now [v(x), Xj ] ∈ [b, b] = z and thus, from (18.17b) we have N1 N1 N1 N2 N2 [v(x), Xj ]2 = Zi , [v(x), Xj ]2 = JZi (v(x)), Xj 2 j =1

=

j =1 i=1

j =1 i=1

N2 N1

N2

i=1 j =1

i=1

JZi (v(x)), Xj 2 =

|Zi |2 |v(x)|2 = N2 |v(x)|2 .

18.3 The Fundamental Solution on H-type Groups

701

(In the fourth equality, we used (18.17a) and (18.4c).) As a consequence, N1

φj (0) = 8 |v(x)|2 + 4 N1 (|v(x)|2 + ε 2 ) + 8 N2 |v(x)|2

j =1

= 16(k + 1) |v(x)|2 + 4 N1 ε 2 , and then L(Φε )(x) = c k φ(0)−k−2 16(k + 1) |v(x)|2 φ(0)

− 16(k + 1) |v(x)|2 φ(0) − 4 N1 ε 2 φ(0)

= −4 N1 c k ε 2 φ(0)−k−1 (−2−Q)/4 = −4 N1 c k ε 2 (|v(x)|2 + ε 2 )2 + 16|z(x)|2 .

(18.20b)

For ε = 0, we clearly have LΦ(x) = 0

∀ x = 0.

(18.20c)

Moreover, from Φ = c d 2−Q we immediately get Φ ∈ L1loc (G).

(18.20d)

In order to prove Theorem 18.3.1, we are only left to show that LΦ = −Dirac0 (Dirac0 denotes the Dirac mass at the origin) in the sense of distributions. Let us first observe that, since v and z are δλ -homogeneous of degree one and two, respectively, we immediately get (18.20e) L(Φε )(δε (x)) = ε −Q L(Φ1 )(x). We now observe that (−2−Q)/4 L(Φ1 )(x) = −c (Q − 2) N1 (|v(x)|2 + 1)2 + 16|z(x)|2 is a C ∞ function on G whose norm is bounded at infinity by a function δλ homogeneous of degree −2 − Q. Thus L(Φ1 )(x) has finite integral on G, and we can choose c > 0 such that G

L(Φ1 )(x) dx = −1.

(18.20f)

Let us fix f ∈ C0∞ (G). We want to prove that G

Φ(x) Lf (x) dx = −f (0).

By dominated convergence, since |Φε | ≤ Φ ∈ L1loc , we immediately get lim

ε→0+ G

Φε (x) Lf (x) dx =

G

Φ(x) Lf (x) dx.

(18.20g)

702

18 Groups of Heisenberg Type

On the other hand, since Φε ∈ C ∞ (G), f has compact support and L is self-adjoint, we have G

Φε (x) Lf (x) dx

=

G

f (x) LΦε (x) dx

(by (18.20e) with the change of variable x = δε (z)) =

G

ε→0+

−→

G

f (δε z) LΦ1 (z) dz f (0) LΦ1 (z) dz = −f (0)

(by (18.20f)).

This proves (18.20g) and completes the proof of Theorem 18.3.1. Remark 18.3.5. In the above proof (see (18.20b)), we have incidentally proved that, for every ε > 0, the function (2−Q)/4 x → (|v(x)|2 + ε 2 )2 + 16|z(x)|2 is, up to a multiplicative constant, a solution of the semilinear differential equation −Lu = u(Q+2)/(Q−2) .

18.4 H-type Groups of Iwasawa-type We begin with the following definition. Definition 18.4.1 (H-type Iwasawa group). Let H be an H-type group, and let, as usual, h = b ⊕ z be the decomposition of its algebra, as in Definition 18.1.1 (page 681). For any Z ∈ z, let JZ be the endomorphism of b defined in (18.1). Then H is called an H-type Iwasawa group (in the sequel, an Iwasawa group) if, for every B ∈ b and for every Z, Z ∈ z with Z⊥Z , there exists Z ∈ z (depending on B, Z, Z ) such that (18.21) JZ (JZ (B)) = JZ (B). In literature, condition (18.21) is referred to as the “J 2 -condition” (see, e.g. [CDKR91,CK84]). Remark 18.4.2 (Hn is an Iwasawa group). Any Heisenberg–Weyl group Hk is an H-type group of Iwasawa-type. Indeed, we know that (see Remark 3.6.5, page 172) the classical Heisenberg–Weyl group Hk on R2k+1 is canonically isomorphic to the (prototype) H-type group H corresponding to the case m = 2k, n = 1 and 0 −Ik (1) B = . Ik 0 In turn, H is obviously an Iwasawa group since its center z is one-dimensional, and if Z, Z ∈ z are orthogonal, then at least one of them is the null vector, so that (for every B ∈ b) we have

18.4 H-type Groups of Iwasawa-type

703

JZ (JZ (B)) = 0 and (18.21) is satisfied by choosing Z = 0. This proves that any H-type group with one-dimensional center is an Iwasawa group. We now explicitly write the J 2 -condition (when H is a prototype H-type group; see Theorem 18.2.1 and Remark 18.2.2 for all the details) in terms of the usual matrices U (r) ’s defining the composition law in H. First, we fix orthonormal bases {B1 , . . . , Bm }, {Z1 , . . . , Zn } of b and z, respectively, and we recall that, if Z = n r=1 zr Zr ∈ z, then we have J : b → b, m Z n m m (r) vj Bj = vi Uj,i zr Bj . JZ j =1

j =1

(18.22)

r=1 i=1

More explicitly, JZ is represented (w.r.t. the basis {B1 , . . . , Bm } of b) by the m × m matrix n zr U (r) . r=1

Now, if we write B=

m

vj Bj ,

Z=

j =1

n

zr Zr ,

Z =

r=1

n

zr Zr ,

Z =

r=1

n

zr Zr ,

r=1

then it is easy4 to see that (18.21) writes as ! n m n zr zr = 0 ∀ v ∈ R , ∀ z, z ∈ R : ⇒ ∃ z ∈ Rn :

r=1

(zs zr − zr zs )U (s) U (r) v =

1≤s

n

! zr U (r) v .

r=1

(18.23) Example 18.4.3 (An H-type non-Iwasawa group). Let us consider type group H on R6 defined by taking m = 4, n = 2 and ⎛ ⎞ ⎛ 0 1 0 0 0 0 −1 0 ⎜ −1 0 0 0 ⎟ ⎜0 0 (1) (2) U =⎝ ⎠, U = ⎝ 0 0 0 1 1 0 0 0 0 −1 0 0 −1 0

the prototype H⎞ 0 1⎟ ⎠. 0 0

4 The reader is invited to verify (18.23) by making use of the well-known properties of the matrices U (r) ’s

(U (r) )2 = −Im , and (since Z⊥Z ) nr=1 zr zr = 0.

U (r) U (s) = −U (s) U (r) (r = s)

704

18 Groups of Heisenberg Type

More explicitly, H has the composition law ⎞ ⎞ ⎛ ⎞ ⎛ x 1 + ξ1 ξ1 x1 x 2 + ξ2 ⎟ ⎜ x2 ⎟ ⎜ ξ2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ x 3 + ξ3 ⎟ ⎜ x3 ⎟ ⎜ ξ3 ⎟ ⎜ ⎟. ⎜ ⎟◦⎜ ⎟=⎜ x 4 + ξ4 ⎟ ⎜ x4 ⎟ ⎜ ξ4 ⎟ ⎜ ⎝ ⎠ ⎝ ⎠ ⎝ t + τ + 1 (x ξ − x ξ + x ξ − x ξ ) ⎠ t1 τ1 1 1 2 1 1 2 4 3 3 4 2 t2 τ2 t2 + τ2 + 12 (−x3 ξ1 + x4 ξ2 + x1 ξ3 − x2 ξ4 ) ⎛

Then it is easy to see that (18.23) has no solution z ∈ R2 if we choose v = (0, 0, 1, 0) ∈ R4 ,

Z = (1, 0) ∈ R2 ,

Z = (0, 1).

Hence H is an H-type group which is not an Iwasawa group. Note 18.4.4. If g is a simple Lie algebra of rank one and n is the Lie algebra in the so-called Iwasawa decomposition k⊕a⊕n of g (also referred to as the KAN decomposition), then (equipped with a suitable inner product) n is an H-type algebra satisfying the “J 2 -condition” (18.21), and vice versa. More precisely, by [CDKR91, Theorem 1.1], the Iwasawa n-components of simple Lie groups of real rank one are exactly the H-type algebras which satisfy the “J 2 -condition”. This motivates the name of Iwasawa-type. The importance of the Iwasawa-type groups appears naturally in dealing with Kelvin-type transforms, which we shall briefly treat in the next section (see [CDKR91]).

18.5 The Inversion and the Kelvin Transform on H-type Groups In the classical theory of harmonic functions, the map RN \ {0} x → σ (x) := −x/|x|2 ∈ RN \ {0} is the so-called inversion map (with respect to the unit sphere centered at the origin). This map is used to define the Kelvin-transform of a function u (defined, for simplicity, on RN with N ≥ 3), namely u∗ : RN \ {0} → R,

u∗ (x) := |x|2−N u(σ (x)).

As it is well known (a simple but tedious computation is enough to the purpose), if u is a solution to the classical Laplace equation on RN , then the same is true of u∗ on RN \ {0}. Unfortunately (at present), a well-behaved analogue of the Kelvin transform for an arbitrary Carnot group is not known, except for the case of H-type groups of

18.5 The H-inversion and the H-Kelvin Transform

705

Iwasawa-type. Indeed, in this section we explicitly write a suitable inversion map for a general H-type group: it turns out that the relevant Kelvin transform is wellbehaved (namely, preserves the L-harmonicity) only for H-type groups of Iwasawatype (hence, in particular, for Heisenberg–Weyl groups). We fix the notation: throughout the remaining of this section, H = (Rm+n , ◦, δλ ) is a fixed prototype H-type group (any general H-type group is naturally isomorphic to a prototype one, as we proved in Theorem 18.2.1) with usual coordinates (x, t), x = (x1 , . . . , xm ) ∈ Rm , t = (t1 , . . . , tn ) ∈ Rn , and the composition map 1 (1) 1 (n) (x, t) ◦ (ξ, τ ) = x + ξ, t1 + τ1 + U x, ξ , . . . , tn + τn + U x, ξ 2 2 where U (1) , . . . , U (n) are fixed m × m matrices with the following properties: (H1) U (r) is an m × m skew-symmetric and orthogonal matrix for every r ≤ n; (H2) U (r) U (s) = −U (s) U (r) for every r, s ∈ {1, . . . , n} with r = s. Moreover, the dilation group is given by δλ (x, t) = (λ x, λ2 t). As usual, ΔH denotes the canonical sub-Laplacian for H. We recall that (see Theorem 18.3.1 and Remark 18.3.3) the fundamental solution for ΔH has the form Γ = c d 2−m−2n for a suitable positive constant c and the following gauge on H d(x, t) = (|x|4 + 16 |t|2 )1/4 . It is known that d defines an actual distance on H, i.e. d satisfies the pseudo-triangle inequality (see Proposition 5.1.7, page 231) with the constant c = 1 (see, e.g. [Cyg81]). We are now ready to explicitly write the inversion map and the Kelvin transform. Definition 18.5.1 (H-inversion and H-Kelvin transform). Let H be an (prototype) H-type group. Following all the above notation, we set σ : H \ {0} → H \ {0}, |x|2 x − 4 nk=1 tk U (k) x t σ (x, t) := − ,− 4 . |x|4 + 16|t|2 |x| + 16|t|2

(18.24)

If σ is as above, for any function u : H → R we let u∗ : H \ {0} → R, u∗ (x, t) := d(x, t)2−m−2n u(σ (x, t)). We call σ the H-inversion map on H and u∗ the H-Kelvin transform of u.

(18.25)

706

18 Groups of Heisenberg Type

The remarkable rôle played by the H-Kelvin transform on Iwasawa-type groups is comparable to that of the classical Kelvin transform. The definitions of the inversion map and the Kelvin transform in the abstract setting of an (general) H-type group (not necessarily written in the exponential coordinates) are slightly more complicated. Our (non-restrictive) choice of prototype coordinates makes it much simpler to give the very explicit definitions in (18.24) and (18.25). We collect some easy properties of the H-inversion map in the following proposition. Proposition 18.5.2. Let σ be the H-inversion map defined in (18.24). We also write σ (x, t) = (σ (1) (x, t), σ (2) (x, t)) with the usual stratified splitting of the coordinates. Then, for every (x, t) ∈ H \ {0}, we have: 1) if | · | denotes the Euclidean norm (in Rm or Rn , accordingly), it holds |σ (1) (x, t)| =

|x| d 2 (x, t)

,

|σ (2) (x, t)| =

|t| d 4 (x, t)

;

(18.26)

2) for every λ > 0, we have σ (δλ (x, t)) = δ1/λ σ (x, t) ; 3) the inversion is involutive, i.e. σ (σ (x, t)) = (x, t); in particular, σ is a bijection of H \ {0} onto itself; 4) it holds 1 , (18.27) d(σ (x, t)) = d(x, t) whence σ maps the punctured d-ball Bd (0, 1) \ {0} = {(x, t) ∈ H : 0 < d(x, t) < 1} onto H \ Bd (0, 1), and vice versa, and maps the d-sphere ∂Bd (0, 1) onto itself. Proof. Throughout the proof, (x, t) is a fixed point of H \ {0} and (for the sake of brevity) we write d instead of d(x, t). 1) From the very definition of σ we have |σ (1) (x, t)| 2 |x| x − 4 nk=1 tk U (k) x = |x|4 + 16|t|2 n −4 2 (k) = d |x| x − 4 tk U x k=1 1/2 n n = d −4 |x|2 Im − 4 tk U (k) x, |x|2 Im − 4 tk U (k) x k=1

k=1

18.5 The H-inversion and the H-Kelvin Transform

=d

−4

|x| Im − 4 2

|x|2 Im + 4

tk U

n

|x|4 Im + 4

n

− 16

n k=1

tk U (k) · |x|2 Im − 4

n

tk U

(k)

1/2

x, x

1/2

tk U (k) x, x

k=1

tk U (k) − 4

k=1 n

· |x| Im − 4 2

k=1

= d −4

T (k)

k=1

= d −4

n

707

n

tk U (k)

k=1

1/2

tj tk U (k) U (j ) x, x

j,k=1

=d

−4

(|x|4 Im + 16 |t|2 Im )x, x1/2 = d −4 d 2 |x| = |x|d −2 ,

thus proving the first identity in (18.26). In the fifth equality, we used the skewsymmetry of the U (r) ’s; in the seventh equality, we used the following fact n j,k=1

= = =

tj tk U (k) U (j )

+

+

tj tk U (k) U (j )

1≤j =k≤n 1≤j
(18.28)

by the assumptions (H1)–(H2) on the matrices U (r) ’s defining an H-type group. As for the second identity in (18.26), we immediately have t (2) = |t| d −4 . |σ (x, t)| = 4 2 |x| + 16|t| This proves 1). 2) A simple computation. 3) From the results in (18.26), we infer σ

(1)

|σ (1) (x, t)|2 σ (1) (x, t) − 4 nk=1 σ (2) (x, t)k U (k) σ (1) (x, t) (σ (x, t)) = − |σ (1) (x, t)|4 + 16|σ (2) (x, t)|2 |x|2 d −4 σ (1) (x, t) + 4 nk=1 tk d −4 U (k) σ (1) (x, t) =− |x|4 d −8 + 16|t|2 d −8 n 2 (1) (k) (1) = − |x| σ (x, t) + 4 tk U σ (x, t) k=1

708

18 Groups of Heisenberg Type

=d

−4

+4

|x|

2

n

|x| x − 4 2

tk U

k=1

=d

|x| x − 16 4

tj U

|x| x − 4 2

−4

(j )

x

j =1

(k)

n

n

tj U

(j )

(k)

(j )

x

j =1 n

tj tk U

! U

x

j,k=1

= d −4 (|x|4 x + 16 |t|2 x) = d −4 d 4 x = x. To derive the sixth equality, we used the computations in (18.28). Moreover, it holds σ (2) (x, t) |σ (1) (x, t)|4 + 16|σ (2) (x, t)|2 −t d −4 = − 4 −8 = t. |x| d + 16|t|2 d −8

σ (2) (σ (x, t)) = −

This proves 3). 4) From (18.26) we have 1/4 d(σ (x, t)) = |σ (1) (x, t)|4 + 16 |σ (2) (x, t)|2 1/4 = |x|4 d −8 (x, t) + 16 |t|2 d −8 (x, t) = d −2 (x, t) (|x|4 + 16 |t|2 )1/4 = d −2 (x, t) d(x, t) = d −1 (x, t), which proves (18.27). The second part of what asserted in 4) immediately follows from (18.27) and the results in 3). This ends the proof. We end the section by stating the following result on the H-Kelvin transform in the setting of Iwasawa-type groups. We denote by ΔH the canonical sub-Laplacian of the H-type group H fixed at the beginning of the section. Theorem 18.5.3 (Cowling et al., [CDKR91], Theorem 4.2). The H-Kelvin transform of every ΔH -harmonic function is always ΔH -harmonic if and only if H is of Iwasawa-type. In this case, the following formula holds ΔH (u∗ (x, t)) = d −2−m−2n (x, t) (ΔH u)(σ (x, t))

(18.29)

for every u ∈ C 2 (H \ {0}) and every (x, t) ∈ H \ {0}. Remark 18.5.4. We explicitly remark that formula (18.29) and the “if” part of Theorem 18.5.3 could also be proved by a direct computation, since both the canonical sub-Laplacian ΔH and the H-Kelvin transform are explicitly written for our prototype groups (see (18.14), page 692, for the expression of ΔH ).

18.6 Exercises of Chapter 18

709

Bibliographical Notes. H-type groups have been introduced by A. Kaplan in 1980 (see [Kap80]). The definition we have given in this chapter is not exactly the original one given by Kaplan, but it is the one that is usually adopted in the most recent literature. It is not difficult to prove the equivalence of the two definitions. The inversion map and the Kelvin transform were first introduced on Heisenberg–Weyl groups by A. Korányi [Kor82] and then generalized to the H-type groups, see [CK84,CDKR91]. It is far from our scopes here to provide complete results and an exhaustive list of references from the existing literature on H-type groups and the related topics, which is nowadays still growing fast. The result asserting that the H-Kelvin transform of a ΔH -harmonic function is ΔH -harmonic if and only if H is of Iwasawa-type (which is not proved in this chapter) is demonstrated in the paper by M. Cowling, A. H. Dooley, A. Korányi and F. Ricci [CDKR91]. Some of the topics presented in this chapter also appear in [BU04a].

18.6 Exercises of Chapter 18 Ex. 1) Prove that (18.21) writes as (18.23) when H is an (prototype) H-type group. Ex. 2) Prove in details the assertions made in Example 18.4.3, page 703. Ex. 3) Verify that the group introduced in Example 18.4.3 is indeed an H-type group and that an orthonormal basis for its algebra is the Jacobian basis 1 X1 = ∂x1 + (x2 ∂t1 − x3 ∂t2 ), 2 1 X2 = ∂x2 + (−x1 ∂t1 + x4 ∂t2 ), 2 1 X3 = ∂x3 + (x4 ∂t1 + x1 ∂t2 ), 2 1 X4 = ∂x4 + (−x3 ∂t1 − x2 ∂t2 ), 2 T1 = ∂t1 , T2 = ∂t2 . Ex. 4) Verify that the H-inversion map σ for the group introduced in Example 18.4.3 is given by ⎛ 2 ⎞ |x| x1 + 4(−t1 x2 + t2 x3 ) ⎜ |x|2 x2 + 4(t1 x1 − t2 x4 ) ⎟ ⎜ 2 ⎟ ⎜ |x| x3 + 4(−t1 x4 − t2 x1 ) ⎟ σ (x, t) = −(|x|4 + 16 |t|2 )−1 ⎜ ⎟. ⎜ |x|2 x4 + 4(t1 x3 + t2 x2 ) ⎟ ⎝ ⎠ t1 t2

710

18 Groups of Heisenberg Type

Ex. 5) Write the explicit expression of the H-inversion map σ for the Heisenberg– Weyl group H1 . Ex. 6) Prove formula (18.29) for the Heisenberg–Weyl group H1 (using the calculus formulas given at the end of Ex. 6, Chapter 1, page 77). Ex. 7) Let X := {X1 , . . . , Xm } be a given set of smooth vector fields on RN . If Ω ⊆ G is an open set with smooth boundary, we recall that the characteristic set (related to X) of Ω is the set X-Char(Ω) := {x ∈ ∂Ω | Xi (x) ∈ Tx (∂Ω), i = 1, . . . , m}, Tx (∂Ω) being the tangent space to ∂Ω at the point x. Provide the details for the following result. Let G be an H-type group. Choose coordinates on G as in Theorem 18.2.1. 2 Let ΔG = m j =1 Xj be the canonical sub-Laplacian of G, as in Proposition 18.2.4. Finally, let X = {X1 , . . . , Xm }. Consider the general half-space Π of G Π = {(x, t) ∈ G | a, x + b, t > c}, where a ∈ Rm , b ∈ Rn and c ∈ R are fixed. Then Π possesses Xcharacteristic points if and only if b = 0. Moreover, any half-space with characteristic points can be left-translated into a half-space of the type {(x, t) ∈ G |b, t > 0}. Indeed, a point (x, t) ∈ ∂Π is characteristic if and only if (why? use also (18.13)) m n 1 (s) xi Ui,j bs = 0 for every j ≤ m, aj + 2 s=1

or equivalently,

(Ch)

i=1

⎧ a, x + b, t = c, ⎪ T ⎨ n (s) bs U x = −2a. ⎪ ⎩ s=1

If b = 0, then a = 0, and the second equation in (Ch) has clearly no solution (why?). If b = 0, then (why?) the matrix ( ns=1 bs /|b|U (s) )T is nonsingular. As a consequence, the second equation in (Ch) admits a solution x ∈ Rm . The characteristic set for Π is then given by {(x, t) ∈ G | x = x, b, t = c − a, x}. i.e. Π is Finally, let b = 0. If (ξ, τ ) ∈ G is fixed, we set Π = (ξ, τ ) ◦ Π, the set of all points ( x , t) ∈ G such that (why?) m n m m n 1 (s) bj ( tj + τ j ) + aj ξj + xj aj + bs ξi Ui,j > c. 2 j =1

j =1

j =1

s=1

i=1

18.6 Exercises of Chapter 18

711

There exists (why?) ξ ∈ Rm such that aj +

1 (s) bs ξ i Ui,j = 0 2 n

m

s=1

i=1

for every j ≤ m. Then, there exists (why?) τ ∈ Rn such that n j =1

bj τ j = c −

m

aj ξ j .

j =1

= {( = (ξ , τ )−1 ◦ Π, we have Π x , t) ∈ G |b, t > 0}. If Π Ex. 8) Let (G, ◦) be a H-type group. Fix z0 ∈ G. We denote by Z z0 the following vector field on G d Z z0 I (z) = ((hz0 ) ◦ z). dh h=0 Prove that, if z0 = (x 0 , t 0 ), then the above vector field Z z0 I (x, t) is given by (we follow our usual notation on ◦) xj0 , j = 1, . . . , m z0 Z I (x, t) = . tj0 + 12 x 0 , U (j ) x, j = 1, . . . , n Ex. 9) Denote by ξ = (z, t) = (x + iy, t) ≡ (x, y, t) the points of the Heisenberg group Hn = Cn × R ≡ R2n+1 . The group law on Hn is given by ξ ◦ ξ = (z + z , t + t + 2 Im(z¯z )). Define F : Hn → S2n+1 \ {(0, . . . , 0, −1)}, −t + i(1 − |z|2 ) 2iz , . F (ξ ) = F (z, t) = t + i(1 + |z|2 ) t + i(1 + |z|2 ) Here S2n+1 denotes the unit sphere of Cn+1 . The map F is a CR equivalence between the Heisenberg group and the sphere minus the south pole, which is the analogue in CR geometry of the stereographic projection. Define now K : Hn \ {0} → Hn \ {0} to be the map conjugate to the inversion in Cn+1 of S2n+1 through the equivalence F , i.e. K := F ◦ (−IdCn+1 ) ◦ F −1 . Prove that K has the following expression −iz t where d := (|z|4 + t 2 )1/4 . K(z, t) = ,− 4 , t + i|z|2 d

712

18 Groups of Heisenberg Type

Compare with the H-inversion map defined in (18.24). Finally compute the Jacobian determinant of K and recognize that det(JK (z, t)) =

1 d 2Q

,

where Q = 2n + 2 is the homogeneous dimension of Hn . Ex. 10) Let us consider the Heisenberg group Hn with the usual notation. An application ρ : Hn → Hn is called a rotation around the t-axis if there exists a complex unitary transformation F : Cn → Cn such that ρ(z, t) = (F (z), t). Let Ω be a half-space of Hn , and let us define Ω1 = {(x, y, t) ∈ R2n+1 | x1 > 0} and Ωt = {(x, y, t) ∈ R2n+1 | t > 0}. Prove that one of the two following cases always occurs: (i) there exists ξ0 ∈ Hn such that either Ω = τξ0 (Ωt ) or Ω = τξ0 (−Ωt ), (ii) there exist ξ0 ∈ Hn and a rotation ρ around the t-axis such that Ω = ρ(τξ0 (Ω1 )). Prove, in particular, that the first case occurs when ∂Ω and the t-axis intersect, the second one when they are parallel. Finally prove that ΔHn is invariant with respect to rotations around the t-axis, i.e. ΔHn (u ◦ ρ) = (ΔHn u) ◦ ρ for every rotation ρ around the t-axis. Ex. 11) Let us consider the Heisenberg group Hn with the usual notation, and let Ω = {ξ = (x, y, t) ∈ R2n+1 | x1 > 0}. For every ξ ∈ Ω, set r(ξ ) =

x1 , Bξ = Bd (ξ, r(ξ )), 2

K : Ω × Ω → R,

K(ξ, ξ ) = md

ψ(ξ, ξ ) χ (ξ ), r(ξ )Q Bξ

where χBξ is the characteristic function of Bξ and ψ(ξ, ξ ) =

|z − z |2 . d(ξ, ξ )2

For u ∈ L1loc (Ω), define K(ξ, ξ ) u(ξ ) dξ .

T u : Ω → R, (T u)(ξ ) = (Mr(ξ ) u)(ξ ) = Ω

(Compare to (5.50f) on page 259 where md was also introduced.)

18.6 Exercises of Chapter 18

713

a) Observe that T u = u for any ΔHn -harmonic function u and deduce that K(ξ, ξ ) dξ = 1 ∀ ξ ∈ Ω. Ω

b) Prove that the integration of K(ξ, ξ ) with respect to the variable ξ also yields a constant value, i.e. there exists a constant β such that K(ξ, ξ0 ) dξ = β Ω

for every ξ0 ∈ Ω. c) Recognize now that T is a bounded linear operator in Lp (Ω) for every p ∈ [1, +∞]. More precisely, T u∞ ≤ u∞ ∀ u ∈ L∞ (Ω), p p T up ≤ β up ∀ u ∈ Lp (Ω), ∀p ∈ [1, +∞[. Moreover, Tu = β Ω

u

∀ u ∈ L1 (Ω).

Ω

d) Prove that β > 1. e) Prove the following Liouville type theorem. If u is a non-negative ΔHn superharmonic function such that u ∈ L1 (Ω), then u ≡ 0 in Ω. f) Using Ex. 10, prove that the Liouville-type theorem stated in (e) holds for any half-space Ω of Hn whose boundary does not intersect the center of Hn . (Hint: In order to prove (b), use the invariance of Ω (and of r) under the left translations “parallel” to the boundary. To prove (d), test the solid mean value formula (5.51), page 259, with a suitable ΔHn -subharmonic function v (for instance, the function d −Q , suitably left translated) in order to obtain that, chosen a compact subset K of Ω with |K| > 0, there exists ε > 0 such that T v ≥ v + ε in K. Use then (c) and deduce (d). To prove (e), observe that u ≥ T u gives u≥ Ω

Tu = β Ω

u, Ω

and then use (d). All the details of this exercise can be found in [Ugu99].) Ex. 12) a) Prove that the Liouville-type theorem stated in Ex. 11 no longer holds on every Hn if we replace the hypothesis u ∈ L1 (Ω) with u ∈ Lp (Ω). b) Prove that the function −∂t (d(ξ )2−Q ) = n t (|z|4 + t 2 )−1− 2

n

714

18 Groups of Heisenberg Type

is ΔHn -harmonic away from the origin, and in the half-space {t > 1} it is positive and belongs to Lp for all p > 1 and also to the weak L1 . 1 and consider the funda(Hint: To prove (a), take p ∈ ]1, +∞], n > p−1 mental solution of −ΔHn with pole outside Ω.)

19 The Carathéodory–Chow–Rashevsky Theorem

The aim of this chapter is to furnish the proof of the so-called Carathéodory–Chow– Rashevsky theorem, i.e. the result asserting the connectedness of RN with respect to a family X of Hörmander vector fields. We shall restrict to the particular but significant case when the given vector fields are the generators of the first layer of a homogeneous Carnot group G. To this aim, we shall make use of two crucial tools. First, it will intervene the Carnot–Carathéodory distance related to X (already introduced in Section 5.2, page 232). Second, we shall exploit the deep properties of the Campbell–Hausdorff operation recalled in Section 14.2.1 (page 584). In particular, we shall make use of a result asserting that, roughly speaking, we can approximate any nested commutator [Xq , · · · [X2 , X1 ] · · ·] of the vector fields Xj ’s in the algebra of G by means of a -composition of the vector fields ±X1 , . . . , ±Xq (up to commutators of length ≥ q + 1). As an application of the connectivity theorem, we shall prove that any Carnot group G admits the decomposition G = Exp (g1 ) ∗ · · · ∗ Exp (g1 ) into a precise number of factors which are the images under the exponential map of the first layer g1 of the stratification of g (see Theorem 19.2.1). We shall make a crucial use of this result again in Chapter 20, where a suitable version of Lagrange mean value theorem in the stratified setting will be provided.

19.1 The Carathéodory–Chow–Rashevsky Theorem for Stratified Vector Fields In this section, we give a proof of the Carathéodory–Chow–Rashevsky connectivity theorem in the case of stratified vector fields. First of all, we recall the definitions of X-subunit path and of X-connectedness, given in Section 5.2, page 232.

716

19 The Carathéodory–Chow–Rashevsky Theorem

Definition 19.1.1 (X-subunit path). Let X = {X1 , . . . , Xm } be any family of vector fields in RN . A piecewise regular path γ : [0, T ] → RN is said to be X-subunit if γ˙ (t), ξ 2 ≤

m Xj I (γ (t)), ξ 2

∀ ξ ∈ RN ,

j =1

for almost every t ∈ [0, T ]. We denote by S(X) the set of all X-subunit paths, and we put l(γ ) = T if [0, T ] is the domain of γ ∈ S(X). We explicitly remark that every integral curve of ±Xj , j ∈ {1, . . . , m} is X-subunit. Definition 19.1.2 (X-connectedness. Carnot–Carathéodory distance). We say that RN is X-connected if and only if, for every x, y ∈ RN , there exists γ ∈ S(X), γ : [0, T ] → RN

such that γ (0) = x and γ (T ) = y.

If RN is X-connected, the following definition makes sense dX (x, y) := inf T > 0 | ∃ γ : [0, T ] → RN , γ ∈ S(X), γ (0) = x, γ (T ) = y for every x, y ∈ RN . Moreover, the function (x, y) → dX (x, y) is a metric1 on RN , called the X-control distance or the Carnot–Carathéodory distance related to X. In what follows, we shall simply write d instead of dX . We now specialize to the case of stratified vector fields. From now on, throughout this section, we always denote by G = (RN , ◦, δλ ) a fixed homogeneous Carnot group with the Lie algebra g. The dilations {δλ }λ>0 of G are given by δλ (x) = δλ (x (1) , . . . , x (r) ) = (λx (1) , . . . , λr x (r) ),

x (i) ∈ RNi ,

1 ≤ i ≤ r.

We shall denote by m := N1 the number of generators of G and by Q = N1 + 2 N2 + · · · + r Nr the homogeneous dimension of G. The following theorem then holds. Theorem 19.1.3 (Carathéodory–Chow–Rashevsky theorem). Let the preceding hypotheses and notation hold. Let us suppose that Z1 , . . . , Zm are vector fields in g such that g = Lie{Z1 , . . . , Zm } 1 See Proposition 5.2.3, page 232.

19.1 The Carathéodory–Chow–Rashevsky Theorem for Stratified Vector Fields

717

(for example, we can choose Z1 , . . . , Zm to be the first m vector fields of the Jacobian basis for g, i.e. the left invariant vector fields on G such that Zj (0) = (∂/∂xj )|0 , j = 1, . . . , m). Then, setting Z = {Z1 , . . . , Zm }, we have that G is Z-connected. Moreover, the Z-connectedness is given by paths that are piecewise integral curves of the vector fields Z1 , . . . , Zm . In particular, the Z-control distance d is well-defined. Finally, setting d0 := d(·, 0), d0 turns out to be a homogeneous norm2 on G. (Note. We explicitly remark that X-connectivity also holds when X is a system of vector fields in RN satisfying Hörmander’s rank condition. See the Bibliographical Notes at the end of this chapter for references.) A main step in the proof of Theorem 19.1.3 is Lemma 19.1.4 below which, roughly speaking, states that we can approximate any nested commutator [Xq , · · · [X2 , X1 ] · · ·] of vector fields Xj ’s in g by means of a composition of the vector fields ±X1 , . . . , ±Xq (up to commutators of length ≥ q + 1). The composition is meant in the sense of iterated application of the Campbell–Hausdorff operation (see Definition 2.2.11 on page 128; see also Section 14.2.1, page 584). More precisely, we have [Xq , · · · [X2 , X1 ] · · ·] = Xj1 · · · Xjc(q) + R(X1 , . . . , Xq ), where c(q) is an integer depending only on q (namely, c(q) = 3 · 2q−1 − 2) and not depending on the Xj ’s, and Xj1 , . . . , Xjc(q) ∈ {±X1 , . . . , ±Xq } can be chosen in a “universal” way, again not depending on the Xj ’s. Finally, the remainder R(X1 , . . . , Xq ) is a linear combination of brackets of height ≥ q +1 of the vector fields X1 , . . . , Xq , and the form of R can, once more, be chosen independently of the Xj ’s. In order to be more precise, we need to introduce some more notation and definitions from algebra. We leave the details to the interested reader. (See also Definition 15.2.1, page 600, and Section 1.1, page 3.) Let q ∈ N, let x1 , . . . , xq be non-commuting letters and let us denote by a the set of the formal words in the letters x1 , . . . , xq . For example, an element of a is x13 x2 x3 x14 x3 . Let also a be equipped with its natural structure of associative algebra3 (a, ·) over R and then of a Lie algebra (a, [·, ·]) by setting [x, y] := xy − yx for every x, y ∈ a. 2 See Section 5.1, page 229, for the definition of homogeneous norm on G. 3 For example, x x 2 x · x 3 x x = x x 2 x 4 x x and x x 2 · x 4 = x x 2 x 4 , etc. 1 3 4 1 3 4 1 7 1 3 1 3 7 7 4 1 7

718

19 The Carathéodory–Chow–Rashevsky Theorem

For every k ∈ N and any given multi-index J = (j1 , . . . , jk ) ∈ {1, . . . , q}k , we set xJ := [xj1 , · · · [xjk−1 , xjk ] · · ·]. We say that xJ is a commutator of length (or height) k of x1 , . . . , xq . If J = j1 , we also say that xJ := xj1 is a commutator of length 1 of x1 , . . . , xq . For any q, k ∈ N, we shall use the notation Pk [x1 , . . . , xq ] := span xJ = [xj1 , . . . [xjk−1 , xjk ] . . . ] : J ∈ {1, . . . , q}k to denote the vector space generated by the symbols xJ , J ∈ {1, . . . , q}k , i.e. the set of linear combinations with real coefficients of the formal objects xJ . We shall refer to Pk [x1 , . . . , xq ] as the set of formal Lie-polynomials homogeneous of length k on the (non-commuting) indeterminates x1 , . . . , xq , and xJ will be called a formal (nested) Lie-monomial (homogeneous of length k on x1 , . . . , xq ). For a fixed q ∈ N, any sum of elements in Pk [x1 , . . . , xq ] with a finite number of k’s will be referred to as a formal Lie-polynomial on the indeterminates x1 , . . . , xq . Let us now denote by h any (abstract) nilpotent Lie algebra. For example, h will be soon equal to g, the Lie algebra of a homogeneous Carnot group G. For any q, k ∈ N and every P ∈ Pk [x1 , . . . , xq ], we wish to give a precise meaning to P (X1 , . . . , Xq ), whenever X1 , . . . , Xq ∈ h. Let P ∈ Pk [x1 , . . . , xq ]. There exist a subset I of {1, . . . , q}k and real numbers aJ ’s (for every J ∈ I) such that aJ xJ . P = J ∈I

We set P (X1 , . . . , Xq ) :=

aJ X J

J ∈I

if X1 , . . . , Xq ∈ g, where ()

XJ := [Xj1 , · · · [Xjk−1 , Xjk ] · · ·]

if xJ = [xj1 , · · · [xjk−1 , xjk ] · · ·]. Note that the signs [·, ·] in () denote the Lie brackets in h. For example, if h = g, then, given P = [x1 , [x2 , x3 ]] − [x3 , [x4 , x1 ]] + 2[x2 , [x3 , x3 ]] ∈ P3 [x1 , . . . , x4 ] and given vector fields X1 , . . . , X4 ∈ g, we have P (X1 , . . . , X4 ) = [X1 , [X2 , X3 ]] − [X3 , [X4 , X1 ]] ∈ g. If r is the step of nilpotency of h, and s is any fixed integer in {1, . . . , r}, we denote by r Pk [x1 , . . . , xq ] k=s

19.1 The Carathéodory–Chow–Rashevsky Theorem for Stratified Vector Fields

719

the set of formal sums rk=s P (k) , where P (k) ∈ Pk [x1 , . . . , xq ]. If h = g, X1 , . . . , Xq ∈ g and Rs ∈ rk=s Pk [x1 , . . . , xq ], then Rs (X1 , . . . , Xq ) will denote a sort of “remainder”, a linear combination of brackets of height ≥ s of the vector fields X1 , . . . , Xq . Vice versa, recalling Proposition 1.1.7, page 12, any linear combination of brackets of height ≥ s ofthe vector fields X1 , . . . , Xq ∈ g can be written as Rs (X1 , . . . , Xq ) for some Rs ∈ rk=s Pk [x1 , . . . , xq ]. With the above notation at hand, the Campbell–Hausdorff operation in every nilpotent Lie algebra h can be written as X Y = X + Y + H2 (X, Y ) 1 = X + Y + [X, Y ] + H3 (X, Y ) 2 where H2 ∈

r

Pk [x1 , x2 ],

k=2

H3 ∈

r

∀ X, Y ∈ h,

(19.1)

Pk [x1 , x2 ]

k=3

depend4 only on the step of nilpotency r of h. Let us now henceforth fix h = g, the Lie algebra of the homogeneous Carnot group G, nilpotent of step r. We want to show some notable algebraic properties of the operation on g in the simple cases q = 2 and q = 3. • For q = 2, we claim that there exists R3 ∈

r

Pk [x1 , x2 ]

k=3

such that [Y, X] = Y X (−Y ) (−X) + R3 (X, Y )

∀ X, Y ∈ g

(19.2)

(we recall that is associative!). Indeed, (19.1) gives (Y X) ((−Y ) (−X)) 1 = Y X + (−Y ) (−X) + [Y X, (−Y ) (−X)] 2 + H3 (Y X, (−Y ) (−X)) 1 1 = Y + X + [Y, X] + H3 (Y, X) + −Y − X + [Y, X] + H3 (−Y, −X) 2 2 1 + [Y + X + H2 (Y, X), −(Y + X) + H2 (−Y, −X)] + P3 (X, Y ) 2 4 The “universality” of H , H follows from the “universality” of the Campbell–Hausdorff 2 3

operation (14.2), page 585.

720

19 The Carathéodory–Chow–Rashevsky Theorem

= [Y, X] + H3 (Y, X) + H3 (−Y, −X) 1 + ([Y + X, −(Y + X)] + P3 (X, Y )) + P3 (X, Y ) 2 = [Y, X] + P3 (X, Y ) for some P3 , P3 , P3 ∈ rk=3 Pk [x1 , x2 ] depending only on the step r of G. • For q = 3, we claim that there exists R4 ∈

r

Pk [x1 , x2 , x3 ]

k=4

such that [Z, [Y, X]] = Z Y X (−Y ) (−X) (−Z) X Y (−X) (−Y ) + R4 (X, Y, Z) for every X, Y, Z ∈ g. (19.3) In order to prove (19.3), we start by recalling that −W is the -inverse of W (for any W ∈ g), i.e. −W is the unique element of g such that (−W ) W = 0 = W (−W ). As a consequence, we get X Y (−X) (−Y ) = −(Y X (−Y ) (−X)),

(19.4)

since (X Y (−X) (−Y )) (Y X (−Y ) (−X)) = (X Y (−X)) ((−Y ) Y ) (X (−Y ) (−X)) = (X Y ) ((−X) X) ((−Y ) (−X)) = · · · = 0, and analogously (Y X (−Y ) (−X)) (X Y (−X) (−Y )) = 0. We now introduce the notation α(X, Y ) := Y X (−Y ) (−X)

∀ X, Y ∈ g,

(19.5)

so that (19.2) reads [Y, X] = α(X, Y ) + R3 (X, Y ) From (19.4) and (19.6) it follows that

∀ X, Y ∈ g.

(19.6)

19.1 The Carathéodory–Chow–Rashevsky Theorem for Stratified Vector Fields

721

Z (Y X (−Y ) (−X)) (−Z) (X Y (−X) (−Y )) = Z α(X, Y ) (−Z) (−α(X, Y )) = α(α(X, Y ), Z) = [Z, α(X, Y )] − R3 (α(X, Y ), Z) = [Z, [Y, X] − R3 (X, Y )] − R3 ([Y, X] − R3 (X, Y ), Z) = [Z, [Y, X]] + P4 (X, Y, Z), for some P4 ∈ rk=4 Pk [x1 , x2 , x3 ] depending only on the step r of nilpotency of g. This proves (19.3). Let us now consider the case of general q. Further (in particular, in (19.7) below) we use the following notation: if X1 , . . . , Xq is a family of vector fields, we agree to let ∀ i ∈ {1, . . . , q}. X−i := −Xi Lemma 19.1.4. Let the notation in the preceding paragraphs be fixed. Let q ∈ N, q ≥ 2, and set c(q) = 3 · 2q−1 − 2. Recall also that r denotes the step of the (fixed) Carnot group G. Then there exists an “indexing map” jq : {1, . . . , c(q)} → {−q, . . . , −1, 1, . . . , q} and there exists r

Rq+1 ∈

k=q+1 r

Rq+1 =

Pk [x1 , . . . , xq ], (q)

Rk ,

(q)

Rk

∈ Pk [x1 , . . . , xq ]

k=q+1

(we agree to let Rq+1 = 0 if q ≥ r) such that [Xq , · · · [X2 , X1 ] · · ·] = Xjq (1) · · · Xjq (c(q)) + Rq+1 (X1 , . . . , Xq ) (19.7) for every X1 , . . . , Xq ∈ g. For example: • when q = 2, we have (see (19.2)) c(2) = 4 and j2 : {1, 2, 3, 4} → {−2, −1, 1, 2}, j2 (1) = 2, j2 (2) = 1, j2 (3) = −2, j2 (4) = −1, so that Xj2 (1) Xj2 (2) Xj2 (3) Xj2 (4) = X2 X1 X−2 X−1 = X2 X1 (−X2 ) (−X1 );

722

19 The Carathéodory–Chow–Rashevsky Theorem

• when q = 3, we have c(3) = 10 and (see (19.3)) j3 : {1, . . . , 10} → {−3, −2, −1, 1, 2, 3}, j3 (1) = 3, j3 (2) = 2, j3 (3) = 1, j3 (4) = −2, j3 (5) = −1, j3 (6) = −3, j3 (7) = 1, j3 (8) = 2, j3 (9) = −1, j3 (10) = −2. Proof (of Lemma 19.1.4). (We explicitly remark that the proof makes a crucial use of the “universality” of the Campbell–Hausdorff operation .) We argue by induction on q ≥ 2. The case q = 2 follows from (19.2). Let us now prove the statement for q + 1 assuming it to be true for q. First of all, let us observe that, arguing as in (19.4), we get (−Xjq (c(q)) ) · · · (−Xjq (1) ) = −(Xjq (1) . . . Xjq (c(q)) ). Hence, recalling (19.5) and using (19.6), we have Xq+1 (Xjq (1) · · · Xjq (c(q)) ) (−Xq+1 ) ((−Xjq (c(q)) ) · · · (−Xjq (1) )) = α(Xjq (1) · · · Xjq (c(q)) , Xq+1 ) = [Xq+1 , Xjq (1) · · · Xjq (c(q)) ] − R3 (Xjq (1) · · · Xjq (c(q)) , Xq+1 )

= Xq+1 , [Xq , · · · [X2 , X1 ] · · ·] − Rq+1 (X1 , . . . , Xq ) + Pq+2 (X1 , . . . , Xq+1 ) = [Xq+1 , [Xq , · · · [X2 , X1 ] · · ·]] + Pq+2 (X1 , . . . , Xq+1 ), for some Pq+2 , Pq+2 ∈ of G. Since

r

k=q+2 Pk [x1 , . . . , xq+1 ]

depending only on the step r

2c(q) + 2 = 2(3 · 2q−1 − 2) + 2 = 3 · 2q − 2 = c(q + 1), the proof is complete. We are now ready to prove Theorem 19.1.3. Proof (of Theorem 19.1.3). Using g = Lie{Z1 , . . . , Zm } and recalling again Proposition 1.1.7 on page 12, we conclude that there exist multi-indices J1 ∈ {1, . . . , m}q1 , . . . , JN ∈ {1, . . . , m}qN ,

1 ≤ qi ≤ r,

such that g = span{ZJ1 , . . . , ZJN }.

(19.8a)

For every i ∈ {1, . . . , N }, we also consider the multi-index J−i := (Ji (1), . . . , Ji (qi − 2), Ji (qi ), Ji (qi − 1)) obtained by interchanging the last two entries of Ji , so that ZJ−i = −ZJi .

(19.8b)

19.1 The Carathéodory–Chow–Rashevsky Theorem for Stratified Vector Fields

723

By means of Lemma 19.1.4, for every i ∈ {−N, . . . , −1, 1, . . . , N } there exists an indexing map hi : {1, . . . , c(qi )} → {−m, . . . , −1, 1, . . . , m} (q−i := qi ) and there exists Wki ∈ Pk [x1 , . . . , xm ]

for every k ∈ {qi + 1, . . . , r}

such that, setting r

W i (t) := =

(q )

Rk i (t 1/qi ZJi (qi ) , . . . , t 1/qi ZJi (1) )

k=qi +1 r

t k/qi Wki (Z1 , . . . , Zm )

∀t ≥0

(19.8c)

k=qi +1

(we agree to let W i ≡ 0 if qi ≥ r), we have t ZJi = [t 1/qi ZJi (1) , . . . [t 1/qi ZJi (qi −1) , t 1/qi ZJi (qi ) ] . . .] = (t 1/qi Zhi (1) ) · · · (t 1/qi Zhi (c(qi )) ) + Rqi +1 (t 1/qi ZJi (qi ) , . . . , t 1/qi ZJi (1) ) = (t 1/qi Zhi (1) ) · · · (t 1/qi Zhi (c(qi )) ) +

r

(q )

Rk i (t 1/qi ZJi (qi ) , . . . , t 1/qi ZJi (1) )

k=qi +1

= (t

1/qi

+

Zhi (1) ) · · · (t 1/qi Zhi (c(qi )) )

r

(q )

t k/qi Rk i (ZJi (qi ) , . . . , ZJi (1) )

k=qi +1

= (t

1/qi

+

Zhi (1) ) · · · (t 1/qi Zhi (c(qi )) )

r

t k/qi Wki (Z1 , . . . , Zm )

k=qi +1

= t 1/qi Zhi (1) · · · (t 1/qi Zhi (c(qi )) ) + W i (t)

(19.8d)

for every t ≥ 0. As usual, we agree to let Z−j := −Zj for every j ∈ {1, . . . , m}. For every i ∈ {−N, . . . , −1, 1, . . . , N}, we now define Ei : G × [0, +∞[ → G, Ei (x, t) := (exp(t 1/qi Zhi (c(qi )) ) ◦ · · · ◦ exp(t 1/qi Zhi (1) ))(x).

(19.8e)

(Here ◦ denotes the usual composition of maps.) Let us recall that, by means of Theorem 15.1.1, we have

724

19 The Carathéodory–Chow–Rashevsky Theorem

exp(Y ) exp(X)(x) = exp(X Y )(x)

∀ X, Y ∈ g, ∀ x ∈ G.

(19.8f)

As a consequence, we can write

Ei (x, t) = exp t 1/qi Zhi (1) · · · t 1/qi Zhi (c(qi )) (x)

= exp tZJi − W i (t) (x). We have used here (19.8d). We now recall5 formula (1.9), page 8, exp(X)(x) =

r 1 k X I (x) k!

∀ X ∈ g, ∀ x ∈ G.

(19.8g)

k=0

Taking X = tZJi − W i (t), we obtain r

k 1 Ei (x, t) = x + tZJi − W i (t) I (x) + tZJi − W i (t) I (x) k! k=2

= x + tZJi I (x) + ωi (x, t),

(19.8h)

for some remainder function ωi : G × [0, +∞[ → G. From the very definition (19.8c) of W i it follows that such remainder function has the following regularity properties:

ωi ∈ C 1 G × [0, +∞[, G ,

ωi (x, t), ωi ∈ C G × [0, +∞[, G , (19.8i) ωi (x, t) = t 1+1/qi (∂t ωi )(x, 0) = 0. Roughly speaking, formula (19.8h) above states that a translation along a commutator ZJ of arbitrary length can be approximated by a composition of a finite number of elementary translations along ±Z1 , . . . , ±Zm , at least for small positive times. We now want to take into account negative times too (see formula (19.8l) below). For every i ∈ {1, . . . , N }, we set if t ≥ 0, ωi (x, t) ri (x, t) := ri : G × R → G, ω−i (x, −t) if t < 0. From (19.8i) it follows that ri has the following regularity properties: ri ∈ C 1 (G × R, G), ri (x, t), ri (x, t) = |t|1+1/(2qi ) (∂t ri )(x, 0) = 0.

ri ∈ C(G × R, G),

(19.8j)

5 We explicitly remark that (19.8g) is a particular case of (1.9), page 8, because X j I ≡ 0 for every j > r, since (for those j ’s) X j is a sum of differential operators δλ -homogeneous of degree > r, whereas the component functions of the identity map I are δλ -homogeneous

of degree at most r.

19.1 The Carathéodory–Chow–Rashevsky Theorem for Stratified Vector Fields

725

For every i ∈ {1, . . . , N }, we now define Bi : G × R → G,

Bi (x, t) :=

Ei (x, t) E−i (x, −t)

if t ≥ 0, if t < 0.

(19.8k)

Recalling (19.8h) and (19.8b), we can write Bi (x, t) = x + tZJi I (x) + ri (x, t)

∀ (x, t) ∈ G × R.

(19.8l)

As a consequence, it turns out that Bi ∈ C 1 (G × R, G). Moreover, we have Bi (x, 0) = x, JBi (x, 0) = (IN , ZJi I (x)).

(19.8m)

We now fix x0 ∈ G, and we define F : RN → RN ≡ G,

F (t) := BN . . . B2 (B1 (x0 , t1 ), t2 ) . . . , tN .

(19.8n)

We have F ∈ C 1 (RN , RN ), JF (t) = JBN (BN −1 (. . . B2 (B1 (x0 , t1 ), t2 ) . . . , tN −1 )) · diag[. . . diag[JB2 (B1 (x0 , t1 ), t2 ) diag[∂t1 B1 (x0 , t1 ), 1], 1] . . . , 1], where, for any p × q matrix A, we have denoted by diag[A, 1] the (p + 1) × (q + 1) matrix A 0 diag[A, 1] := . 0 1 Using (19.8m), we obtain JF (0) = JBN (x0 , 0) diag[. . . diag[JB2 (x0 , 0) diag[ZJ1 I (x0 ), 1], 1] . . . , 1] = (IN , ZJN I (x0 )) · diag[. . . diag[(IN , ZJ2 I (x0 )) diag[ZJ1 I (x0 ), 1], 1] . . . , 1] = (ZJ1 I (x0 ), . . . , ZJN I (x0 )). As a consequence, recalling the choice of J1 , . . . , JN (see (19.8a)), we get det JF (0) = 0, and F is a diffeomorphism from a neighborhood V of the origin to a neighborhood W of x0 = F (0).

726

19 The Carathéodory–Chow–Rashevsky Theorem

In particular, for every y ∈ W , there exists t ∈ V such that y = F (t). We now observe that (by the definition of F , see (19.8n), (19.8k), (19.8e)) F (t) is the final point of a (Z1 , . . . , Zm )-subunit path (recall Definition 19.1.1 of subunit path) starting from x0 . Furthermore, such a path is piecewise an integral curve of the vector fields Z1 , . . . , Zm . As a consequence, there exists a neighborhood of x0 whose points are (Z1 , . . . , Zm )-connected to x0 along paths that are piecewise integral curves of the vector fields Z1 , . . . , Zm . A standard argument now allows to pass from local to global connectivity. We set Z = (Z1 , . . . , Zm ) and, once x0 ∈ RN is fixed, we consider the set E of points Z-connected to x0 along paths that are piecewise integral curves of the vector fields of Z. Such a set E is non-empty, since x0 ∈ E. Moreover, E is open. Indeed, if y ∈ E, from the above arguments it follows that there exists an open neighborhood of y whose points are Z-connected to y along paths that are piecewise integral curves of the vector fields of Z. Therefore, x0 is Z-connected to any point of such a neighborhood of y. Finally, E is closed. In fact, let yn be a sequence in E such that yn → y. By the above argument, there exists an open neighborhood W of y whose points are Z-connected to y along paths that are piecewise integral curves of the vector fields of Z. Moreover, there exists n¯ ∈ N such that yn¯ ∈ W . As a consequence, x0 is Zconnected to y (passing through yn¯ ) along paths that are piecewise integral curves of the vector fields of Z, and then y ∈ E. This completes the proof of the (Z1 , . . . , Zm )connectivity of G. We now want to prove the second statement of Theorem 19.1.3, i.e. that d0 = d(·, 0) is a homogeneous norm on G. By means of Proposition 5.2.6, page 234, we only need to prove that d0 is continuous. For every i ∈ {−N, . . . , −1, 1, . . . , N }, we have d(x, Ei (x, t)) ≤ c(qi ) t 1/qi .

(19.8o)

Indeed, for every fixed t ≥ 0, the point Ei (x, t) is connected to x by a (Z1 , . . . , Zm )subunit path, which is piecewise an integral curve of one of the vector fields Z1 , . . . , Zm . Each piece of this path connects two points which have d-distance ≤ t 1/qi . Hence, from the triangle inequality for d we get d(x, Ei (x, t)) ≤

c(q i )

t 1/qi = c(qi ) t 1/qi .

j =1

Recalling the definition of F (see (19.8n), (19.8k)) and setting x1 := B1 (x0 , t1 ),

x2 := B2 (x1 , t2 ),

...,

xN := BN (xN −1 , tN ) = F (t),

from the triangle inequality we also get (for |t| ≤ 1) d(x0 , F (t)) ≤ d(x0 , x1 ) + d(x1 , x2 ) + · · · + d(xN −1 , xN ) ≤ c(q1 ) |t1 |1/q1 + c(q2 ) |t2 |1/q2 + · · · + c(qN ) |tN |1/qN ≤ c |t|1/r .

19.2 An Application of Carathéodory–Chow–Rashevsky Theorem

727

Therefore, |d0 (x) − d0 (x0 )| = |d(x, 0) − d(x0 , 0)| ≤ d(x0 , x) = d(x0 , F (F −1 (x))) ≤ c|F −1 (x)|1/r −→ c|F −1 (x0 )|1/r = 0 as x → x0 . This completes the proof.

19.2 An Application of Carathéodory–Chow–Rashevsky Theorem As an application of Theorem 19.1.3, we have the following result, which has an interest in its own. Theorem 19.2.1. Let (G, ∗) be a homogeneous Carnot group, let g1 be the first layer of a stratification of the algebra g of G and let Z = {Z1 , . . . , Zm } be a basis for g1 . Finally, fix a homogeneous norm d on G. Then there exists an absolute constant M ∈ N (only depending on G) such that G = Exp (g1 ) ∗ · · · ∗ Exp (g1 ) . M times

More precisely, for every x ∈ G, there exist x1 , . . . , xM ∈ Exp (g1 ) with the following properties: x = x1 ∗ · · · ∗ xM ,

(19.9a)

d(xj ) ≤ c0 d(x) for all j = 1, . . . , M, (19.9b) for every j = 1, . . . , M, there exist tj ∈ R and ij ∈ {1, . . . , m} such that xj = Exp (tj Zij ). (19.9c) Here c0 > 0 is a constant depending on G, d and Z, but not depending on x and of the xj ’s. Before proving Theorem 19.2.1, we need a simple preliminary remark (used also in the proof of Theorem 20.3.1, page 746). Remark 19.2.2. Let us recall Definition 2.2.6, page 125. Let g = g1 ⊕ · · · ⊕ gr be a stratification of the algebra g of a homogeneous Carnot group G. For every (k) (k) k = 1, . . . , r, let us fix a basis for gk , say X1 , . . . , XNk (where Nk = dim(gk )). As usual, we say that (1)

(1)

(r)

(r)

{X1 , . . . , XN } := {X1 , · · · , XN1 , · · · , X1 , · · · , XNr } is a basis of g adapted to the stratification. For every ξ = (ξ1 , . . . , ξN ) = (ξ (1) , . . . , ξ (r) ), we set

728

19 The Carathéodory–Chow–Rashevsky Theorem

ξ ·X =

N

ξj X j =

j =1

Nk r

(k)

(k)

ξj X j .

(19.10a)

k=1 j =1

If the dilations on G are indifferently denoted by δλ (x) = (λσ1 x1 , . . . , λσN xN ) = (λx (1) , . . . , λr x (r) ), we can define a dilation on g (see Definition 1.3.24, page 46) still denoted by δλ , extending by linearity the map such that δλ (Xj ) := λσj Xj for every j = 1, . . . , N . In other words, we have N Nk N r δλ ξj X j = ξj λσ j X j = ξj(k) λk Xj(k) . j =1

j =1

k=1 j =1

Following (19.10a), this can be rewritten as δλ (ξ · X) = δλ (ξ ) · X.

(19.10b)

Now, in (4.12) (page 196), we have proved that, for every x ∈ G, ξ ∈ RN and λ > 0, we have

δλ exp(ξ · X)(x) = exp((δλ ξ ) · X)(δλ (x)). Following (19.10b), this also reads as

δλ exp(ξ · X)(x) = exp δλ (ξ · X) (δλ (x)).

(19.10c)

In particular, if we take as ξ the i-th coordinate vector of the standard basis for RN , this gives

δλ exp(Xi )(x) = exp λσi Xi (δλ (x)). (19.10d) Here λ > 0, x ∈ G and (due to the arbitrariness of the adapted basis) Xi can be any left-invariant (non identically vanishing) vector field in g which is δλ -homogeneous of degree σi . As a consequence, if λ = t 1/σi , we have

δt 1/σi exp(Xi )(x) = exp t Xi (δt 1/σi (x)) for all t > 0 and x ∈ G, so that, if d is any homogeneous norm on G, we have

d exp(t Xi )(δt 1/σi (x)) = t 1/σi d exp(Xi )(x) .

(19.10e)

Proof (of Theorem 19.2.1). We claim that the proof of this theorem is contained in the proof of the Carathéodory–Chow–Rashevsky Theorem 19.1.3. We closely analyze that proof. We know that the following map (take x0 = 0 in (19.8n)) F : RN → RN ≡ G,

F (t) := BN . . . B2 (B1 (0, t1 ), t2 ) . . . , tN

19.2 An Application of Carathéodory–Chow–Rashevsky Theorem

729

is a diffeomorphism from a neighborhood V of the origin to a neighborhood W of the origin. It is not restrictive to replace V by 1/q V1 := V ∩ t ∈ G : |t1 |1/q1 + · · · + tN N < 1 and take W1 := F (V1 ). (We notice that the qi ’s and the map F itself depend only on the structure of G.) We have (see (19.8k) and (19.8e)): for every 1 ≤ i ≤ N, Bi (x, ti ) is the composition of c(qi ) = 3 · 2qi −1 − 2 exponential maps (starting from x) along one of the following vector fields 1/q 1/q ±ti i Z1 , . . . , ±ti i Zm . But since 1 ≤ qi ≤ r for every 1 ≤ i ≤ N, we can infer that F (t) is the composition of at most M := N (3 · 2r−1 − 2) ≥

N N (3 · 2qi −1 − 2) = c(qi ) i=1

i=1

vector fields of the above type. Thus (setting for brevity τi := 0 if needed) any ζ ∈ W1 can be written as ζ = exp(τM ZiM )(· · · exp(τ2 Zi2 )(exp(τ1 Zi1 )(0)) · · ·),

(19.11a)

where i1 , . . . , iM ∈ {1, . . . , m} and |τi | < 1 are suitably chosen. We set ζ1 := exp(τ1 Zi1 )(0),

ζk := exp(τk Zik )(ζk−1 ),

k = 2, . . . , M. (19.11b)

By the well-known formula exp(Z)(x) = x ∗ Exp (Z), from (19.11b) we have ζk := ζk−1 ∗ Exp (τk Zik ) (here ζ0 := 0), so that from (19.11b) we also derive ζ = ζM = ζM−1 ∗ Exp (τM ZiM ) = ζM−2 ∗ Exp (τM−1 ZiM−1 ) ∗ Exp (τM ZiM ) = [· · ·] = Exp (τ1 Zi1 ) ∗ · · · ∗ Exp (τM−1 ZiM−1 ) ∗ Exp (τM ZiM ). As a consequence, we have the decomposition ζ = ξ1 ∗ · · · ∗ ξM ,

where ξj = Exp (τj Zij ).

(19.11c)

We explicitly remark that, since the Zij ’s all belong to the first layer of g, from (19.10e) we get

d(ξj ) = d Exp (τj Zij ) = |τj | d Exp (±Zij ) . Recalling that |τj | < 1 and setting

αd := max d Exp (±Z1 ) , . . . , d Exp (±Zm ) , we have proved the estimate

730

19 The Carathéodory–Chow–Rashevsky Theorem

d(ξj ) ≤ αd

for every j = 1, . . . , M.

(19.11d)

Thus, (19.11c) and (19.11d) demonstrate (19.9a)–(19.9c) for points in W1 . Finally, take any x ∈ G. We shall use a simple δλ -homogeneity argument. Since W1 is a neighborhood of the origin, there exists r > 0 such that Bd (0, r) ⊂ W1 . Consequently, we have ζ := δr/d(x) (x) ∈ W1 . Thus, from what we proved above, one gets (setting λ := d(x)/r) x = δλ (ζ ) = δλ (ξ1 ∗ · · · ∗ ξM ) = δλ (ξ1 ) ∗ · · · ∗ δλ (ξM ) =: x1 ∗ · · · ∗ xM . We notice that (by using (19.10d))

xj = δλ (ξj ) = δλ Exp (τj Zij ) = Exp (λτj Zij ),

whence (19.9c) follows by setting tj = λτj . Moreover, it holds d(x) αd d(ξj ) ≤ d(x) r r (using (19.11d)), so that (19.9b) follows as well by taking c0 := αd /r. This ends the proof of the theorem. d(xj ) = d(δλ (ξj )) = λ d(ξj ) =

Bibliographical Notes. The connectivity Theorem 19.1.3 (indeed, a more general version of it) is due to W.L. Chow [Cho39] and P.K. Rashevsky [Ras38] (the version of this theorem in the R3 case and for two Hörmander vector fields is also known as Carathéodory’s theorem, see [Car09]). For modern proofs, see also [Bel96,Gro96, Her68,NSW85,VSC92]. For applications, see e.g., P. Hajłasz and P. Koskela [HK00]. We explicitly remark that X-connectivity also holds when X is a system of vector fields in RN satisfying Hörmander’s rank condition (see the references above). For a proof of Theorem 19.2.1, see also G.B. Folland and E.M. Stein [FS82, Lemma 1.40] and P. Pansu [Pan89, Corollary]. Theorem 19.2.1 is called “generating property” by V. Magnani [Mag06, Proposition 3.12] and is applied in [Mag06] to derive properties of convex functions in Carnot groups.

19.3 Exercises of Chapter 19 Ex. 1) Let Ω ⊆ RN be an open and connected set, and let X = {X1 , . . . , Xm } be any family of locally Lipschitz-continuous vector fields defined in Ω. Obviously, the Definition 5.2.1 of X-subunit path γ ∈ S(X) makes sense (see page 232). Given x, y ∈ Ω, we let, by definition, dX (x, y) = ∞ iff there does not exist any γ ∈ S(X), γ : [0, T ] → Ω, such that γ (0) = x and γ (T ) = y. Otherwise, dX (x, y) is defined as in Definition 5.2.2, page 232. Prove that Ω splits into a (possibly infinite and not denumerable) family of sets Ω = i Ωi such that:

19.3 Exercises of Chapter 19

731

a) x, y ∈ Ωi iff there exists γ ∈ S(X) connecting x and y; b) for every i, (Ωi , dX |Ωi ×Ωi ) is a metric space; c) if x ∈ Ωi and y ∈ Ωj with i = j , then dX (x, y) = ∞. Are the Ωi ’s open sets? With the above notation, let Ω = R2 , X = {∂x1 }. Prove that dX (x, y) < ∞ iff x and y both lie on the same line parallel to the x1 -axis. Find the relevant Ωi ’s. Ex. 2) Write down and prove a simplified version of Lemma 19.1.4 in the case of step two Carnot groups. Ex. 3) Write down explicitly the indexing map jq of Lemma 19.1.4 in the cases q = 4 and q = 5.

20 Taylor Formula on Homogeneous Carnot Groups

The aim of this chapter is to prove the Lagrange mean value theorem and, consequently, several versions of the Taylor formula for smooth functions on homogeneous Carnot groups. We begin with a brief sketch of polynomial functions and derivatives of a homogeneous Carnot group. Then, we introduce and investigate the Taylor polynomials and we prove suitable versions of the Lagrange mean value theorem and of the Taylor formula. The latter will be derived from the former by an adaptation of standard arguments, whereas the proof of the Lagrange mean value theorem will deeply rely on the Carathéodory–Chow–Rashevsky connectivity theorem in Chapter 19. Due to the importance of the Lagrange mean value theorem on stratified groups, we provide two proofs of it both relying on the cited Theorem 19.1.3. The first one makes also use of the Carnot–Carathéodory distance dZ , the second one does not involve dZ , but it makes use of some precise estimates in the connectivity theorem. Finally, Taylor formulas with remainder of Lagrange-type, of Peano-type and in integral form will be given. Throughout the chapter, we fix the following notation. G = (RN , δλ , ∗) is a homogeneous Carnot group nilpotent of step r, whose points are denoted indifferently by (20.1) x = (x1 , . . . , xN ) = (x (1) , . . . , x (r) ), where x (i) ∈ RNi . Following our usual notation for the dilations δλ : RN → RN , δλ (x) = (λσ1 x1 , . . . , λσN xN ) = (λ x (1) , . . . , λr x (r) ).

(20.2)

α will always denote a multi-index with N entries, i.e. α = (α1 , . . . , αN ) with α1 , . . . , αN ∈ N ∪ {0}. As usual, if α is a multi-index, we set α

x α = x1α1 · · · xNN ,

(Dx )α = ∂xα11 · · · ∂xαNN ,

and α! = α1 ! · · · αN !.

We shall also use the following notation |α| = α1 + · · · + αN ,

|α|G = α1 σ1 + · · · + αN σN

(20.3)

734

20 Taylor Formula on Carnot Groups

to denote, respectively, the Euclidean length (also said isotropic length) of α and the homogeneous length (also said δλ -length or G-length) of α. Furthermore, we denote by {Z1 , . . . , ZN } the Jacobian basis for the algebra g of G, i.e. for every j ∈ {1, . . . , N }, Zj is the left invariant vector field on G such that Zj |0 = ∂xj |0 . When ξ = (ξ1 , . . . , ξN ) ∈ RN , we use the useful notation ξ ·Z to denote N j =1 ξj Zj , the vector field in g whose Jacobian coordinates are given by (ξ1 , . . . , ξN ). Finally, we denote by Exp : g → G, Log : G → g, respectively, the usual exponential and logarithmic maps related to (G, ∗).

20.1 Polynomials and Derivatives on Homogeneous Carnot Groups 20.1.1 Polynomial Functions on G We begin with a very natural definition; although, despite this naturalness, we shall abandon it very shortly (after some due remarks). G-polynomial). A function P : G → R is called a G-polynomial Definition 20.1.1 (G if p := P ◦ Exp is a polynomial function on the vector space g, i.e. p is a polynomial function when expressed in coordinates w.r.t. any (or equivalently, w.r.t. at least one) basis for g. Since {Z1 , . . . , ZN } is a basis for g, the set of G-polynomials is given by P = p ◦ Log | p (ξ ) := p(ξ · Z) is a polynomial in ξ . We now recall formulas (1.75a) and (1.75b) (page and Log have the following coordinate form ⎛ ξ1 ξ2 + B2 (ξ1 ) ⎜ Exp (ξ · Z) = ⎜ .. ⎝ .

(20.4)

50), where we proved that Exp ⎞ ⎟ ⎟ ⎠

ξN + BN (ξ1 , . . . , ξN −1 ) x1 x2 + C2 (x1 ) ⎜ Log (x) = ⎜ .. ⎝ . ⎛

∀ ξ ∈ RN ,

⎞ ⎟ ⎟·Z ⎠

∀ x ∈ G,

(20.5)

xN + CN (x1 , . . . , xN −1 ) where the Bi ’s and Ci ’s are polynomial functions on RN (δλ -homogeneous of degree σi ) completely determined by the composition law on G. It is then immediately seen that

20.1 Polynomials and Derivatives on Homogeneous Carnot Groups

735

the set of G-polynomials coincides with the set of polynomial functions with respect to the fixed coordinate system (x1 , . . . , xN ) on G. Indeed, if Q is any polynomial function on G, then the function q := Q ◦ Exp is a polynomial on g, for (thanks to (20.5)) q(ξ · Z) = Q(Exp (ξ · Z)) = Q(ξ1 , ξ2 + B2 (ξ1 ), . . . , ξN + BN (ξ1 , . . . , ξN −1 )) is evidently a polynomial function in ξ . Vice versa, let P be a G-polynomial. Then (setting p = P ◦ Exp and following the notation in (20.4)) P (x) = (P ◦ Exp ◦ Log )(x) = p((x1 , . . . , xN + CN (x1 , . . . , xN −1 )) · Z) =p (x1 , x2 + C2 (x1 ), . . . , xN + CN (x1 , . . . , xN −1 )) is a polynomial function in x, since p and the Ci ’s are polynomials. If p = p(x) is any polynomial function in the fixed coordinates (x1 , . . . , xN ) on G, we set (also following notation (20.3))

deg(p) := max |α| : p(x) = cα x α with cα = 0 for every α , α

α cα x with cα = 0 for every α degG (p) := max |α|G : p(x) = α

to denote, respectively, the Euclidean degree (also said isotropic degree) of p and the homogeneous degree (also said δλ -degree or G-degree) of p. For any n ∈ N ∪ {0}, we give the following definition Pn := {p polynomial on G : degG (p) ≤ n}.

(20.6)

Obviously, Pn is a vector space over the field R. We denote its dimension by μn := dimR (Pn ).

(20.7)

For every n ∈ N ∪ {0}, we introduce the set of multi-indices In := {α multi-index in (N ∪ {0})N : |α|G ≤ n}.

(20.8)

With the above definitions, we have the following proposition, whose simple proof is left as an exercise. Proposition 20.1.2 (On the vector space Pn ). For every n ∈ N ∪ {0}, it holds Pn = span{x α : α ∈ In }, so that μn equals the cardinality of In . Moreover, if In has been ordered in some fixed way, then the map Pn → Rμn ,

p → ((Dx )α p(0))α∈In

is a vector space isomorphism. More explicitly, if β ∈ In , the above map sends x β into (0, . . . , β!, . . . , 0), where the only non-vanishing entry appears at the same place where β appears in the fixed ordering of In .

736

20 Taylor Formula on Carnot Groups

Example 20.1.3. If G = H1 is the Heisenberg–Weyl group on R3 (whose points are denoted by (x, y, t)) with dilations δλ (x, y, t) = (λx, λy, λ2 t), we have I0 I1 I2 I3 P0

= = = = =

{(0, 0, 0)}, {(0, 0, 0), (1, 0, 0), (0, 1, 0)}, I1 ∪ {(0, 0, 1), (2, 0, 0), (0, 2, 0), (1, 1, 0)}, I2 ∪ {(1, 0, 1), (0, 1, 1), (3, 0, 0), (0, 3, 0), (2, 1, 0), (1, 2, 0)}, span{1}, μ0 = 1,

P1 = span{1, x, y}, μ1 = 3, 2 2 P2 = span{1, x, y; t, x , y , xy}, μ1 = 7, 2 2 P3 = span{1, x, y; t, x , y , xy; xt, yt, x 3 , y 3 , x 2 y, xy 2 },

μ1 = 13.

20.1.2 Derivatives on G We now relate the usual Euclidean derivatives on RN ≡ G to the derivatives along the Jacobian basis related to G. We recall formulas (1.37) and (1.38) (page 22) where we showed that, for any smooth function u on G and every x ∈ G, it holds Z1 u(x), . . . , ZN u(x) = ∂x1 u(x), . . . , ∂xN u(x) · Jτx (0), (20.9) ∂x1 u(x), . . . , ∂xN u(x) = Z1 u(x), . . . , ZN u(x) · Jτx −1 (0), ⎛

where

1

⎜ ⎜ a2,1 (x) Jτx (0) = ⎜ ⎜ .. ⎝ . aN,1 (x)

0

··· .. . .. .

1 .. . · · · aN,N −1 (x)

⎞ 0 .. ⎟ .⎟ ⎟ ⎟ 0⎠ 1

is the Jacobian matrix of the left translation by x on G (i.e. τx (y) = x ∗ y) and the ai,j ’s are polynomial functions on G. More explicitly, the usual partial derivatives w.r.t. the fixed coordinates xi ’s on G and the vector fields of the Jacobian basis are related by the following “triangle-shaped” formulas: Z1 = ∂x1 + a2,1 (x) ∂x2 + a3,1 (x) ∂x3 + · · · + aN,1 (x) ∂xN , Z2 = ∂x2 + a3,2 (x) ∂x3 + · · · + aN,2 (x) ∂xN , .. .. . . ∂xN ZN = and, setting bi,j (x) := ai,j

(20.10)

(x −1 ),

∂x1 = Z1 + b2,1 (x) Z2 + b3,1 (x) Z3 + · · · + bN,1 (x) ZN , ∂x2 = Z2 + b3,2 (x) Z3 + · · · + bN,2 (x) ZN , .. .. . . ∂xN = ZN .

(20.11)

20.1 Polynomials and Derivatives on Homogeneous Carnot Groups

737

Taking into account the δλ -degrees of the Zh ’s and of the ∂xh ’s we have the formulas

Zh = ∂xh + aj,h (x) ∂xj , j : σj >σh

∂xh = Zh +

(20.12)

bj,h (x) Zj .

j : σj >σh

For example, if G is the Heisenberg–Weyl group on R3 (see Example 20.1.3), Z1 = ∂x + 2y∂t ,

Z2 = ∂y − 2x∂t ,

Z3 = ∂t

is the Jacobian basis. It is immediately seen that Z1 = ∂x + 2y ∂t , Z2 = ∂y − 2x ∂t , Z3 = ∂t , ∂x = Z1 − 2y Z3 , ∂y = Z2 + 2x Z3 , ∂t = Z3 . For any multi-index α, we introduce the higher order Z-derivative setting αN . Z α := Z1α1 · · · ZN

(20.13)

Obviously, Z α is a left invariant differential operator (not necessarily a vector field) and, as a differential operator, it is δλ -homogeneous of degree α1 σ1 + · · · + αN σN = |α|G (for Zj is δλ -homogeneous of degree σj ). We explicitly observe that, a priori, Z2 Z1 is not necessarily contained in the set of the Z α ’s. Nonetheless, we have the following proposition. Proposition 20.1.4. For every k ∈ N, every i1 , . . . , ik ∈ {1, . . . , N }, and every β1 , . . . , βk ∈ N ∪ {0}, we have (Zi1 )β1 · · · (Zik )βk ∈ span{(Z1 · · · ZN )α : α multi-index}. More precisely, the multi-indices in the right-hand side run over the set of the α’s such that1 |α|G = σi1 β1 + · · · + σik βk . Proof. First, we write (Zi1 )β1 · · · (Zik )βk = Zi1 · · · Zi1 · · · Zik · · · Zik = Zi1 · · · (Zi Zj ) · · · Zik , β1 times

1 Here, the σ ’s are the exponents in (20.2). ij

βk times

738

20 Taylor Formula on Carnot Groups

where by i and j we have indicated an arbitrary couple of contiguous indices. Now, since Zi Zj = [Zi , Zj ] + Zj Zi , we have Zi1 · · · (Zi Zj ) · · · Zik = Zi1 · · · [Zi , Zj ] · · · Zik + Zi1 · · · (Zj Zi ) · · · Zik . (20.14) Since [Zi , Zj ] is a left invariant vector field, it can be written as a linear combination ] = of the Zk ’s, say [Zi , Z j r cr Zr , so that the first summand in the right-hand side of (20.14) equals r cr Zi1 · · · Zr · · · Zik . Moreover, the Zr ’s can be chosen so that σr = σi + σj . As a consequence, we have written the left-hand side of (20.14) as the sum of a term with Zi and Zj interchanged plus a linear combination of summands each with β1 + · · · + βk − 1 factors of the type Zh ’s. Also, these two terms are δλ -homogeneous operators of the same degree as the left-hand side (namely, σi1 β1 +· · ·+σik βk ). Now, this argument can be iterated and, arguing by induction, the assertion can be easily proved.

For example, if G is the Heisenberg–Weyl group on R3 (see Example 20.1.3) and Z1 = ∂x + 2y∂t , Z2 = ∂y − 2x∂t , Z3 = ∂t is the Jacobian basis, then Z2 Z1 does not belong to the set of the Zα ’s in (20.13). Indeed, it holds Z12 = ∂x,x + 4y ∂x,t + 4y 2 ∂t,t , Z22 = ∂y,y − 4x ∂y,t + 4x 2 ∂t,t , Z1 Z2 = ∂x,y − 2 ∂t − 2x ∂x,t + 2y ∂y,t − 4xy ∂t,t , Z2 Z1 = ∂x,y + 2 ∂t + 2y ∂y,t − 2x ∂x,t − 4xy ∂t,t . Though, as asserted in the above proposition, it holds Z2 Z1 = Z1 Z2 + 4 Z3 = (Z1 Z2 Z3 )(1,1,0) + 4 (Z1 Z2 Z3 )(0,0,1) . We now provide a higher-order version of formula (20.12). Proposition 20.1.5 (Z α ’s and (Dx )α ’s). For every multi-index α, we have

Z α = (Dx )α + aβ,α (x) (Dx )β ,

(20.15)

β: β=α, |β|≤|α|, |β|G ≥|α|G

where the aβ,α ’s are polynomials, δλ -homogeneous of degree |β|G − |α|G . Example 20.1.6. If G is the Heisenberg–Weyl group on R3 (see Example 20.1.3) and Z1 = ∂x + 2y∂t , Z2 = ∂y − 2x∂t , Z3 = ∂t is the Jacobian basis, we have (Z1 Z2 Z3 )(1,1,0) = ∂x,y − 2∂t − 2x∂x,t + 2y∂y,t − 4xy∂t,t = (Dx,y,t )(1,1,0) − 2(Dx,y,t )(0,0,1) − 2x(Dx,y,t )(1,0,1) + 2y(Dx,y,t )(0,1,1) − 4xy(Dx,y,t )(0,0,2) . We compare this to (20.15). We notice that the far right-hand side contains (Dx )α plus 4 summands having (Dx )β ’s terms with β = α: the first term satisfies |β| < |α| and |β|G = |α|G , the second, third and fourth terms satisfy |β| = |α| and |β|G > |α|G .

20.1 Polynomials and Derivatives on Homogeneous Carnot Groups

739

Proof (of Proposition 20.1.5). We argue by induction on the (isotropic) length of α. If |α| = 1, then α = eh (the h-th vector of the standard basis of RN ) and Z α = Zh . In this case, (20.15) immediately reduces to (20.12) (see also Proposition 1.3.5, page 34). Suppose now that (20.15) has been proved for every multi-index α with |α| ≤ k. Let us prove it for a multi-index α with |α| = k + 1. We have α = (0, . . . , 0, αh , . . . , αN ),

with αh = 0.

α , where In particular, it holds Z α = Zh Z α = α − eh , so that | α | = k. Moreover, we have | α |G = |α|G − σh . As a consequence, by the inductive hypothesis, we have

α β Z = Zh aβ, α (x) (Dx ) β: |β|≤| α |, |β|G ≥| α |G

=

σj ≥σh

=

aj ∂xj

aβ (Dx )β

|β|≤k, |β|G ≥| α |G

aj (∂xj aβ ) (Dx )β

σj ≥σh , |β|≤k, |β|G ≥| α |G

+

aj aβ (Dx )β+ej .

σj ≥σh , |β|≤k, |β|G ≥| α |G

Now we treat each of the two sums appearing in the above far right-hand side. First, α |G = |β|G − |α|G + σh we notice that aβ is δλ -homogeneous of degree |β|G − | and that aj is δλ -homogeneous of degree σj − σh . Then the G-degree of aj (∂xj aβ ) is aβ ≡ 0 whenever the σj − σh + |β|G − |α|G + σh − σj = |β|G − |α|G . Moreover, ∂xj G-degree of aβ is < σj , i.e. whenever |β|G − |α|G < σj − σh . So we can suppose that |β|G − |α|G ≥ σj − σh ≥ 0. aβ is σj − σh + |β|G − |α|G + Finally, as for the second sum, the G-degree of aj σh = σj + |β|G − |α|G = |β + ej |G − |α|G . Moreover, |β + ej |G = |β|G + σj ≥ | α |G + σh = |α|G . We now isolate the coefficient of (Dx )α . In the first sum, this term does not appear, since the sum runs over the β’s with |β| ≤ k, whereas |α| = k + 1. In the second sum, this term appears only when j and β (satisfying also σj ≥ σh , |β| ≤ k and |β|G ≥ |α|G − σh ) are such that β + ej = α. In particular, β = α − ej , so that |β|G = |α|G − σj , which is ≥ |α|G − σh only if σj ≤ σh . But since the sum runs over the j ’s such that σj ≥ σh , it must be σj = σh . Now, from the first equality in (20.12), we see that the only summand with σj = σh is the one with j = h, and α , so that in this case ah = 1. This gives β = α − eh = aj aβ = aj aβ, α = aj a α , α = 1, also using the inductive hypothesis a

α , α = 1. This completes the proof. A completely analogous proof (this time making use of the second formula in (20.12)) demonstrates the following result.

740

20 Taylor Formula on Carnot Groups

Proposition 20.1.7 ((Dx )α ’s and Z α ’s). For every multi-index α, we have

(Dx )α = Z α + bβ,α (x) Z β ,

(20.16)

β: β=α, |β|≤|α|, |β|G ≥|α|G

where the bβ,α ’s are polynomials, δλ -homogeneous of degree |β|G − |α|G . From Propositions 20.1.5 and 20.1.7 we derive the following corollary. Corollary 20.1.8. Let u be a smooth real-valued function on G. Then u is a polynomial if and only if there exists k ∈ N such that Z α u ≡ 0 for every multi-index α with |α|G ≥ k (or with |α| ≥ k). The same assertion holds if there exists k ∈ N such that Z α u ≡ 0 for every multi-index α with |α|G = k. Proof. First, we remark that (see (20.3)) σ1 |α| ≤ |α|G ≤ σN |α|.

(20.17)

As a consequence, the “only if” part follows from (20.15). Vice versa, if Z β u ≡ 0 for every multi-index β with |β|G ≥ k, then (20.16) proves that (Dx )α u ≡ 0 whenever |α|G ≥ k. Thanks to (20.17), if |α| ≥ k/σ1 , then |α|G ≥ k, so that (Dx )α u ≡ 0. Hence u is a polynomial. As for the last statement of the corollary, we only need to prove that if there exists k ∈ N such that Z α u ≡ 0 whenever |α|G = k, then there exists k ∗ ∈ N such that Z α u ≡ 0 whenever |α|G ≥ k ∗ . Now, it suffices to take k ∗ large enough depending on k and σ1 , . . . , σN . Indeed2 , we argue by induction noticing that α1+i

αi

αi −1 αi0 −1 α1+i0 Zi0 Z1+i0

αN Z α = Zi0 0 Z1+i00 · · · ZN = Zi0 0

αN · · · ZN

(here i0 is the first index such that αi0 = 0) and (0, . . . , 0, αi − 1, α1+i , . . . , αN ) = |α|G − σi . 0 0 0 G This ends the proof.

We now recall that, since G is a Carnot group, then the elements of the Jacobian basis laying in the first layer of the stratification of g Lie-generate the whole g. Setting m := N1 in (20.1), we denote these vectors by Z1 , . . . , Zm . We thus have Lie{Z1 , . . . , Zm } = span{Z1 , . . . , ZN } = g.

(20.18)

The following result holds. 2 Notice that the choice k ∗ = k will not do. For instance, in the Heisenberg group on R3 of Example 20.1.6, if we know that Z α u ≡ 0 whenever |α|G ≥ 5, then we cannot immediately deduce by induction that Z α u ≡ 0 whenever |α|G = 6. For example, Z33 = Z (0,0,3) and |(0, 0, 3)|G = 6, but Z33 = Z3 Z32 = Z3 (Z (0,0,2) ) and |(0, 0, 2)|G = 4.

20.2 Taylor Polynomials on Homogeneous Carnot Groups

741

Proposition 20.1.9. It holds span{Zi1 · · · Zik : i1 , . . . , ik ∈ {1, . . . , m}, k ∈ N} = span{(Z1 · · · ZN )α : α multi-index}. Proof. The inclusion ⊆ follows from Proposition 20.1.4. In order to prove the opposite inclusion, it suffices to prove that Z1 , . . . , ZN can all be written as linear combinations of terms of the type Zi1 · · · Zik . This fact now easily follows from (20.18), since [· · · [[Zi1 , Zi2 ], Zi3 ] · · · Zik ] ∈ span{Zi1 · · · Zik : ij ∈ {1, . . . , m}, k ∈ N}. This ends the proof.

With the aid of Lemma 20.1.11 below (whose proof is left as an exercise), it can immediately be proved the following result (compare it to Corollary 1.5.5, page 68). Corollary 20.1.10. Let u be a smooth real-valued function on G. Then u is a polynomial if and only if there exists k ∈ N such that Zi1 · · · Zik u ≡ 0 for every i1 , . . . , ik ∈ {1, . . . , m}. Proof. It follows by collecting together Corollary 20.1.8, Proposition 20.1.9 and Lemma 20.1.11 below.

Lemma 20.1.11. If H1 , . . . , Hn are (not identically vanishing) differential operators on RN ≡ G which are δλ -homogeneous of degrees d1 , . . . , dn , respectively, with di = dj for every i = j , then they are linearly independent. Example 20.1.12. Let G be the Heisenberg–Weyl group on R3 (see Example 20.1.6). Then, for example, we know that u is a polynomial provided that Z12 u, Z1 Z2 u, Z22 u,

Z3 u

all vanish identically: this is stated in the last part of Corollary 20.1.8. Also, we know that u is a polynomial function provided that Z12 u, Z1 Z2 u, Z2 Z1 u, Z22 u all vanish identically: this is stated in Corollary 20.1.10. These conditions are equivalent, though different in appearance.

20.2 Taylor Polynomials on Homogeneous Carnot Groups We begin with another result on the derivatives on G, helpful for our Taylor formula. We follow the notation in the previous sections (see in particular (20.6)–(20.8) and (20.13)).

742

20 Taylor Formula on Carnot Groups

Theorem 20.2.1. Let n ∈ N ∪ {0} and consider the space Pn of the polynomials δλ -homogeneous of degree at most n. Moreover, let In be ordered in some fixed way. Then the map p → ((Z α p)(0))α∈In Pn → Rμn , is a vector space isomorphism. Proof. We consider (20.15) of Proposition 20.1.5. Since aβ,α (x) is δλ -homogeneous of degree |β|G − |α|G , we have aβ,α (0) = 0 whenever |β|G > |α|G and aβ,α (x) =constant (say, aβ,α ) whenever |β|G = |α|G . Setting aα,α = 1, from (20.15) we thus derive

aβ,α (Dx )β |x=0 . (20.19) Z α |x=0 = β: |β|≤|α|, |β|G =|α|G

Now, we recall the dual formula

(Dx )α |x=0 =

bβ,α Z β |x=0

(20.20)

β: |β|≤|α|, |β|G =|α|G

implied by (20.16) of Proposition 20.1.7. These facts prove the following result: If we indicate by Φ : Pn → Rμn the map in Proposition 20.1.2 and by Ψ : Pn → Rμn the map in Theorem 20.2.1, then (20.19) shows that there exists a linear map φ : Rμn → Rμn such that Ψ (p) = φ(Φ(p))

for every p ∈ Pn ;

analogously, (20.20) shows that there exists a linear map ψ : Rμn → Rμn such that Φ(p) = ψ(Ψ (p))

for every p ∈ Pn .

These facts together demonstrate that ψ and φ are inverse to each other, so that Ψ = φ ◦ Φ is a linear isomorphism since Φ is (see Proposition 20.1.2).

Theorem 20.2.1 has the following important corollary. Corollary 20.2.2. Let n ∈ N ∪ {0}. Let, as usual, Pn be the space of polynomials, δλ -homogeneous of degree at most n, and let μn be the dimension of Pn . Then, for any μn -tuple of real numbers z, there exists one and only one polynomial p in Pn such that (20.21) ((Z α p)(0))α∈In = z. Here, as usual, we have fixed an ordering on In , the set of multi-indices α such that |α|G ≤ n. Corollary 20.2.2 proves that the following definition is well-posed. Definition 20.2.3 (Z-Mac Laurin polynomial). Let f ∈ C ∞ (G, R). Let, as usual, Z1 , . . . , ZN denote the Jacobian basis related to G. Then, for every n ∈ N ∪ {0}, there exists one and only one polynomial p, δλ -homogeneous of degree at most n, such that

20.2 Taylor Polynomials on Homogeneous Carnot Groups

(Z1 · · · ZN )α (p)(0) = (Z1 · · · ZN )α (f )(0)

for every α ∈ In .

743

(20.22)

We say that p(x) = Pn (f, 0)(x) is the Z-Mac Laurin polynomial of δλ -degree n, related to f . Indeed, it suffices to apply Corollary 20.2.2 with z = ((Z α f )(0))α∈In . Remark 20.2.4. In (20.56) of Section 20.3.2, we shall provide an explicit formula for Pn (f, 0). An example is in order. Example 20.2.5. Let us consider G = H1 , the Heisenberg–Weyl group on R3 (see also Example 20.1.3), with its Jacobian basis Z1 = ∂x + 2y∂t , Z2 = ∂y − 2x∂t , Z3 = ∂t . Let f ∈ C ∞ (H1 , R). Let us find p = P1 (f, 0). Since p ∈ P1 , there exist a, b, c ∈ R such that p(x, y, t) = a + b x + c y. Then (20.22) gives a = p(0) = f (0), b = Z1 p(0) = Z1 f (0), c = Z2 p(0) = Z2 f (0), so that P1 (f, 0)(x, y, t) = f (0) + Z1 f (0) x + Z2 f (0) y. Let us now find q = P2 (f, 0). Since q ∈ P2 , there exist a, b, c, d, e, f, g ∈ R such that q(x, y, t) = a + b x + c y + d t + e x 2 + f y 2 + g xy. Then (20.22) gives a = q(0) = f (0), b = Z1 q(0) = Z1 f (0), c = Z2 q(0) = Z2 f (0), d = Z3 q(0) = Z3 f (0), 2e = (Z1 )2 q(0) = (Z1 )2 f (0), 2f = (Z2 )2 q(0) = (Z2 )2 f (0), g − 2d = Z1 Z2 q(0) = Z1 Z2 f (0), so that P2 (f, 0)(x, y, t) = f (0) + Z1 f (0) x + Z2 f (0) y + Z3 f (0) t 1 1 + (Z1 )2 f (0) x2 + (Z2 )2 f (0) y2 + (2Z3 f (0) 2 2 + Z1 Z2 f (0)) xy. Since in this case Z3 = − 14 (Z1 Z2 − Z2 Z1 ), this formula can be rewritten as

744

20 Taylor Formula on Carnot Groups

P2 (f, 0)(x, y, t) 1 = f (0) + Z1 f (0) x + Z2 f (0) y − (Z1 Z2 − Z2 Z1 )f (0) t 4 1 1 1 2 2 2 + (Z1 ) f (0) x + (Z2 ) f (0) y2 + (Z1 Z2 + Z2 Z1 )f (0)xy. (20.23) 2 2 2 Notice that, though unique, the Z-Mac Laurin polynomial can be written in many different symbolic ways. For instance, observe that (20.23) can be rewritten in the suggesting form x P2 (f, 0)(x, y, t) = f (0) + ∇G f (0) · y 1 x + Z3 f (0) t + (x, y) · Hesssym f (0) · , (20.24) y 2 where ∇G f (0) = (Z1 f (0), Z2 f (0)) is the horizontal gradient of f at 0 and 1 (Z1 Z2 f (0) + Z2 Z1 f (0)) Z12 f (0) 2 Hesssym f (0) := 1 Z22 f (0) 2 (Z1 Z2 f (0) + Z2 Z1 f (0)) is the so-called symmetrized horizontal Hessian of f at 0. See Ex. 4 for the computation of P3 (f, 0) in the present case. See Ex. 6 for a suitable generalization of (20.24) to other Carnot groups and see Section 20.3.2 for a general formula for Pn (f, 0) for every n and every Carnot group.

We leave as an exercise (see Ex. 7 at the end of the chapter) to prove the following assertion. Remark 20.2.6. Let Pn be the vector space of the polynomials with G-degree at most n. Let also P be the set of polynomial functions on G. Since the set of all multi-indices I = (N ∪ {0})N can be decomposed as I = n≥0 In , the following map is well-defined p → pn , πn : P −→ Pn , where p(x) =

α

cα x α =

α∈In

cα x α +

cα x α =: pn (x) + pn∗ (x).

α∈ / In

Then, if Qn (f, 0)(x) denotes the usual Euclidean Mac Laurin polynomial of f , it is easy to see that Pn (f, 0)(x) = πn Qn (f, 0)(x) . In other words, the Z-Mac Laurin polynomial of f coincides with the sum of the terms, in the usual Euclidean Mac Laurin polynomial of f , which are δλ homogeneous of degree at most n. The following result is not so evident, a priori.

20.2 Taylor Polynomials on Homogeneous Carnot Groups

745

Proposition 20.2.7 (Nested property of the Z-Mac Laurin polynomials). Let n ∈ N and f ∈ C ∞ (G, R). Then Pn (f, 0) − Pn−1 (f, 0) is a δλ -homogeneous polynomial of degree n. In other words, Pn (f, 0) = Pn−1 (f, 0) + {a polynomial of G-degree = n}. Proof. Let us write Pn (f, 0) = p + q, where p has G-degree ≤ n − 1, so that q has G-degree = n (unless it vanishes identically). We have to prove that p = Pn−1 (f, 0). By Definition 20.2.3, this follows if we show that Z α (p)(0) = Z α (f )(0)

for every α ∈ In−1 .

(20.25)

If α ∈ In−1 , then α ∈ In , so that by the very definition of Pn (f, 0), we have Z α (f )(0) = Z α (Pn (f, 0))(0) = Z α (p + q)(0) = Z α p(0) + Z α q(0). Then, (20.25) will follow if we show that Z α q(0) = 0. This fact is a consequence of the fact that Z α q is a polynomial, δλ -homogeneous of positive degree, precisely n − |α|G (recall that q has G-degree = n and |α|G ≤ n − 1, since α ∈ In−1 ). This ends the proof.

The following definition is definitely not unexpected. Definition 20.2.8 (Z-Taylor polynomial). Let f ∈ C ∞ (G, R), x0 ∈ G and n ∈ N ∪ {0} be fixed. Let us consider the Z-Mac Laurin polynomial of the function x → f (x0 ∗ x), i.e. the polynomial Pn (f ◦ τx0 , 0). We say that the polynomial Pn (f, x0 )(x) := Pn (f ◦ τx0 , 0)(x0−1 ∗ x)

(20.26)

is the Z-Taylor polynomial of G-degree n related to f and centered at x0 . Remark 20.2.9. In (20.56) of Section 20.3.2, we shall provide an explicit formula for Pn (f, x0 ). The following characterization of Pn (f, x0 ) holds. Proposition 20.2.10 (Characterization). With the notation of the above definition, Pn (f, x0 ) is characterized by being the only polynomial p, δλ -homogeneous of degree at most n, such that (Z1 · · · ZN )α (p)(x0 ) = (Z1 · · · ZN )α (f )(x0 )

for every α ∈ In .

(20.27)

Proof. An immediate consequence of Definitions 20.2.3, 20.2.8 and the left invariance of Z1 , . . . , ZN .

746

20 Taylor Formula on Carnot Groups

With reference to Example 20.2.5, setting z0 = (x0 , y0 , t0 ), we have P2 (f ◦ τz0 , 0) = f (z0 ) + Z1 f (z0 ) x + Z2 f (z0 ) y + Z3 f (z0 ) t 1 1 + (Z1 )2 f (z0 ) x2 + (Z2 )2 f (z0 ) y2 2 2 1 + (Z1 Z2 + Z2 Z1 )f (z0 ) xy, 2 so that P2 (f, z0 )(x, y, t) = P2 (f ◦ τz0 , 0)(z0−1 ∗ (x, y, t)) = f (z0 ) + Z1 f (z0 ) (x − x0 ) + Z2 f (z0 ) (y − y0 ) 1 + Z3 f (z0 ) (t − t0 − 2y0 x + 2x0 y) + (Z1 )2 f (z0 ) (x − x0 )2 2 1 2 2 + (Z2 ) f (z0 ) (y − y0 ) 2 1 + (Z1 Z2 + Z2 Z1 )f (z0 ) (x − x0 )(y − y0 ). 2

20.3 Taylor Formula on Homogeneous Carnot Groups The aim of this section is to provide our G-version of Taylor formula. In order to do this, we use a suitable version of the mean value theorem, using paths along vector fields of the first layer of the stratification of G. For this reason, we shall call it the stratified Lagrange mean value theorem. To derive this theorem, we need some important prerequisites. Indeed, we shall make a crucial use of the results in Chapter 19 (and in particular, of Lemma 19.1.4, a subtle consequence of the Campbell– Hausdorff formula). Throughout the section, (G, ∗) is a fixed homogeneous Carnot group with m generators. We take Z1 , . . . , Zm (the first m vectors of the Jacobian basis) as generators for the algebra of G. Moreover, we shall denote by Exp : g → G the usual exponential map related to (G, ∗) and by d any homogeneous norm on G. Theorem 20.3.1 (The stratified Lagrange mean value theorem). There exist absolute constants c1 , b > 0, depending only on the Carnot group G and on the homogeneous norm d, such that |f (x ∗ h) − f (x)| ≤ c1 d(h)

sup

|(Z1 f (x ∗ z), . . . , Zm f (x ∗ z))|,

z: d(z)≤b d(h)

(20.28) for all f ∈ C 1 (G, R) and every x, h ∈ G. Proof. Due to the importance of this result, we provide two proofs of it, both relying on the Carathéodory–Chow–Rashevsky connectivity Theorem 19.1.3 in Chapter 19.

20.3 Taylor Formula on Homogeneous Carnot Groups

747

The first one is very simple, but it makes also use of the Carnot–Carathéodory distance dZ related to the set of vector fields Z = {Z1 , . . . , Zm } (see Section 5.2, Chapter 5, page 232). The second one does not involve dZ but it makes use of the precise estimate in Theorem 19.2.1, page 727. First proof. Since all homogeneous norms are equivalent (see Proposition 5.1.4, page 230) we can replace d by the Carnot–Carathéodory distance dZ related to Z = {Z1 , . . . , Zm }. By the very definition of dZ (see (5.6), page 232), there exists a sequence Tn ↓ dZ (x ∗ h, x) = dZ (h, 0) and, for every n ∈ N, there exists an absolutely continuous curve γn : [0, Tn ] → G connecting x to x ∗ h (i.e. γn (0) = x and γn (Tn ) = x ∗ h) which is {Z1 , . . . , Zm }-subunit, i.e. γ˙n (t), ξ 2 ≤

m

Xj I (γn (t)), ξ 2

∀ ξ ∈ RN

j =1

almost everywhere in [0, T ]. Then we have (also setting Zf (y) = (Z1 f (y), . . . , Zm f (y))) Tn Tn d |f (x ∗ h) − f (x)| = (f (γn (s))) ds = (∇f )(γn (s)), γ˙n (s) ds ds 0 0 1/2 Tn m 2 ≤ Zj I (γn (s)), (∇f )(γn (s)) ds 0

=

j =1

Tn

(Zf )(γn (s)) ds ≤ Tn

0

sup |Zf (γn (s))|.

s∈[0,Tn ]

(20.29)

Now, by the very definition of dZ , for every s ∈ [0, Tn ] we obviously have dZ (x, γn (s)) ≤ dZ (x, γn (Tn )) = dZ (x, x ∗ h) = dZ (h, 0), so (20.29) yields |f (x ∗ h) − f (x)| ≤ Tn

|Zf (y)|.

sup

(20.30)

y: dZ (x,y)≤dZ (h,0)

Let now n → ∞ in (20.30), and recall that Tn ↓ dZ (h, 0). This gives3 |f (x ∗ h) − f (x)| ≤ dZ (h, 0)

sup

|Zf (y)|.

(20.31)

y: dZ (x,y)≤dZ (h,0)

Finally, (20.31) implies (20.28) by setting y = x ∗z and by taking b = c2 and c1 = c, where c is the constant such that c−1 d ≤ dZ ≤ c d on G × G. Second proof. First, suppose that h ∈ Exp (span{Zj }) for some j ∈ {1, . . . , m}. We can suppose that h = Exp (tZj ) for some t > 0 (the case t < 0 follows by replacing Zj by −Zj ). Then (19.10e) gives d(h) = t d(Exp (Zj )) (recall that {Z1 , . . . , Zm } span the first layer of the stratification, i.e. σj = 1 for 1 ≤ j ≤ m). Let us set 3 (20.31) shows that the stratified mean-value theorem in (20.28) holds with constants c = 1 1

and b = 1 when d is the Carnot–Carathéodory distance.

748

20 Taylor Formula on Carnot Groups

c := max d −1 (Exp (Z1 )), . . . , d −1 (Exp (Zm )) . With this position, we have t = d(h) d −1 (Exp (Zj )) ≤ c d(h). Thus, for every x ∈ G, it holds |f (x ∗ h) − f (x)| = f x ∗ Exp (tZj ) − f (x) = f exp(tZj )(x) − f (x) (set γ (t) := exp(tZj )(x)) t t d = (f (γ (s))) ds = (Zj f )(γ (s)) ds ds 0

0

≤ t sup |Zj f (x ∗ Exp (sZj ))| ≤ c d(h) s∈[0,t]

sup

|Zf (x ∗ z)|. (20.32)

z: d(z)≤d(h)

Here we have set Zf (y) = (Z1 f (y), . . . , Zm f (y)), and we have used again the fact that, for every s ∈ [0, t], one has d(Exp (sZj )) = s d(Exp (Zj )) ≤ t d(Exp (Zj )) = d(Exp (tZj )) = d(h). In the case of an arbitrary h ∈ G, we first remark that by Theorem 19.2.1 (page 727), we have h = h1 ∗ · · · ∗ hM , where any hj ’s belong to Exp (span{Zi }) for some i ∈ {1, . . . , m} and d(hj ) ≤ c0 d(h) for all j = 1, . . . , M. Thus, using (20.32) repeatedly, one gets (also setting h0 := 0) |f (x ∗ h) − f (x)| = f x ∗ h1 ∗ · · · ∗ hM − f (x) ≤

M

f x ∗ h1 ∗ · · · ∗ hj − f (x ∗ h1 ∗ · · · ∗ hj −1 ) j =1

(by (20.32)) ≤

M

j =1

c d(hj )

sup z: d(z)≤d(hj )

|Zf (x ∗ h1 ∗ · · · ∗ hj −1 ∗ z)|. (20.33)

By the pseudo-triangle inequality for d, d(x ∗ y) ≤ c (d(x) + d(y)), we have d(h1 ∗ · · · ∗ hj −1 ∗ z) ≤ cj −1 (d(h1 ) + · · · + d(hj −1 ) + d(z)) (since d(z) ≤ d(hj )) ≤ cj −1 (d(h1 ) + · · · + d(hj )) (since d(hj ) ≤ c0 d(h)) ≤ cj −1 j c0 d(h) ≤ cj −1 M c0 d(h) =: b d(h). As a consequence, (20.33) proves that |f (x ∗ h) − f (x)| ≤ M c c0 d(h)

sup

ζ : d(ζ )≤b d(h)

so that the position c1 := M c c0 finally gives (20.28).

|Zf (x ∗ ζ )|,

20.3 Taylor Formula on Homogeneous Carnot Groups

749

With Theorem 20.3.1 at hand, we can prove the following theorem. Note that here and in the proof below, α is a multi-index with m entries (not N, as in the previous paragraphs). Note that the Euclidean length |α| coincides with |(α, 0, . . . , 0)|G , the 0’s occurring N − m times. Theorem 20.3.2 (Stratified Taylor inequality). Let (G, ∗) be a homogeneous Carnot group and d a homogeneous norm on G. For every n ∈ N ∪ {0}, there exists a constant cn > 0 (depending only on n, G and d) such that |f (x ∗ h) − Pn (f, x)(x ∗ h)| ≤ cn d n (h) (Z1 , . . . , Zm )α f (x ∗ z) − (Z1 , . . . , Zm )α f (x) : |α| = n × sup d(z)≤bn d(h)

(20.34) for all f ∈ C n (G, R) and every x, h ∈ G. Here b is as in Theorem 20.3.1. Proof. We fix x ∈ G and n ∈ N ∪ {0}, and we let Φ(h) := f (x ∗ h) − Pn (f, x)(x ∗ h). By Proposition 20.2.10, it holds Z β (f − Pn (f, x))(x) = 0 whenever β is a Ndimensional multi-index with |β|G ≤ n, i.e. (by using the left-invariance of the Z β ’s) (Z1 , . . . , ZN )β Φ(0) = 0

N ∀ β ∈ N ∪ {0} ,

|β|G ≤ n.

By taking a multi-index β with 0 entries from the (m + 1)-th one, we get (Z1 , . . . , Zm )α Φ(0) = 0

∀ α : |α| ≤ n.

(20.35)

Let us prove by induction on j = 0, 1, . . . , n that if |α| = n − j then (Z1 , . . . , Zm )α Φ(h) ≤ cj d j (h) (Z1 , . . . , Zm )α f (x ∗ z) − (Z1 , . . . , Zm )α f (x) : |α| = n . × sup d(z)≤bj d(h)

(20.36) When j = n (whence |α| = 0, so that (Z1 , . . . , Zm )α Φ(h) = Φ(h)) this will obviously prove (20.34). = (Z1 , . . . , Zm ) (to distinguish it Step zero: For the sake of brevity, we set Z α Pn (f, x)(·) has from the former Z = (Z1 , . . . , ZN )). If j = 0, i.e. |α| = n, then Z G-degree = 0, i.e. it is constant, so that α α Pn (f, x) (x ∗ h) = Z Pn (f, x) (x) = (Z α f )(x), Z whence the far left-hand side of (20.36) equals

750

20 Taylor Formula on Carnot Groups

α α f (x). Z f (x ∗ h) − Z In this case (20.36) is trivially true (for the left-hand side is one of the elements in the curly brackets when z = h). Step of induction: Suppose that (20.36) holds true for indices 0 ≤ · · · ≤ j −1 and let us prove it for j . Let α be a multi-index with |α| = n − j . We apply the stratified α Φ and the points x = 0, h ∈ G mean-value Theorem 20.3.1 to the function f = Z α (so that f (x) = (Z Φ)(0) = 0 thanks to (20.35)). Then from (20.28) we infer α Φ(h)| ≤ c1 d(h) |Z

α Φ(z), . . . , Zm Z α Φ(z))| =: (). (20.37) |(Z1 Z

sup z: d(z)≤b d(h)

β ’s with |β| = |α| + 1 = α , . . . , Zm Z α are linear combinations of the Z But Z1 Z n − (j − 1) (see the proof of Proposition 20.1.4 and apply, if needed, a simple homogeneity argument). As a consequence, we can apply the inductive hypothesis to estimate the right-hand side of (20.37). We get (here c(Z) denotes a structural constant depending only on the algebraic structure properties of the set of vector fields {Z1 , . . . , ZN }; see the proof of Proposition 20.1.4) () ≤ c1 d(h)

sup z: d(z)≤b d(h)

×

sup

c(Z) cj −1 d j −1 (z)

{|(Z1 , . . . , Zm )α f (x ∗ ζ ) − (Z1 , . . . , Zm )α f (x)|}

ζ : d(ζ )≤bj −1 d(z) |α|=n

≤ c1 c(Z) cj −1 bj −1 d j (h) × sup {|(Z1 , . . . , Zm )α f (x ∗ ζ ) − (Z1 , . . . , Zm )α f (x)|}. ζ : d(ζ )≤bj d(h) |α|=n

This is precisely (20.36). If we set cj := c1 c(Z) cj −1 bj −1 , the proof is complete.

Starting from Theorems 20.3.1 and 20.3.2 it is a standard matter to prove the following stratified Taylor formula with remainder. Here, α is a multi-index with m entries (not N, as in the paragraphs preceding Theorem 20.3.2). Theorem 20.3.3 (Stratified Taylor formula). Let (G, ∗) be a homogeneous Carnot group. For every n ∈ N ∪ {0}, there exists a constant cn > 0 (depending only on n and G) such that |f (x ∗ h) − Pn (f, x)(x ∗ h)| ≤ cn d n+1 (h) ×

sup

{|(Z1 , . . . , Zm )α f (x ∗ z)| : |α| = n + 1}

d(z)≤bn+1 d(h)

for all f ∈ C n+1 (G, R) and every x, h ∈ G. Here b is as in Theorem 20.3.1.

(20.38)

20.3 Taylor Formula on Homogeneous Carnot Groups

751

Proof. First, apply the stratified Taylor inequality in Theorem 20.3.2, and then estimate the right-hand side of (20.34) by means of the stratified mean value Theorem 20.3.1. This ends the proof.

Example 20.3.4. Consider the Heisenberg–Weyl group H1 on R3 . The relevant Jacobian basis is Z1 = ∂x1 + 2x2 ∂x3 ,

Z2 = ∂x2 − 2x1 ∂x3 ,

Z3 = ∂x3 ,

and a homogeneous norm on H1 is, for example, d(x) = |x1 | + |x2 | + |x3 |1/2 . Hence formula (20.38) in Theorem 20.3.3 gives the estimate u(x) − u(0) + x1 Z1 u(0) + x2 Z2 u(0) ≤ c1 |x1 |2 + |x2 |2 + |x3 | × sup {|Z12 u|(z), |Z22 u|(z), |Z1 Z2 u|(z)}. d(z)≤b2 d(x)

As an application, if u ∈ C 2 (G, R) is such that Z12 u, Z1 Z2 u, Z22 u ≡ 0 in H1 , then u(x) = u(0) + x1 Z1 u(0) + x2 Z2 u(0). In particular, Z3 u ≡ 0 whence also Z2 Z1 u = Z1 Z2 u + 4Z3 u ≡ 0. This is quite surprising, since we have derived an information on {Z3 , Z2 Z1 } starting from an information only on {Z12 , Z1 Z2 , Z22 }, but / span{Z12 , Z1 Z2 , Z22 }. Z3 , Z2 Z1 ∈ See Ex. 7 for a generalization of the above comments. 20.3.1 Stratified Taylor Formula with Peano Remainder From Theorem 20.3.3 we immediately derive the following corollary. Corollary 20.3.5 (Stratified Taylor formula–Peano remainder). Let (G, ∗) be a homogeneous Carnot group. For every f ∈ C n+1 (G, R), x0 ∈ G and n ∈ N ∪ {0}, we have (20.39) f (x) = Pn (f, x0 )(x) + Ox→x0 d n+1 (x0−1 ∗ x) . There is another way to prove the stratified Taylor formula with Peano remainder in (20.39), by starting from the usual Euclidean Taylor formula. This alternative proof, which we now describe, is much simpler than the proof given in Section 20.3. Nonetheless, it does not provide any estimate of the remainder (in terms of the sole Z α ’s, |α|G = n + 1) as in Theorem 20.3.3. We begin with a simple lemma.

752

20 Taylor Formula on Carnot Groups

Lemma 20.3.6. Let f be a smooth function on RN such that f (x) = Ox→0 (|x|n+1 ), where | · | is the Euclidean norm on RN . Then Z α f (0) = 0 for every multi-index α with |α|G ≤ n. Proof. It is known4 that the assertion holds in the Euclidean setting, i.e. when Z α = (Dx )α . In the present setting, note that, by (20.15),

aβ,α (0) (Dx )β |0 . Z α |0 = (Dx )α |0 + β: β=α, |β|≤|α|, |β|G =|α|G

Hence, Z α f (0) is a linear combination of terms (Dx )β f (0) with |β| ≤ |α|. Note that if |α|G ≤ n, then |α| ≤ n, for (in general) it holds |α| ≤ |α|G . Consequently, Z α f (0) is a linear combination of terms (Dx )β f (0) with |β| ≤ n, and the proof is complete, thanks to the cited Euclidean version of the lemma.

By the classical Mac Laurin formula, for a given f ∈ C ∞ (RN , R), we have f (x) = Qn (f, 0)(x) + R(x),

where R(x) = Ox→0 (|x|n+1 ).

(20.40)

Here Qn (f, 0)(x) =: q(x) is the (only) polynomial in x of ordinary degree n such that (Dx )α q(0) = (Dx )α f (0) for all |α| ≤ n. Now, we decompose (in a unique way) q as q1 +q2 , where q1 contains only monomials which are δλ -homogeneous of degree ≤ n. Note that q2 contains only monomials of δλ -degree ≥ n + 1. Consequently, f = q1 + (q2 + R).

(20.41)

Next, we take any multi-index α with |α|G ≤ n. Then Z α f = Z α q1 + Z α q2 + Z α R, so that Z α f (0) = Z α q1 (0) + Z α q2 (0) + Z α R(0).

(20.42)

Now, Z α q2 is a polynomial of δλ -degree ≥ n + 1 − |α|G ≥ 1, so that Z α q2 (0) = 0. Moreover, since R(x) = Ox→0 (|x|n+1 ), by means of Lemma 20.3.6, it follows that Z α R(0) = 0, whence (20.42) yields Z α f (0) = Z α q1 (0)

whenever |α|G ≤ n

i.e. by definition, q1 is the Z-Mac Laurin polynomial of f . 4 One of the simplest way to see this is to apply the Euclidean Taylor formula. Note that

when N = 1, a Cauchy-type theorem is needed.

20.3 Taylor Formula on Homogeneous Carnot Groups

753

Consider now the following δλ -homogeneous norm on G = d(x (1) , x (2) , . . . , x (r) ) = |x (1) | + |x (2) |1/2 + · · · + |x (r) |1/r d(x) (here | · | denotes the Euclidean norm on any RNi , i = 1, . . . , r). We have n+1 ). q2 (x), R(x) = Ox→0 (d(x)

(20.43)

Indeed, the estimate of q2 follows by recalling that q2 contains only monomials of δλ -degree ≥ n + 1. Moreover, noticing that (for every i = 1, . . . , r) it holds |x (i) | ≤ |x (i) |1/ i near the origin, we can use (20.40) to estimate R. By collecting together (20.41) and (20.43), we have proved the formula f (x) = q1 (x) + Ox→0 (dn+1 ),

(20.44)

where q1 is the Z-Mac Laurin polynomial of f or, equivalently, the “projection” on Pn of the usual Mac Laurin polynomial of f , Pn being the vector space of the polynomials of δλ -degree ≤ n. so that Finally, if d is any homogeneous norm on G, then d is equivalent to d, (20.44) gives back (20.39). (Incidentally, we have also demonstrated the assertion in Remark 20.2.6.) The following remark shows that there is only one polynomial of δλ -degree at most n which plays the role of Pn (f, x0 ) in (20.39). Remark 20.3.7. Suppose f ∈ C n+1 (G, R), x0 ∈ G and n ∈ N ∪ {0}. If p(x) is a polynomial of δλ -degree at most n such that () f (x) = p(x) + Ox→x0 d n+1 (x0−1 ∗ x) , then p coincides with Pn (f, x0 ). Indeed, () and (20.39) give q(x) := Pn (f, x0 )(x) − p(x) = Ox→x0 d n+1 (x0−1 ∗ x) . q (y) := q(x0 ∗ y), We remark that q is a polynomial of δλ -degree at most n. Setting the above is equivalent to (20.45) q (y) = Oy→0 d n+1 (y) . Now, since q is a polynomial of δλ -degree at most n and the i-th component function of x0 ∗ y is a polynomial function in y of δλ -degree5 at most σi (see Theorem 1.3.15-(3), page 39), then q (y) = q(x0 ∗ y) is a polynomial function in y of q ≡ 0, i.e. q(x0 ∗ y) = 0 δλ -degree at most n. Hence, (20.45) can hold if and only if for every y ∈ G. Recalling the definition of q, this is in turn equivalent to p(x0 ∗ y) = Pn (f, x0 )(x0 ∗ y)

∀ y ∈ G.

Obviously, this means p = Pn (f, x0 ). 5 Here, we are writing, as usual, the dilation δ on G as λ

δλ (x) = (λσ1 x1 , . . . , λσN xN ).

754

20 Taylor Formula on Carnot Groups

20.3.2 Stratified Taylor Formula with Integral Remainder In this section, we derive a stratified Taylor formula with integral remainder which will furnish yet another proof of the Taylor formulas in the previous sections. The resulting Taylor polynomials will be apparently different from the Pn (f, x0 )’s introduced in Section 20.2, but they will prove to be equivalent, thanks to the uniqueness result in Remark 20.3.7. Thus, we shall obtain another way to represent the Z-Taylor polynomials. As usual, G = (RN , ∗, δλ ) is a homogeneous Carnot group of step r and m generators and Z1 , . . . , ZN is the relevant Jacobian basis of g, the Lie algebra of G. We denote by Exp and Log the exponential and logarithmic maps related to G. Furthermore, we denote, as usual, the dilation δλ on G by δλ (x) = (λσ1 x1 , . . . , λσN xN ). We begin by recalling a known result. Let X ∈ g be given, and suppose that γ (t) is an arbitrary integral curve of X, i.e. γ˙ (t) = XI (γ (t)) for every t ∈ R. Then, for every u ∈ C n+1 (G, R), we have (see Ex. 8 in Chapter 1) u(γ (t)) =

n

tk k=0

k!

1 + n!

(X k u)(γ (0))

t

(t − s)n X n+1 u (γ (s)) ds.

(20.46)

0

Indeed, it suffices to apply the ordinary Taylor formula with integral remainder to t → u(γ (t)), by noticing that dk u(γ (t)) = (X k u)(γ (t)). k dt We thus immediately get the following assertion. Lemma 20.3.8. Let x, h ∈ G. Let also u ∈ C n+1 (G, R). Then we have n

1 ((Log h)k u)(x) u(x ∗ h) = k! k=0 1 1 + (1 − s)n ((Log h)n+1 u)(x ∗ Exp (s Log h)) ds. (20.47) n! 0

Proof. We know that (see Corollary 1.2.24, page 24) the integral curve γ of X := Log h ∈ g starting at x is given by s → γ (s) = exp s Log h (x) = x ∗ Exp s Log h . Note that γ (0) = x, γ (1) = x ∗ Exp Log h = x ∗ h, so that, by applying (20.46) with t = 1, we get (20.47).

20.3 Taylor Formula on Homogeneous Carnot Groups

755

A restatement of the lemma above gives the following corollary. Corollary 20.3.9. Let x, h ∈ G and u ∈ C n+1 (G, R). Let (X1 , . . . , XN ) be any basis of g. Then we have u(x ∗ h) = u(x) +

n

N

k=1 i1 ,...,ik =1

ζi1 (h) · · · ζik (h) Xi1 · · · Xik u(x) k!

N

ζi1 (h) · · · ζin+1 (h) n! i1 ,...,in+1 =1 N 1

Xi1 · · · Xin+1 u x ∗ Exp × s ζi (h) Xi (1 − s)n ds. (20.48) +

0

i=1

Here, we have used the following notation Log h = ζ1 (h) X1 + · · · + ζN (h) XN

∀ h ∈ G,

(20.49)

i.e. ζ1 (h), . . . , ζN (h) are the components of Log h w.r.t. (X1 , . . . , XN ). Proof. Obviously, there exist (polynomial) functions G h → ζi (h) ∈ R,

i = 1, . . . , N,

such that (20.49) holds. For every k ∈ N, we thus obtain (Log h) = k

N

k ζi (h) Xi

N

=

ζi1 (h) · · · ζik (h) Xi1 · · · Xik .

i1 ,...,ik =1

i=1

Now, (20.48) follows directly from (20.47).

Next, we recall that the Lie algebra of G is equipped with the stratification (i+1) if i = 1, . . . , r − 1, V (1) (r) (1) (i) g = V ⊕ ··· ⊕ V with [V , V ] = {0} if i = r. We say that a basis of g is adapted to the stratification if it has the form (1)

(1)

(r)

(r)

X = (X1 , . . . , XN1 ; . . . ; X1 , . . . , XNr ),

(20.50)

where, for every i = 1, . . . , r, (i)

(i)

X 1 , . . . , X Ni

is a basis of V (i) .

If X is a basis for g adapted to the stratification as in (20.50), then we can define on g a group of dilations, still denoted by {δλ }λ>0 , such that

756

20 Taylor Formula on Carnot Groups (i)

(i)

δλ (Xj ) = λi Xj

for every i = 1, . . . , r and every j = 1, . . . , Ni .

We then know that (argue as in Theorem 1.3.28 on page 49) δλ Log (x) = Log δλ (x) , δλ Exp (X) = Exp δλ (X) ∀ x ∈ G, X ∈ g. (20.51) In the proof of Proposition 20.3.11 below, we shall need the following lemma having an interest in its own. Lemma 20.3.10. Let d be any homogeneous norm on G. Then there exists a constant c = c(d, G) such that, for every x ∈ G and every s ∈ [0, 1], we have d(γ (s)) ≤ c d(x), where γ is the integral curve of Log (x) starting at the origin. More explicitly, d Exp (sLog (x)) ≤ c d(x) for every x ∈ G and every s ∈ [0, 1]. (20.52) Proof. We know that the integral curve γ of Log (x) starting at the origin is s → Exp (sLog (x)), hence we have to prove (20.52). It is not restrictive to suppose x = 0, otherwise (20.52) holds for any c. Let X be any basis for g adapted to the stratification (for example, the Jacobian basis), and let δλ denote both the dilation on g defined above and the usual dilation on G. Then, thanks to (20.51), the linearity of δλ and the fact that d is δλ -homogeneous of degree 1, we have d(Exp (sLog (x))) = d(δ1/d(x) Exp (sLog (x))) = d(Exp (δ1/d(x) (sLog (x)))) d(x) = d(Exp (sδ1/d(x) (Log (x)))) = d(Exp (sLog (δ1/d(x) (x)))). As a consequence, being d(δ1/d(x) (x)) = 1, we have d(Exp (sLog (x))) ≤ d(x)

max ξ ∈G: d(ξ )=1

d(Exp (sLog (ξ ))) =: c.

s∈[0,1]

Note that c is finite, for {ξ : d(ξ ) = 1} is compact and d, Exp , Log are continuous functions.

We are in a position to prove the following result. Proposition 20.3.11 (Stratified Mac Laurin formula—integral remainder). Let n ∈ N ∪ {0}, and let u ∈ C n+1 (G, R). Suppose X = (X1 , . . . , XN ) is any basis of g adapted to the stratification of g. Then, following the notation in (20.49), we have u(x) = u(0) +

n

h=1

k=1,...,n i1 ,...,ik ≤N σi1 +···+σik =h

ζi1 (x) · · · ζik (x) Xi1 · · · Xik u(0) + Rn (x), (20.53) k!

20.3 Taylor Formula on Homogeneous Carnot Groups

where

Rn (x) =

n

k=1

i1 ,...,ik ≤N σi1 +···+σik ≥n+1

757

ζi1 (x) · · · ζik (x) Xi1 · · · Xik u(0) k!

N

ζi1 (x) · · · ζin+1 (x) n! i1 ,...,in+1 =1 N 1

Xi1 · · · Xin+1 u Exp s ζi (x) Xi (1 − s)n ds . × +

0

(20.54)

i=1

Moreover, for every fixed homogeneous norm d on G and every n ∈ N ∪ {0}, there exists cn > 0 (depending on n, G, d and the basis X ) such that, for x near 0, |Rn (x)| ≤ cn d n+1 (x) ×

sup d(z)≤c d(x)

{|Xi1 · · · XiM u(z)| :

n + 1 ≤ M ≤ r(n + 1), i1 , . . . , iM ∈ {1, . . . , m}}.

(20.55)

Finally, when X is the Jacobian basis related to G, the polynomial function qn (x) = u(0) +

n

h=1

ζi1 (x) · · · ζik (x) Xi1 · · · Xik u(0) k!

(20.56)

k=1,...,n i1 ,...,ik ≤N σi1 +···+σik =h

coincides with the Z-Mac Laurin polynomial of δλ -degree n related to u. Proof. Let us begin by taking x = 0 and h = x in (20.48). We get u(x) = u(0) +

n

N

k=1 i1 ,...,ik =1

ζi1 (x) · · · ζik (x) Xi1 · · · Xik u(0) k!

N

ζi1 (x) · · · ζin+1 (x) n! i1 ,...,in+1 =1 N 1

(Xi1 · · · Xin+1 u) Exp s ζi (x) Xi × (1 − s)n ds. (20.57) +

0

i=1

If Z = (Z1 , . . . , ZN ) is the Jacobian basis of G, then it is easily seen that (with the obvious notation), for every i = 1, . . . , r, (i)

(i)

Z1 , . . . , ZNi

is a basis of V (i) .

Hence, there exist r non-singular square matrices M (1) , . . . , M (r) such that, for i = 1, . . . , r, M (i) has order Ni and such that the matrix

758

20 Taylor Formula on Carnot Groups

⎛

⎞ ··· 0 .. ⎠ .. . . · · · M (r)

M (1) ⎝ ... 0

represents the transformation of the coordinates between the bases X and Z. Hence, by the corresponding results for the Jacobian basis in Theorem 1.3.28 (page 49), we can infer that, following the notation in (20.49), ζi (x) is a polynomial function in x, δλ -homogeneous of degree σi . In particular, if d is any homogeneous norm on G, there exists c = c(d, Log ) > 0 such that c−1 d σi (x) ≤ |ζi (x)| ≤ c d σi (x)

∀ x ∈ G, i ≤ N.

(20.58)

As a consequence, for every k ∈ N∪{0}, ζi1 (x) · · · ζik (x) is a polynomial function in x, δλ -homogeneous of degree σi1 + · · · + σik . Observe that this is also the δλ -degree of Xi1 · · · Xik as a differential operator. In particular, ζi1 (x) · · · ζin+1 (x) is δλ -homogeneous of degree ≥ n+1 and, in the integral summands, there appear only higher order derivatives of u with δλ -height ≥ n + 1. We rewrite (20.57) pointing out, in the right-hand side, the polynomial of δλ degree ≤ n u(x) = u(0) +

n

h=1

ζi1 (x) · · · ζik (x) Xi1 · · · Xik u(0) k!

k=1,...,n i1 ,...,ik ≤N

+

σi1 +···+σik =h n

k=1

i1 ,...,ik ≤N

ζi1 (x) · · · ζik (x) Xi1 · · · Xik u(0) k!

σi1 +···+σik ≥n+1 N

ζi1 (x) · · · ζin+1 (x) n! i1 ,...,in+1 =1 N 1

n (Xi1 · · · Xin+1 u) Exp s ζi (x) Xi × (1 − s) ds +

0

i=1

=: qn (x) + {Rn (x)}.

(20.59)

As we remarked above, the function qn (x) =

n

h=0

qn(h) (x),

20.3 Taylor Formula on Homogeneous Carnot Groups

759

where

qn(h) (x)

=

⎧ u(0) ⎪ ⎪

⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ζi1 (x) · · · ζik (x) Xi1 · · · Xik u(0) k!

k=1,...,n

if h = 0, if h = 1, . . . , n, (20.60)

i1 ,...,ik ≤N σi1 +···+σik =h (h)

is a polynomial function of δλ -degree ≤ n and each qn (x) is a homogeneous polynomial of δλ -degree = h. We now turn our attention to Rn (x). The first sum in braces (ranging over k = 1, . . . , n) is a polynomial function of δλ -degree ≥ n + 1. The remaining summand can be estimated as follows (for x in a neighborhood of 0) N ζi1 (x) · · · ζin+1 (x) n! i1 ,...,in+1 =1 N 1

n × (Xi1 · · · Xin+1 u) Exp s ζi (x) Xi (1 − s) ds 0 i=1

N

≤

i1 ,...,in+1 =1

×

|ζi1 (x)| · · · |ζin+1 (x)| (n + 1)!

sup d(z)≤c d(x)

{|Xi1 · · · Xin+1 u(z)| : i1 , . . . , in+1 ∈ {1, . . . , N }}.

Indeed, we recall that, by definition, it holds N

s ζi (x) Xi = s Log (x).

i=1

Then, we can apply Lemma 20.3.10 to derive that N

Exp s ζi (x) Xi ∈ Bd (0, c d(x)), i=1

so that N

s ζi (x) Xi (Xi1 · · · Xin+1 u) Exp i=1

≤

sup d(z)≤c d(x)

{|Xi1 · · · Xin+1 u(z)| : i1 , . . . , in+1 ∈ {1, . . . , N }}.

As a consequence, Rn (x) can be estimated as follows (for x near 0).

760

20 Taylor Formula on Carnot Groups

|Rn (x)| ≤

n

k=1

|ζi1 (x)| · · · |ζik (x)| Xi1 · · · Xik u(0) k!

i1 ,...,ik ≤N σi1 +···+σik ≥n+1

+

N

i1 ,...,in+1 =1

×

sup d(z)≤c d(x)

|ζi1 (x)| · · · |ζin+1 (x)| (n + 1)!

{|Xi1 · · · Xin+1 u(z)| : i1 , . . . , in+1 ∈ {1, . . . , N }}

≤ cn d n+1 (x) ×

sup d(z)≤c d(x)

{|Xi1 · · · Xik u(z)| :

k ≤ n + 1, i1 , . . . , ik ≤ N, σi1 + · · · + σik ≥ n + 1} ≤ cn d n+1 (x) ×

sup d(z)≤c d(x)

{|Xi1 · · · XiM u(z)| :

n + 1 ≤ M ≤ r(n + 1), i1 , . . . , iM ∈ {1, . . . , m}}. Here, we have used the fact that (thanks to the stratification of g) any Xj (for j ∈ {1, . . . , N }) is a precise linear combination (with scalars fixed together with g and the basis (X1 , . . . , XN )) of terms like Xi1 · · · Xih with 1 ≤ h ≤ r (r being the step of nilpotency of g) and i1 , . . . , ih ∈ {1, . . . , m}. Since the above estimate of Rn gives Rn (x) = O(d n+1 (x))

as x → 0,

the decomposition u(x) = qn (x) + Rn (x) in (20.59) rewrites as u(x) = qn (x) + O(d n+1 (x))

as x → 0.

Thus (being qn a polynomial of δλ -degree ≤ n) via an application of what has been proved in Remark 20.3.7, we infer that qn (x) is necessarily the Z-Mac Laurin poly nomial of δλ -degree n related to u. This ends the proof. For an improvement of (20.55) for large x’s, see Exercise 9 at the end of the chapter. For the future references, we state the explicit formula for Pn (u, 0) proved above in the following corollary. Corollary 20.3.12 (Explicit form of the Z-Mac Laurin polynomial). Let n ∈ N ∪ {0}, and let u ∈ C ∞ (G, R). Let Z = (Z1 , . . . , ZN ) be the Jacobian basis related to G. We set Log (x) = ζ1 (x) Z1 + · · · + ζN (x) ZN

∀ x ∈ G,

where Log is the logarithmic map, i.e. ζ1 (x), . . . , ζN (x) are the components of Log h w.r.t. Z or, equivalently,6 6 See Remark 1.2.20 on page 21.

20.3 Taylor Formula on Homogeneous Carnot Groups

ζj (x) = Log (x)Ij (0)

761

∀ j ∈ {1, . . . , N }.

Then the Z-Mac Laurin polynomial of δλ -degree n related to u, i.e. the only polynomial p of δλ -degree ≤ n such that (Z1 · · · ZN )α (p)(0) = (Z1 · · · ZN )α (u)(0), for every multi-index α with |α|G ≤ n, is given by the following formula Pn (u, 0)(x) = u(0) +

n

h=1

ζi1 (x) · · · ζik (x) Zi1 · · · Zik u(0). k!

k=1,...,n i1 ,...,ik ≤N σi1 +···+σik =h

For example, if n = 2, the above formula for P2 (u, 0) rewrites as

P2 (u, 0)(x) = u(0) + ζi (x) (Zi u)(0) + ζi (x) (Zi u)(0) i: σi =1

i: σi =2

+

i,j : σi =σj =1

ζi (x) ζj (x) (Zi Zj u)(0). 2

(20.61)

Example 20.3.13. Consider the Heisenberg–Weyl group H1 on R3 . In order to obtain the expansion of a smooth u on H1 by Proposition 20.3.11, we must take n = 1. Hence, if we consider the Jacobian basis Z1 = ∂x1 + 2x2 ∂x3 ,

Z2 = ∂x2 − 2x1 ∂x3 ,

Z3 = ∂x3 ,

then we have Exp (ξ1 Z1 + ξ2 Z2 + ξ3 Z3 ) = (ξ1 , ξ2 , ξ3 ), Log (x1 , x2 , x3 ) = x1 Z1 + x2 Z2 + x3 Z3 ), so that formula (20.53) gives u(x) = u(0) + x1 Z1 u(0) + x2 Z2 u(0) 1 3

xi xj (Xi Xj u)(sx) (1 − s) ds . (20.62) + x3 Z3 u(0) + i,j =1

0

This shows that formula (20.53), despite its very simple proof, is not as “optimal” as formula (20.38) in Theorem 20.3.3. Indeed, note that the remainder in braces in (20.62) contains also derivatives of u of order 4 (w.r.t. Z1 , Z2 ) such as 1 Z2 Z1 Z2 Z1 − Z2 Z12 Z2 − Z1 Z22 Z1 + Z1 Z2 Z1 Z2 u. 16 Instead, formula (20.38) gives the estimate u(x) − u(0) + x1 Z1 u(0) + x2 Z2 u(0) ≤ c1 |x1 |2 + |x2 |2 + |x3 | × sup {|Z12 u|(z), |Z22 u|(z), |Z1 Z2 u|(z)}. Z3 Z3 u =

d(z)≤b2 d(x)

762

20 Taylor Formula on Carnot Groups

If x0 ∈ G is any given point, by replacing u and x in the above results respectively with u ◦ τx0 and x0−1 ∗ x, we obtain the corresponding results for the stratified Taylor formula. We state them without further comment, recalling only that Pn (u, x0 ) = Pn (u ◦ τx0 , 0) ◦ τx −1 . 0

Proposition 20.3.14 (Stratified Taylor formula-integral remainder). Let n ∈ N ∪ {0}, u ∈ C n+1 (G, R), x0 ∈ G. Suppose X = (X1 , . . . , XN ) is any basis of g adapted to the stratification of g. Then, following the notation in (20.49), we have n

ζi1 (x0−1 ∗ x) · · · ζik (x0−1 ∗ x) u(x) = u(x0 ) + k! h=1

k=1,...,n i1 ,...,ik ≤N σ +···+σ =h

i1 ik × Xi1 · · · Xik u (x0 ) + Rn (x, x0 ),

where Rn (x, x0 ) n

= k=1

i1 ,...,ik ≤N σi1 +···+σik ≥n+1

(20.63)

ζi1 (x0−1 ∗ x) · · · ζik (x0−1 ∗ x) k!

ζi1 (x0−1 ∗ x) · · · ζin+1 (x0−1 ∗ x) n! i1 ,...,in+1 =1 1 N

−1 n s ζi (x0 ∗ x) Xi Xi1 · · · Xin+1 u x0 ∗ Exp × (1 − s) ds . × Xi1 · · · Xik u (x0 ) +

N

0

i=1

(20.64) Moreover, for every fixed homogeneous norm d on G and every n ∈ N ∪ {0}, there exists cn > 0 (depending on n, G, d and the basis X ) such that (for x near x0 ) sup {|(Xi1 · · · XiM u)(x0 ∗ z)| : |Rn (x, x0 )| ≤ cn d n+1 (x0−1 ∗ x) × d(z)≤c d(x0−1 ∗x)

n + 1 ≤ M ≤ r(n + 1), i1 , . . . , iM ∈ {1, . . . , m}}.

(20.65)

Finally, when X is the Jacobian basis related to G, the polynomial function n

ζi1 (x0−1 ∗ x) · · · ζik (x0−1 ∗ x) qn (x, x0 ) = u(x0 ) + k! h=1

k=1,...,n i1 ,...,ik ≤N σi1 +···+σik =h

× (Xi1 · · · Xik u)(x0 )

(20.66)

coincides with the Z-Taylor polynomial of δλ -degree n related to u and x0 . For an improvement of (20.65) for large x’s, see Exercise 9 at the end of the chapter.

20.3 Taylor Formula on Homogeneous Carnot Groups

763

Corollary 20.3.15 (Explicit form of the Z-Taylor polynomial). Let n ∈ N ∪ {0}, u ∈ C ∞ (G, R) and x0 ∈ G. Let Z = (Z1 , . . . , ZN ) be the Jacobian basis related to G. If the ζi ’s are as in Corollary 20.3.12, then the Z-Taylor polynomial of δλ degree n related to u and x0 , i.e. the only polynomial p of δλ -degree ≤ n such that (Z1 · · · ZN )α (p)(x0 ) = (Z1 · · · ZN )α (u)(x0 ), for every multi-index α with |α|G ≤ n, is given by the following formula Pn (u, x0 )(x) = u(x0 ) +

n

h=1

ζi1 (x0 ∗ x) · · · ζik (x0 ∗ x) k!

k=1,...,n i1 ,...,ik ≤N σi1 +···+σik =h

× (Zi1 · · · Zik u)(x0 ). By suitably choosing the “increment” h in Corollary 20.3.9 to belong to the “first layer” of G, we obtain the following “horizontal” Taylor formula. Corollary 20.3.16 (Horizontal Taylor formula). Let x ∈ G, and let u ∈ C n+1 (G, R). Let (X1 , . . . , Xm ) be any basis of the first layer of the stratification of g. Then, for every ξ1 , . . . , ξm ∈ R, we have m

u x ∗ Exp ξj X j j =1

= u(x) +

n

m

k=1 i1 ,...,ik =1

+

ξi 1 · · · ξi k Xi1 · · · Xik u(x) k!

m

ξi1 · · · ξin+1 n! i1 ,...,in+1 =1 1

× 0

(Xi1 · · · Xin+1 u) x ∗ δs Exp

m

ξj X j

(1 − s)n ds.

(20.67)

j =1

Proof. It suffices to apply (20.67) with h = Exp

m

ξj X j

j =1

and to observe that, in this case, Log (h) = m j =1 ξj Xj , so that (with the notation in (20.67)) ξj if j ∈ {1, . . . , m}, ζj (h) = 0 if j ∈ {m + 1, . . . , N }. Finally, notice that (see, e.g. (19.10d) on page 728)

764

20 Taylor Formula on Carnot Groups

Exp

m

s ξj X j

m m

ξj X j ξj X j = Exp δs = δs Exp .

j =1

j =1

This ends the proof.

j =1

For example, if we take n = 1 in the above corollary, we get the horizontal Taylor formula with the integral remainder of order two m

ξj X j u x ∗ Exp = u(x) + +

j =1 m

ξj Xj u(x)

j =1 1

m

ξh ξk (Xh Xk u) x ∗ δs Exp

0 h,k=1

m

(1 − s) ds (20.68)

ξj X j

j =1

for every ξ1 , . . . , ξm ∈ R. Since, for a given arbitrary m × m matrix A = (ah,k )h,k , it holds # $ m m

A + AT ah,k + ak,h = ξ, ξ , ξh ξk ah,k = ξh ξk 2 2 h,k=1

h,k=1

(20.68) also rewrites as u(x ∗ Exp (ξ · X)) = u(x) + (ξ · X)u(x) 1 + Hesssym u(x ∗ δs (Exp (ξ · X))) ξ, ξ (1 − s) ds, 0

(20.69)

where we have set ξ ·X =

m

ξj X j

j =1

and Hesssym u =

Xh Xk u + Xk Xh u 2

, h,k=1,...,m

the so-called symmetrized horizontal Hessian of u. As an application of Taylor’s formula, we give the following result. Proposition 20.3.17 (L-harmonicity of Taylor polynomials). Let (G, ∗) be a homogeneous Carnot group, and let Z = (Z1 , . . . , ZN ) be the Jacobian basis related to G. Let also L be a sub-Laplacian on G. Suppose that u ∈ C ∞ (G, R) is an Lharmonic function. Then, for every n ∈ N ∪ {0} and every x0 ∈ G, the function x → Pn (u, x0 )(x) is L-harmonic in G.

20.3 Taylor Formula on Homogeneous Carnot Groups

765

Proof. Suppose we have proved the assertion for x0 = 0. Then, by applying the assertion for u ◦ τx0 (which is L-harmonic on G since u is) we infer that y → Pn (u ◦ τx0 , 0)(y) is L-harmonic on G. As a consequence, since (20.26) gives Pn (u, x0 ) = Pn (u ◦ τx0 , 0) ◦ τx −1 , 0

we infer that x → Pn (u, x0 )(x) is L-harmonic in G. The above argument shows that it suffices to prove the assertion when x0 = 0. By Lemma 20.3.18 below, we know that, for every n ≥ 2, L(Pn (u, 0)) = Pn−2 (Lu, 0) = Pn−2 (0, 0) = 0. Since L(Pn (u, 0)) = 0 also for n = 0, 1 (since P0 and P1 are polynomials of Gdegree ≤ 1), this completes the proof.

Lemma 20.3.18. Let u ∈ C ∞ (G, R). Then we have L(Pn (u, 0)) = Pn−2 (Lu, 0)

for every n ≥ 2.

(20.70)

Proof. By Proposition 20.3.11, we have u(x) = Pn (u, 0)(x) + Rn (x), so that Lu(x) = L(Pn (u, 0))(x) + LRn (x).

()

If we show that, for every n ≥ 2, one has ()

LRn (x) = Ox→0 (d n−1 ),

then, by Remark 20.3.7, () will give (20.70). We are then left with the proof of (). Denote the expression of Rn in (20.54) as follows Rn (x) =

n

k=1

Qk (x) +

AI (x) × BI (x),

I =(i1 ,...,in+1 )

where BI is the integral in the far right-hand side of (20.54). Clearly, each Qk is a δλ -homogeneous polynomial function of degree ≥ n + 1, so that LQk is a δλ -homogeneous polynomial function of degree ≥ n − 1, whence LQk (x) = Ox→0 (d n−1 ). Dropping the subscript I in AI and BI , we have (see Ex. 6, Chapter 1) L(A B) = B LA + A LB + 2 ∇L A, ∇L B. Since A(x) = ζi1 (x) · · · ζin+1 (x)/n!, it clearly holds LA = Ox→0 (d n−1 ),

A = Ox→0 (d n+1 (x)),

|∇L A| = Ox→0 (d n ).

This proves that L(A B) = Ox→0 (d n−1 ), since B, LB and ∇L B are bounded. This completes the proof.

766

20 Taylor Formula on Carnot Groups

Bibliographical Notes. For the topics presented in Section 20.2, we are much indebted to the presentation of the same subject in [FS82] by G.B. Folland and E.M. Stein. For results concerning Taylor’s formula on Carnot groups of step two, see G. Arena, A. Caruso, A. Causa [ACC06]. For other results of calculus on Carnot groups, see [CM06,Hei95a].

20.4 Exercises of Chapter 20 Ex. 1) Prove Proposition 20.1.2, page 735. (Hint: Show that the map introduced therein is linear and sends a basis into a basis.) Ex. 2) Retracing the proof of Proposition 20.1.5, prove in details Proposition 20.1.7. Ex. 3) Prove Lemma 20.1.11, page 741. (Hint: If j cj Hj ≡ 0, then 0 = ( j cj Hj )× (u(δλ (x))) = j cj λdj (Hj u)(δλ (x)). Now take as u any function of the type α x and complete the argument.) Ex. 4) With reference to Example 20.2.5, let us now find r = P3 (f, 0). Since r ∈ P3 , there exist real numbers, a, b, . . . , m such that r(x, y, t) = a + b x + c y + d t + e x2 + f y2 + g xy + h xt + i yt + j x3 + k y3 + l x2 y + m xy2 . Then (20.22) gives a = r(0) = f (0), b = Z1 r(0) = Z1 f (0), c = Z2 r(0) = Z2 f (0), d = Z3 r(0) = Z3 f (0), 2e = (Z1 )2 r(0) = (Z1 )2 f (0), 2f = (Z2 )2 r(0) = (Z2 )2 f (0), g − 2d = Z1 Z2 r(0) = Z1 Z2 f (0), h = Z1 Z3 r(0) = Z1 Z3 f (0), i = Z2 Z3 r(0) = Z2 Z3 f (0), 3!j = (Z1 )3 r(0) = (Z1 )3 f (0), 3!k = (Z2 )3 r(0) = (Z2 )3 f (0), 2l − 4h = (Z1 )2 Z2 r(0) = (Z1 )2 Z2 f (0), 2m − 4i = Z1 (Z2 )2 r(0) = Z1 (Z2 )2 f (0), so that P3 (f, 0)(x, y, t) = f (0) + Z1 f (0) x + Z2 f (0) y + Z3 f (0) t 1 1 + (Z1 )2 f (0) x2 + (Z2 )2 f (0) y2 + (2Z3 f (0) 2 2

20.4 Exercises of Chapter 20

767

+ Z1 Z2 f (0)) xy + Z1 Z3 f (0) xt + Z2 Z3 f (0) yt 1 1 + (Z1 )3 f (0) x3 + (Z2 )3 y3 + (2Z1 Z3 f (0) 3! 3! 1 1 2 2 + Z1 Z2 f (0)) x y + (2Z2 Z3 f (0) + Z1 Z22 f (0)) xy2 . 2 2 Ex. 5) Consider the homogeneous Carnot group G on R3 with the composition law x ◦ y = (x1 + y1 , x2 + y2 , x3 + y3 + x1 y2 ). Prove that the related Jacobian basis is given by Z1 = ∂1 ,

Z2 = ∂2 + x1 ∂3 ,

Z3 = [Z1 , Z2 ] = ∂3 .

Show that P2 (f, 0)(x1 , x2 , x3 ) = f (0) + Z1 f (0) x1 + Z2 f (0) x2 + Z3 f (0) x3 1 1 + (Z1 )2 f (0) x21 + (Z2 )2 f (0) x22 2 2 + (Z2 Z1 )f (0)x1 x2 (20.71) is the Z-Mac Laurin polynomial of G-degree 2. Deduce that formula (20.24) does not apply in this case. (Indeed, this would hold true iff 12 (Z1 Z2 + Z2 Z1 ) = Z2 Z1 , i.e. [Z1 , Z2 ] = 0 which is false.) This shows that (20.24) strongly depends on the properties of the fixed coordinate system on G or, equivalently, on the properties of the composition law. Notice that the inverse on G differs from −x (and the coordinate system on G is not the logarithmic one). See also Ex. 6 below. Finally, compare to (20.61). Notice that here we have Exp (ξ1 Z1 + ξ2 Z2 + ξ3 Z3 ) = (ξ1 , ξ2 , ξ3 + ξ1 ξ2 /2), so that Log (x1 , x2 , x3 ) = x1 Z1 + x2 Z2 + (x3 − x1 x2 /2)Z3 . With the notation of Corollary 20.3.12, this means that ζ1 (x) = x1 , ζ2 (x) = x2 , ζ3 (x) = x3 − x1 x2 /2, whence (20.61) gives P2 (f, 0)(x1 , x2 , x3 ) = f (0) + Z1 f (0) x1 + Z2 f (0) x2 1 1 + Z3 f (0) (x3 − x1 x2 /2) + (Z1 )2 f (0) x21 + (Z2 )2 f (0) x22 2 2

768

20 Taylor Formula on Carnot Groups

1 1 + (Z1 Z2 )f (0)x1 x2 + (Z2 Z1 )f (0)x2 x1 2 2 = f (0) + Z1 f (0) x1 + Z2 f (0) x2 + Z3 f (0) x3 1 1 + (Z1 )2 f (0) x21 + (Z2 )2 f (0) x22 2 2 1 + (Z1 Z2 + Z2 Z1 − Z3 )f (0)x1 x2 , 2 which is (20.71), for Z1 Z2 + Z2 Z1 − Z3 = Z1 Z2 + Z2 Z1 − [Z1 , Z2 ] = 2Z2 Z1 . Ex. 6) (Symmetrized Z-Mac Laurin formula of δλ -degree two). Suppose that G is a homogeneous Carnot group (of step r ≥ 2) for which the inversion equals −x. Then7 the Z-Mac Laurin polynomial of δλ -degree two is given by P2 (f, 0)(x(1) , x(2) , x(3) , . . . , x(r) ) = f (0) + ∇ (1) f (0) · x(1) & 1% (20.72) + ∇ (2) f (0) · x(2) + x(1) , Hesssym f (0) · x(1) . 2 Here

Hesssym f (0) :=

1 (1) (1) (1) (1) {Z Z f (0) + Zk Zh f (0)} 2 h k

h,k=1,...,N1

is the so-called symmetrized horizontal Hessian of f at 0. For example, a sufficient condition for x −1 = −x to hold (and hence, for (20.72) to hold) is that G is equipped with logarithmic coordinates. Note that (20.72) holds for the Heisenberg–Weyl group Hn since the inversion on Hn is −x (even if Hn is not generally equipped with logarithmic coordinates). Compare also to the previous Ex. to show that (20.72) does not hold in general unless suitable hypotheses on the coordinates are made. Hint: In order to prove (20.72), by Definition 20.2.3, it suffices to show that the polynomial (say q) in the right-hand side of (20.72) has δλ -degree two (obvious) and the following facts hold: (i)

q(0) = f (0),

(ii)

Zh q(0) = Zh f (0)

(iii) (iv)

(1)

(1)

for h = 1, . . . , N1 , (2) (2) Zh q(0) = Zi f (0) for h = 1, . . . , N2 , (1) (1) (1) (1) (Zh Zk )q(0) = (Zh Zk )f (0) for 1 ≤

h ≤ k ≤ N1 .

7 Here we used the following notation. The usual notation (1.79a) (page 56) for the “strati-

fied” coordinates on a homogeneous group; moreover, for i = 1, 2, ' ( (i) (i) ∇ (i) f (0) = Z1 f (0), . . . , ZN f (0) , i

(i) i.e. Zj is the left-invariant vector field coinciding with ∂ (i) at 0 or, equivalently, xj (i) (i) Z1 , . . . , ZN are the vector fields of the Jacobian basis which are δλ -homogeneous of i degree i. Notice that ∇ (1) f (0) is the horizontal gradient of f at 0.

20.4 Exercises of Chapter 20

769

Now, (i) is trivial, whereas (ii) and (iii) are simple consequences of the fact (i) (i) that Zh |0 = (∂/∂ xh )|0 . In order to show (iv), recall (20.10), so that

(1)

Zh = ∂x (1) + h

h = 1, . . . , N1 ,

aj,h (x) ∂xj ,

j : σj >1

where ⎛

1

⎜ ⎜ a2,1 (x) aj,h j,h=1,...,N = ⎜ ⎜ .. ⎝ . aN,1 (x)

1 .. . ···

⎞ 0 .. ⎟ .⎟ ⎟ = Jτ (0) x ⎟ 0⎠ 1

··· .. . .. .

0

aN,N −1 (x)

is the Jacobian matrix of the left translation by x on G (i.e. τx (y) = x ∗ y) and the aj,h ’s are polynomial functions, δλ -homogeneous of degree σj − σh . (1) (1) As a consequence, Zh Zk coincides with ∂2 (1) (1)

∂ xh xk

+

∂ aj,k j : σj =2

(0) ∂xj = (1)

∂ xh

∂2 (1) (1)

∂ xh xk

+

(2) N2

∂ aj,k j =1

(1)

∂ xh

(0) ∂x (2)

at 0. Here, we used a “stratified” notation for Jτx (0), namely ⎛

IN1

0

⎜ (2) ⎜ A1 (x) Jτx (0) = ⎜ .. ⎜ ⎝ .

IN2 .. . ···

(r)

A1 (x) Notice that

··· .. . .. . (r) Ar−1 (x)

⎞ 0 .. ⎟ . ⎟ ⎟. ⎟ 0 ⎠ INr

(2) (aj,k (x))1≤j ≤N2 ,1≤k≤N1 = A(2) 1 (x).

This shows that (iv) is equivalent to (2)

2 ∂ a

1 (1) (1) j,k (1) (1) (2) {Zh Zk f (0) + Zk Zh f (0)} + (0) Zj f (0) (1) 2 ∂x

N

j =1

?

=

∂ 2 f (0) ∂

(1) (1) xh xk

+

N2

∂

j =1

∂

(2) aj,k (0) ∂x (2) f (0). (1) j xh

Arguing as above, this is in turn equivalent to ∂ 2 f (0) (1) (1)

∂ xh xk

+

(2) N2

∂ aj,k j =1

(1)

∂ xh

(0) ∂x (2) f (0) j

h

j

770

20 Taylor Formula on Carnot Groups

+

∂ 2 f (0)

?

=

(2) (2) N2 ∂ aj,k ∂ aj,h 1 (0) + (0) ∂x (2) f (0) (1) (1) j 2 ∂ xk j =1 ∂ xh (1) (1)

∂ xh xk

+

(2) N2

∂ aj,k j =1

(1)

∂ xh

(0) ∂x (2) f (0). j

Thus, we are left to prove that, for all j ≤ N2 and all h, k ≤ N1 , it holds (2)

(2)

∂ aj,k

(0) + (1)

∂ aj,h (1) xk

?

(0) = 0.

∂ xh ∂ (2) (2) Now, the matrix aj,k = A1 involves the (derivatives of the) components of the left translation along the second layer. We know that the left translation is given by τx (y) = x ∗ y = x (1) + y (1) , x (2) + y (2) + Q(2) (x, y), . . . where Q(2) (x, y) is a polynomial in x (1) , y (1) of (ordinary) degree 2, mixed in x, y. Hence, for every j ∈ {1, . . . , N2 }, there exists a square matrix B j of order N1 such that (2)

(2)

(2)

(τx (y))j = xj + yj + B j · x (1) , y (1) . We now use the hypothesis x −1 = −x to derive that B j is skew-symmetric. Consequently, this gives N1 (2) ∂ j (2) τ (x) = (y) = xi(1) Bi,k . aj,k x j (1) ∂ yk 0 i=1 Hence, (2) ∂ aj,k (1)

∂ xh

(0) +

(2) ∂ aj,h (1)

∂ xk

j

j

(0) = Bh,k + Bk,h = 0,

for B j is skew-symmetric. The assertion is proved. Ex. 7) Suppose f, g ∈ C n+1 (G, R) are such that (Z1 , . . . , Zm )α f ≡ (Z1 , . . . , Zm )α g

∀ α ∈ (N ∪ {0})m : |α| = n + 1.

Show that there exists a polynomial function p, δλ -homogeneous of degree at most n, such that f = g + p on G. (Hint: Apply Theorem 20.3.3 to f − g and derive that f (h) − g(h) = Pn (f − g, 0)(h) for every h ∈ G.) Derive that (Zi1 , . . . , Zim )α f ≡ (Zi1 , . . . , Zim )α g for every α ∈ (N ∪ {0})m such that |α| = n + 1 and for every i1 , . . . , im ∈ {1, . . . , m}. This proves that the (Z1 , . . . , Zm )α f ’s with |α| = n + 1 determine all the (Zi1 , . . . , Zim )α f ’s with |α| = n + 1.

20.4 Exercises of Chapter 20

771

Ex. 8) Provide a detailed proof of Theorem 20.3.3. Ex. 9) Prove the following improvement of (20.55) and (20.65). For every x ∈ G, we have |Rn (x, x0 )| ≤

n+1 k

C k=1

k! ×

d(x0−1 ◦ x)σi1 +···+σik

i1 ,...,ik ≤N, σi1 +···+σik ≥n+1

×

sup d(z)≤C d(x0−1 ◦x)

|Zi1 · · · Zik u(x0 ◦ z)|,

where C is a constant depending only on G, d and the basis X . Here, the σi ’s are the same as in (20.2).

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Index of the Basic Notation1

∂j , ∂xj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 X= N j =1 aj ∂j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Xf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 T (RN ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 X ≡ XI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 ∇ .........................................................................5 γX (t, x), D(X, x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 X (k) , X h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 exp(tX)(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 exp(X)(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 [X, Y ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10, 103 [Zj1 , · · · [Zjk−1 , Zjk ] · · ·] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 ZJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Lie{U } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 rank(Lie{U }(x)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 [V , W ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 G := (RN , ◦) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 τα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14, 107 Jτα (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 η → J (η) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 H1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 {e1 , . . . , eN } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 π : g → RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 (Z1 u, . . . , ZN u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 X Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 δλ : RN → RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1 The page number refers to the first time the symbol is introduced within the text. In case of

multiple numbering, this means that the symbol is employed in multiple contexts.

790

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σ = (σ1 , . . . , σN ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31 G = (RN , ◦, δλ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31, 56 |α|σ , |α|G , degG (p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 X ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 div(A · ∇ T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Jτx (0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 |E| (E ⊆ RN ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Q= N j =1 σj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44, 126 g = g1 ⊕ · · · ⊕ gr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 δλ : g → g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ξ · Z, E · ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50, 133 C-H(G) := (RN , ∗, δλ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 x (1) , . . . , x (r) , x (i) ∈ RNi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 r and m = N1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56, 126 W (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59 g = W (1) ⊕ · · · ⊕ W (r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ΔG , L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62, 144 ∇G , ∇L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 L = div(A(x)∇ T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 A(x) = σ (x) σ (x)T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 qL (x, ξ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Isotr(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 u → J (u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 X β := Xi1 ◦ · · · ◦ Xik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 (X1 , X2 )-connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 πi : RN −→ R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 ϕ : U → RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 (Uα , ϕα ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 v(f ), Mm , T (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 M, N, M , N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 ∂/∂ xi |m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 dm ψ : Mm → Mψ(m) dψ : T (M) → T (M ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 X : Ω −→ T (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 π(m, v) := v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 X, Y , Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Xm , X(f )(m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 f → Xf , Xf : M → R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 X (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 dψ : X (M) → X (M ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 X and X μ(t) ˙ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 [X, Y ] : M → T (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 G, H, F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 [·, ·] : g × g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

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791

g, h, f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14, 107 ◦, ∗, • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 τα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 x −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 α : g → Ge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 ϕ : G → H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 dϕ : g → h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 expX (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Exp : g → G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24, 49, 118 Log : G → g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26, 49, 120 h = V1 ⊕ · · · ⊕ Vr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 (1) (r) B = (E1 , . . . , ENr ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 ad X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 πE : h → RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131, 139 Ψ := Exp ◦ (πE )−1 : RN → H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131, 139 Δλ : h → h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 δλ := πE ◦ Δλ ◦ πE−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 H∗ := (RN , E , δλ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Exp h : h∗ → h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 HL (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Im(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .155 Hn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 hn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 ΔHn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 (x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Bx, ξ , B (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .158 (Fm,2 , ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 γi,j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 H = (Rm+n , ◦, δλ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169, 681 ρ : N → N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 HM-group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Bη , g∗2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 E = (RN , +, δλ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 (B, ◦) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 B = (R1+N , ◦, δλ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 h1 , . . . , hk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 d, |x|G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 d0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 dX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 S(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 l(γ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

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X-connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Dirac0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 uε , u ∗G Jε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Jε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 B (x, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Bd (x, r), B˙d (x, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252, 459 D(x, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 -dist(x, A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 ωd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 KL , ΨL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Mr , Nr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Mr , Nr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Φr , Φr∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 βd , md , nd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255, 259 a d , ad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257, 261 −D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Iα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 ML (f )(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 F ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 ν⊥F at y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 AL(u)(x), AL(u)(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 lim infy→x , lim supy→x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 l.s.c., u.s.c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .338 u, uˇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 (E, T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 F : V → F(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 HϕV , HfV , HfΩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341, 361, 388 μVx , μΩ x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341, 367, 388 H∗ (Ω), H∗ (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 F ↑, F ↓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 Tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 (E, H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 H(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346, 388 B-H∗ (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 K ↓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 B(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 S(Ω), S(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353, 389 uV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 (H-D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Ω Uf , UΩ f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Ω

Hf , HΩ f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 R(∂Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

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793

R∞ (∂Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 + S c (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 sx0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 sxΩ0 := HfΩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 u0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 f f RA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 RA , L H(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Lε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 (G, H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 X 2 u(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 V := Exp (V1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 [x ◦ h−1 , x ◦ h] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 Vx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 V -Convv (Ω), V -ConvH (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 GΩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425, 427 Γ ∗ μ, GΩ ∗ μ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 k, K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 mr [u], Mr [u] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 λx,r (y), Λx,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 μ, μ[u], μu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 S b (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 + S (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 I (μ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 V (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 UE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 RuE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 ΦEu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 RuE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 WK , VK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 μK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 C(K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502, 508 C∗ (E), C ∗ (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 GΩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 g Ω (x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 M, M0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 M(E), M0 (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 A B, μ|A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 νE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 μ, ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 μ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 E + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 (ρ) Mφ (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 mφ (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558

794 (ρ)

Index of the Basic Notation

M(α) , m(α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 α(E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 n(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 I (φ, Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 fm,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 H (m, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 eX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 LA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 F = {F1 , . . . , FH } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .650 z, z⊥ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 Jz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 U (1) , . . . , U (n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 v : G → b, z : G → z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 σ (x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 u∗ (x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 Pk [x1 , . . . , xq ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 r k=s Pk [x1 , . . . , xq ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 jq , c(q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 Pn = span{x α : α ∈ In } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 α Z α := Z1α1 · · · ZNN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 Pn (f, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 πn : P → Pn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 Pn (f, x0 )(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 ζ1 (h), . . . , ζN (h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 Hesssym u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764

Index

B-hyperharmonic function, 348–350, 352 minimum principle, 351 B-invariant set, 349–351 δλ -degree (of a polynomial), 33 δλ -homogeneous degree (of a polynomial), 33 G-cone, 480 G-degree (of a polynomial), 33 G-polynomial, 734 S-harmonic space, 363, 367 positivity axiom (in a), 363 S∗ -harmonic space, 370 on a Carnot group, 382 L-maximal function, 277 q-set, 477 associative algebra, 600 asymptotic solid sub-Laplacian, 261 asymptotic surface sub-Laplacian, 257 balayage, 376 balayage (w.r.t. a sub-Laplacian), 501 barrier function (in a S∗ -harmonic space), 371 Bôcher-type theorem, 460, 461 Bouligand theorem, 371, 391, 544 bounded above L-subharmonic functions, 474 Brelot convergence property, 268 Caccioppoli–Weyl’s lemma for subLaplacians, 408 Campbell–Hausdorff formula, 128, 129, 194, 585–587, 600, 659

formula for formal power series, 600 formula for homogeneous vector fields, 194, 593, 594 operation, 29, 128, 130, 131, 142, 143, 585, 586, 595, 626, 652 canonical sub-Laplacian, 621, 622 of a Heisenberg-type group, 171, 692 capacitable set (w.r.t. a sub-Laplacian), 509 capacitary distribution (w.r.t. a subLaplacian), 502, 505 capacitary potential (w.r.t. a sub-Laplacian), 502, 508 capacity of a compact set (w.r.t. a sub-Laplacian), 502 and polarity, 507 characterization of the-, 503, 505 monotonicity, 506 right continuity, 506 strong subadditivity, 506 capacity (w.r.t. a sub-Laplacian), 509 and polarity, 509 exterior-, 509 interior-, 508 of Borel sets, 509 Carnot group, 56, 122, 131, 198 B-groups, 184 canonical sub-Laplacian of a-, 62 classical definition, 122 Euclidean group, 183, 280 filiform-, 207

796

Index

free-, 578, 584, 587, 588, 625, 658, 661, 662 harmonic space, 381 Heisenberg–Weyl group, 155 HM-groups, 174 homogeneous-, 121 K-type groups, 186 of Heisenberg type, 169, 681 of Iwasawa type, 702 of step r, 56, 198 of step two, 158, 163, 661, 666 of type HM, 174 stratified change of basis on a-, 61, 165 sum of-, 190 with homogeneous dimension Q ≤ 3, 184 central series (lower-), 149, 207 characteristic set (of a smooth open set), 710 Choquet lemma, 339 commutator, 103 nested-, 11 of homogeneous fields, 37 of length k, 11 continuity principle for potentials, 489 convergence axiom, 345, 347 in a Carnot group, 382 convex function equivalence between H- and v-, 417 H-, 416 horizontally-, 416 v-, 411 Cornea theorem, 344, 347 covering lemma, 566 decomposition theorem, 303 derivatives on G, 740 differentiable manifold, 88 differential, 95 of a homomorphism, 113, 114, 121 of a smooth map, 95 of the exponential map, 119 dilation, 31, 48, 56, 121, 132, 191, 593, 627, 638, 649, 656, 669 -invariance of a sub-Laplacian, 63 differential operator homogeneous with respect to-, 32, 34 function homogeneous with respect to-, 32, 34 invariance of the Lebesgue measure with respect to-, 44 on the Lie algebra g, 46

Dini–Cartan theorem, 343 Dirichlet problem (in a harmonic space), 359, 361, 371 doubling measure, 277 down directed (family of functions), 342, 343, 349, 356, 357 down directed (family of sets), 349 eikonal equation, 466 energy w.r.t. a sub-Laplacian, 528 equilibrium distribution (w.r.t. a subLaplacian), 497, 505 equilibrium potential (w.r.t. a sub-Laplacian), 497 as barrier function, 514 fundamental theorem on-, 498 uniqueness of the-, 508 equilibrium value (w.r.t. a sub-Laplacian), 497 characterization of the-, 505 exponential function generated by a system of vector fields, 194 map, 24, 49, 118, 129, 131, 626 inverse function of-, 27, 49, 626 of a homogeneous group of step 2, 166 of a vector field, 8, 23, 117 extended L-Green function, 516, 518 extended maximum principle for Lsubharmonic functions, 493 extended Poisson–Jensen’s formula, 518 exterior L-capacity, 509 filiform Carnot group, 207, 209, 285 fine topology (w.r.t. a sub-Laplacian), 537 fractional integral (in a Carnot group), 277 free Carnot group, 578 free homogeneous Carnot group, 584, 586–588, 625, 629, 656, 658, 659, 661, 662 free nilpotent Lie algebra, 577, 579, 583, 586, 624, 650, 656 Hall basis for the-, 579 fundamental solution, 236, 425, 456, 477, 649, 661–663, 665 H-type group, 696 fundamental theorem on L-equilibrium potentials, 498

Index gauge function (w.r.t. a sub-Laplacian), 247 generalized solution (in the sense of Perron–Wiener–Brelot), 359, 361, 390 generators of a homogeneous Carnot group, 56 geodesics (for a homogeneous Carnot group), 309, 314 gradient (canonical G-), 62 Grayson–Grossman theorem, 582 Green function extended-, 516, 518 of a general domain, 427 approximation of the-, 429 symmetry of the-, 431 of an L-regular set, 425, 426, 445, 446, 448 potential of a measure related to a-, 432 H-groups (in the sense of Métivier), 174 H-inversion map, 705 H-Kelvin transform, 705 H-type algebra, 681 H-type group, 681 canonical sub-Laplacian, 692 fundamental solution, 696 H-inversion map, 705 H-Kelvin transform, 705 Iwasawa group, 702 prototype-, 169, 687 Hall basis, 579, 581, 583, 586, 587 Hardy–Littlewood–Sobolev theorem, 277 harmonic function (w.r.t. a harmonic sheaf), 340, 346, 347, 353, 357 harmonic function (w.r.t. a sub-Laplacian), 146, 381, 388–390, 408, 433, 441–445, 458–461 Brelot convergence property for a-, 268 decomposition theorem for a-, 303 removable (or isolated) singularity, 458 harmonic majorant (least-), 358 harmonic measure (w.r.t. a harmonic sheaf), 341, 367 harmonic measure w.r.t. a sub-Laplacian, 388, 391, 426, 445, 518 of a polar set, 515 harmonic minorant (greatest-), 358, 427 harmonic sheaf, 340 harmonic space, 345, 347, 348, 353 in a Carnot group, 381

797

Harnack inequality, 265–267 on rings, 267, 460 Hausdorff dimension w.r.t. d, 558, 568 Hausdorff measure w.r.t. d, 557 height δλ -height of a multi-index, 32 G-height of a multi-index, 33 Heisenberg group polarized, 180 Heisenberg–Weyl group, 155, 156, 172, 580, 625, 630, 632, 659, 702 Heisenberg-type algebra, 681 Heisenberg-type group, 681 canonical sub-Laplacian, 692 fundamental solution, 696 H-inversion map, 705 H-Kelvin transform, 705 Iwasawa group, 702 prototype-, 687 Hessian (horizontal-), 412 higher-order derivatives on G, 738 HM-groups, 174 homogeneous Carnot group, 56, 121, 131, 198, 206, 586, 630, 637, 649, 656, 659 generators of a-, 56 homogeneous dimension, 44, 126, 128, 184, 303, 477, 559, 663 homogeneous Lie group on G homogeneous dimension of a-, 477 homogeneous Lie group on RN , 31, 196 homogeneous dimension of a-, 44 of step two, 158, 163 homogeneous norm, 229 pseudo-triangle inequality for a-, 231, 484 homomorphism, 114, 130 of Lie algebras, 112 of Lie groups, 112, 121 Hopf-type lemma, 297 horizontal Hessian, 412 horizontal segment, 415 horizontal Taylor formula, 763 horizontally convex function, 416 equivalence with v-convex function, 417 Hörmander condition, 12, 69, 185, 193, 202, 210, 281 hyperharmonic function (w.r.t. a harmonic sheaf), 341, 348, 352, 353, 355, 356

798

Index

hypoelliptic-(ity), 188, 193, 280, 434, 441, 633 analytic, 280 hypoharmonic function (w.r.t. a harmonic sheaf), 342 interior L-capacity, 508 isolated singularity, 458–461 Iwasawa-type group, 702 Jacobi identity, 11, 103, 107 Jacobian basis, 19–22, 26, 45, 50, 58, 59, 69, 115, 143, 144, 156, 159, 185, 187, 588, 622, 625, 627–629, 638, 639, 643, 649, 653, 656, 658–662, 666 of a homogeneous Lie group, 42, 43 Kelvin transformation, 704 l.s.c. function, 338 Lagrange mean-value theorem (on a Carnot group), 746 Laplace operator, 100, 183, 445, 621, 622 left-translation, 106 length δλ -length of a multi-index, 32 G-length of a multi-index, 33 Levi–Cartan theorem, 343, 347, 348 Lie algebra, 11, 14, 107, 197 filiform-, 207 free nilpotent-, 577 generated by a set, 11 Jacobian basis, 19 of a Carnot group, 59 of a Lie group, 108 Lie group, 106, 195 Carnot group, 56 composition law of a homogeneous-, 39, 41, 50, 58 homogeneous on RN , 31 on RN , 13 structure on a Lie algebra, 130 Lie polynomial, 600 Lifting, 653–656, 659, 661, 665, 666 Liouville-type theorems, 269, 270, 274, 461, 633 asymptotic-, 274–276 lower functions, 359 lower regularization, 339, 501

lower solution, 359 Lusin-type theorem, 495 Mac Laurin polynomial (on a Carnot group), 742 Maria–Frostman domination principle, 495, 499, 503 maximum principle, 474 -set, 474 for L-subharmonic functions, 409 extended-, 493 on unbounded open sets, 474 strong version, 426 weak version, 388, 389, 474 mean value formula, 391, 447 mean value operator, 397, 456 solid-, 399, 404, 405, 432, 441, 447, 458 superposition formula, 457, 458 surface-, 401, 404, 410, 447, 456 measure function, 557 minimum principle for B-hyperharmonic functions, 351, 360 mollifier, 239, 240, 401, 455 MP set, 474 mutual L-energy, 528 non-characteristic exterior ball condition, 384, 385, 387 peaking function at a point, 542 Perron family, 356–358, 362 Perron–Wiener–Brelot operator, 359 Perron–Wiener–Brelot solution (related to a sub-Laplacian), 390 Perron-regularization, 355 Poisson–Jensen’s formula, 445, 448 extended-, 518 polar set (w.r.t. a sub-Laplacian), 491, 493, 495, 508, 544, 559, 568 and harmonic measure, 515 characterization in terms of capacity, 507, 509 polar∗ set (w.r.t. a sub-Laplacian), 498, 499, 508 the L-irregular points are a-, 514 polarized Heisenberg group, 180 polynomial functions on G, 734 positivity axiom, 345, 351, 354 in a S-harmonic space, 363 in a Carnot group, 382

Index potential of a measure (related to the fundamental solution Γ ), 445, 451, 458 continuity principle (for the), 489 potential of a measure (related to the L-Green function GΩ ), 432, 433, 441, 443, 444 prototype H-type group, 169, 687 pseudo-triangle inequality, 231, 400, 484 improved-, 306 PWB function, 361, 428 quasi-continuity of L-superharmonic functions, 528 reduced function, 376 reduced function (w.r.t. a sub-Laplacian), 501 regular point, 371 regular point w.r.t. a sub-Laplacian, 518, 542, 544 and L-polarity∗ , 514 regular set (w.r.t. a harmonic sheaf), 341, 345, 346, 348, 353, 355 regular set w.r.t. a sub-Laplacian, 385, 387, 388, 391 approximation by-, 430 Green function of a-, 425 regularity axiom, 345, 346 in a Carnot group, 383 removable singularity, 458–461 resolutive function, 361 characterization of the-, 367 Riesz measure (of an L-subharmonic function), 441–443, 445, 447, 451, 502, 518 Riesz-type representation theorem, 441, 443–445, 451, 458 Riesz-type representation theorem, 484 segment (horizontal-), 415 separation axiom, 345, 352 in a Carnot group, 382 sheaf (of functions), 340, 353 harmonic-, 340 Sobolev–Stein embedding theorem, 279 Stone–Weierstrass theorem, 366 stratification, 45, 122, 131, 583 of a Carnot group, 60, 309, 314

799

stratified change of basis, 61, 165 stratified group, 122, 131 harmonic function, 146 sub-Laplacian of a-, 144 stratified Lagrange mean-value theorem, 746 stratified Taylor formula, 750 stratified Taylor inequality, 749 strong maximum principle, 296 sub-Laplacian, 62, 66, 198, 623, 625, 637, 641, 650 arising in control theory, 205 Caccioppoli–Weyl’s lemma (for-), 408 canonical-, 62, 198, 625, 637, 641, 650, 663 on a Heisenberg-type group, 692 characteristic form of a-, 65 degenerate-ellipticity of a-, 66 harmonic measure related to a-, 388 harmonic space related to a-, 381 invariance with respect to the dilation, 63 of a stratified group, 144 of Bony-type, 202, 223, 285 of Kolmogorov-type, 204 Perron–Wiener–Brelot solution related to a-, 390 q-set w.r.t. a-, 477 regular set w.r.t a-, 388 Riesz measure of a function subharmonic w.r.t. a-, 441 subharmonic function w.r.t. a-, 389 superharmonic function w.r.t. a-, 389 thin set w.r.t. a-, 474 sub-mean function, 397–399, 401 sub-mean properties, 399 subharmonic function (w.r.t. a harmonic sheaf), 353 criterion (for subharmonicity), 354 subharmonic function (w.r.t. a subLaplacian), 389, 401, 402, 404, 405, 411, 441, 442, 445, 447, 451, 456, 494, 516, 518 bounded above in G, 451 bounded above-, 474 extended maximum principle for-, 493 Riesz-type representation theorem for a-, 441, 443 subharmonic smoothing for a-, 456 subharmonic minorant, 356, 358

800

Index

superharmonic function (w.r.t. a harmonic sheaf), 353 characterization of the-, 353 superharmonic function (w.r.t. a subLaplacian), 389, 410, 432, 457, 458, 491 superharmonic majorant, 358 surface mean value theorem, 391, 404 symmetrized horizontal Hessian, 764 tangent bundle, 91 tangent space, 91, 109 tangent vector, 91 Taylor formula (on a Carnot group), 750 Taylor inequality (on a Carnot group), 749 Taylor polynomial (on a Carnot group), 745 thin set w.r.t. a sub-Laplacian, 474 thinness of a set at a point (w.r.t. a sub-Laplacian), 538, 550, 553 total gradient, 22 total set in C0+ (G), 529 u.s.c. function, 338 up directed (family of functions), 342–344, 347, 348, 354

upper functions, 359 upper regularization, 339 upper solution, 359 v-convex function, 411 equivalence with H-convex function, 417 vector field, 4, 97 complete-, 102, 116 generating a Carnot group, 191 integral curve (of a), 6, 101 left-invariant, 14, 17, 107, 197 completeness of the-, 116 Lie-bracket, 10 related-, 99, 105, 106, 114 smooth-, 98 weak maximum principle, 295 Wiener resolutivity theorem, 364, 390 Wiener’s criterion for sub-Laplacians, 547, 550 Wiener’s regularity test for sub-Laplacians, 553 Zorn’s lemma, 350

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