Topics in Harmonic Analysis, Related to the Littlewood-Paley Theory

TOPICS IN HARMONIC ANALYSIS Related to the Littlewood-Paley Theory BY

ELIAS M. STEIN

PRINCETON UNIVERSITY PRESS AND THE UNIVERSITY OF TOKYO PRESS

PRINCETON, NEW JERSEY 1970

Copyright© 1970, by Princeton University Press ALL ItlOHTS RESERVED

L.C. Card: 72-83688 S.B.N.: 691-08067-4 A.M.S. 1968: 2265, 4201, 4750

Published in Japan exclusively by the University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of America

PREFACE This monograph contains essentially the material presented in a course I

given during the spring semester of 1968 at Princeton University. My purpose in these lectures was two-fold: First, to give a new approach to that part of harmonic analysis which for .tl:re-sake of simplicity we refer to as the "Littlewood-Paley Theory." The techniques that are used lead to a wide generalization of results hitherto restricted to

an

and other special

contexts. My second aim was to give the interested student a rapid (although admittedly sketchy) introduction to various areas in analysis, in particular some elements of Lie groups, almost everywhere limit theorems in the context of martingales, and complex interpolation of operators. If I have succeeded in my two aims it is because the main tools used in Chapters III and IV come from martingale theory and interpolation theory, while interesting examples of the results may be obtained in the setting of compact and semi-simple groups. I am deeply indebted to R. Gundy for several enlightening conversations and to C. Fefferman for his great care and effort in preparing the lecture notes.

CONTENTS Preface •••....•••..•..•.•.•....•... '· . . . . . . . . • . • • • • • • • . .

v

Introduction . . . • . • • • • • • . . • . . . . . • . • •.. . . . . . . . . . • . • . • • . • . • .

1

Chapter I. Lie Groups (A Review) §1. Compact groups .•..........•.....•....-............

5

§2. The Peter-Weyl theorem ........................ , . . 12 §3. The Peter-Weyl theorem (Concluded) ..........•••... 15 §4. Lie groups; examples .••••.••.........••.•••.••... 20 §5. Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 §6. Universal enveloping algebra •.•................•••. 28 §7. Laplacian ......••••.•..••..•.•.....•..•.••...... 33 Chapter II. Littlewood-Paley Theory for a Compact Lie Group §1. The

heat~iffusion

semi-group ...................... 38

§2. The Poisson semi-group; the main theorem . . . . . . . . . . . 46 §3. Proof of Theorem 2. . • . . . . . . . . . . . . . . . . . . • . . . . • . . . . . 50 §4. Applications: Riesz transforms, etc.

. . . . . • . . • . . • . . . 57

Bibliographical remarks .••..••.......•••.•......... 64 Chapter III. General Symmetric Diffusion Semi-groups §1. General setting .....•.•..•.•.........•.....•....... 65 §2. Analyticity of these semi-groups .........•••......... 67 §3. The maximal theorem .•.••.•••.......•............. 73 §4. A digression: L 2 theorems ••••.......•.•.•.••••••.. 82 Bibliographical remarks . . . . . . . . . . . . . • . . . . . . . . . . . . . 88

vii

viii

CONTENTS

Chapter IV. The General Littlewood-Paley Theory §1. Conditional Expectation and Martingales .....••••.....••. 89 §2. The inequalities for martingales ................. , . . . . . . 94 §3. An additional "max" inequality ••..•...•....•••..••••• 103 §4. The link between margingales and semi-groups

, , .. , .. , .. 106

§S. The Littlewood-Paley inequalities in general ..........••. 111 §6. Denouc_ement ........•.•... , •..........•.•.•..•••..... 120 Bibliographical remarks .....••.•..•.....•....••.•••••. 122 Chapter V. Further Illustrations §1. Lie groups ..•.•••..•...........•.••..•.•....•..•••.. 123 §2. Semi-simple case .......•....•••..•.••......•... , ..... 128 §3. Sturm-Liouville ••.••..••.....••....•••••..••••..• , .••• 136 §4. Heuristics ••..•.............•.•••..•..•...••••........ 137 Bibliegraphical remarks •••..••••...........••••.•..•• 141 References

143

TOPICS IN HARMONIC ANALYSIS Related to the Littlewood-Paley Theory

INTRODUCTION

Background

We shall use the phrase "Littlewood-Paley theory" rather loosely, to denote a variety-of related results in classical hannonic analysis whose extension to a general setting is our mai.n goal. In its one-dimensional form the theory goes back to the 1930s and may be said to contain the Hardy-Littlewood maximal theorem, Hilbert transforms, the work of Littlewood-Paley in (161, and capped in effect with the multiplier theorem of Marcinkiewicz [35}. 1 This theory may be described in terms of the Poisson integral, which in R 1 is given by the family of transfonn ations f(x) .... .!. 11

f

oo

-oo

f(x- y)dy t2 + Y2

u(x, t)

As is well known the behavior of the hannonic function u(x, t) closely reflects the behavior of its boundary values. Now the main point of the socalled "complex method" is to pass to the analytic function F(z) whose real part is u and to exploit complex function theory to study F and thus f . An example of this far-reaching idea arises in the Hilbert transfonn, which gives in effect the passage from the real to the imaginary parts of the boundary values of F. The next stage of the development of this type of analysis which culminated in the 1950's, saw the primacy of complex function theory give way to real-variable methods, and this led to an extension of many of these

1

See also Zypund [20, Chapters 14 and 15].

2

INTRODUCTION

results to Rn 2 . Characteristic of these techniques were various "covering lemmas" and certain singular integral transforms, whose kernels had a quite explicit description, all rather specific to Rn.

Main results Our approach is essentially different from the above two and allows for a very 'general formulation of an essential core of the subject; it can be applied, in particular, to various new and interesting situations. Our starting point is the semi-group of Poisson integrals, that is we assume that we are dealing with a family of operators !Ttlt> 0 , defined simultaneously on Lp(M)

1 ~ p ~ "", with the'property thatTtl. Tt2 = Ttl+t2.

To= I,

and in. addition to the usual measureability in t , it satisfies the following basic assumptions: (I) (U)

Tt are contractions on Lp(M), i.e.,

II T~ll p :5 II fll P'

1 S P :5 ""·

Tt are symmetric, i.e., each Tt is self-adjoint on L 2(M).

(Ill)

Tt are positivity preserving, i.e., T~ ~ 0, if f?: 0.

(IV)

Tt(l) = 1.

We refer to the above as symmetric diffusion semi-groups. The task we set ourselves is to develop, as far as is possible, the analogues of the Littlewood-Paley theory in the context of these semigroups. The interest in this arises from the multiplicity of examples of symmetric diffusion semi-groups and the consequence this theory has for the eigenfunction expansions of their infinitesimal generators. Besides the usual Poisson integrals for R 0 a variety of examples may be found in Chapter 2, Section 2 of Chapter 3, and in Chapter 5. The main tools that are used are three-fold:

2 See the bibliographic indications at the end of Chapter

2.

3 It is to be noted that the assumptions (I)- (IV) are to some extent

redundant.

3

INTRODUCTION

(i) The spectral representation in L 2 (M) of Tt as Tt = [ 000 e-At dE{A.),

for an appropriate spectral family E(A). This is, of course, the direct substitute for the Fourier transfonn in Rn. (ii) Connections of the semi-group Tt! and certain auxiliary martingales

and ergodic theorems. (iii) Convexity properties of holomorphic families of operators which allow one to mediate between (i) and (ii). '\

A curious fact that should not be ov-erlooked'J,s that the theorems for martingales required in the technique (ii) were to a significant extent already anticipated by Paley in his paper[ 17] dealing with the Walsh-Paley series. Among the results we obtain are: The maximal theorem (in Chapter 3), to wit II sup ITtf(x~ lip< Apllfll t

>0

-

p

,

In Chapter 4 we prove the Littlewood-Paley type inequality

r. ~~F'I dr1/

1(

2

Apll'ip •

l
and its converse. We also obtain the multiplier theorem to the effect that if Tt =

f 0oo e-At dE(A),

and T m = f 0oo m(x) dE(x), then l < p < oo,

whenever

where M is bounded on (O,oo). An interesting example arises if in(A) = Aiy, for y real.

4

INTRODUCTION

The theorem for T m may be viewed as a variant of the Marcinkiewicz multiplier theorem, but instead of requiring conditions on a finite number of of derivatives of m(A), it requires a specific analyticity. 4

Another approach Besides the general theory just described, to which ChapteiS 3 and 4 are devoted, still another approach is given, less general but on the other hand more· fruitful in various instances. It arises whenever, roughly speaking, the square of the infinitesimal generator of Tt is a second-order elliptic operator, i.e., a "Laplacian." In that case the interplay between the x-derivatives and the t-derivative are also of interest, and it is possible to

~tudy

the natural generalization of the Hilbert transform (the so-

called Riesz transforms). This situation occuiS in the context of compact Lie groups, and to some extent also in the non-compact Lie groups (in particular for the zonal functions of semi-simple groups); also for SturmLiouville expansions. This attack has as its starting point the universality of the identity

where u > 0, .!\u = 0, and .!\ is an appropriate Laplacian with V its associated gradient.

Organization Since our purpose is also pedagogical we have included much illustrative material, in particular in Chapters II and V. Chapter I is intended merely as a brief review of some basic facts about Lie groups, to be read as an introduction for Chapters 2 and 5. The core of the monograph, Chapters II and III, however, stands independently of the other chapters. This is what is to be expected from a universal conclusion of this kind. Since it requires the same condition for m(A) for e.g., the case of Rn (all n ), it would require that m(A) is C"'. Other examples of symmetric diffusion semi-groups notably the one constructed in Section 2 of Chapter v, when G = SL(2, R) shows that it is even necessary that m be analytic in A. 4

CHAPTER II LIE GROUPS (A REVIEW)

This chapter is a review consisting of various statements, often with only a bare indication of proof, which serve as the requisite background for what follows. In Sections 1 through 3 the main goal is the discussion of the Peter-Weyl theorem for compact groups. Secti-ons 4 through 6 deal with Lie groups, their Lie algebras and the resulting universal enveloping algebra. Finally in Section 7 we describe the Laplacian for compact Lie groups which will be fundamental in Chapter II. Section 1. Compact groups

We begin by summarizing some facts from the theory of integration of locally compact groups. For details see Weil [8, Chapter II] and Loomis [3, Chapter VI]. A topological group is a group G which is also a topological space, and whose group operations · : G x G .... G and -l: G .... G are continuous. A left Haar measure on the locally compact topological group G is a Borel measure 11 on G, satisfying the properties (1) 11CK) <+oo forK a compact subset of G; (2) 11 (0) > 0 for 0 f,

cp an open subset of G. (We ignore the fact

that 0 might not be Borel if G is too big.) (3) p. (aE)

o:

p. (E) for any a

f

G and Borel set E.

The crucial property is, of course, (3), the "left-translation invariance" of 11· Here and elsewhere aE denotes lax! x

5

f

El.

6

LIE GROUPS (A REVIEW)

Every locally compact topological group has a left Haar measure, unique up to multiplication by a positive constant. For example, Lebesgue measure is a left Haar measure on the circle group S1 , being invariant under the left group-action, i.e., rotation; and every invariant measure on S 1 is a mUltiple of Lebesgue measure. Haar measure on the circle grou-p is not only left-invariant, but also right-invariant, i.e., IL (Ea) = IL (E). A group with a left and right-invariant Haar measure is said to be unimodular. That the circle, or any abelian locally compact group, is unimodular, is quite trivial, for left and rightinvariance mean the same thing. We can also see without difficulty that a compact topological group G is unimodular. For let IL be a left Haar measure on G, and pick any a f G. The measure ~'a on G, defined by lla(E) =' IL (Ea) is obviously a left Haar measure. By the essential uniqueness of

left Haar measure, IL a since ll (G)

=

t,. ·IL for some positive constant t,.. But t,.

= ll (G ·a) = lla(G) = t,. ·ll (G).

=l

Therefore ll (Ea) = ll (E) for all

E, so that the left-invariant measure is also right-invariant. Not all groups are unimodular. A standard example is the group of all transformations x

->

ax + b from the real line to itself, where a f, 0. Suppose that a belongs to the unimodular locally compact group G, and

that f is any function defined on G. The (left) translate of f by a is just

= f(a- 1x), x f G; and the right translate is defined (p (a)f)(x) = f(xa). Because of our choice of conventions, ,\

the function (,\(a)f)(x) similarly, by

and p satisfy the composition laws ,\(a)(,\(b)f)

=

,\(ab)f and p(a)(p(b)f)

=p(ab)f. Obviously ,\(l)f=p(l)f= f. Let us regard ,\(a) and p(a) as operatom taking functions to functions. Then ,\(a) and p (a) are isometries of Lp(G) where the Lp -spaces are defined with respect to the Haar measure on G, and ,\(a)- 1 = ,\(a- 1 ). We shall prove the useful fact that ,\(a)f

->

f and p (a)f

->

f in Lp(G), as

a _. 1 in G, for each fixed f f Lp(G). Since the operatom ,\(a) and p(a) are uniformly bounded, we need only consider f f C 0 (G), the space of continuous functions with compact support. For f E c 0 (G), however, ,\(a)f and

§ 1.

7

COMPACT GROUPS

p(a)f converge uniformly to f, and have their supports contained in a fixed compact set, provided that a varies in a small enough neighborhood of 1. So A(a)f .... f and p (a)f .... f in Lp for f £ C0 (G), which completes the proof. The result is, of course, valid ~nly when p < oo • . The convolution of two functions f al;td g on the group G, is defined by the formula (f * gXx)

=

j 0 f(y)g(y-_tx)dy, whe,re dy denotes Haar mea-

sure. If f and g belong to L 1(G) then th7 integral defining f * g(x) con-

verges absolutely for almost every x f G, so that f * g is defined-in fact wehave !If* giiL 1(G) S llf!IL 1{G)IIg\IL 1(G)" This follows easily from Fubini's theorem. Convolution is a well-behaved product operation. For instance (f * g) * h f*g = g*f

= f * (g * h) and (af + a 'f ') * g = a (f * g)+ a '(f' * g).

when the group G is commutative, but not in general.

=

Nevertheless we can still guarantee f * g

g * f on a non-commutative

group, if f is a central (or invariant, or class) function, which means that f(a- 1 xa) = f(x) for a, x l G. Equivalently, f is a central function if and only if f(xy) = f(yx) for x, y £ G.

A 11 of the above elementary observations follow from changes of variables and manipulations with multiple integrals. The reader may reconstruct the proofs or consult the cited references. Hence~orth,

G denotes a compact topological group with its Haar mea-

sure dx normalized so that f0 dx = l. We now tum to the representations of compact groups. Let V be a finite-dimensional vector space over R or C. A finitedimensional representation of G on V is a continuous homomorphism g .... Rg which maps G into the group of non-singular linear transformations on V. Thus, a representation R of G on V satisfies (1) Rg 1Rg}v) = Rg 1g 2 (v) for gl' g 2

£

G and

Vf

V ;

(2) R 1 = I, the identity mapping on V; (3) g .... Rgv is a continuous mapping of G into V for each fixed v £ V.

8

LIE GROUPS (A REVIEW)

The representations R 1 on V1 , and R2 on V2 , are called equivalent if there is an isomorphism of vector spaces A: V1 --. V2 (onto), such that

R~ = A- 1 R~A for each g£ G. Two equivalent representations are, in an

obvious sense, different manifestations of the same object. A urtitary representation of G on the

finite~imensional

Hilbert space

H is a continuous homomorphism g ... Rg which maps G into the group of unitary transfonnations of H. Two unitary representations R 1 on H 1 , and R 2 .on H2·, are unitarily equivalent if there is an isomorphism of Hilbert spaces A: H1 ... H2 (onto), such that R~"' A- 1 R~A for each g t G. Every

finite~imensional

representation of G is equivalent to a unitary

representation. In other words, if R is a representation of G on a finitedimensional vector space V, we may provide V with Hilber-space structure, so that the transformations Rg are all unitary. To prove this, we pick an arbitrary (strictly) positive-definite inner product ( · , · )0 on V. Define a new inner product by setting (cp,t/J) = JG(Rgcp, Rgtfr) 0 dg, for cp,t/J £ V. The reader may easily verify that ( . , · ) is an inner product and that (Rgcp, Rgtfr)

=

(cp,t/J), so that Rg is, in fact, unitary.

The reason for the above lemma is that unitary representations are often easier to work with than ordinary representations. We shall see applications shortly. Suppose that R and S are representations of G on vector spaces V and W respectively. The direct sum R mS is the representation on V E& W given by (R ms)g([v, w]) = [Rgv, Sgw l, for gc G and [v, wl tV mW. The tensor product representation R EllS on the vector space V ® W is defined by(R®S)g(v®w) = (Rgv) ®(Sgw). Finally,thecontragredient representation to R is the representation g ... (R~ )- 1 where "t" denotes the transpose. We leave to the reader the interpretation of these operations in tenns of the matrices for the transformations Rg, Sg, etc. Note that if R is a unitary representation given by matrices Rg = I Rij(g)l then its contragredient is given by Rg = I R ij(g)l, where the term stands for complex conjugation.

9

§1. COMPACT GROUPS

An invariant subspace for a representation R of G on V, is a subspace Ws; V, such that Rg(W) f W for each g E G A representation R on the space is irrf!ducible if V has no non-trivial R-invariant subs paces, and it is completely reducible if it is (equivalent to) a direct sum of irreducible representations. Every unitary representation (R, V) _of G is completely reducible. The proof is by induction on dim V, whiclris

calle~

i

-the degree of R. If R is

equivalent to a non-trivial direct sum S E&T, then R is completely reducible, by an application of the inductive hypothesis to S and T. But if R does not split as a direct sum, then it is irreducible, hence completely reducible. For if W is any non-trivial R-invariant subspace of V, then the orthocomplement W1 is also R-in variant. Therefore R = R Iw E9 Rl W.... , contradicting the indecomposability. We proved, using the compactness of G, that every finite-dimensional representation is unitary. So every finite-dimensional representation of G is completely reducible; which reduces the study of representations to that of irreducible representations. Later on we will use the notion of characters. The character of a representation R of G, is the function XR(x) = trace Rx taking G into the complex numbers. XR depends only on the equivalence class of R, and XR(x) = XR(a xa- 1 ) for x, a l G, i.e., XR is a central function. Eventually we shall show that XR determines R uniquely up to equivalence. Let us illustrate some of the concepts we have defined, byconsidering ing the circle group G = [0, 2n.J. The finite-dimensional irreducible representations are given by () -+ eik(J for () f G, k E Z. The tensor product eik1(J ® eik2(J is the representation ei(k1 + k2> 9 , and the contragredient representation to eikO is e-ikO. Thus, all the irreducible representations of G arise from the single representation () ... ejf) by successive tensor products and contragredients.

10

LIE GROUPS (A REVIEW)

(To prove that every irreducible representation is an eik6, argue as follows. An irreducible representation, of G is equivalent to an irreducible unitary representation R.of G on the space V. Now IRgl g t Gl is a commuting family of unitary operators, and can therefore be diagonalized simultaneously. Hence R is equivalent to a representation

'

But such a representation is the direct sum of the one-dimensional representations Sii(g). Therefore every irreducible representation of G is onedimensional.) Observe that the irreducible representations eikO of G are precisely the fundamental objects of Fourier analysis on G. In particular, the famous Parseval theorem states that we can decompose L 2 (G) as

I""

k=-<><>

e (eikl\ where leikOI is the one-dimensional space spanned by

eili1, and the direct sum means the usual concept-that the leikOI are pairwise orthogonal, and that every f t L 2 (G) is a countable linear combination of eikO. The appearance of representations in the analysis of the circle group is no coincidence. We shall prove that L 2 of any compact group splits as an infinite direct sum of finite-dimensional spaces arising from the irreducible representations of the group. This result is contained in the Peter-Weyl theorem, one of the goals

~f

the present chapter.

As a first step to generalizing Fourier analysis on the circle, we shall prepare to prove an analogue of the orthonormality of leikOI. SCHUR'S LEMMA: (a) Let R 1 and R2 be finite-dimensional repre-

sentations of G, on vector spaces V1 and V2 which are irreducible. Suppose that the linear transformation A: V2 .... V1 satisfies R~ A = AR~ for all g t G (A is called an intertwining operator). Then either A = 0 or A is an isomorphism onto.

11

§1. COMPACT GROUPS

(b) Let R 1 be a finite~imensional representation of G on a complex vector space V. If the transformation on A: V .... V satisfies AR~

=

R~ A

for all x £ G, then A is a multiple of the identity operator. Proof: (a) The hypothesis R~A = A~~ shows immediately that ker A

is an R 2 -invariant subspace of V 2 and tbat im A is an R 1-invariant subspace of V 1 • Since R 1 and R2 are irre'ducible, ~er A = 0 or V 2 , and im A = 0 or V1. (b) Since Rg(A- AI)

=

(A- AI)Rg for any g £ G, A € C, part (a) shows

that A- AI is 0 or else is invertible. A- AI is not invertible if A is an eigenvalue of A. THEOREM

QED.

(Schur's Orthogonality Relations): Suppose that R 1 and R 2

are irreducible finite-dimensional unitary representations of G. Let

R~ = IR/j(g)l and R~ = IR~e(g)l. Then if R 1 and R 2 are inequivalent

j

(a)

R{j(x)R~e(x)dx

= 0 for each i', j, k, f;

G

(b)

JG R{j(x)~~e(x)dx

=

stf/degree (R 1 ), where

8~

is

the Dirac delta, set equal to one when i == k, j = f; 0 elsewhere. Recall that Haar measure on G is normalized so that

JG dx = I.

Proof: (a) Let B be any linear transformati~n from V2 into V1 , and

set A=

f0

R 1(x)B R 2(x- 1)dx. Nothing could be easier than to verify that

R! A = AR~. Since R 1 and R2 are not equivalent, A could not be an isomorphism. From Schur's Lemma we conclude that A= 0, By taking B a matrix with a single non-zero entry, we can easily show that

12

LIE GROUPS (A REVIEW)

(b) Let B be a linear transformation: V 1

....

V2 and set A =

JGR 1(x)BR 1(x- 1 )dx. As above, R!A = AR~. By Schur's lemma A= AI, so JGR 1(x)BR 1(x- 1 )dx =AI. Taking the trace of both sides, we find that tr(B) = A · deg (R) 1• If we take B to have a single 1 as its only non-zero entry, then trace (B)= 0 or 1 depending on the location of the non-zero entry. Recalling that RJ!e(x- 1)

=

RJ!e(x), we conclude as in part (a), that

f0 R.{j(x)R~e(x)dx = stf/degR 1 • QED As advertised, the Schur orthogonality relations generalize the orthonormality of leikOI on the circle group. As a consequence we see that if R 1 and R 2 are two irreducible unitary representations then

according to whether R 1 is or is not equivalent with R2 • In particular the irreducible representation R 1 is determined, up to equivalence, by its character.

Section 2. The Peter-Weyl Theorem For a detailed discussion see Weil [8, Chapter V ], Loomis [3, Chapter VIII], and Pontrijagin [6, Chapter IV}. In order to formulate the Peter-Weyl theorem, we need to generalize our notion of unitary representation. Suppose that H is any complex Hilbert space, finite or infinite dimensional. A map which associates to every g E G, a linear operator Rg: H-+ H, is aunitary representation of G on H if and only if 1. each Rg is unitary;

2. g .... Rg is a homomorphism of G into the group of unitary operators on H;

13

§2. niE PETER-WEYL niEOREM

3. For each fixed vector

rp £

H, the mapping g-> Rgr/J is continuous

from G to H (with the norm topology). The notions of equivalence of arbitrary representations, direct sums any number of unitary representations, and of irreducibility of a unitary representation of G, are defined in the obvious way. Our old friends ,\(a) and p(a) : L:fG)-> L 2(G) are unitary representations of G on L 2(G). ,\ (respectively, p) is called the left (right) regular representation of G.

The space

& of entry

functions is the linear space of functions on G

spanned by the entries of finite-dimensional irreducible representations of

& is spanned by all functions of the form x-> (Rxr/J, 1/J), where (R, H) is an irreducible representation, and rp, if! £ H.

G. Equivalently,

If G is the circle group, then

& is

just the space of trigonometric

polynomials. We can now state the Peter-Weyl Theorem. THEOREM:

(1) Every irreducible unitary representation of G is finite-dimensional. (2) The right (or left) regular representation of G is the direct sum of finite-dimensional irreducible representations, and every equivalence class a of finite-dimensional irreducible representations appears in this direct sum as many times as da, the degree of that equivalence class. (3) There are sufficiently many finite-dimensional representations to separate the points of G. That is, for any two different points x, y (4)

E

G, there is a suitable representation R such that Rx

~

Ry.

& is dense in C(G) and in LiG).

(5) Let A be the set of equivalence classes of finite-dimensional irreducible representations. For each a tation R'1 in the class a, If f

E

£

A, pick a unitary represen-

LiG), set F(a) equal to the

14

LIE GROUPS (A REVIEW)

finite-dimensional matrix

JG f(x) R'i dx.

(F(a) is analogous to a

Fourier coefficient.) Then llfll~

Ia£Adatr{Fa)F*(a))=Ia£Ada111F(a)iii 2 , IIIAIII de-

=

noting the Hilbert-Schmidt norm of the matrix A = (aij), given by Ill Alll

2

=

i

I.1,]-la1] .. 2•

,

We shall sketch a proof of the Peter-Weyl theorem. Fo~

the time being, let us assume that G has a faithful f.d. represen-

tation. (A representation Rx is faithful if x f. y implies Rx f. Ry .) This assumption greatly simplifies the proof. Later we will return to the general case. (Incidentally, although not all groups have such representation, most interesting groups are given as matrix groups already and obviously have faithful f.d. representations.)

Begin with (4). Note that

algebra. To prove this, we need only show that f • g R 11 (x) and g(x)

=

£

& is

an

& when f (x) =

S 11 (x), where R and S are f.d. irreducible representa-

tions of G. But obviously f · g is the (1, 1) entry in the representation R ®'S. By decomposing R ® S into irreducible representations, we may easily express f · g as a linear combination of entries of

irr. rep.

&, so & is an algebra. A similar argument, this time with the contragredient representation, shows that & is closed under complex

Thus f · g

£

conjugation. The trivial representation puts all constants into the algebra ly,

& separates

&. Final-

the points of G, by our assumption that G has a faithful

f .d. representation. By the above, the Stone-Weierstrass theorem applies to

&. Hence & is dense in C(G), and therefore also in L 2(G). This

proves (4), and we noted (3) just a moment ago. Let us turn to (2). Set H = L 2 (G), and suppose that a is an equivalence class of irreducible representations, of degree da. We define Ha as the subspace of H, spanned by the d; entries of a representation R(x) of class a. Ha is independent of the choice of R, and has dimension exactly d;, by the Schur orthogonality relations.

15

§2. 'nfE PETER·WEYL THEOREM

Now Hal H~ in H if a f.~. again by the Schur orthogonality relations. Furthermore, the space & is precisely the space of all finite linear combinations of vectors in the various Ha. From (4), we conclude that, in fact, H co~plete

= IaEA

e Ha.

This decomposition is canonical. To

the proof of (2), we shall have to'decompose Ha' and this can-

. not be done canonically. For each a, pick a unitary repres,entation (R~, va) in the class a,

~end an orthonormal base e 1

•· •

ed

a

of

'v«.

Set cpij(x) = (~ei, ej) the

tnatrix entry. lc!Jijl forms an orthogonal base for Ha· Since Ra is a representation of G, we obtain lcpij(x)llcpij(a)l = lcpij(xa)l as matrices. In other words, Ikcpik(x)cpkj(a)

=

cpij(xa). This equation shows that on

the subspace Hai of Ha, spanned by the vectors cpi 1 , cpi 2 , ... , cpida, the right regular representation of G coincides with the representation Ra, i.e., p (a)\ H . notation.

al

= Raa.

This is a simple matter of disentangling

So for each i ~ da, we have a subspace Hai on which the right regular

representation coincides with Ra. Since H = Ia EA e Ha

=

Ia fA e Hai, "
we have proved the decomposition of (2). (5) is just the Parseval identity for the orthonormal base ld~Yzc/Jij(x)l of H.

( 1\ d~*c!Jff ·) 11 2

=

1 follows from the Schur relations.)

Finally, (1) is left as a good exercise for the reader. Thus, the Peter-Weyl theorem is proved, under the assumption that G has a faithful finite-dimensional representation. Section 3. The Peter-Weyl Theorem (Concluded)

In this section we remove the restriction of a faithful f.d. representation from the proof of the Peter-Weyl theorem. To do so, we shall need some results from the elementary theory of Hilbert-Schmidt operators. Let (:!R, dx) be a measure space, H = L 2 (lll, dx), and T be the operator on H defined by (Tf)(x)= J:mK(x, y)f(y)dy, where K is some fixed

16

LIE GROUPS

(A

REVIEW)

function in L 2 (}JJ x )It). T is called a Hilbert-Schmidt operator. Every Hilbert-Schmidt operator is completely continuous, and the norm of a HilbertSchmidt operator, is dominated by its so-called Hilbert-Schmidt norm


=

K(y, x) for almost

all x, y £:ln .. If T is self-adjoint, then of course its eigenvalues are real, and if A. 1 ~ A. 2 then the eigenspaces {¢>£HI T = A. 1¢>1 and {¢>

HI T

£

=

A. 2 ¢>1 are orthogonal.

We shall use the following spectral theorem for Hilbert-Schmidt operators. THEOREM. Let T be a self-adjoint Hilber-Schmidt operator given by a kernel K(x, y), let A. 1 , ~' •.• be the non-zero eigenvalues, counted according to their multiplicies (the multiplicity of an eigenvalue is the dimension of its eigenspace). Then I A. 2 < oo: Let ¢>1' ¢> 2 , ••• be an orthonormal sequence of eigenvectors such that Ti


= \

Ij "-j
X

=

m>.

See Riez-Nagy [34, Chapt. VIJ. Note that for any appropriate T, we can find a sequence i as above, by taking an orthonormal base for each eigenspace. Now we can prove the Peter-Weyl theorem. First of all, we need only show that

& is dense in

C( G), for that was the only place in our first proof

of the theorem where we used the faithful representation. To apply the theory of Hilbert-Schmidt operators, let K be any continuous complexvalued function on G, and define the ooerator T on H = L 2(G), by (T0(x)

=

J0 f(y)K(y- 1 x)dy.

Thus Tf = f

* K.

T is a Hilbert-

Schmidt operator on H, and a bounded linear transformation from LiG) to C(G). In addition, T is self-adjoint if K(x) .. K(x- 1). The connection between the operator T and the Peter-Weyl theorem is that for any p. ~ 0, the eigenspace Hp.

=I¢>£ HI

T

=

p.¢>1 is contained in

17

§3. 'IHE PETER-WEYL THEOREM (CONCLUDED).

&.

To prove this, observe that HIL is finite-dimensional, since it an eigen-

space of a completely continuous operator. Furthermore, HIL is invariant under the left regular representation. For if f A.(a)(f

* K)

* K =ILf,

then (A.(a)f)

*K=

= A.(a)(/Lf) = IL(A.(a)f). Therefore, picking an orthonormal base

f 1 :·· fe of HW we have for each a£ G, fj(a- 1x) = Ik Rkj(a)fk(x), for a unique Rkla). Obviously lRkja}l-.is a f.d. unitary representation of G. Setting a= y- 1 and-~= 1 in the last equation, we have fj(y) = Ik Rkly- 1) fk(l). So, ~-.f

&, which implies HIL f &.

Next we will show that if ¢

H, then T¢

f

C(G) may be uniformly ap-

f

proximated by an element of [i,. In fact, the spectral theorem for HilbertSchmidt operators, which we have quoted above, shows that we can write ¢ = Ij,., 1 aj cpj + t/1 where I aJ < +oo and t/1 £ ker T. (The cpj are as in the spectral theorem.) ForE > 0 we may find N so lare that Ij> N af < E 2 • Let us write T¢"' T(If= 1 ajcpj) + T(Ij>N ajcpj) + 0. Since Tis bounded from L 2(G) to C(G), it follows that IIT(Ij2:,N ajcpj)llcco> .:S C E , where C is the norm of T: L 2 (G)

liT¢-~= j

1

A·a·¢·11 = liT¢J J J C (G)

T(~= j

-+

1

C(G). Hence

a·¢·) J J

II C (G)

.:S liT( I a-¢·) II j >N J J C (G)

< CE • Since

I~= 1 (Ajaj)cpj

f

&

and E > 0 is arbitrary, we conclude that T¢

may be uniformly approximated by an element of We have just proved that if f then f

*K

f

L 2(G), K

f

&.

C(G), and K(x) = K(x- 1),

may be uniformly approximated by functions belonging to

&.

But given any f in C(G), we have only to take K above with small s,upport, and

J K(x)dx

= 1, then f

*K

uniformly approximates f. This completes

the proof of the Peter-Weyl theorem. The proof just given shows that

&

C C(G), a fact implicit in state-

ment (4) of the Peter-Weyl theorem. If G is a Lie group, we can also show

18

LIE GROUPS (A REVIEW)

that & s; C00 (G). For, we have only to modify our proof that & s; C(G) by using K £ COO(G) instead of merely K c C( G). Closing Remarks.

The notation of this section is that of section 2. A.

The decomposition (1) of section 1 induces projections E : H .... H . a a We shall now show that Ea has an explicit form as a convolution operator; that, in fact, Ea(f)(x) = da

f0

X(z(y)f (y- 1 x)dy, where

Xa

is the charac-

ter of the representations in the class a. To verify this fact, it is enough to look at f = ¢fl since {d~ ¢fl1 form an orthonormal base for H. Thus, we must show that

By definition of 1¢nl and

Xa,

xu(x) = tr
{¢if I is a unitary representation of G,

q,J (y) ¢kf (x). So da /G xa
¢!/ (y- x) 1

= Ik¢{t (y- 1 )~ix)

= Ik

by Schur's orthogonality relations. The verification is complete.

B.

By part (5) of the Peter-Weyl theorem, there is one-to-one correspon-

dence between f c L 2 and the corresponding sequence of "Fourier coefficients" {F(a)la

£

A• It may easily be shown that iff"" IF(a)la £ A•

then f(a- 1 x) "" IR~ (a) F(a) Ia

£

A and f( xa) "" {F(a) R~ (a )Ia

that if f ,... {F(a) I and g ,... IG(a)J then f

*g

,... {F(a) G(a)l.

£

A , and

19

§3. THE PETER-WEYL THEOREM (CONCLUDED)

C.

The basic operators of Fourier analysis are those which commute

with translations. These operators are characterized in the following result. THEOREM:

Let T be a bounded operator from L 2 (G) to L 2 (G).

(a) T commutes with the right

re~ular

representation if and only if there

is a collection !Mala£ A of matrices such that g .. Tf with g "'"' IG(a)J, f ,.... IF(a)} means preeisely that G(a) = MaF(a).

(N.B. Ma acts on the

left.) The norm of T is ,supa I Ma II· A similar result characterizes lefttranslation invariant operators. (b) T commutes with both the left and right regular representation if and only if all the above matrices Ma are constant multiples of the identity matrix. Let T¢{j

= 1/J{j(x) f

L 2 (G). Since ¢{j(xa) .. Ik ¢fk(x)¢kj (a) and

T commutes with the right regular representation,

Setting x = 1, a = y, we now have

Part (a) now follows easily, with Ma = IY,i~(l)l. Part (b) is immediate from part (a) and Schur's lemma. D.

Using the preceding theorem we may characterize the central functions

on G. For if f is a central function in L 1(G), then g -.. f

*g

commutes

with the left and right regular representations. From our theorem, we de-

duce that f "'"' !F(a)la

f

A where each F(a) is a multiple of the identity

matrix. What does it mean that F(a) is a constant multiple of I, the identity?

20

UE GROUPS (A REVIEW)

° ~ff a of {3{3 , then f ia just the character of the

If, for example, F(a) = I 1

1

a""

representation {3. So in general, a central function is, at least fonnally, a countable linear combination of characters on G. We list two concrete applications of this heuristic principle. Every continuous central function f on G is a unifonn limit of linear combinations of characters on G. Every

L2

central function f on G is the L 2 limit of linear combina-

tions of characters on G. These results can be proved by noting that f may be approximated in the appropriate norm by a finite linear combination i Aij ¢>ij(x) (see statement (4) of the Peter-Weyl theorem). Now f(x) while

f at G

= faE G

f(axa- 1)da ,

1 ) da = d- 1 8. . )( (x). Details are left to the reader. lj>~(axa1J a 11 a

These observations show that although the theory of compact topological groups is not essentially commutative, there is an important commutative part of the theory, namely the study of central functions and characters. Fourier analysis will be especially concerned with this part of the theory.

Exercise for the reader: Show, using Schur orthogonality, that a f.d. representation is uniquely determined by its character.

Section 4. Lie groups; examples. The purpose of the next three sections is to sketch some portions of the theory of Lie groups which we shall need later on. Useful references are: 1. Nomizu, Lie Groups & Differential Geometry, [ 4]. 2. Chevalley, Theory of Lie Groups, [1]. 3. Pontrjagin, Topological Groups (1st and 2nd editions), [6] and [7]. 4. Helgason, Differential Geometry and Symmetric Spaces, [2].

21

§4. LIE GROUPS; EXAMPLES

A Lie ~roup is a group G, which is also a COO-manifold, such that the group operations (a, b) c G x G .. ab £ G and a .. a- 1 are CC"-functions. It can be shown that every Lie group has a real-analytic structure which

makes the group operations real-analytic . . Two Lie groups G1 x G2 are isomorphic (G 1 ~ G2) if there is a group isomorphism j: G1 -. G 2 (onto) which is also a diffeomorphism. G1 and G2 are locally isomorphic (G 1 "' G2 ) ir' there are neighborhoods N1 f G1 and N2 f G2 of the identity and a diffeomorphism j: N1 ... N2 (onto) such that a. If x, y, x · y c N1 then j(x)j(y) = j(xy) ; b. If x, y, x · y £ N2 then j- 1(x)j- 1(y) = j-1(xy). For example, the real line R 1 is locally isomorphic to the circle group. We shall use the notion of local isomorphism to classify compact connected Lie groups. First we give the easy part of the classification. THEOREM.

In each equivalence class of locally isomorphic connected

Lie groups, there is a unique simply-connected group G. Every group G in the equivalence class is of the form G = G/Z, where Z is a discrete central subgroup of G.

Conversely, G = G/Z is locally isomorphic to

G

if Z is a discrete central subgroup.

G is called the rmiversal covering group of

G.

The theorem is proved by picking any group G of the equivalence class and setting

Gequal to the universal covering space of

G. It is very

easy to impose a group structure on G. Details may be found in Chevalley [1). Notice that the fact that Z is central is immediate from the observation that the fibre Z is a discrete normal subgroup of G. For if z £ Z is arbitrary, aza- 1 will be close to z if a belongs to a small enough neighborhood of l.in G. On the other hand aza- 1 £ Z, which implies that aza- 1 = z for a

close to 1, since Z is discrete.

Gis connected and is

therefore generated by any neighborhood of 1. Hence aza- 1 = z for any a

£

G. Thus Z is central.

22

LIE GROUPS (A REVIEW)

Having determined the structure of each equivalence class of connected Lie groups, we are left with the immensely more difficult task of classifying (connected) Lie groups up to local isomorphism. For compact groups, the solution is given in terms of the following: 1. The circle group T 1 = R 1 /Z; more generally, then-torus Tn = T 1 x ... x T 1 (n factors). These are the only compact connected abeJian Lie groups (see Chevalley [l], p. 212-213). 2. The group SO(n)

(n ~ 3) of all orthogonal n-dimensional matrices

of determinant + 1. SO (n) is called the special

ortho~onal ~roup.

When n = 2k, SO (n) is called a Dk group; when n = 2k + 1, SO (n) is called a Bk group. 3. The special unitary group SU (n) (n ~ 2), the group of all unitary n-dimensional complex matrices of determinant 1. This gives the A -series of groups, An-t= SU(n). 4. The symplectic group Sp(n), the quaternionic analogue of the real group SO (n), and its complex version SU (n). More explicitly, let Qn denote quatemionic n-space, a vector space over the quaternion field Q. Recall that for each quaternion a= a 1 + a 2 i + a 3 j + a 4 k, the quatemion conjugate a is defined as a 1 - a 2 i - a 3 j - a 4 k. Using the quaternion conjugate, we can define the "inner product" on Qn by (a, b)=~= 1

ae he

for a= {al··· an}, b = {bl··· bn} ( Qn.

Sp(n) consists exactly of those transformations of Qn, linear over Q, and preserving the inner product. Sp(n) is also designated Cn. S. The exceptional groups E6 , E7 , E 8 , F 4 , G2 , which we cannot describe here. The classification of compact connected groups is given by the following result. THEOREM. Every compact connected Lie group is locally isomorphic to a finite product of the groups listed above. The proof is too difficult and long to be included here.

§4. LIE GROUPS; EXAMPLES

23

A few remarks are in order. First of all, the product of basic groups, mentioned in the theorem is unique, except for the following redundancies:

1. SO (3) "' SU (2)

~

2. S0(4)"' S0(3)

X

Sp (1) S0(3)"' SU(2)

X

SU(2)

3. SO (5) "' Sp (2) 4.

so (6)

"'

su (4)

Secondly, it is possl'~e to show that except for list have compact

univers~l

-rn, all the groups in our

covering groups. In particular, Spin (n), the

spinor group, defined as the universal covering group of SO (n), is compact. It can be shown that SO (n) = Spin (n)/Z 2 •

Finally, the reader is entitled to an explanation of the cryptic symbols Ak, ~· Ck, Dk, E 6 , E 7 , E8 , F 4 , G2 . The notation is based on the concept of rank. The rank of a compact Lie group is the largest integer k such that the group has a k-torus embedded in it. Equivalently, the rank of a group G is the highest dimension of any abelian subgroup of G. The groups Ak, ~· Ck, and Dk all have rank k, E 6 has rank six, and so forth. The importance of the classification theorem for us, is that it gives us a bird's eye view of what Fourier Analysis on compact Lie groups might be like. For among the basic groups 1-5, Tn is the setting for classical Fourier series, and on SO (n) much of the classical theory has been carried over.

Section 5. Lie algebras We shall (eventually) introduce the Lie Algebra of a Lie group G.

Re-

call first that if M is an n-dimensional C"" -manifold and p is any point of M, the tangent space at p, T p(M) is the n-dimensional vector space of all linear functionals L on C00 (M) which satisfy L(fg) = L (f) g (p) + f (p) L(g). L(fg) = L(f)g(p)+ f(p)L(g) . A vector field is a linear mapping X: C00 (M) .... COO(M) which satisfies the

24

LIE GROUPS (A REVIEW)

condition X(fg) = (Xf)g + f(Xg). This is equivalent to the usual definition. The bracket operation assigns to any two vector fields X and Y, a vector field [X, Y] defined by [X, Y](f) = X(Yf)-Y(Xf). The bracket operation is bilinear, anti-symmetric, and satisfies Jacobi's identity [X, [Y, Z]] + [Z,[X, Y]] + [Y, [Z, X]]

= 0.

The space of all vector fields is infinite-dimensional, and so too big for our purposes. We therefore restrict our attention to the space of all left invariant vector fields. The vector field X on G is called left inva-

riant if for any a£ G, A.(a)X = XA.(a), where A denotes the left regular representation of G. In other words, a left-invariant vector field commutes with left translations. The space of left-invariant vector fields is closed under the bracket operation. Every vector field X on a Lie group G determines an element X1 of the tangent space toG at 1, defined by X1 (£) = (X(£))(1) for f

£

C""(G).

It is easy to show that X -o X 1 is an isomorphism of the space of left-

invariant vector fields on G, onto the tangent space T 1(G). This isomorphism induces a bracket product [ · , · ] on T 1 (G). T 1 (G) with its bracket product, is called the Lie algebra g of the Lie group G. The process which defines the Lie algebra is essentially differentiation. In particular, a vector field, being a section of the tangent bundle is really nothing but a first-order differential operator on G. We shall define a process, analogous to integration, which takes us from the Lie algebra back to the Lie group. A family {cptl is called a one-parameter group of diffeomorphisms of

the n-manifold M if

1. cf>t is a diffeomorphism of M onto itself, for each t, -oo < t < +oo. 2. The map (t, p) ... cf>t(P) taking R 1 x M into M is smooth. 3. cf>t

0

cps

=

cf>t+ s and

cp 0

is the identity.

We can associate a vector field X to each one-parameter group {cptl, by setting

25

§s. LIE GROUPS

(Xf)(p) = lim f(cpt{p))- f (p) t ... 0

for f

E

C""(M), p

E

t

M.

The converse problem is not, in general, solvable. That is, given a vector field X, ther,e may not be any one-parameter group l!f>tl which satisfies (Xf )(p) = ~ f 0 if>t(p)l t= 0

•

But there does exist a local one-

parameter group for which it holds. More precisely, given any point p c M, there is a neighborhood N f M of p, an E > 0, and a family lcfot II tl =5 E of mappings, defined only on N, such that l '. cfot is a diffeomorphism of N into M, for each t (j tl < E ). 2'. (t, q)

-o

cfot(q) is smooth.

3'. If !t11 < E, !t 2 1 < E, !t1 + t 2 1 <£and q cfot
2

4'. (Xf)(q)

1

oc

Jl dt

2

cfoo(q)

l

N, cfot (q)

l

N; then

2

=q·

(focfot(q))l for qc N, fc C""(M). [ t= 0

Moreover, X (essentially) uniquely determines lcfotl. The proof is an easy application of the Cauchy-Lipschitz existence theorem of ordinary differential equations. For complete details, see Nomizu [4], Chapter I. Returning to Lie groups, let X be an element of the Lie algebra of the connected Lie group G. We may think of X as a left-invariant vector field on G. By the above, there is an essentially unique local one-parameter group {cfot I satisfying l' to 4' .above. Using the uniqueness of lcfot I and the left-invariance of X, we can easily show that cfot commutes with left translations, i.e., a . '¢t(x)

=

cfot(ax) for a, x E G, whenever this equation makes

sense. Setting x = l we find that cfot(a)

=a

. cfot(l). Thus cfot is nothing

but a multiplication on the right by an element a(t)

£

G. The group prop-

erty of {cfotl say that a(t 1)a(t 2) = a(t 1 + t 2). (Also a(O) = l.) So far a(t) is only defined for t in a neighborhood of 0, but we can easily extend a

26

LIE GROUPS (A REVIEW)

to a homomorphism a: t ... a(t) from the real line into G. We have proved: For any left-invariant vector field X on G, there is a unique homomorphism a: R ... G such that (X f) (x) =

...!!_ f (x · a(t dt

We

denot~

))I t= 0

a(t) by exp(tX).

The exponential map exp from the Lie algebra g to the Lie group G is now easy to define. We simply set exp(X)

=

exp(l· X).

The exponential is a powerful tool in the study of Lie groups, for it tells very precisely, the relations between a Lie group and its Lie algebra. Thus, one can show using the exponential, that two groups are locally isomorphic if and only if their Lie algebras are isomorphic. A calculation of jacobians shows that exp: g ... G is a local diffeomorphism. Hence any base X1 ordinates at 1

· ••

Xn of g induces a system of local co-

G, given by (t 1 , ... ,tn)-> exp(t 1 X1 + ··•·+ tnXn) f G; the system is called the canonical co-ordinate system with respect to the base f

xl ... xn. The following formulas are useful for the application of canonical coordinates. 1. (Taylor's formula) If f: G -+ R is real-analytic, and x f G, then 00

f(x · exp tX)

=I

t~ (Xnf)(x), for X in a neighborhood of zero in g.

n-o n.

2. For X, Y f g , exp tX · exp Y = 2 = exp(t(X+ Y) +.!_[X, Y] + O(t 3 ))

2

3. Fc;>r X, Y E g, (exp tX)(exp tY)(exp tx)- 1(exp tY)- 1 = exp (t 2 [X, Y] + O(t3 ))

as t ... 0.

as t ... 0.

§s. LIE GROUPS

27

Formula 1. follows if we apply Taylor's theorem to the auxilliary function F(t) = f(x · exp tX), keeping in mind that (d/dt)f(x·exptX)\t=O = (Xf )(x) by definition of the exponential.

To prove formula 2., we select any real-analytic function f defined on G. Using formula 1., we can easily check that the Taylor expansions of .

2

t ... f(exp tX · exp tV)··and t ... f(e~p(t(X+ Y) + ~ [X, Y])) agree up to second order.· Since·.f is essentially arbitrary, we conclude that 2

exp tX · exp tY and exp(t(X+ Y) + t2 [X, Y]) differ by a third-order quantity, in the obvious sense. Formula 2. now follows from the fact that the exponential map is a local diffeomorphism. Finally, fonnula 3. is a trivial consequence of formula 2. Details of these hastily sketched proofs are left to the reader or to the cited literature. Let us now consider some examples. First of all, let G = GL(n, R), the general linear group. Although G has two connected components, the Lie algebra and exponential map still make sense for G. Since G is an open subset of the vector space M(n, R) of all real n x n matrices, we can identify the tangent space at I

£

G with M(n, R). Thus, the Lie alge-

bra of G is canonically isomorphic to M(n, R), as a vector space. To comthe exponential map, we shall find all one-parameter groups in G. If t .... N (t) is a one-parameter group, then of course N (s + t) = N (s)N (t). Differentiating this equation in s, we obtain

~t! (t) dt

=

I

AN(t), where A = dN (t) dt t= 0

£

M(n, R).

This differential equation, together with the initial condition N (0)

= I,

has only one solution. 00

N(t)

= etA

=I

tn An . n=O n!

Therefore the one-parameter subgroups of G are t ... etA, A £ M(n, R).

28

LIE GROUPS (A REVIEW)

An easy computation shows that the tangent vector (i.e., element of the Lie algebra) induced by etA is just A. So the exponential map must be given by exp(tA)

= etA,

i e., exp(A)

= eA,

whieh justifies the name

"exponential." Fonnula 2. above, now shows that the bracket operation on the Lie algebra M(n, R) is just [A, B] = AB - BA. Our next example is G = SO (n), the special orthogonal group. Since G is a suhgroup of Gl (n, R), it follows from the first example, that the Lie algebra is a subalgebra of M(n, R) (given the natural bracket product [A, B] "" AB - BA). In particular, the one-par~eter subgroups of SO (n) are all of the form t -. etA. But from elementary linear algebra we know that eA is orthogonal if and only if A is skew-symmetric, i.e., (Ax, y) = - (x, Ay), So the Lie algebra of SO(n) consists of all real skew-symmetric n x n matrices, under the natural bracket product; and the exponential map is exp(A) ... eA. Both for GL (n, R) and for SO (n), we see vividly that the exponential map is a local diffeomorphism. For comic relief, we consider the examples G = ~ and G .. Tn, the n-torus. Since Rn and ~ are locally isomorphic, they have the same Lie algebra. We leave to the reader the task of verifying that the Lie algebra of Rn is Rn with the bracket product [X, Y)

=0,

and that the expo-

nential is the identity map. The exponential map for Tn is the natural projection of Rn onto Tn. Note in passing that in Tn, some one-parameter groups are closed, while others are not. In Rn, however, all oneparameter subgroups are closed.

Section 6. Universal enveloping algebra As we have already seen, the Lie algebra of a group G consists exactly of all left-invariant first-order differential operators on G which anihalate constants. We shall now study the universal enveloping algebra of G, which is just the (non-commutative) algebra of all left-invariant differential operators on G.

§6. UNIVERSAL ENVELOPING ALGEBRA

29

To be precise, let M be an n-manifold. A kth·order differential operator on M is a linear mapping D: CI)O(M) -+ CI)O(M) which can be written in terms of local co-ordinates (x1 · · · xn) for M defined in a neighborhood of p l M, in the form (Df)(x 1 •·· xn) for t

£

=

Ilal ~k aa(x 1 ••• xn) ::.~ (x 1 •·· xn)•

CI)O(M), where aa are fixed cr"' -functions defined in a neighborhood

of p. A differential qperator D on a Lie group G is said to be left- invaril61t if D(Aaf)

= Aa(Of)

for any a £ G, f £ C""(G); where Aa is the left-

regular representation. The left-invariant differential operators on G fonn an algebra (seldom commutative), which we denote by ~(G). For the third time, we note that every X belonging to the Lie algebra g is a left-invariant differential operator, i.e.~ belongs to ~(G). It can be shown that g generates the algebra ~(G). In fact we shall prove a far stronger result: Regard g as a vector space, and let Tg denote the (non-commutative) tensor algebra of g, i.e., Tg ...

Ik=

0 sk

g. Equivalently, Tg is the algebra of real polynomials

in the non-commuting variables X1 · • • Xn, where the Xi form a base for g. Let g(G) denote the two-sided ideal in Tg, generated by all expressions X® Y- Y ®X- [X, Y) where X, Y £g. The quotient algebra Tg,1(G) is written U (G) and is called the universal enveloping algebra of G. We shall exhibit a canonical isomorphism of ~(G) with U (G). THEOREM: g

f Tg-+ U (G) is an injection of g into U (G). If we iden-

tify g f U (G) with g f ~(G), then the resulting correspondence extends uniquely to an isomorphism of U (G) onto ~(G).

Proof: We can easily define an algebra homomorphism from Tg into ~(G), by mapping X® Y ®

••• ®

W into the differential operator

f-+ X(Y(···(Wf)) ··· ), for X, Y, ... , W £ g . By definition of the bracket product, our homomorphism sends X s Y- Y s X- [X, Y) to 0 .

30

LIE GROUPS (A REVIEW)

Hence we obtain a homomorphism j of algebras from the quotient U (G) into :D(G). LEMMA: Let X1, ... ,Xn be a base for IJ, and let m = (m 1, ••• ,mn) be a multi·index, lml = ~~= 1 lmkl· Define X(m) f U(G) to be the coefficient of

t~l

··· t:;'n in the expression

I~~ ! (X 1t 1 + ••:.·+ Xn tn)l ml

(the t's are

supposed to commute with each other and with the X's, but x. and x. do •

J

1

not commute). We set X(O) = 1. Thus X(m) is a "symmetrization" of m m X1 1 ··· Xn n . Then the elements X (m) span the vector space U (G). Proof: We must show that each monomial Z = Xk Xk · ·· Xk 1

2

e

l

U (G)

is in the span of the X(m). This is clear for f ~ 1, and we use induction on f, the degree of Z. Suppose that appearing in Z are m1X1 's, m2 X2 's, ... , mnXn's. Set m = (m 1, m2 , ••• , mn), so that lml =f. It is not difficult to see that Z

=

X (m) +

lower~egree

terms, in U (G). (This is because

X (M) is a linear combination of all monomials obtained by rearranging the order of the Xk. within

z.

But any such monomial Z' differs from Z by

1

terms of degree lower than that of Z. For example, consider the monomial Z'

=

xk2 xk1 xk3 xk4 xjs ... xkf . Then

which has degree f- 1. The case of a general Z' is handled in the same way.) Since the lower degree terms are in the span of the X(m) by indue· tive hypothesis, we have proved that Z is in the span of the X (m). This completes the proof of the Lemma. Now consider j(X (m))

E

QED

:D (G). We show next that these elements are

linearly independent. This will show that the X(m) form a base for the vector space U (G), and that the map j is injective. To prove the linear independence, we use canonical co·ordinates for the base X1 ·· • Xn of g. Let f: G .. R be any real-analytic function.

31

§6. UNIVERSAL ENVELOPING ALGEBRA

Then in the spirit of formulas 1., 2., and 3. above, we can show that F(tl' ... ,tn)

=f(exp(t 1X1 +····+ tnXn)) •Imtm(jX(m)f)(x).

Hence, in

canonical co-ordinates (t 1 , •.• , tn), the elements jX(m) E ~(G) correspond to the operators al ml ;attl at:2 ... i and the latter are obviously

at:n

linearly independent. So the jX(m) are also linearly independent. To finish the prdOf of the theorem, we have only to show that j is onto. Nothing could be·simpler. First of all, a left-invariant differential operator D is uniquely determined by the functional f ... (Df)(l). On the other hand, a monent's thought !eveals that we can write (Df)(O)

=

I

aaXJf

\al::;k

o ••• o

1

(where aa are fixed constants, and Xka

E

Xf a (0),

11

g) for any differential operator

i

D. So if D is left-invariant, ~

Df"'

I

1

lkisa Thus D f j (U (G)), so that j is onto.

QED

The center Z (G), of the universal envoloping algebra ~ consists of all those differential operators D which are bi-invariant, i.e., commute with both the left and the right regular representation. For suppose that D is hi-invariant. To show that D E Z (G), we need only check that D commutes with all X f g, since g generates the universal enveloping algebra. Any X

f

g may be written in the form (Xf) (x)

= d~

(x · y (t ))\t= 0 , where

y(t) is the one-parameter group corresponding to X. Therefore (X

o

D)f(x)

= _E_ dt

= _c!_ D (f(x · y(t))l

(Df)(x · y(t))l t=O

dt

t=O

,. D ...<.Lf(x · y(t))/ = (Do X)f(x), dt t,., 0

32

LIE GROUPS (A REVIEW)

by the right-invariance, so X and D commute. Conversely, suppose D ( Z(G). With X and y(t) as above, we have

I

-d D p f dt y(t) t= 0

= -p d

dt y(t)

Df

It=

0

where p is the right regular representation; for this equation simply means DXf

= XDf~

Applying the last equation to the function f = Py(s)g' we ob-

tain

for any s. Since Dpy(O)

=

Py(O)D, it follows that Dpy(t)g = Py(tpg.

This proves that D commutes with Pa if a is in the image of the exponential map, which contains a neighborhood N of 1 in G. Under our standing assumption that G is connected, we conclude that N generates G, which implies that D commutes with the right regular representation. So D is hi-invariant. As a corollary to the above remark, we note that the algebra of hi-

invariant differential operators on G is commutative. A reasonable way to generalize non-trivial Fourier analysis to a Lie group G, would be to study the simultaneous spectral decomposition of all the hi-invariant differential operators. We shall not follow this approach strictly, but will rather study a single hi-invariant operator /)., which will play the role of the Laplacian. The base IX (m)l for U (G) provides another useful isomorphism. In fact, let S(G) be the symmetric tensor algebra of g,, i.e., the quotient algebra of ~(G) with respect to the ideal generated by elements of the form X® Y - Y ®X. Then IX (m)l is also a base for S(G), as may be verified easily. So there is an induced isomorphism of vector spaces (not algebras) c: S(G) ~ U(G) (onto). This isomorphism is actually independent of the

§6. UNIVERSAL ENVELOPING ALGEBRA

base X1

••• ~

33

relative to which the constructions were made. We have

thus the Birkhoff-l'itt Theorem:

S(G) is canonically isomorphic to

U(G). The Birkhoff-Witt theorem allows us to formulate an interesting and imp~rtant

characterization of Z(G}, -the hi-invariant operators, in terms

of the adjoint represemation of G, defined as follows. If a

l

G, the inner

automorphism x .... a xa~ 1 of G, induces a linear mapping Ad (a) from the tangent space toG at 1, to itself. Thus Ad(a): 1J .... IJ. Since Ad(ab) = Ad(a)Ad(b) and Ad(1) is the identity mapping on IJ, we see that Ad is a representation of G on g; Ad is called the adjoint representation. The adjoint representation allows us to make a "transition'' between left and right multiplication in G. For example, a exp X= exp(Ad(a)X)a. Ad extends uniquely to a representation on S(G), the symmetric algebra-we just set Ad(a)(X®Y® .. ·®W)- Ad(a)X®Ad(a)Y®···®Ad(a)W. Under the Birkhoff-Witt isomorphism i, the hi-invariant differential operators correspond precisely to those elements Z

l

S(G) which are fixed by

the transformations Ad(a). The easy proof, which uses the fact that D l ~(G) = U (G) is uniquely determined by the functional f ... Of (1), is left to the reader.

Section 7. Laplacian We have now built up enough machinery to define a "Laplacian" on a general compact Lie group. Our Laplacian will mimic the behavior of the Laplacian ~ = I~== 1 a 2;axi2 on Rn, in the following respects: 1. ~ is a second-order hi-invariant differential operator on Rn. 2.

~

has no zero-order term. In other words,

~

takes constant func-

tions to be zero. 3. ~ is formally self-adjoint, This means that for any f, g l C'; (Rn),

J f(x)~g(x)dx I =

~n

M(x)g(x)dx

Rn

which follows from integration by parts.

34

LIE GROUPS (A REVIEW)

4. n

is elliptic. In general, a second-order differential operator on

f).

.

n

R , gJ.Ven by D = Ii,j= 1 aij(x)

ax.a2ax. 1

aij(x)

=

+ lower order terms, with

J

aji(x), is called elliptic if for any complex numbers ~1 ,

••• , ~n,

we have n

.!

aij(x)~i (j

.! I~i \2

:;: c

i ,j= 1

i

with a fixed constant c, independent of x and ~These concepts all make sense on a Lie group. Certainly the notion of a second-order hi-invariant differential operator presents no problems, and the condition D (1)

=0

is well-defined. We must be a little more care-

ful with ellipticity and self-adjointness. A second-order differential operator D on a manifold M is called elliptic, if for any local co-ordinate on M, given by the diffeomorphism ¢ taking a neighborhood in M to a neighborhood U in Rn, the differential operator (D(f

o¢)) o¢- 1 ,

0 on U s;

Rn, defined by Of

=

is elliptic. This definition agrees with our previous no-

tion of ellipticity for open sets in Rn. For if ¢ is a diffeomorphism of the neighborhoods U and V of Rn, D is a second-order differential opera-

.

.

tor on V, and D is the differential operator on U, defined by Df ( D(f o ¢))

o ¢- 1 ;

then D is elliptic if and only if

=

D is elliptic.

We have defined ellipticity of an operator on a COO -manifold, hence on any Lie group. Let us now turn to self-adjointness. A differential operator D on a Lie group G is formally self-adjoint if for any f, g E c~ (G),

f G

f(x)Dg(x)dx =

f

Df(x)g(x)dx,

G

where dx denotes left Haar measure on G. The subtlety is that if we used right Haar measure for dx, we would not end up with the same class of "formally self-adjoint operators." If G is compact, this problem disappears, since left and right Haar measure coincide.

35

§7. LAPLACIAN

THEOREM:

Let G be a compact Lie group. There is a second-order

differential operator D on G, such that (a) D is hi-invariant; (b) D is elliptic; (c) D is formally self-adjoint; (d) D maps constant functions to zero. n

Moreover, D may betaken to be the form Ii,j= 1 aij ~ Xj, where X1 ... Xn form a base of the Lie algebra, and the constant matrix (aij ) is strictly positive-definite. Proof: Start with any operator D0 =

I~,j= 1 ai~ Xi Xj, where (ai~ ) is

a constant, strictly positive-definite matrix. D0 belongs to the universal enveloping algebra, and corresponds under the Birkhoff-Witt isomorphism, with the symmetrization of the element I?,j,., since

1

aij Xi

EB

Xj£ g(G). But

(ai~ ) is symmetric, D0 = I?,j= 1 ai~ xi EB xj is already symmetrized.

So the Birkhoff-Witt isomorphism carries Do to D0 mation Ad(a) fixed 0°, for a

l

l

S(G). If the transfor-

G, then D0 would be hi-invariant. In the

general case, we make 0° into an Ad (a)-invariant element by forming the "average," 0 1 = fG Ad(x)(D 0)dx. Here, dx is normalized so that fG dx = l. This is the only place where we use compactness of G. By the usual computation, Ad(a)D 1 = D1 . Each Ad(x)(D 0 ) is strictly positive-definite, since D0 is; hence 0 1 =

I~,j= 1 aijXi $Xj £ S(G) with (aip strictly

positive-definite. The Birkhoff-Witt isomorphism takes D1 to the differenn tial operator D = Ii,j= 1 aij Xi Xi , which must be hi-invariant, since Ad(a)D 1 = 0 1 . D is elliptic, since in canonical co-ordinates it takes the form

I,?.

l,J=

1

a 1•1.

at.1aat.J 2

. Clearly, D maps constant functions to zero. It

remains only to show that D is formally self-adjoint. To do so, we shall need integration by parts on a Lie group. More pedantically, if X E

and

p,l{uC~(G) then fG(Xcp)t/Jdx =-fGcp(Xt/J)dx. Theproofoftheformula

is quite simple. We set y( t) = exp( t X), and differentiate the equation

36

LIE GROUPS (A REVIEW)

fa (x)l{l(x)dx = facfJ(x

· y(t)l{l(x · y(t))dx in t . Setting t

= 0,

we ob-

tain the desired formula. Now for , 1{1

1

E

C""(G) we have

D(x) l{l(x) dx

=

~ 1,J

G

=I i,j

a.· 1J

a-·J(x)x.x.yi(x) 1J J 1 G

(x)(

G

X·1 x.J (x) 1{1 (x)dx

G

J =J

=

f

~-

aijXjXi)l{l(x)dx

1,J

l{l(x)DI{I(x)dx, dy

G

the symmetry of (aij) and two applications of the integration-by-parts formula. Thus D is formally self-adjoint.

QED

Only in rare cases is the operator D unique. On R2 , for instance, (o 2 /ox 2) + 3(o 2 /oy 2) is just as good a Laplacian as (o 2 /ox 2 ) + (o 2 /oy 2 ). It is not hard to show that D is unique up to constant multiple if and only if the Lie algebra g has no proper "ideals." A Lie algebra (other than the

trivial one-dimensional algebra) without proper ideals, is called a simple Lie algebra. Henceforth, we shall work with a single differential operator 1l (on G), which satisfies the properties of D above. ll will be called a Laplacian on the Lie group. Problem: Show that ll is the Laplace-Beltrami operator on G corresponding to a suitably defined hi-invariant Riemannian metric on G. (For the properties of the Laplace-Beltrami operator see e.g., Helgason [2, Chapter X].)

CHAPTER II LITTLEWOOD-PALEY THEORY FOR A COMPACT LIE GROUP

In this chapter we present the analogue of the Littlewood-Paley theory for any compact Lie group. This development is independent of the more general one given in Chapters III and IV, but has distinct advantages of its own. It is based, in effect, is the following special circumstances. The functions u (x, t) = (Ptf)(x), where pt is the Poisson semi-group are solutions of the equation /Au (x, t) = 0; various special properties of the Laplacian /A come into play (for more details on this point see Section 3 below). The special approach followed here can also be used in other contexts, such as the case of non-compact semi-simple groups G treated in Chapter

v. The present chapter is organized as follows. Section 1 constructs the

heat-diffusion semi-group on G, represe:;ting the solutions of the heatequation au/at=

~u.

Among the general semi-groups that we consider

later in Chapters III and IV, this gives probably the simplest and most easily constructed example.

In Section 2 we pass from the heat semi-

group, to the Poisson semi-group, (which represents the solutions of (a 2 u/at 2 ) + ~u

= 0), by

a familiar process fo "subordination." In terms of

this latter semi-group we define the g-function and study its properties in Section 3. Various applications are pointed out in Section 4.

37

38

LITTLEWOOD-PALEY THEORY FOR A COMPACT LIE GROUP

Section 1. The heat-diffusion semi-group Our construction of the Poisson integral will be made in terms

~f

simpler object, namely the semigroup {Ttl of operators defined by Tt

a

=

etA, in a sense which we shall make precise. (A denotes the Laplacian.) More rigorously, we shall show THEOREM 1: There exist operators Tt, 0 < t < +oo, such that

-

.

i) each Tt is a bound operator of nonn 1 on Lp(G), l_:s p .:S +oo

and also on C(G); ii) !Ttl form a semigroup, i.e., Ttl Tt2

= Tt1+t2,

and T 0 is the iden-

tity operator; iii) {Ttl is strongly continuous, i.e., the mapping t .... Ttf is continuous

from [0, oo) into Lp(G), for each fixed f f Lp(G) (p

I= oo ). In the

case p = +oo we have instead that t .... Ttf mapping [0, oo) into C(G) is continuous, for each f f C(G); iv) each Tt is a self-adjoint operator on LlG); v) Tt is positive, i.e., f ~ 0 implies Ttf ~ 0; vi) Ttl = 1; vii) the Laplacian A is the infinitessimal generator of the semigroup {Ttl, i.e., limt .... 0 (Tt-I)f/t

=

M, whenever f belongs to a suitable

dense subset of C(G); viii) Tt may be written in the form Ttf = Kt * f, t > 0, where Kt(x): G X (0, oo) .... R is a C00 function; ix) if for f f L 1(G) we set u(x, t) = (Ttf)(x), then u(x, t)

satisfies the heat equation

at1__u (x, t) = Au (x, t)

f

C (Gx(O, oo)) 00

and the boundary

condition u( ·, t) .... fin L 1(G) as t .... 0+.

Condition ix) is the raison-d'etre of the semigroup {Ttl. As soon as we have proved Theorem 1, we shall be able to write down an expression

§1. THE HEAT-DIFFUSION SEMI-GROUP

39

for the Poisson integral in terms of !Ttl. Much machinery could be brought to bear in the proof of Theorem 1. There is an extensive theory of semi groups from which it is possible to prove the existence (and uniqueness) of !Ttl satisfying conditions (i)-(viii) above. The smoothness conditions (viii) and (ix) really express the fact i~

that the heat equation

hypoelliptic.

We shall not hav$ to use any of this machinery because we have concrete information, namely an explicit eigenfunction expansion of 1\ (see below), which will enable us to give an elementary proof of Theorem 1. But using the theory of semigroups and of partial differential equations, we can generalize Theorem 1. from a Lie group to a Riemannian manifold M. In this case, 1\ is the Laplace-Beltrami operator on M, and again there is a semigroup !Ttl as above such that Ttf(x)

= u(x, t)

solves the

heat equation. For the general theory of semigroups, see Feller [10], Yosida [40], and Hille-Phillips [11]. Semigroups of operators arising from Lie groups are treated in the papers Semigroups of Measures on Lie Groups by G. Hunt (T.A.M-S., 1956) [12], and Analytic Vectors, by E. Nelson (Ann. Math.,

1959), [14]. Proof of Theorem 1: We shall first produce an eigenfunction expansion

of 1\, using the Peter-Weyl theorem. For purposes of this proof we use the (slightly non-standard) terminology of an eigenfunction ¢ of 1\ with eigenvalue .\, which means 1\¢ =- .\¢ (instead of the usual .\¢). With this con-

vention, all eigenvalues of 1\ are non-negative, because, (1\f, f) ::; 0 for any f

f

C"'(G), ( ·, ·) denoting the usual inner product on L 2(G). In fact,

(M, f) =

f (I G

=

~

ij

a lJ..

~

aij

!

X-1 X-J f(x)f(x)dx

G

ij

=

Xi X/) (x)f(x)dx

-I ij

aij

1

xj f . xi f dx = _

G

f

I

j G ij

a .. X.fX.f dx < 0 1J

1

J

-

'

40

LITTLEWOOD-PALEY THEORY FOR A COMPACT LIE GROUP

with equality if and only if Xif = 0, i.e., f = constant. (We have used the strict positivity of the matrix (aij)). Now recall the Peter-Weyl decomposition L 2 (G) =I

al

A e Ha , where

A denotes the set of equivalence classes of finite-dimensional irreducible representations of G, and Ha denotes the vector space spanned by the (degree a ) 2 entries cpij of a representation Ra = lcpij (x)l of class a. Each Ha is-contained in C00 (G); so it makes sense to take the Laplacian of any function in @, the algebraic direct sum of the H . We shall prove a that each cp l Ha is an eigenvector of fl, and that all cp l Ha have the .same . eigenvalue Aa . In fact, let r/Jij = flcpij, where lc/Jij(x)l is a representation of class a. Then since cpij(xa) = Ik cpik(x) cpkj(a), we have r/Jij(xa)

= Ik r/Jik(x)cpkj(a)

by the right-invariance of fl. Setting x = l, we obtain r/Jij(y) =

~ tPjk(1) cpkj · (y). Similarly cpij(ax) = Ik cpik (a) cpkj(x) so r/Jij (y)

=

Ik cpik(y)r/Jkj(l), by the left-invariance of fl. Thus, the matrix lr/lij(l)l commutes with the irreducible representation lcpij(x)l, so that by Schur's lemma, lr/lij(l)l = -Aai' where I denotes the identity matrix. We conclude that r/Jij(y) = Ik cpik(y)r/Jkj(1) = -Aacpij(y). which shows that flcp = -Aacp for

,~., l

Ha . Recall that Aa must be non-negative. The decomposition L 2( G) = Ia l A E9 Ha is therefore an eigenfunction

~

decomposition with respect to fl. For convenience, we pick orthonormal bases of all the Ha, and list all the base elements 1, cp 1(x), cp 2 (x), .... Each cpk is an eigenfunction of fl, with eigenvalue ILk (note that ILo = 0, and Ilk > 0 for k ~ 1). The space @ consists exactly of all finite linear combinations of the cpi. If f =I ai cpi l@ (ai finitely non-zero) then of course M =I- ILiaicpi. This suggests a definition for Tt = etfl: for f = I a i cpi

l

@, we set

Ttf =I e-ILitaicpi . Obviously IITtfll~ = Ii le'""1Litail 2 .$ Ii lai1 2 = II fll ~ for f l @, since "-i ~ 0 and t ~ 0. Thus Tt extends from @, to a bounded linear operator on L 2 (G) of norm 1-the extension is also denoted by Tt. The reader may check that !Ttl is a strongly continuous semigroup of self-adjoint operators on L 2 , Ttl = 1; and if f

l

@, the function

41

§1. THE HEAT-DIFFUSION SEMI-GROUP

u(x, t) = (Ttf)(x) belongs to C00 (G x (0, t)) and satisfies the heat equation. Hence, the L 2-theory of the operators Tt is well in hand. Next, we show that Tt is a positive operator. From this fact the Lpproperties of Tt will be easy. Define the resolvent R(.\, 11) to be the operator (.\1 -11)- 1, for .\ > 0. The analysis of R(A, 11) trivializes here, for f "" I ai ¢i(x) t li;, R(.\,11)£ =~ - 1- ai¢i(x) ; (.\+~t;>

which shows when.\> 0, R(.\,11) is a bounded operator on L 2 (G). This is typical of the usefulness of li; in avoiding all technical difficulties. A standard fact from semigroup theory is that each Tt is positive if

and only if R(A, 11) is positive for .\ > 0. On the one hand, for

(A)

f£

&,

since we can write f = ~ ai ¢i(x) (finite sum) and then equation (A) reduces to

~ e-~tita·¢·(x) 1

1

... lim

n .... oo

So if R(.\,11) is positive, then (

~(

n/t ) n a·¢·(x) , (n/t)+ILi) 1 1

f R( ~. 11))n is positive, which implies

that Tt is positive. Similarly, that T\?.. 0 implies R(A, 11) ?.. 0, follows from the identity (B)

R(.\,11)£ •

foo e-.\t Ttf dt

(f f ff,),

0

which in tum comes from the same kind of routine computation as (A).

&. Let f f li;, and suppose that f?.. 0 and R(.\,11)£ = g £ li;, Then .\g-l1g = f ?.. 0. We must We can now show that R (.\, 11) is positive on

show g?.. 0. If this did not hold, then at the point x 0 £ G at which g is a minimum (recall that G is compact) we have g(xo> < 0. On the other hand,

42

LJTTLEWOOD-PALEY THEORY FOR A COMPACT LJE GROUP

!ig(x0 ) ~ 0 since g takes its minimum at x 0 • Hence A.g(x 0) - !ig(x0 ) < 0, contradicting f ~ 0. This completes the proof that R(A., ~) and therefore Tt is positive. We have used tacitly the fact that R(A,

til f is real if f is

real, which we leave as an exercise to the reader. Since the positive operator Tt:

& -. & maps

1 into 1, it follows that

Tt extends to a positive Tt: C(G) -. C(G) of .norm 1. For fixed t > 0 and x0

E

G, .the positive linear functional f

f

C(G) ... (Ttf)(x0 ) is of the

form

where ~~

o

is a positive measure with total mass 1, by the Riesz repre-

sentation theorem. On the other hand, the operator Tt is hi-invariant, since for f E &, Ttf

=I

means that Ttf(xol = (f

tn!inf/n!. Therefore IL~ (E)

* l!i)(x 0 )

0

= IL~ (x01 E),

which

for every f E C(G). Since

we have verified property i) in the statement of the theorem. Property iii) follows from property i) and the density of 6; in Lp. It remains to check properties viii) and ix). Now ix) is clear in the case f E &, from which we deduce by a routine limiting argument that u (x, t) (Ttf)(x) satisfies the heat equation, for any f

£

Ll' once we have proved

viii). To prove viii) we require a simple form of the Sobolev lemma; which we state as an a priori inequality: Let f

£

=

<=;
43

§1. THE HEAT-DIFFUSION SEMI-GROUP

sup n lf(x)l XfR

~

~

A

o~lai~N

II ~

a

fll

, 2

where N is any integer > n/2, and A depends only on n. The lemma is most easily proved by means of the Plancherel theorem. In fact, llflloo ~ llfll 1 ~ II(~+ IY\>-NII 2

~

K

1\(l + IYI)N f(y)ll 2

I 111Yiaf(y)ll 2 = K I II lal ~N lal ~N

:a

a

£112

Next, we shall extend our Lemma, and transfer it to the setting of the compact Lie group G: LEMMA.

(a)

Let f

€

C00 (G). Then

llflloo

~

N

~ ll~ff\12

A

f= 0 where A depends only on G, and N is any integer > n/2. To prove this, first observe that (b)

where Xk belongs to our basis for the Lie algebra g . For II M 11 2 11 f ]\ 2 2: -(M,f)

=

Iij aij(Xjf, Xif) 2: CIIXkfll 22 by the strict positivity of (aij).

Combining (b) with the inequality 2ab ~ a 2 + b 2 , we obtain II Xkfl\ 2 ~ C (\1M 11 2 + II f 1\ 2). Repeated application of this inequality yields N

IIP(X 1 ·•• Xn)f11 2

~ C~ ~ ll~ffll 2 f= 0

for P(X) in the universal enveloping algebra of G, of degree

~

N. To com-

plete the proof of (a), we have only to show that II fll 00 ~ IPf All P(X 1 • ·• Xn)~l 2 for some finite A of P(X) of degree at most N. If f has small enough

44

LITTLEWOOD-PALEY THEORY FOR A COMPACT LIE GROUP

support, this follows from our Sobolev lemma and an application of canonical co-ordinates-the general case then follows from a partition of unity argument. Thus, inequality (a) holds. '

Let us apply the above estimates to the study of the operator Tt and

ist eigenvectors cpk. Since !1~

=-

p.i cpi and ~ cf>iil 2 = l, inequality (a)

shows that supxeG lcf>i(x)l .S C(l + IP.ii)N for any N > '!:: dim G. Consider any f

= I ak cpk(x) e &, and let u (x, t) = Ttf(x) = I e'iLkt ak cpk(x). Then

tie u (x, t) "' I

('iLk)e e "'iic:t ak cpk(x), so that

ll!ieu(·,t)il~ =Ip.~ee- 2 P.ktlakl 2

.s Cc 2ei

lakl 2 = ct- 2ellflli

(We make use of the elementary inequality p. 2ee- 21Lt _s

cc 2e,

valid for

~ 0 ). We have proved an a priori estimate for llt1eu ( · , t )11 2 • Since (a/at) u (x, t) = !1 u (x, t) for f £ &, we also obtain the a priori inequality

p., t

where C and M depend on k and

e alone.

By the Sobolev lemma and

familiar limiting arguments, u(x, t) = (Ttf)(x) belongs to C"(G x (0, oo)) for every f

l

L 2 (G), and the map f .... u(x, t) is a continuous operator from

L 2 (G) to C (G 00

X

(0, oo)).

Property viii) above is now easy to prove. For, as a very special case of what we just showed, we have Tt: L 2 (G) .... L00 (G) is a bounded operator. The usual duality argument shows that Tt is also bounded as a mapping from L 1(G) into L 2 (G). But then Tt = Tt/ 2 Tt/2, the composition of continuous operators from L 1 .... L 2 and from L 2 .... C"'. Thus, for any f £ L 1(G), Ttf(x) l COO(G x (O,oo)), which completes the proof of Theorem 1. QED We can even give an explicit representation for the operator Tt, in terms of the docomposition L 2 (G) = IalA E9Ha. As usual, let us select a unitary representation lcpfj (x)l of class a, for each a £A, and use as

Y: c/>ij a Ia,i,j• where our orthonormal base {cpil' the family of vectors I da·

45

§1. THE HEAT-DIFFUSION SEMI-GROUP

the factor d~ ... (degrees a)lf;. is put in to normalize vectors. The eigen· value of !':! corresponding to

d~ cpfj is >.a in our previous notation. Hence

Tt(d~ cp~j) • e-Aatd~ ¢\j, so we can write formally, Ttf(x)

(c)

=

f

(Ia e->.at da Xa(xy-I)) f(y)dy ,

G

since

-A t 'h. a d a

a (x) cf>·. lJ

I-l,J,a . e

'h. cf>·. (y) = lJ

· da

~

k

a

-.>.a t d

e

a

~

k· .

l,J

cf>·alJ.(x) ¢·aJl.(y-1 )

Ia e-'Aat da Xa(xy- 1), which follows because lcpfj(x)l is unitary, with character Xa·

We shall verify that this identity holds, not furs formally,

but literally, by showing that the series

converges in the strongest possible sense. Namely any order partial derivative, with respect to x and t, of the series Ia e-Aatda Xa(x) converges absolutely absolutely and uniformly for x

E

G, t

> o > 0. In fact, since f .... l':!e Ttf(l)

is a bounded linear functional on L/G) for each f and since (c) holds for· mally, we have

aEA for each N > 0. Since t

> 0 is arbitrary, da ~ l, and

:xa<·>

has ck-norm

at most some fixed power of 'Aa (again by our form of the Sobolev lemma), we conclude that sup

~

.-4

s~chOaEA

e

-'As a da11Xa11 k C(G}

for each k, which implies the desired conclusion.


46

LITTLEWOOD-PALEY THEORY FOR A COMPACT LIE GROUP

Section 2. The Poisson semi-group; the main theorem As indicated above, we are not mainly concerned with ITt~ but rather with the Poisson semigroup !Ptl, which is defined roughly as saying that Ptf(x) • u(x, t) is the solution of Laplace's equation ((a 2;at 2) + A)u .. 0 with the boundary condition u (x, 0) "" f (x). It is easy to find such a u for f

£

&.

In fact, if f = Ik ak cpk

£

&,

we can set

t -.c -(ILk)'h t P f -. ~ e . akcpk k

(t 2:. 0) ,

where the notation is as in the proof of Theorem 1. Thus, formally,

P t • e -t(-/j.)IJ2 . THEOREM

1 '. The operator pt just defined on & extends to a bound-

ed operator on L

p

(1

< p -< +oo) and·on

-

C(G). Moreover

i) IIPtfllp~ llfllp• forall f (t2:0); ii) pt is a self-adjoint operator on L 2(G);

iii) f 2:. 0 implies ptf ~ 0; iv) lim

ptf_f

= (-A)lf2f

for f

£

&.

(-A)¥2 is defined by (-A)Ylcpk ...

t

t .. 0

IL;:cfok ; v) u(x, t)

';!;

ptf(x) belongs to C00 (G x (O,oo)) and satisfies the equa-

tion ((a 2;at2) + ,!\)u • 0 and the boundary condition u(x, t)-+ f(x) in the L 1(G) norm if f

£

L 1(G).

Proof: We shall prove the properties of pt from analogous properties of "ft. by means of the heuristic principle of subordination. From the wellknown identity

f

oo

0

e

-u

vu

2

e-f3 /4u du

47

§2. TilE POISSON SEMI-GROUP; TilE MAIN TIIEOREM

valid for

f3 > 0,

(*)

(t

>

0) •

Hence

~!!flip( _l_ roo e-u Vrr lo Vu

du) "'!!flip'

which shows that pt extends to a bounded operator on Lp(G) and that (*) holds for all f

£

Lp(G). Properties i, ii, iii, and v follow from the corre-

sponding facts about Tt and (*).

QED

The function ptf(x) is called t~e Poisson integral of f, just as in the case G .. the circle group. Using the Poisson integral, we can formulate and prove a generalization of the classical Littlewood-Paley inequality for the g-function. Recall that for f, say, a real-valued function in C00 (G), .M • Iij aij xi xj f. Define

Iv f\ 2(x)

to be Iij adXif)(Xjf). Similarly, for

f c C 00 (G x (0, oo)) real-valued, set M(x, t)

1VI f\" 2(x, t)

.... ((a/at)£) 2 +

= (a 2 f!at 2) +

11i and

1v i1 2 •

Now let f £ Lp(G), and let u (x, t) be its Poisson integral. Then we define the Littlewood-Paley g-function for f to be

THEOREM 2:

If f £ Lp(G)

and have the inequality

(l

I g (Oil p

< p < +oo), then

g(f) is also in Lp(G),

II I

~ Ap f p where Ap depends only on p.

Conversely, if [Gf(x)dx = 0,. then !!flip~ Bpl\g(f)\lp.

48

LITTLEWOOD-PALEY THEORY FOR A COMPACT LIE GROUP

Proof: Note first that

I fll p

fa f (x)dx

= 0 or something similar is needed if

;5 Bp II g (Oil p is to hold, for the g·function of every constant is zero.

Our proof develops a recent approach for the classical case, and re· quires a series o( Lemmas, the first of which generalizes the Hardy-Little· wood maximal theorem. LEMMA

1: For any (reasonable) function f, define the maximal func-

tion £* as f*(x) • supt> 0 Iptf(x)l. Then if 1 < p S +oo,

ll~lp SAp I £11 p

for every f t Lp(G), with Ap independent of f. Proof of Lemma 1: The sledge-hammer with which we crack this pro~

lem is the Hopf-Dunford-Schwartz ergodic theorem, one of the most powerful results in abstract analysis. The form in which we shall use the theorem is this: (Here

Let !Ttl be a (measurable) semigroup of operators on LPOR' d,L).

OR. d,L)

denotes an arbitrary measure space.) Suppose that IITtfllp

;5 I flip for all f t LPOJl' df.L), and for each p t [l,oo]. Then the "maximal

function" Mf, defined as

satisfies the inequalities (a) IIMfll p :5 Ap II fll p for each p with 1 < p ;5 +oo ; (b) f.L(IxdRi Mf(x) > al) :5:11£11 1 foreach a> 0 and feL 1 , where A is independent of f and a. The proof of this theorem is in Dunford-Schwartz [9] Linear Operators, Chapter VIII. Although (b) is the basic inequality of the theorem, we shall only use (a) here. For

OJl, df.L)

we take G with its Haar measure; for Tt we take

the heat semigroup of Theorem 1. Inequality (a) gives us control of the averages of Ttf. This is exactly what we need, since(*) in the proof of Theorem 1 ', shows that pt is nothing but a weighted average of the Tt. In fact, changing variables in (*), we obtain

§2. THE POISSON SEMI~ROUP; THE MAIN TIIEOREM

49

(**) where ¢>(y)

5

_1_ e-l/4y y-3/2 .

2y-; The reader may verify that f/>(y) and y¢ '(y) belong to L 1(0,oo), Integration by parts in (**) shows that !Ptf(x)l = l-t- 2

~oo(/oy Tlr(x)dt)~;

+•-2 { (~ { S supy> 0

I~ / 0

:S A sup

y>

where A

= II y¢> '(y) 11 1 •

0

(y/t 2)dyl

T'f(x)dt)(

y Ttf(x)dtl . (t- 2

I~ f

y

d~ ¢(~

)H

~ jy :~ (~ )dy)

YTlr(x)dtl ,

0

Hence ptf(x) S AMf(x) for each t > 0, so that

f*(x) :S AMf (x). The lemma now follows from inequality a). QED LEMMA 2:

Let u(x, t) be a harmonic function on G x (O,oo), i.e.,

A\ u '"" 0; and suppose that u > 0. Then lot any p > 1, .l uP p(p-l)uP- 2 1Wu(x, t)1 2• Proof: Since Xj corresponds to

=

a;atj

in canonical co-ordinates, we have X. uP,.. puP- 1 X-u. So X. X.uP .. puP- 1 X· X-u+ p(p-l)uP- 2(X-u)(X.u). J J 1J 1J 1 J Similarly (a 2;at 2)uP .. puP- 1(a 2;at 2)u + p(p-l)uP- 2(au/at) 2• Combining these identities, we obtain ,luP,..

a2 uP+t at2 ij +

a .. x.x.uP,. puP- 1 ( a 2u 1J 1 J at2

p{p-l)uP- 2 ~ at~~ j2 + t. . 1J

a ..

1J

+tij

a .. x.x.u) 1J 1 J

(X.u)(X.u)~ 1 J

= puP-t.l u + p(p-l)uP- 2 l'u 12 • p(p-l)uP- 21Wu 12 .

QED

50

LITTLEWOOD-PALEY THEORY FOR A COMPACT LIE GROUP LEMMA

3: Let F(x, t)

£

C""(G x [0, oo]) md assume that tl ~FI ... 0

at

as t .... 0 or as t ... +oo. Then

Jt f 0

AF{x, t)dx dt ..

G

F(x, O}dx-1 F(x, +oo)dx •

G

Proof: For any f

I

J

f

G

C00 (G),

I

Xjf(x)dx •

G

1· Xl(x)dx • -

G

f

(Xj1) · f(x}dx • 0 •

G

Hence

foot f AF(x,t}dxdt =f 0

G

00

t

0

f ot

02 ; (x,t)dxdt

I

G

so Lemma 3 is obtained by integrating the identity 2 b at

00

(***)

t

/

0

F(x, t)dt • F{x, 0}-F(x,oo}

over the group G. To prove (***), we integrate by parts, finding

1

00

0

a2

t -;:2F(x,t}dt ... taF -

at

at

00

0

Joo -(x,t}dta aF F(x,O)-F(x,+oo). 0

at

QED

Section 3. Proof of Theorem 2

Now we can proceed to the proof of Theorem 2. Part 1: We will prove that II g (f) liP .$ Ap II f II P' for 1 < p .$ 2. To do so, we need only consider f

£

C00 (G) which are strictly positive. f

~ E > 0.

Since f £ C00 (we could also get away with assuming f £ G;) the Poisson integral u(x, t) ( C00(G

X

[O,oo])., In addition, u(x, t)

F{x, t) = (u(x, t))P also belongs to C00(G

X

~ E

> 0, so that

[O,oo]), The verification that

51

§3. PROOF OF THEOREM 2

t OF/at (x, t) .... 0 as t .... 0 or as t .... +oo is left to the reader. Lemma 2 shows that

AF > 0. 00 ;ow (g(f)(x)) 2 • / t1Wul 2dt

a

0

::; Ap(f*(x)) 2 -P

by Lemma 2

f

f

00

_1_ tu(x,t) 2 -P&uP(x,t)dt p(p- 1) 0 00

t & F{x, t) dt, (since u(x, t) ::; f*(x) ).

0 00

· Let I(x) '"' !0 t & F(x, t) dt 2: 0. We have just proved that

On the other hand,

1I(x)dxz f F(x,O)dx -1 F(x, +oo)dx 5:1 F(x,O)dx G

G

.f

G

G

lf(x)IPdx ,

G

by Lemma 3. Hence I g(

Oil~ •

f

g(f)P dx ::; A

G

$

A(~

(by H6lder's inequality),

1

(f*(x)) 2 -p/pl(xl12 dx

G

lf*(x)iPdx)'-p/

2(/G

I(x)dx)"

12

5: A'll£11~( 2 -p)/2 ·llfll~·p/ 2 = Allfll;

Thus llg(f)li~ ~ All fll~, which completes part I of the p·oof. Note that the above arbument simplifies in the case p '"' 2, since then f* does not enter the calculations, and Lemma 1 becomes unnecessary. Part II: Using part I, we shall show that lig(f)llp::; Apllfllp for

4

5: p < +oo. From this and part I we deduce that II g(f)ll p::; Ap II f II p for

every p (1

< p < +oo). In fact, we have only to apply the M. Riesz or Mar·

cinkiewicz interpolation theorem.

52

LITTLEWOOD-PALEY THEORY FOR A COMPACT LIE GROUP

Suppose

th:~t

f is a positive C"" function on G, p is a number 2: 4,

and q is the exponent conjugate to p/2. Thus 1 < q Now lig(f)11p 2

= sup JG{g(f)(x)) 2 ¢>(x)dx,

where the supremum is taken

over all positive CO" functions ¢> such that II¢> II q

1

g(f) 2 (x)¢>(x)dx

(*)

G

:S Ap[

•

f

f(x) 2 ¢>(x)dx +

G

:S 2. Assume II f II P = l.

:S I. Our plan is to prove

f

f*{x)g(f)(x)g(¢>)(x)dx]

G

From this inequality part II is not hard. The reason is that the left-hand side involves g(f) 2 , while the right-hand side involves only g(f).

f*(x)

and g(¢>) are under control, the latter by part I and the fact that II¢> llq :S L So the size of g(f) will be controlled. The crucial step in proving (*) is to make the estimate

(**)

J

(g(f)(x)) 2¢>(x)dx =

G

1 A!J

/oo 0

:S

tl 'f u(x,t)1 2 ¢>(x)dxdt

G

0

tl 'fu(x,t)1 2¢>(x,t)dt

G

where ¢> (x, t) is the Poisson integral of ¢>. To prove (**) we will use a "subharmonicity" argument. We remark first, and this is basic, that our Laplacian !':! is hi-invariant, and so commutes with the Xj. Thus any function of it also has this property, and so in particular for Tt = et!:l, t > 0, and the Poisson semi-group. This commutitivity is obvious on the formal level and immediately justifiable on the subspace

&. The passage to general

Coo

functions is then by

a routine limiting argument Since I Ptl is a semi group, u (x, s 1 + s 2 ) = Pslu (x, s 2 ); this shows that Xju(x, s 1 + s 2 ) Hence Xju(x, t)

=

S

dU au as (x, s 1 + s 2 ) = P 1 ~x, s 2 ).

X;u(x, t/2) and

a~

P lX;u (x, s 2 ) and

= pt/ 2

S

u(x, t)

=

Pt/ 2 :tu(x, t/2).

53

§3. PROOF OF THEOREM 2

)

Next, note that since I~ ui 2 = Iij aij (Xiu)(Xju) + ( g~ 2, we can, by a change of bases in the Lie algebra, assume that

i~ui2 =I (X.u)2+(aa u)2. i

t

1

So if we prove that IXju(x, t )1 2 ~ pt/2q Xju(x, t/2)1 2) and similarly for


=

JGPt; 2 (xy- 1)Xju(y, t/2)dy (where Pt denotes

the Poisson kernel) ~ (JG pt/ 2 (xy- 1)1 Xju(y,t/2 )! 2dy)~ (Holder's inequality)

= (pt/ 2(Xju)2)~.

Hence, l~u(x,t)i 2 ~ pt/ 2 (1~u(x,t/2)1 2) -which says,

more or less, that the square of a (particular) harmodc function is subharmonic. We can now prove(**). By the above,

(by the self-adjointness of pt/ 2) = f0"" JG t I V u (x, t/2)1 2¢J (x, t/2)dx dt. If we make the change of variables s

= t/2,

we obtain (**).

Putting I Wu (x, t)1 2 = ~! u 2 (x, t) in (**), we obtain

f

(g(f)(x)) 2 ¢J(x)dx

5:

A

f"" ~ t(&u (x,t))¢J(x,t)dxdt 2

G 0 G It is tempting to integrate the right-hand side of this inequality by parts,

since & ¢J

=

0. We shall arrive at the result of such a computation in a

slightly slicker way-namely, we shall use the elementary identity

~(FG) =(&F)· G + F ·(&G)+ 2I a .. (X.FXX·G) + 2(c!E.)(aaG) .. lJ

lJ

1

J

at

t

If we take F = (u (x, t )) 2 , G = ¢J; and apply our identity to the last in-

equality, we obtain

54

LITTLEWOOD-PALEY THEORY FOR A COMPACT LIE GROUP

f

(***)

fool _2/ool [~ o

(g(f)(x)) 2¢(x)dx .S A[

G

t.l(FG)dxdt

0

G

G

aijxj

(u2(x,t))Xi¢(x,t)

lJ

•(~ x, t~(.~ ¢(x, tl)}xdt J =A[tfrl- t/12]

¢ 1 and ¢ 2 separately.

~stimate

We shall

.

By Lemma 3, ¢ 1 = JGF(x)G(x)dx- positive= JGf 2 (x)¢(x)dx- positive .:S JG f 2(x)¢ (x)dx. Since Xju 2 = 2uXju and

¢2 =

4/""/ 0

a~

u 2 = 2u

a~

u, we have

tu(x,t){I.ij aij(Xju(x,t)(Xi¢(x,t))+

(~I (x,t)X~x,t))]dxdt

G

.:S

4! ! 41 41 00

0

(***)

.:S

f*(x)/""tj'fu(x,t)Ji'f¢(x,t)jdtdx

G

.:S

tu(x,t)i.V u(x,t)IIV¢(x,t)Jdxdt

G

0

f*(x)g(f)(x)g(¢) (x)dx, by Schwarz's inequality.

G

Combining(***) with other estimates for ¢ 1 and ¢ 2 , we obtain (finally!) the basic inequality (*). Now we can prove part II. Applying

H~lder's

inequality to (*), we ob-

tain

II ,l~,n,

~(g(f)(x)) 2 ¢(x)dx ~ Ap[llfll~ + llf*llp~lg(¢)llql. By part I, II g(¢)11 q .:S Aq for II¢ II q .:S 1, so

55

§3. PROOF OF THEOREM 2

I

(g(f}(x)) 2¢(x)dx

~ A~ [ llf\1~ +II f*llp II g(f)ll ~

G

(We have used Lemma 1 to deduce the last inequality.) Taking the sup over all¢ (with ll¢11q ~ 1) yields llg(f)ll~ S A;'[ilfll~+llfllpllg(f)llp], which shows that lig(f)l\p ~ AP"IIfllp· This completes part II. We hav-e! just shown that i\g(f)llp ~ Apllfl\p for 1 < p < +oc, which is half of Theorem 2. Using a standard duality argument, we can prove the converse inequality, and even a bit more. In fact, for f

f

Lp, define the

g 1-function for£ by g 1 (f)(x) = (J000 t1a~ u(x,t)\ 2 dt)'h. Obvio;Jsly g 1 (f)(x)

=5 g( f)(x). We shall show that II fll p ~ Bp II g 1 COli p if JG f (x)dx = 0. To do so, we observe that f..., g 1(£) is essentially an isometry on L 2(G). More precisely, let f

E

L 2 and let c = JG f(x)dx. Then

(A)

Formally, this identity is a consequence of the Parseval theorem for the complete orthonormal system Y,k of eigenvectors of /j.. For if f ,.., t -<11 k>v.. t ak 1/Jk(x), so u (x, t) ,.., ~ ak Y,k(x), then u = P f ,...., ~k e

ata

~

-""k;io(llk

)'h -Crtk)'-h t ( ) e akt/Jjx.Hence

1 au

(at (x, t )) 2dx

G

so

~

= ~

k;io

Ilk e

-2(11k)'12 t

2

ak ,

56

LITTLEWOOD-PALEY 'lliEORY FOR A COMPACT LIE GROUP

This shows that 4/lg 1 (f)ll~ + c 2 = c 2 + :Ik~ 0

a' =llfll

2•

Justification

for these formal manipulations is routine and is left to the reader. By polarizing identity (A) we obtain the equation

where ui(x, tj = P\(x) and ci f

£

L2

n Lp

(1

< p <. +

=

4 sup

oo) and c

ff f G

~

= J0 fj,(x)dx,

4. sup

oo t

=J0 f(x)dx

valid for f 1 , f 2 =

£

LaCG). If

0, then by the above,

atJt. u 1(x , t) . at__q_ u 2 (x, t)dxdt

0

g 1(f)g 1 (f 2 )dx

G

(by Holder,s

inequality)~

(4

sup llg 1(f2 )\lq) llg 1 (f)11p f 2 c L 2 n Lq

II f 2 llq

sI

by virtue of the inequality llg 1(f2 )llq ~ IJg(f 2)llq S Apl!f2 llq, which we which we already know. Hence II flipS Bp11g 1 (f)/lp' and Theorem 2 is proved. QED. The following observation is in order. The arguments just given are possible because of the following two properties of the Laplacian A, which incidentally are not shared by general Laplace-Beltrami operators. (1) /). is a quadratic expression in first order differential operators. (2) /). commutes with these first order operators. We shall return to this special situation in Chapter V.

57

§4. APPLICATIONS: RIESZ TRANSFORMS, ETC.

Section 4. Applications: Riesz transforms, etc. The following examples, chosen for their simplicity, illustrate the applications of the Littlewood-Paley inequalities to the Fourier analysis on the group G.

1. Riesz Transforms. Let X1 ... Xn be a base for the left-invariant vector fields on G. The j-th Riesz trmsform of f is given by the formal equation R/

= Xj(-L\)-'h f

4t ak 1/Jk(x)

= (-L\)-'h X{ This makes sense, because for f

=

if> we can define (- L\)-'h f rigorously as

f

~

~

k~o

ak
As in the "classical" case G = Rn (which strictly speaking, recall, doesn't fall into our general frame"'ork, since it isn't compact), the Riesz transforms may be used to prove miscellaneous Lp inequalities involving derivatives. For example, we can show that IIXiX/IIp ~ Apll A flip (1

<

p

< +oo),

By using our Littlewood-Paley inequalities we can easily prove the basic property of Riesz transforms, namely that Rj is a bounded operator on Lp (1

< p < +oo). In fact, let f = I ak lfrk(x) belong to &, and let

fj = R/ Consider the Poisson integrals u and uj corresponding to f and fj respectively. We have au/at = - Pt((- L\ )-'h f) since both expressions are equal to - Ik~ 0 (1-'k) 'h e-llkt ak 1/Jk(x) -all the equation really says is that -(-A)'h is the infinitessimal generator of IP 11. Applying R, to both sides and noting that everything commutes we obtain

au.J

at=-

Therefore

=

Kg(f) .

Xju(x,t) .

.

58

LITTLEWOOD-PALEY THEORY FOR A COMPACT LIE GROUP

Since I g 1 (~/)llp ~ !IRjfll p and II g(f)llp ~ II flip' we conclude that Rj is a bounded operator on LP(G).

Problems: (1) (possibly not difficult). Show that the Rj are pseudo-differential oper-

ators on G. More particularly, show that the Rj are obt8inable from the standard Eu_clidean singular integral operators by patching together. (For an introduction to the latter see e.g. Palais et al [5].) (2) Make a systematic study of the pseudo-differential operators on G which commute with the left action of G. 2. Rie~z Potentials. The operator (-~)iy (y real), which as usual is rigorously defined on functions in

&,

is of interest in analysis. We shall

prove THEOREM 3: 11(-A)iYfllp ~ Ap(y) · llf lip It would be interesting to know whether (-A}iY is also a pseudo-differ-

ential operator. This problem is analogous to the same question for Rj. To prove Theorem 3, we note that Aiy = KA J0 oo e->.s · s-iy ds for A > 0. A function m (A) is said to be of Laplace-transform type if we can write m(A) = A J0 00 e-AS a(s)ds, A> 0, for some function· a(s) uniformly bounded on (O,oo)·-say la(s)[ ~ M (all

S£

(O,oo)). Thus Theorem 3 is a

special case of THEOREM 3 ': Let m (A) be a function of Laplace-transform type. De-

fine the operator T on Tf = I

& by setting

m((l-lk)lh) · akt/lk(x) iff= I

k Then I[Tfllp~ Apllfllp

ak¢k(x)

t"

& .

k

0 < p <+oo). lfwetake m(A)

=

A2 iy in Theo-

rem 3' we obtain Theorem 3.

Theorem 3' corresponds to a weakened form of the classical Marcinkiewicz multiplier theorem for the n-torus Tn. The connection is that if

59

§4. APPUCATIONS: RIESZ TRANSFORMS, ETC.

111 (.).)

is of Laplace transform type, then \Akm(k)(A)\ .S Ak for every k ~ 0,

while a special case of the Marcinkiewicz multiplier theorem says that if j.).km(k)(A)\.S Ak foreach k.S[n+l/2],then m(\xi) isamultiplieron Lp(T0 ) (l < p < +oo). Proof of Theorem 3 ': For a (reasonable) function f, define the Littlewood-Paley g2 -function of f by the equation

g2 (f)(x) =

(!

0

00

t 3 \ ;t22 u(x,t)\ 2 dt

)'n

We shall prove that \\g 2 (0\lp ~\\flip and that g 1(Tf)(x) .S Kgif)(x). Firstofall, \\f\\p.S Bp\l~(f)\\p (ifJGf(x)dx= 0) isquiteeasy. For

1

00

t

by

Holder~s

\(

2 Ff)\dr ~

ar

=

f."" t

\

2 pTf\ r dr ~

ar

T

inequality. Hence

by Fubini's theorem. Thus g 1(f) .S g2(f), so we have 1\fllp.S Bp\\g2(f)\\p by Theorem 2. The converse inequality \\g 2(f)\1p .S Apl\fllp is more difficult, and we proceed with stealth. We prove it here only for p

~

2, and this suffices for our immediate

purposes. The full case 1 < p < oo is contained in the general treatment of Chapter IV.

60

LITTLEWOOD-PALEY THEORY FOR A COMPACT LIE GROUP

Consider the Poisson kernel Pt defined by the identity ptf = Pt*f. We note that 1. t

JG

Pt(x)l dx S C for all t > 0, as follows immediately from the

I;

argument given in 2. below. 2. For a reasonable function f on G, define f**(x) '"' sup (tlaat Ptl * f Xx). t

>0

Then II f**H p S Ap II £11 p (1 < p < +oo). This is proved in the same way as

£*.

we proved the corresponding inequality for

In fact, let Kt be the ker-

nel for the heat semi group (i.e., Ttf "' Kt * f). By the subordination of pt to Tt we have

.P == _l_ /

y2rr

t

""

2

te-t /4sK . s-3/2 ds

o

s

'

so that tl 2._ p (x)l <

at

t

-

l

2rr

1 at 00

t

0

2

I .Q_te-t / 48 IK (x) s-3/2 ds. s

The last inequality allows us to deduce

II f**ll p S

Apll f II

Hopf-Dunford-Schwartz ergodic theorem. Now set

(as usual u (x, t) == ptf(x)). We shall prove that (A)

and (B)

(A) and (B) together imply II g2(f)11p

S Apllfllp (2 < P < +oo).

from the

61

§4. APPLICATIONS: RIESZ TRANSFORMS, ETC.

To prove (A) we take arbitrary ¢ ~ 0 a satisfying II ¢11 q ,S l, where q , is the exponent conjugate to p 2 . By (2) above, II ¢41 q S K. So

J (g*(f)(x)) 2 ¢~(x)dx G

=I! If 1 G

foot 2 11t Pt(xy- 1)11Wu(y,t)l 2 ¢(x)dydxdt

G 00

:S

G

0

tl 'f'u(y, t)1 2 ¢*"(y)dtdy

0

(g(f)(y) 2 ¢*"(y)dy S

=

sup llif!llq:SK

G =

K II g(f)ll ~

:s Apll fll ~

f

(g(f)(y)) 2 if!(y)dy

G

.

Taking the sup over all
Statement (B) is even easier. If t

= t1 +

so

S {oot3 (t-1 {l(lt Pt)

)0

)G

(by Schwarz's inequality) :S

If

t/2

00

1

2

*

f,

(xy-l)ll
t 2 ·(1t Pt{ 12(xy- 1)1 Vu(y, t)1 2dt dy

G 0

=

t 2, then ptf = Pt * Pt

k(~(f)(x))2

62

LITTLEWOOD-PALEY THEORY FOR A COMPACT LIE GROUP

by virtue of the change-of-variable s

=

t/2. This completes the proof of (B).

Finally, we have indeed shown that hif)llp S

II flip'

at least for

p > 2.

Remember why we have gone through this elaborate contortion? We are trying to prove that if m(A) = A f0ooe-Ata(t)dt is of Laplace-transform type, then the operator T defined by

is bounded on Lp(G). Very well, then. Let f

E

8', have Poisson integral u and let F = Tf.

We claim that F (x) =

-1

00

0

au (x, t)a(t)dt . at l

For it suffices to check this identity foi f u (x, t)

= t/Jk; in that case

'h

= e-(1£k) t ¢k(x) ,

so that

Applying the Poisson kernel to both sides of our identity yields

p

t

1 F(x)

=-

fO()t P 1 ( g_

= -

f

0

ptf(x))a(t)dt

at

0 00

.Q_ pt+tlf(x)a(t)dt. at

§4. APPLICATIONS: RIESZ TRANSFORMS, ETC.

63

Hence

_Q_ P t lF(x)

=-

at 1

J"" ata2

~ P t 1 +t f(x)a(t)dt .

0

So

<

!

00

IL

o

< M

2

P

t+t

1 f(x)Jia(t)ldt

at2

f

2

00

t+ t

I Lp

!

1 f(x)ldt

at 2

o

2

00

M t t

la~2

u(x,t)l

~t

1

Repeating the arguments proving that g 1 (f)

Since lig1 (F) I p ~

I Fil p

and

II gif)ll p

~

~

C g/f), we find that

II f II p

(p > 2), it follows that T

is a bounded operator on Lp(G) for p > 2. The usual duality argument now shows that T is bounded on Lp(G) for p < 2, and T is clearly bounded on L 2 (G).

QED

Open problem: Obtain a strengthened multiplier theorem, which reduces to the Marcinkiewicz multiplier theorem in the classical case G = Tn, the n-torus.

64

LITTLEWOOD· PALEY niEORY FOR A COMPACT GROUP

BIBLIOGRAPHICAL COMMENTS FOR CHAPTER II

Section 1. Most of the results stated in Theorem 1 (the construction of heat-diffusion semi-group) could be read off from Hunt's paper [12]. Secfion 2, 3, and 4. The idea of this approach to the Littlewood- Paley theory originates in the author's course at Orsay [19], where it is carried out for R". For some related ideas. see Gasper [411. The original theory for R" and its related realated real-variable approach owes much to the pioneering work of Besicovitch, Marcinkiewicz, and the paper of Calder6n and Zygmund, "On the existence of certain singular integrals,'' Acta. Math., 88 (1952), 85-139. For the inequalities for the g-functions see the author's paper [lBJ.

CHAPTER III GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

Section 1. General setting. We shall now develop the main theme of these lectures. Our purpose is to obtain such basic Fourier-analytic tools as the Hardy-Littlewood maximal function, the Littlewood-Paley inequalities, and conditions for LP multipliers, in a very general setting. An appropriate context seems to be that of the symmetric diffusion semi-group. This notion is defined as follows. C}R,dx) is some positive measure space. (Ttlo
(I)

(II)

IITtfllp

S llfllp

(l

S

P

S +oo)

(Contraction property)

Each Tt is a self-adjoint operator on L 2(m) (Symmetry property)

(III)

Ttf 2!: 0 if f 2!:. 0

(Positivity property)

(IV)

Ttl "' 1

(Conservation property)

(Ttl is called a symmetric diffusion semigroup. Sometimes we shall drop assumptions (III) and (IV), but (I) and (II) are absolutely essential for what follows. Our introductory example is of course the Poisson semigroup. But symmetric diffusion semigroups occur often in analysis. We content ourselves

65

66

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

here by noting some other examples:

m= (a 1 , a 2)

1) Let

be some finite, half-infinite, or infinite interval,

and consider the second-order differential operator Lf = a(x) d 2f + b(x) d~ + c(x)f dx 2 dx Assume that a(x) > 0, c(x) :S 0, and that a, b, and c satisfy minimal smoothness conditions. It is not hard to find a non-trivial measure q (x)dx on

m, which makes

L formally self-adjoint, i.e.

{ f(x)Lg(x)q(x)dx

1m for f, g

f

=

I

1m

Lf(x)g(x)q(x)dx

Co00 (al I a2). By imposing certain boundary conditions on func-

tions to which we apply L, we can make L into a self-adjoint (unbounded) operator on Li'm, q(x)dx). One can then show that L is the infinitessimal generator of a semi group Tt .. etL, satisfying (1), (II), and (III). If c (x) _ 0 then (IV) also holds, with appropriate boundary conditions.

This example is very general. It includes the theory of expansions in Hermite polynomials, Legendre polynomials, trigonometric polynomials, and many other kinds of expansions. For details, the reader may consult Titschmarsh's book Eigenfunction Expansions [36] and a paper Element-

ary Solutions for Certain Parabolic Partial Differential Equations by H. P. McKean, Jr. [13]. For further discussion see Chapter V.

2) Parts of the theory of second-order operators extend to Rn. Let D he a domain in Rn with a reasonable boundary, or more generally a smooth manifold. We consider an elliptic second-order partial differential operator of the form

n

Lf = e(x)- 1

I

i =I

where we assume c(x) :S 0 and e(x) > 0.

L

is then formally self-

67

§1. GENERAL SETTING

adjoint with respect to the measure e(x)dx. As before, L subject to boundary conditions, generates a semi group I Ttl satisfying (1), (II), and (III); and if c(x) :. 0 then (IV) holds. For details, see Phillips, T.A.M.S.

1961, p. 62-84, [15]. An important special case is the Laplace-Beltrami operator on a compact Riemannian manifold. If the Riemannian manifold is a Lie group, we obtain the heat semigroup of Chapter II (and Chapter V for the non-compact case). 4) Given symmetric diffusion semigroup we can construct other sym-

metric diffusion semi groups, by "subordination." (For the facts concerning subordination, see Bochner [22] and Feller [ 10, Chapter 13].) For now, we shall abandon examples, and study diffusion semigroups for their own sake.

Section 2. Analyticity of these semi-groups. Our first result already shows that despite the misleadingly simple appearance of the assumptions I) - IV) some rather far-reaching implications may be drawn from them. THEOREM 1. Let (:)R, dx) be a si~ma-finite measure space, and sup-

pose that the semi~roup !Ttl satisfies axioms (I) and (II). Let 1 < p

< +oo.

Then the map t ... Tt has an "analytic continuation," i.e., it extends to an analytic Lp-operator-valued function t + ir ... Tt+ir, defined in the sector Sp: \arg(t+ir)\ < ~

0-1}

-1\).

The terms used in the statement above are explained by the following DEFINITION: A function f mapping an open subset {l of

C, into the

complex Banach space B, is called ooalytic (or holomorphic) if for each continuous linear functional L on B, z ... L(f(z)) is an analytic function of z

l

fl. A map z ... Tz from

{l

is called analytic if, for each b

0 to B.

into the space of bounded operators on B £

B, the function z ... Tzb is analytic from

68

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

Proof of the theorem: We shall first prove the result for p spectral theorem, the bounded self-adjoint operator T 1 f""

f

T1

=2

By the

has the form

\dEA(f) ,

-1

where I EA I is a resolution of the identity in L 2(lll) . Since T 1 = TlhTlh

= (Tlh)*
it follows that the resolution lEAl

is concentrated on [0, 1], i.e., EA = 0 if A T 1f =

!

1

< 0. Hence

"Ad EA(f)

0

for f £ L 2 <)K ). Repeated application of the uniqueness of positive definite square roots of operators (for a proof of uniqueness, see Riesz-Nagy f35], section 104) shows that Tlhkf =

f

1

AlhkdEA(f)

0

Therefore Tm/2k f =

I

1

Am/2 k d EA(f) ,

0

and by the strong continuity of the family !Ttl we conclude finally that Ttf ..

I

1 AtdEA(f)

0

for every f £ L 2 <'1l) and every t > 0. It is now child's play to define Tt+ ir in the half-plane t = Re(t+ ir) ~ 0.

We merely set Tt+irf •

j

1 At+irdEA(f) .

0

This concludes the proof of our theorem for the case p • 2. Note that 11Tt+irll 2 :S 1 fort~ 0.

69

§2. ANALYTICITY OF THESE SEMI-GROUPS

The L 1 ( L.,,,) result is (vacuously) true since the "sector"

s 1 (or

S.,.,)

is just the positive real axis, where !Ttl is already defined. Finally, we shall interpolate between the L 2 result and the L 1 (or L00 ) result, to obtain the theorem for Lp, 1 < p < 2, {2 < p < + oo). To perform the interpolation, we make use of a "convexity" theorem of the author, which generalizes the Riesz convexity theorem. The extension of the Riesz convexity theorem is the following: Let

mbe a measure space and let

U (z) be a linear operator for each

complex z in the strip 0 :S Re z :S l. For every z in the strip, we suppose

m(i.e.• finite linear combinations of sets of finite measure) to measurable functions on mwhich are integrable that

u (z)

maps simple functions on

over all sets of finite measure. Suppose further that for any simple functions f and g on

m. the function z ....

fm

{U{z) f) gdx

< Re z < 1, and continuous in the closure. Finally, assume that \IU{z)fllp 0 :S M0 \lf11p 0 for Rez =0, and is bounded and analytic in t_he region 0

IIU{zHI\p 1 :S M 111fllp for Rez = 1, (f simple). Then if we set p between p 0 and p 1 by 1/p =h-t)/p 0 + t/p 1 , we have 1\U{t)fllp:S M~-tM~\\f\\p, (f simple). This theorem reduces to the Riesz-Thorin convexity theorem if we put U(z) equal to the single operator U for every z. Sketch of Proof of the Convexity Theorem. We shall make use of the

so-called "Three-Lines Lemma": Let ~{z) be a bounded analytic function in the strip 0 < Re z < l, continuous in the closure. Suppose that \~{t)\

:S M0 for Rez ,. 0 and 0 :S t :S 1, l~{t)\ :S M~-t M\ .

l~z)l

:S M1 for Rez

== l. Then for

Proof: The auxilliary function 'l'{z) .. {M 0 JM 1 l~{z) is bounded, an-

alytic in the strip, continuous in the closure, and satisfies \'l'(z)\ :S M0 for

70

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

Re z '"' 0 or l. By the Phragmen-Lindelt>f maximum principle for the strip, l'l'(t)l ~ M0 , i.e., lcl>(t)l ~ M~-tM~ .

QED.

Now let f be a simple function with II flip= l. To show that IIU(t)fltp ~ M~-t M~, we need only prove that

I/'Jn

(*)

(U(t)f)gdxl

~ M~-tM~

for every simple function g for which II g II q ~ l, where q is the exponent conjugate to p. Inequality (*) follows from the three-lines lemma if we set up the right functi~n Ill. In fact, we can write f "' F f and g = G i- where F, G ~ 0 and lf(x)l fz

=F

= li-(x)l =

l for almost all x. Put

p ( 1-z + _.!.) Po p1 f

and

gz

=G

q ( 1-z + .!.._) Qo q 1 g.

where q0 and q 1 are the exponents conjugate to p0 and p 1 , respectively. Obviously ft = f, gt = g; II fzll , II gz II ~ l for Re z = 0, and Po

qo

\\fz\lp 1 , \\gzllq 1 ~ l for Rez = l. Hence the bounded analytic function tl»(z)

=

1m

(U(z)fz)gzdx

satisfies l«<>(z)l ~ (norm of U(z) as an operator on Lp 0) ~ M0 if Re z

=

and similarly \«<>(z)l ~ M1 if Rez .. 1. The three-lines lemma now shows that

This completes the proof of (*).

QED.

More detailed expositions of convexity theorems may be found in Stein [22], Dunford-Schwartz Linear Operators [9, Chapter 6, Section 10] and in Zygmund Trigonometrical Series, Vol. II, Chapter XII, [20). Notice that the above argument has to be patched up for p difficult.

= + oo ;

the problem is not

0,

71

§2. ANALYTICITY OF THESE SEMI-GROUPS

Now we can return to semigroups and finish off the proof of Theorem 1. Recall that we have defined a family of operators {Ttl for complex t in the right half-plane Ret

~

0, satisfying the properties

(a)

I\Ttfll 2 :5l\fll 2 (all t intherighthalf-plane).

(b)

If f, g f L 2 then t

-o

~ ( Ttf) g dx is an analytic function of t,

bounded in Ret > 0 and continuous in the closure.

We want to interpolate between (a) and (c). So let 71 > 0 be arbitrary, let

-77/2 < (} < 77/2, and define U(z)f

=

T71ei(Jzf. By (b) above {U(z)l is

an analytic family of operators, in the sense of the hypothesis of the above convexity theorem. Furthermore, (a) shows that II U (z)f 11 2 :5 II f 11 2 for Rez ~ l, and (c) shows that i[U(z)f[[ 1 :5 llfli 1 :5 llfll 1 for Rez = 0. Therefoie, by the convexity theorem, IITtfllp :5 llfllp (l < p < 2) where t = 71e

t

i0(2-2)

p . Hence liT flip :5 llfllp whenever \arg t I < !! (2- ~) • 1L - 2 p 2

0-1

~

p

-

1[)

'

for 71 > 0 and (} c [- ~, ~] are arbitrary. We can prove an analogous result for 2 < p < +oo by interpolation between L 2 and L00 •

It remains to show that

is bounded, analytic, and continuous in the closure of the sector Sp, for f

£

Lp• g l Lq. This follows from (b) when f and g are simple, so that

letting I fkl (respectively

I gkl) be a sequence of simple functions tending

in LP (respectively Lq) to f, (respectively g), we find that Jm(Ttf)gdx is the uniform limit of the analytic functions

QED.

72

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

As an application of Theorem l, we can show that if f

t

Lp (l < p < +oo)

then for almost every x, t ... Ttf(x) is a very smooth function on (0, "")· This is essential is we are to define a Littlewood-Paley g-function involving

dfat Ttf(x).

LE'MMA: Let f

t

LP(m), 1 < p

< +"".

For each t, we can redefine Ttf

on a set of measure zero, in such a manner that for every fixed x, Ttf(x) is a real-analytic function of t

f

(O,oo),

. Proof: By Theorem 1, the function t -+ Ttf t Lp extends to an analytic

Lp-valued function on the sector Sp. Now, our definition of an analytic function ell from a region 0 s;; C to a Banach space B was that for each continuous linear functional L on B, z-+ L(ell(z)) is a complex-valued analytic function on 0. But it is a standard fact of functional analysis that this definition of analyticity is equivalent to any other "reasonable" definition imaginable, for example. ell is continuous and satisfies Cauchy's integral formula; lim .1.z -+ o

If z 0

ell (Z+.1.z)- ell(z)

t

Az

exists in the norm topology on B.

.1.(z 0 , E)

s:; 0, then ell has a power series expansion 00

Cll(z) ""

!.

bk(z-z 0)k where ~

f

B

k= 0

valid in .1.(z0 ,

€)

and such that

Ik= 0 j~j rk < +oo

for any r < €.

We shall use the last definition of analyticity. So for any t 0 Ttf =

Ik=

0

> 0,

fk(t- t 0 )k, for all t in some neighborhood Mt 0 , 2€), and

Ik'... o jjfkJipek < +oo. Each fk isanequivalenceclassoffunctionspick a particular representative, which we also denote fk. For t

i

we can modify Ttf on a set of measure zero, in such a manner that 00

Ttf(x) =

I

k=O

fk(x)(t- t 0)k

(every x).

ll (t 0 , E),

73

§2. ANALYTICITY OF THESE SEMI-GROUPS

This makes sense, because Ik'= 0 £ k I fk(x)l ~ + oo for almost every x, as follows from Ik=O£kllfllp < +oo. Now cover (0, oo) with countably many neighborhoods .1.(t 0 , £). The rest is quite tirival, and details are left to the reader.

QED.

Section 3. The maximal theorem. Our first main result is the following: MAXIMAL THEOREM: Let the semigroup !.Ttl satisfy (I) and (II}. Then (a) the maximal function, defined by f*(x}

=supt> 0 ITtf(x)l,

satis-

fies the inequality

1 < p (b) if f

£

~

+oo ;

Lp("l), then lim Ttf(x) = f(x) a.e.

(1 < p <+oo).

t .... 0

Proof: First of all, the theorem makes sense, by virtue of the last

lemma. The plan of the proof is as follows: 1. We will prove the theorem for p

£

2.

2. Next, we will give a very simple refinement of the theorem, valid for p .. 2. 3. We will take the result of part 2 as information on p "' 2, use the Hopf-Dunford-Schwartz ergodic theorem as information on p = 1 + £ . Our theorem then follows from interpolation.

Step 1: This is essentially a Tauberian argument for we have (by the Hopf-Dunford-Schwartz ergodic theorem) strong bounds on the averages

and we have some additional information-from this we can prove bounds

74

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

for Ttf(x). The extra information is provided by our good friend, the Littlewood-Paley function g 1 (f)(x)

!, ~tIJ: T

1f(x)j 2

- (

dt) " .

We now show that !ig 1(011 2 ~ cllfll 2 where c is a universal constant. By the spectral theorem, we can write Tt in the form Ttf =

f

00

e-At dEAf

0

where lEAl is a resolution of the identity. (See the proof of Theorem 1 in Section 2. There we had in effect written Tt • [ 01 At dE{ A). We should then have E{A) ,. E(e-A),

0 $ A ~ oo.) Hence

Ergo,

(again by the spectral theorem)

=

f f oo A2

~

$

00

0

te-2Atdt(dEAf' f) '"'

!f

oo (dEAf' f)

~

~ll£115 .

We are in position to prove step 1. Integrating by parts shows that

§3. THE MAXIMAL THEOREM

75

Therefore,

IT$f{x)J

s I~ S

I}

£\tf(x)dtl

+I~

Ia\< a~

fT'f(x)dH }(

Ttf{x)dtl

ftdt ~ {'1 ;,

T't(x)[ 2dt)"

S Mf(x) + g 1(f), where Mf(x)

Hence f*(x)

=ssup >0

I!. J s

s Ttf{x)dt]

0

S Mf(x) + g1(f){x) so that

by the Hopf-Dunford-Schwartz ergodic theorem and the L 2 -boundedness of the g-function. This completes step 1 of the proof. Step 2: For f ! Lp and k

~

0 define

f*0 is the same as the maximal function f*. We shall show that for each k ~ 0,

llfkll2 S AkilfJJ2 ·

The proof copies the argument of step 1. In fact, integration by parts shows that

76

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

so that by Htllder's inequality,

where

The first term on the left-hand side of this inequality has already been estimated in step 1, where we showed that

So in order to complete step 2, for the maximal function ~ , we need only

show that hi011 2 ~ A llfll 2. This follows from the spectral theorem in a manner analogous to the proof in step 1 for II g1(f) 11 2 ~ A II f 11 2. Thus II~ 11 2 ~ A II f 11 2 . The general case II £k 11 2 ~ Akll f 11 2 follows by induction. We begin the inductive step by computing t /

0

sa k+l

Tsf(x)ds

ask+l

by parts. Details are left to the reader.

Step 3: So far, we know that for p > 1 · (think of p very near to 1), and that

II

k sup tk ak Ttf(x)ll2 t >o

at

~ Akll£112

foreach k 2:0.

Clearly, in order to interpolate between these inequalities, we should

77

§3. THE MAXIMAL THEOREM

search for an analytic family of operators f1 which act on functions of one variable t

£ ( 0, oo ),

such that

1 t

I 1(f)(t) =

f(s)ds,

and

0

The formal computation

In(f)(t) =

Jtf 0

s... Jr f(r) dr ... ds dt "'

0

0

1

~!

f

\t- s)n-l f(s)ds

0

suggests a reasonable definition for fl: (a

£

C).

If Rea > 0 then the integral defining Ia converges absolutely for

f

£

L 1(0, ""), but if Rea

~

0, the integral need not be defined. This is as

it should be, for 1-k is supposed to be the k-th derivative. fl(f) is

called the a·th fractional integral of f. To justify our definition of fractional integration, we shall prove the LEMMA:

Let f be in

defined for Rea

> 0,

L1

n C""

on (0, oc), Then the function a .... fl(f)

has an malytic continuation to all of C. F~rther­

more, the functional equations Ia If3f = ~ + f3c and I 0 f = f hold. Pr<;KJf: It is convenient to replace

f1 by the "modified fractional in-

tegral" Ma defined by

As we noted, this integral is absolutely convergent, and in Rea > 0. Write

~

is analytic,

78

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

~(f)(t)

f

= - 1r(a)

f

'h

(1-s)a- 1 f(st)ds + _1_ 0 r(a)

1

(1- s)a-lf(st)ds

¥..

=CD+®. There is no difficulty at all in continuing term plex line a

f

C. Tenn

@

CD

into the whole com-

is not so simple, because of the singularity

of (1-s)a-! at s • l (Rea :S 0). But we can evaluate

@

formally by

integrating by parts: (1)

@

=

1 1 / (1-s)ll_
If Rea > -1, this expression makes good sense-using it as a definition of

@ , we obtain

an analytic continuation of~ into the region Rea > l.

If we integrate by parts (fonnally) once again (i.e., in

(1)~.

we get a defi-

nition of Ma valid for Rea > -2. By successive integration by parts, we continue Ma into the region Rea > - k for any k > 0. Thus Ma (and therefore also Ia) may be continued throughout C. The semi group relation Ia 1{3 = Ia + f3 follows from a routine computation for Re a, Re

f3 :;: :

0, and so holds for all a,

f3

f

C, by virtue of the

analyticity of ~. To show that 1° is the identity, consider JU (a > 0), let a ... 0, and apply a routine "approximation of the identity" argument. Details are left to the reader.

QED.

In particular, since lkl-k =identity, we have 1-kf =(ak;atk)f. The operators JU and Ma are fraught with applications to our maximal functions. For, the inequalities between which we are trying to interpolate can be rephrased (l

and

< p < +oo),

79

§3. TilE MAXIMAL THEOREM

We are trying to show that

Two obstacles stand in the way of interpolation: (a) To interpolate using the given family of operators m : f -o sup I~(T 0 f)(t)\, a t > 0 we need inequalities not merely for m_k + iy for any y. (b) The operators

ma

'"1 and '"-k• but for '"1 + iy and

are unfortunately non-linear.

Neither of these problems is very serious. Let us first take Rea > 0.

'"cp a

=sup _ l _ t> 0 \1\a)\

< sup I 1\Re a) - t >0 1\a) "' I if

cp

1\Re~l

~ 1 f\t-s)a- 1¢(s)dsl

\till

0

l

I·

11\Re a)\

m

1\a)

· -l

f

jtal

t R 1 (t- s) e a- \¢ (s)\ds

o

1¢1 , Rea

is a decent function on ( 0, oo ), If we use the fact that jr(X+iy)\ -

e

-E.Irl 2

·IYI

(x-~)

. ..j2TT

as Y-+.±""

(see Titschmarsh, Theory of Functions, p 259) and apply the last inequality to the function if>(t)

=

Ttf(x), we obtain the inequality ilma II p< Ke"llm al11m··Rea II p

for any a and p, where the constant K' is uniform in Re a , provided Re a varies inside a bounded set.

i/

Therefore, 11m 1+ II p .$ Kpen'\YI II f II p for any p (l < p < +oo) and any real y; and 11'"-k+i/11 2 .$ Ake"IY\11£!1 2 foranypositiveinteger k and any real y, similarly, if we use (1) and the integration by-parts that follows it.

80

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

To handle (b), we linearize our operators as follows: For any reasonable function t(x), mapping our basic measure space

Clll, dx) into (0, oo),

define an operator T~(*) on Lp by setting

T~(*)f(x)

= r(~)

J

t(x)

t(x)-a

T 8 f(x)ds .

0

Obviously ITat(*)f(x)l

s ma(f)(x)

for any function. By our inequalities for

ma, (1)

and (2)

where Ap and Bk are independent of the function

t( · ). Since the opera-

tors T!(•) are linear, we may immediately apply the convexity theorem (of the previous section) to inequalities (1) and (2). To do this, set U(z) =

where a= a(z) '"'0-z)(-k) + z.

eZ 2 Tt(·)

a

The result is IITJ<·>fllpS Kllfllp

where p is determined by the equations

.. (3)

-1 p

1·0 + 0-0)(-k) - 0. The "constant" K, whatever else it may depend on, is independent of t( ·).

Any p (1

< p < oo) arises from equations (3) for some values of k

and p 0 > 1, for we have only to pick k very large. Therefore, we have shown that for every p (1

ClR, dx) ... (O,oo),

< p < oo) for every measurable function t(·):

the inequality

81

§3. THE MAXIMAL THEOREM

(4)

is valid, with Kp independent of t(·). Now we are (essentially) done. We have merely to pick our function t(·) in such a way that ITtf(x)j ~ Y. supt > 0 ITtf(x)j for each x. By inequality (4), ~supt > 0 !Ttf(x)l!p :S 2Kpl!fllp' In other words,

II£*~ p S Apllfll p

{l

< P < oo),

which is exactly what we wanted to prove. It remains only to show that

limt ..

0+ Ttf

= f almost everywhere

(f ( LP, 1 < p <+oo). As always, the almost-everywheretheorem is an easy

consequence of the Lp·boundedness of the maximal function. First let f ( L 2 • Since Ttf(x) is a real-analytic function of t l (0, oo) for almost all x, we have limt .. 0+ T~Tsf) • Tsf almost everywhere, for each s > 0. Hence lim sup !Ttf(x)-Hx)\ .S lim sup t .. O+

t .. O+

\T~f-Tsf)(x)l

+lim sup \Tt(Tsf)(x)- Tsf(x)\ + !Tsf(x}-f(x)l t .. 0+

S sup t >0 =

IT 1U- Tsf)(x)j

+ ITsf(x)- f(x)j

(f- T 8 f)*(x} + jTsf(x)- f(x)j .

So

!I lim sup t

>0

!T~(x)- f(x)l !1 2 < 21!(£- T 8 f)* -

\b

S Kl!f-T8 f\1 2

(by the L 2-boundedness of the maximal function) .. 0 as s .. OT, by strong continuity of !Tsl on L 2 • This proves that lim sup

t .. 0+

=0

ITtf(x)- f(x)l

in other words Ttf .. f a.e. as t .. OT, for f

l

a.e.,

L/lR).

82

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

Now let f belong to L p (1R) (l < p < +oo), and suppose We can find a function g E L 2 n Lp such that II f- gil p ~arne trick as before, we write

+ lim sup

< sup t)

0

t-+ o+

<£

£

> 0 is given.

•

Using the

ITtg(x)- g(x)l + lg(x)- f(x)l

ITt(f_ g)(x)l + l(f- g)(x)l (since Ttg(x)-+ g(x) a.e. as t-+ o+) I

= (f- g),x) + (f- g)(x) .

So by Lp -boundedness of the maximal function, we have II lim supt-+O+ ITtf()- f( )I lip .S Kllf-gllp .S Kc Letting £ .... 0, we get lim sup ITtf(x)- f(x)l "' 0 a.e., which means t t-+ o+ that T f-+ f a.e. QED.

Section 4. A digression: L 2 theorems. Before we continue our development of Littlewood-Paley theory, we shall pursue a digression. The above maximal theorem and ergodic theorem were posed for semi groups {Tt I which were contractions
all p (l :S p .S +oo). It is interesting to consider purely L 2 -variants of the maximal and ergodic theorems. THEOREM

3: Let P be a self-adjoint operator on L 2 (')J{), satisfying

IIPII 2 :S l and the positivity condition Pf ~ 0 if f ~ 0. Then

for any f

E

L 2(JR), where C is a universal constant.

REMARK:

Is the theorem still true if we drop the positivity assump-

tion? A priori the answer would seem to be no, since then the hypotheses

83

§4. A DIGRESSION: L 2 THEOREMS.

would be purely Hilbert space (i.e., unitarily invariant) while notions like almost everywhere convergence are not unitarily invariant. In fact Burkholder has pointed out that the answer is indeed no, using consideration of his "semi-Gaussian subspaces," see [33]. That the condition of self-adjointness cannot be much modified is evident if we take instead of P an appropriate unitary operator. For example if U f( x) = f( x + 0) is a shift by an irrational (} on the circle T 1 = R1 /Z, then it is easy to construct an f sup

Iunf( x)l

= ""

f

L 2 (T 1 ), so that

everywhere.

n~O

Proof of the theorem: Define A(n)

=(n+l)-l~~=Opk, We shall first

prove an L 2 -niaximal theorem for Mf(x) = supn A(n) f(x) -then we shall prove our maximal theorem by using the same sort of g-function argument which served us so well in the maximal theorem for ITt I.

If f

~

0 then A(n) A(m)f

(1)

:5 2[A(2n)f + A{2m)f]

To prove this, suppose n :5 m. Then l n m l m +n ~ A (m) A{n)f = --,------pk ~ pif = ILf P f (m+l)(n+l) k=.O j=O (m+l)(n+l) ~=O

l:

I

where ILf is the number of ways in which f can be written as k + j with k

:5 n, j :5 m. Obviously A{m) A(n)f

:5

:5 n + l for any

~.

m +n 1 ~ (n + l) pff (m+ l)(n+ l) e= 0

=

l

ILf

2m

So m+n 1 ~ pff (m+ l) f= 0

~ P~f :5 2A(2m)f :5 2[A(2m)f + A{2n)f].

m+l f=O The situation is analogous if m

:5 n .

84

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

Now suppose that I!PII 2

= 1-8< 1.

Then IIPnll 2 ~

0- 8)n.

So

DO

~ l!fll~ ~ (1- 8) 2n

=

Alllfll~ •

n= 0

It follows easily that II Mf 11 2 ~ As II f 11 2 for f

L 2 . Of course As ... + DO as 8 ... 0. But we shall show, using inequality (1), that 11Mfll 2 ~ Cl!fll 2 where C is independent of 8. f:

Let C be the smallest constant such that II Mf 1! 2 ~ C II f 11 2 . As in the proof of the maximal theorem for semigroups, there is a function n (x) defined on the measure space

m, such that the operator B: f ... A(n(x))f(x)

on L 2 has norm ~ C(l-,g) (€ > 0 is prespecified). B is a bounded operator on L 2 and f :::_ 0 implies Bf ~ 0, but B is not necessarily selfadjoint. Define another operator B 2 on L 2(m) by setting B 2(f)(x) = A(2n(x))f(x). By inequa.lity (1), A(n(x)) A(m) ~ 2[A(2n(x)) + A(2m)] in the obvious sense. In other words BA(m)f ~ 2[B 2 + A(2m)]f for f :::_ 0. We cannot pull a similar trick on A(m), because it is being operated on by B. To overcome this difficulty, we pass to the adjoint. Thus A(m)B*f ~ 2[B~f + A(2m)f). Taking m = n (x), we obtain BB* f ~ 2[B~ + B 2 ]f for f 2: 0. The operators BB* and

B~

+ B2 map positive functions to positive functions, so

IIBB*II 2 ~ 2IIB~I1 2 + 2IIB 211 2 . But by definition of B, B2, and C, IIBB*II 2 = IIBII~?:. c 2(1-e) 2 and IIB2112 ~c. So IIB;II ~ c and c 2(1-e)2 ~ 4C. Since

E

> 0 is arbitrary, C ~ 4.

We have shown that if (2)

II P 11 2

< 1, then

I

Pjf<·>lll2 llsuPn I< 1 1> n+ j=O But this inequality also holds if liP 11 2

=

~

4llfll2 ·

1, for we may apply (2) to the

§4. A DIGRESSION: L 2 THEOREMS

85

operator PeS .. (l- cS)P and let cS ... 1-. Details are left to the reader. Now that we have a maximal inequality for the averages A(n), we can easily prove our theorem. Summation by parts shows that 1

pfn_

n

I

(n+ l) k=O

n

I

pkf = _l_ (n+ l)

k[pk_pk-11f

k=1

which has absolute value less then or equal to

So to prove our theorem we need only show that II g 1(f) \1 2 .S C II f 11 2 for

L2. Suppose that P is positive definite in the Hilbert space sense (Pf, f) ~ 0. By the spectral theorem, we can write P = fo 1 A.dE{A). f

E

Hence

.! kll(pk_pk-1)fll~ =I k

r

lo

k j\Ak-Ak-1)2(dE(A)f,f)

k

1 1

0

I k(Ak _ Ak- 1>2 ](dE(A.)f, o k

1

.S C

Ia (dE(A.)f,f) .S C\lfll~

.

Thus, the maximal theorem II supn 2:. 0 IP"f( ·) 111 2 .S C II f U2 is proved if we assume, in addition to the hypothesis, that P is positive-definite in the Hilbert space sense. If P satisfies the hypothesis of the theorem but is not positive-definite, we apply what we already know to the operator P 2 , which is positivedefinite, to obtain II supn 2:. 0 IP 2 nf( · >111 2 .S C II f 11 2 • But supn> 0 IPnf(x)l ~ supn> 0 \P 2 ne1£)(x)\ + supn> 0 IP2 n(pf)(x)\

-

-

-

86

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

CoROLLARY: limn~oo pnf exists almost everywhere, for every f t L 2, if P is positive-definite in the Hilbert space sense. Proof: By the maximal theorem just proved, we need only show that

li~ ~ oo pnf exists a.e. for f in a dense subset of L 2()Jl}. (The argument

proving a.e. convergence from a maximal theorem was already given twice at the end of the proof of the ITt I maximal theorem, and will not be repeated.) Consider f of the fonn f some t: > 0, g

t

==

(E(l)- E(1 -)}g + E( 1- t: )g for

L 2 (E ( ·) denotes the resolution of the identity for P ).

Since (E(l)-E(l-))g+ E(l-t:)g~g inL 2()R) as t: ... o+, thesetof such f's is dense in L 2 . Let f = (E(l)- E(l-))g +EO- t:)g = h 1 + h 2 . Pnh 1 = h 1 , so Pnh 1 converges almost everywhere as n ... + oo, On the other hand,

so

II Pnh 211

2 :S C (1 - t:)n II& 11 2 . Therefore

i n=O

~I pnh2<x>l2dx :s lJi

c2

~

(1- t:)2n I giJ

~

< +oo ,

n=O

so

Hence l:n":o lpfih 2(x)l 2

< +oo for almost ail x, so Pnh 2 ~ 0 almost

everywhere as n-+ +oo. Finally, then pnf ,... Pnh 1 + Pnh 2 converges almost everywhere as n ... +><> for all f in a dense subset of L 2(ffl).

QED.

§4. A

DIGRESSION: L 2 THEOREMS

87

The arguments used in the above theorem originate in the papers of Kolmogoroff-Selivestroff-Plessner in 1928, in which the authors prove the ·almost everywhere convergence of a Fourier series I: =-oo anein(J for which I;~-oo \ ~\

2

logIn\< +oo. Paley [29], Bochner (in his book Fourier

Analysis and Probability 1955) [32] and E. Stein [31] successively widened the scope of these ideas. The following two theorems are proved by essentially the same technique as Theorem 3. THEOREM 4: (L 2 ergodic theorem). Let U be a unitary operator on L 2(»l) such that f > 0 implies Uf ?. 0. Define

~+(f)

n

= -

1-

(n + 1)

_!

Ujf

j= 0

for f

f

L 2 . Then limn -ooo An+ (f) exists almost everywhere.

1 n · Sketch of Proof: Let An(£) = 2 n + 1 Ij = -n uJf. The operators An

are self-adjoint, and satisfy A(n)A(m)f

:5

2[A(2n) + A(2m)]f iff?. 0.

Proceeding as in the proof of Theorem 3, we obtain the maximal inequality l\supn2:_0A(n)f(·)H 2 :5 C\\f\1 2 . Butfor f ?_0, A;£ :5 2Anf, sothemaximal inequality is proved. Almost everywhere convergence follows easily. THEOREMS: (Martingale Convergence Theorem- L 2 variant). Let E 1 , E2 ,

...

be the orthogonal projection operators to sub spaces X1 f X2 f ...

of L 2(»l). If f > 0 implies all E. f > 0, then limn Enf exists almost J -ooo everywhere, for each f £ L 2(m). Sketch of Proof: Use the inequality Em En f .$ Emf+ En f (if f 2:. 0 ). This inequality is vlaid because Em En

= Emin (m,n).

This implies that

1\supnEn£112 .$ 2\lf\12· Some examples of applications of theorems 4 and 5 are in order. The typical operator U of Theorem 4 arises as follows: Let <)R, dx) be the

88

GENERAL SYMMETRIC DIFFUSION SEMI-GROUPS

interval [0, 1] with Lebesgue measure, and let cp: [0, 1]-+ [0, 1) be some "reasonable" function. Then we define Uf by setting Uf(x)

=(

~

) 1h

f(cp(x)) .

This example and its trivial generalizations include the classical situation in which U is induced by a measure-preseiVing transformation on

OR. dx). Theorem 5 is much more pertinent for our purposes. To see its implications, consider the case (»l, dx)

=

([0, 1], Lebesgue measure), and set

En equal to a "conditional expectation" operator:

For this example, Theorem 5 amounts in effect to the classical differentiation theorem f(x)

= lim .! y... o Y

f

Yf(x + t)dt o

As the next step in our development of Littlewood-Paley theory, we shall analyse sequences of operators I En I, possessing the same essential properties as the above example (which we shall call the dyadic inter-

val example). BIBLIOGRAPHICAL COMMENTS FOR CHAPTER lli

Sections 2 and 3. The Maximal theorem goes back to the author's paper [31). Theorem 1 was implicit in that approach but has not been published before. A later proof of the maximal theorem was given by Rota (30}. but it requires properties (III) and (IV) in addition to (I) and (II). The required convexity for analytic family of operators appears in [22].

Section 4. An outline of the proof of Theorem 3 appears in the paper [31). Theorems 4 and 5 are in the same spirit but were unpublished.

CHAPTER IV THE GENERAL UTTLEWOOD-PALEY THEORY

Section 1. Conditional Expectation Bind Martingales Let (M, dx) be a sigma-finite measure space, and let }R denote the family of measurable subsets of M. Suppose that j= f }R is some smaller si8711a-field, and assume (for technical reasons) that the "restricted" measure space {M, 1, dx) is also sigma-finite. We shall define an operator E which maps the measurable function f on {M, JR, dx) to its conditional ex-

pectatioo E (f I j=). To clarify the meaning of E, we first consider a simple example. Suppose that M is the disjoint union of a sequence S1 , S2 , . . . of measurable sets, where 0 < m (Sj) < +"" for each j . The Sj 's generate a sigma-field

j=

s;

m.

We sltt.ll call j= a special subfield of

JR.

Conditional expectations relative to special subfields of )R are easy to define. If (say)) f

t

L 1 (M, }R, dx) and j= is the special subfield of

:JR.

generated by sl, s2, ... , then we set Ej= f(x) = _l_ m(Sj)

f

f{y)dy

for

X f

sj ,

s. J

and we call Ej=(f) the conditional expectation of f relative to the field j=_ The operators En defined above to illustrate Theorem 5 are obviously conditional expectation operators. Note the fundamental property of E~ : Now suppose that j=

s; }H

Ej: (f) is j= -measurable.

is any (sigma-finite) subfield. If f is a local-

ly integrable function on {M, JR, dx) then there is an induced absolutely

89

90

THE GENERAL LITTLEWOOD-PALEY "ffiEORY

continuous measure 'A on ~ defined by ,\(A) = fA f(x)dx, for A f ~. (The point is that we are restricting our attention to A

€

~.) By the

Radon-Nikodym theorem, there is an essentially unique ~-measurable function g on M, such that ,\(A) ,. fA g(x)dx for A e ~. In other words, for every f on M, we can find a unique ~-measurable function g on M, such that fA f (x)dx

= fA

g (x)dx for every A f ~. g is

called the c;onditional expectation of f relative to ~, and is written g = E(f\ ~). We collect a few trivial properties of conditional expectations in the following list: (1)

E(f\ ~) ~ 0 if f ~ 0 •

(2)

If f is ~-measurable, then E (f I~)

= f.

In particular

E (1\ ~) = 1 . (3)

E(f+g\~) = E(f\~)+ E(g\~) and E(af\~) = aE(f\~).

(4)

E(E(f\~)\ ~) = E(f\ ~)

(5)

If f and g are L 2 functions on M, then

f.

E(f) gdx = ( fE (g)dx M

}M

(6)

II E(f\ ~)lip 5 II flip

(7)

If (say) g is bounded and ~-measurable, then E(gf\ ~) =

0 5

P

5 +oo)

gE(f\ ~). (8) If

§

~ ~ is another sigma-field, then E (E (f \ ~)I

§) = E (f \ §).

Proof: These properties follow immediately from the definition of conditional expectation, but we still agree to writing out a proof

~or

(5) and

(6). (5)

f !

f

E(f) · g dx =

M

and

f· E(g)dx = M

E (E(f) · g)dx =

M

f

f

E (f)· E(g)dx by (7),

M

f

E(f· E(g))dx = E(f)· E(g)dx by (7) M M

91

§1. CONDITIONAL EXPECTATION AND MARTINGALES

(6): Since E (f! j=) is j=-measurable, we have (with !_ + !_ p

f;(f! j=)l!p

= supl =

q

= 1)

fmEf(x)g(x)dx! g is measurable in j=, and hllqS 1}

sup I JMf(x)g(x)dx! g is measurable in j= and hllq S

~sup{ JMf(x)g(x)dx! g is measurable in

11 (by (7)).

ffl and !igi!q S

1} = ~~~~p.

QED.

For the next idea I shall require an increasing sequence j= 1 f j= 2 f ... of sigma-fields contained in If f is a reasonable function on M, let

m.

En(f)

= E(f! j= n>'

and define f*(x)

=

supn~ 1 !En(f)(x)!.

THEOREM 6 ("Martingale Maximal Theorem"):

al ~: l!f1! 1 , i.e., f-+f* hasweak-type(l,l)

(a)

mlx!f*(x) >

(b)

!If* lip ~ Ap !!flip

(1 ( p ~ +oo).

If M = [0, 1] and j= n is the special field generated by the sets

[ l , k·~/], 2n 2

k =·0,1, ... ,2n

then Theorem 6 asserts in effect the Hardy-Littlewood inequalities for the maximal function. Anon, we shall see that Theorem 6 (essentially) implies the maximal theorem for semigroups, in a rather general case, which we have proved with much machinery and effort. To prove Theorem 6 we need the following: LEMMA 1:

Let f 1 , f 2 , ... , fn be a finite sequence of functions on (M, dx),

where fj is measurable with respect to j=.i. Suppose that fj = E (fn I j=j). Then

f~ (x)

= supj ~n fj (x)

satisfies the inequalities

(a)'

mix! f~(x) >

al ~ : !Ifni! I ' with A independent of n.

(b)'

ll~llp ~ Ap!lfnl!p' with A independent of n.

Proof: We may as well suppose that fn > 0, so that all f. are positive. -

Let Sa= {xf M! fri(x) >

al

= lxf M! some fj(x) exceeds

J

al. Then

92

'lliE GENERAL LITTLEWOOD-PALEY THEORY

sa = u~= 1 s~>

where s~n

= IX

E

MI fj (x) > a, but fe (x) $ a for

e < jl,

and the sets s~D are pairwise disjoint. The crucial point is that S~j) is measurable with respect to the field

j= j . From this fact we deduce that

!

fn (x)dx

S

a

=

·

i Js<J>.

fn (x)dx

j= 1

In other words mIx

E

a

=

I f

J. = 1 s
MI f: (x) > a I < !. -a

E (fn 19)(x)dx

J;1xtM I fri(x)>a Ifn (x)dx , which

prove (a).' Assertion (b)' is a trivial consequence of (a)' and the following special case of the Marcinkiewicz interpolation theorem. LEMMA 2: Let T be a mapping from Lp(M, dx) to Lp(M, dx) for all p ( l .::; p .::; +oo). Suppose that T is a sub-linear, i.e.,

for all f 1 , f 2, and all x

E

M. Furthermore, suppose that T satisfies the

inequalities (1)

mlxtMIITf(x)l > al

(2)

IITflloo $ Allflloo •

< ~llfll 1 (i.e., T has weak-type

-

a

(1,1)).

Then for every f E Lp (l < p < +oo), Tf belongs to Lp, and the inequality IITfllp ~ ~llfllp

(3)

is valid, where Ap depends only on A and p. Proof: For convenience, suppose A = ¥., and denote the measure of a set E by IEl . Let F be any function in Lp {M, dx) ( l shows that yields

II F II &=

-

~

p

< + oo), A moment's thought

f 0oo aPd II x E MI IF (x)l > a 11.

Integration by parts

93

§1. CONDITIONAL EXPECTATION AND MARTINGALES

IIFIIC

= (p)f""

aP- 1 11xfMIIF(x)l > aljda.

0

In particular, picking an f ::: 0, we have (4)

IITfiiP

p

= Pf"" aP- 1 11mfMIITf(x)l

> allda.

0

To estimate jlx f M IITf(x)l > all, write f = ~ + ha where g (x) = a

I

f(x) if f(x)
if f(x) 2:: a

a

and h (x) = a

\ 0

if f(x)
f(x)-a if f(x)

za

Clearly ll~iloo ~a and llxfMI ha(x) >All= llxfMI f(x) > A+all. Now, we have

by the sublinearity ofT. The first term on the right-hand side of the inequality is zero, by virtue of ( 2). The second term is at most 1/a fM ha(x)dx (by (1)), which is equal to 1/a f 0"" II xI ha(x) > .\II d ,\ 1/a

J; llx I f (x) > .\II d ,\.

Thus

ilx£M11Tf(x)j >all S} J""llxlf(x) >AlidA a Applying this to equation (4), we obtain

! f 00

,. p

0

0

,\

aP- 2 da

llx

f

Ml f(x) >.\II d,\ (by Fubini 's theorem)

94

THE GENERAL LITTLEWOOD-PALEY THEORY

This completes the proof of (3).

Proof of Theorem 6: By Lemma 1,

II supj _sn f;( ·)lip

.$ A I flip

(1 < p .$ + oo), where A is independent of n. Since the sequence {supj _sn f;( ·)I increases to f* ( · ), Theorem 6, part (b), follows from the monotone convergence theorem. Similarly, we can easily deduce Theorem 6, part (a), from (a)' of Lemma 1.

QED.

REMARKS: 1. From Theorem 6 we can prove, by the usual arguments, that for f

f

Lp (1 <5 p

:S

+oo) {fjl converges almost everywhere. Details are

left to the reader. 2. Lemma 1 also implies a result, analogous to Theorem 6, on "reverse martingales": Suppose we are given an infinite descending sequence of sigma-fields, • ·· ~ j= n +1 ~ ~ n ~ ·· · ~ j= 2 ~ j= 1 . For f f L 1(M, ffl, dx), set En(f) = fn = E (f I j= n>· Then f* (x) supn Ifn(x)l satisfies inequali-

=

ties (a) and (b) of Theorem 6. The proof of this result is exactly the same as the proof of Theorem 6. Later, we shall see that the maximal theorem for martingales, Theorem 6, implies a general case of the maximal theorem for symmetric diffusion semigroups. Furthermore, it will be shown that the Littlewood-Paley inequalities for semigroups can be deduced in part from appropriate results on martingales.

Section 2. The inequalities for martingales. We turn to the problem of formulating and proving a martingale version of the Littlewood-Paley inequalities. As before, we are given an increasing family of sigma-fields j= 1 ~ j= 2 ~ • from which we form the operators En : f .... E (f I j= n>· For convenience, set

E0 •

o.

THEOREM 7: Suppose that a • (a 1 , a 2 , ... ) is any sequence of numbers such that la;l :S 1 for all j. Set

95

§2. THE INEQUALITIES FOR MARTINGALES

Ta
""' ~ ~ ak(~- Ek-l}f

for

f

f

L2 •

k•l

T 8 is well-defined for f (Ek- ~- 1 )£.

f

L 2 , by virtue of the orthogonality of the

Then

1.

1/Taf/lp $ Ap/lf/lp for f

2.

Ta has weak-type (1,1), i.e., m{xj Taf(x) > ,\j $ -~ llf/1 1 for f

E

L1

f

L 2 nLP

(1 < p <+oo).

n L2 .

If we define the "Littlewood-Paley" function G (f)(x)

3.

""'

(Ik .. l

IEkf(x)-

2~

~-lf(x)j ) ' then for f

f

=

L2 n Lp. (1

<

p

< +oo).

The constants A and Ap appearing in 1. and 2. do not depend on the particular sequence (a). REMARKS: Theorem 7 was first proved for the dyadic interval case,

by Paley, who formulated it as a theorem on Walsh-Paley series in his 1931 paper [17]. The imvolved proof he gave can be carried over to the general case. However, we shall present here the proof of Gundy [28], which is modern in viewpoint, and is based on a lemma analogous to the Calder6n -Zygmund decomposition. See also the earlier papers of Austin [23] and Burkholder [24]. Gundy's Lemma is the following: 0 belong to L 1 . Given ,\ > 0, we can write f '"' g + h + k, where

Let f

~

(a)

mlxj supniEn(g)(x)J

(b)

I\I: .. 1 \En(h)-

(c)

likll,., $ K,\ and llkl_l 1 $ K!lf/1 1 •

'>

Ol $

~

\\~11' and \\g\\ 1 $ Kjjf\\ 1 .

En_ 1 (h)jj 1 $ K\\f\\ 1 . In particular l\hil 1 $ K\\fl\ 1 .

96

THE GENERAL LITTLEWOOD-PALEY THEORY

From Gundy's lemma, we can easily prove parts 1. and 2. of Theorem 7. For if f

£

L1

n L 2 , and f

mlxiTi(x)

= g + h + k as in the lemma, then we have

>.\I~ mlxJTag(x) > ~ l+mlxiTah(x) >~I

+ m lxJ T ak(x) > ~I

= I. +II. +III.

Now I.$ mlxl supnlEn(g)(x) ~-01 ~ ~ JJ£11 1 • To estimate II, we note that ITa(h)(x)l ~ In J(En- En_ 1)h(x)J. So II.

~

""

m lxl!n J(En- En_ 1)f(x)l >

~I ~.!}II fJI 1

by (b) of Gundy's lemma. Finally, III. ~ ~ ll fJI 1 • For by (c) of the lemma, we have the inequality II k II

i~

K ,\II f 11 1 • Since T a is a bounded operator

on L 2 ,

Putting together our estimates for I., II., and III., we find that

which is exactly part 2 of Theorem 7. The operator Ta thus has weak-type (1, 1) and strong type (2, 2) (i.e., T a is a bounded operator on L 2 ). By the Marcinkiewicz interpolation theorem, T a is a bounded linear operator on Lp ( 1


~

2). (The basic idea

of the proof of the Marcinkiewicz interpolation theorem is already indicated

' in Lemma 2 in the proof of Theorem 6 above. A detailed discussion of the Marcinkiewicz interpolation theorem may be found in Zygmund's Trigonometric Series, Chapter XII, [20].) That T is bounded on Lp (2 ~ p

< +oo)

follows from the usual duality argument. Proof of Gundy's Lemma: We are given an L 1 function f (say, f 2 0)

and a ,\ > '0. Let fn

= En(f)

• E (f 11n>· The decomposition of f will be

carried out using the IfnI, and the notion of a stopping time, which we now define.

97

§2. THE INEQUALITIES FOR MARTINGALES

Suppose r(x) is a positive integer-valued function on the measure space (M,m, dx) such that lxl r(x) • nl is measurable not only with respect to

m.

s= n,

but with respect to

for each n > 1.

r(.) is then called a

stopping time.

If r(x) is a stopping time, then

f

(*)

f(x)dx

M

=I

fr(x)(x)dx

M

For the 16ft-hand side of(*) is just

=

i

j=l

f

f(x)dx •

lxjr(x)•jl

Equation(*) generalizes the identity

f

f(x)dx.

M

JM

fn(x)dx •

JM f(x)dx.

We can construct a new martingale from Ifni and r(x). Simply set fn' (x) .. f m1n . ( n,r( x ))(x). lfn' I is called the stopped martingale defined by lfn I and r. (For each x, If~ (x)l looks just like I fn(x)l until time r(x), when If~} "stops".) The proof that If~ I form a martingale, i.e., f~ .. En(f~ +1)

resembles the proof of(*), and is left to the reader.

As a slight extension of the definition of stopping time, we allow r(x) to take on the value +oo. Equation(*) still holds if we define f00 (x) - f(x), and If~ I is still a martingale. (In fact f~

= En(fr(x)(x)) .)

Recall that we are trying to prove Gundy's Lemma. The easiest of the three parts of the decomposition is g, which will be defined by g(x) .. f(x)- ft(x)(x) where t(x) is a particular stopping time. To define t, we let r(x)

= inflnl fn(x) > AI. lx\ r(x) = n}

Next write fn(x)

r(x) is a stopping time, since

= (xI f 1(x) •••• , fn_ 1(x) .$A but fn(x) >A}£ S: n.

= Ik= 1

cpk(x) where cpk = fk- fk_ 1 and f0

= 0;

98

THE GENERAL UTTLEWOOD-PALEY THEORY

and set En(x)

= ¢n(x)

Xfy\ r (y)-= nl (x) •

Obviously En ~ 0. (Think

about it for a moment.) Define a new stopping time s by s(x) = infln\ I~ .. 0 E(Ek+ 1 \ j=k)(x) >..\I. (The reader may check that s is, in fact, a stopping time.) Now set t(x) = min(r(x), s(x)). t is a stopping time, since it is the minimum of two stopping times. I claim that mlxlt(x) < +ool ~ ~ \lf\1 1 , where K isauniversalconstant. First of all, I xI r (x) ~ + oo I =- I xI supn fn(x) > AI, so that mIx I r(x) < +ool ~ ~

1\ fl\ 1

by the martingale maximal theorem. Similarly,

lx\I;,.. 0 E(Ek+l\j=k)(x) >..\I and

"' i !,

(fk+l(x)- fk(x))dx

k=O lxlr(x)=k+ll

<

i {

fk+l(x)dx

(since f

~

O)

k=O}Ixlr(x)=k+ll

1

fr(x)(x)dx lx\ r(x) <+ oo I

~

f

fr(x)(x)dx =

M

f

f(x)dx

M

(by(*)) Hence mix\ s(x) < +ool $ ~ 1\f\1 1 with K == l. Finally, m lxj t(x) < +ool ~ mix\ r(x) < +ool +mix\ s(x) < +ool

As advertised, define g(x) .. f(x)- ft(x)(x). Immediately, we see that lx\g(x) ~ Ol

s;

lxlt(x) ~ +ool so that mlx\g(x).;,. Ol ~ ~ 1\fl\ 1 • With

only a little more work we can prove part a of the splitting lemma. For,

99

§2. THE INEQUALITIES FOR MARGINGALES

En(g) = fn- fn, where l{nl is the stopped martingale defined by lfn I and the stopping time t(x). So lxl supnl En(g)(x)l ;I. Ol f lxl t(x).;. +ool, which shows that mIx I supn I En(g)(x) .;. Ol ~ ~

hll 1

Note that

::;

llfll 1 + llft(x)(x)ll 1 .$

II fll 1 •

Part cis proved.

211£11 1 •

Since we have to wind up with g + h + k = f, we are left with the job of constructing functions hand k such that h(x) + k(x) = ft(x)(x) and conditions b and c hold. Shall we have a go at it? Begin with the formula fn =

~~= 1 ¢;Xhlt(y)2:,il, which follows from

decipherment of notation. Now ly It (y) 2:. jl Since ¢; Xjyl r{y)= ;I

= ly I r(y) 2:. jl n ly I s (y) 2:. jl.

'"' Ej' we can write

n

in

=

!

(yj + Ej)Xhl s(y)2:,jl where Y; = ¢;Xhlr(y)> j I

j:d

Set hn(x) = ~;= 1 (E;-E(E;Ij=;_ 1 )) · Xhls(y)2:,il kn(x) ,.

=~~= 1

if!;

and

~;= 1(yj + E(E j I j= j- 1)) • Xl jl s(y)2:, jl . Obviously, hn + kn = fn,

so that in the distant future, when we leam that lhnl and lknl converge in the Lcnorm, the limits h and k will satisfy h (x) + k (x) = ft(x)(x), i.e., g+h+k=·f. I claim that lhn I and Ikn I_ are martingales, i.e., En(hn + 1 ) = hn and En(~+l)

= kn.

Since kn = fn- hn, we need only check the martingalen . = EnC.Pn+ 1 )+~j= 1 if!; (stnce

n+1 I But En(hn+l) = En(~j= dom of ! hn. 1 1/Jj) by inspection,

if!;

is j=j -measurable)

= EnC.Pn + 1)

+ hn. We are thus re-

duced to showing that EnC.Pn +1) = 0. By definition EnC.Pn+l) .. En((En+ 1 - En(En+l)) · Xjyl s(y)2:,n+ 1 J) "'(En(En+ 1 - En(En+ 1)))· Xhls(y)~n+ll (since lyls(y) ~ n+ ll = M- U~ 1 1yl s(y) = jl is measurable with respect to j= ) = (E (E )J= n n n+ 1

En((En+ 1)) · X!yl s(y) 2:. n+ 11 = 0. This proves that lhnl and lknl are martingales.

100

TilE GENERAL LITTLEWOOD-PALEY TIIEORY

Next, we shall verify that 00

~ llr/ljlll

(**)

.$ Kll£111 ·

j .. 1

This will show that {hnl converges in L 1 to a limit h. Enh '"' hn since lhnl is a martingale, so Enh- En_ 1h = r/ln, which implies that

I~= 1 I Ej h -- Ej-l h 11 1 S K II fll 1 . The construction of h and the proof of part b of the lemma are therefore consequences of (**). Written out in full, (**) says that

But since e:. > 0, the left-hand side of this inequality is dominated by J -

I fe:jX{yls(y)~ jldx+~ j

M

J

'" 2I

f

f

E(e:jiS:j-1). X{yls(y):;:: jldx

M

Elx>xt Yl s(y):;:: jjdx (by definition of

~onditional expectation)

M

j

00

~ (fj(x)- fj_ 1(x)) · X{ylr(y)=jl(x)dx (by definition of e:j) j= 1

S2

f

fr(x)(x)dx = 211£11 1 by(*).

M

Therefore (**) is verified with K '"' 2.

§2. THE INEQUALITIES FOR MARTINGALES

101

So far we haven't used any information on s(x) except that it is a

stopping time. We now finish off the proof of the lemma by exploiting the properties of s(x) to prove c. By the properties of g and h already demonstrated, k = f- g- h has

L 1 norm

II kll 1

K II fll 1 ; and we have the representation

~

valid pointwise almost everywhere. Part 3 of the lemma says that 1

k(x)l _$ KA

almost everywhere, so it is surely enough to prove that

""

III

(a)

Yj- X{yj s(y) 2: jl

j= 1

t

.$ KA

and

""

II!

{{3)

E (e) 1 j-t>Xty 1s(y) 2: jl II oo

j= 1

~

KA

(a) follows from the computation 00

00

min (r(x)- 1, s(x}}

=

·

~

J=

cp.{x) = f

J

1

min(r(x}-l,s(x)}

which has absolute value at most A , by definition of r(x). Similarly, ({3) follows from the computation 00

0

~ ~

E(ej11j_ 1 )(x) · X{yis(y)2: jl(x)

j= 1 s(x)

= ~ i= 1

s(x)- 1

E(ej11j_ 1 )(x) ~

~ E(e:e+ 1 11e)(x) ~A,

e .. o

(x)

102

TilE GENERAL LITTLEWOOD-PALEY TIIEORY

by definition of s(x). This completes the proof of Gundy's lemma. QED. Recall that we are trying to prove Theorem 7, and that we already established parts 1 and 2 of the theorem, by using the splitting lemma. To carry on, we need another of the basic tools of Fourier analysis-the family Irk} of Rademacher functions. If k is a non-negative integer, then rk is the function on [0,1] defined by

, =1

1 if j/2k _s t

rk(t)

< O+ 1)/2k, j even

, -1 if j/2k S t < (j+ 1)/2k, j odd.

The Irk} form an orthonormal system on [0,1]. which is, however, very far from being complete. Suppose that F(t) = I.

k

= 0

akrk(t), where I.k = 0 Iaki 2

< + oo

. Then

of course F < L 2 and IIFII 2 = (I.k=O jak1 2) 1h. But we can say much more. For any p (1 S p < +oo), F f Lp[0,1], and

Bpc~. hi')" $I FI p $Ape~. 1·.1

(*)

2 )"

where the constants Ap and Bp depend only on p. For a proof of (*), see Zygmund's Trigonometric Series, Vol. I. Chapter V, [20]. We can now prove part 3. of the theorem, by using part 1. and inequality (*) for the Rademacher functions. Part 1 says that

where T af For t

f

= I.k= 1

fM ITaf(x)jPdx S Ap\lfll/

ak(Ekf- Ek_ 1 f) and a = (ak) is any sequence of norm 1.

[0,1], let ak

= rk(t).

We obtain the inequality

with Ap independent of t. Integrating in t, and changing the order of integration, yields

I [! M

0

1

I

i

k=

(Ekf(x)- Ek_ 1f(x)) · rk(t)jpdtldx $ Ap!/fll: 1

J

103

§3. AN ADDITIONAL "MAX" INEQUALITY

By inequality (*),the expression in brackets is approximately

Ap • (

I

IEkf(x)- Ek_ 1f(x)l 2

k=l

)p/ =

Therefore, fMIG(f)(x)IPdx ~ ApllfiiC for all f

2

£

A;IG(O(x)IP .

Lp(M,dx). This completes

the proof of part 3 of Theorem 7. Part 4 comes from part 3 by the usual duality argument, based on the fact that G is an isometry on L 2 • Thus all parts of Theorem 7 are proved.

Q.E.D. Section 3. An additional "max" inequality. We have proved two big theorems on martingales-the "Paley inequality" and the maximal theorem. There remains one more result, and after we get it out of the way, we can come (finally!) to the link between semigroups and martingales, that will enable us to prove the general semigroup form of the Littlewood-Paley inequality.

1 1 £ 1 2 £ ·· · as before, let Ek denote the conditional expectation operator with respect to 1 k. Suppose that {fkl is any sequence of functions on (M, dx), where fk is not ass111ned to be 1 k-measur able; and let {nk I be any sequence of positive integers. Then THEOREM

8. Given

(l

< p < +oo)

where Ap depends only on p. Proof: The theorem has an easy proof. Let Lp(~q) denote the Banach space of all sequences of functions, {fkl, for which the norm

is finite. (If q • +"" we make the obvious modification

104

THE GENERAL LITTLEWOOD-PALEY THEORY

LP(eq) is really very much like LP. For example, the dual space of LP(eq) is Lp ,(fq,), under the pairing <(fk) ,(gk) > 1/p'+ 1/p

= 1/q' + 1/q = 1,

JM ~k fk(x) gk(x)dx,

=

provided that p

where

f. + oo, q f. + ""'·

We shall use the following generalization of the Riesz convexity theorem: Let T be a linear operator which maps sequences of functions to sequences of functions. Suppose that T is bounded as an operator from LPo(eq 0) to itself, and as an operator from LP 1 (eq 1 ) to itself. Then T is also bounded as an operator from LPt(eqt) to itself, where 1 ph

(1- t)

---

=

Po

+

t Pt

and

= (1-t) + -~

1 qt

qo

ql

(0 ~ t $ 1)

The proof of this theorem is very similar to that of the Riesz convexity theorem. A full proof is found in a paper of Benedeck- Panzone, The Spaces

J.

LP with Mixed Norm (Duke Math.

1961, p. 301- 324), [21]. See also

Calder 6n [39]. Now, consider the operator T, which sends the sequence lfkl of functions, to the sequence lEnkfkl.

f (~ Mk

lEn fk(x)IP) k

p/p dx =

T is a bounded operator on LP(ep), since

~k

f

M

IEn fk(x)IPdx k

~ ~f. lfk(x)jPdx kM

~ ( ~ lfk(x)!P )"'•dx .

•

On the other hand, T is a bounded operator on LP(e 00 ) if l < p $ + oo. This is because

f

M

lsupkEnkfk(x)IPdx $

f

lsupn,k Enfk(x)IPdx $

M

= AP

f

M

(supklfk(x)I)Pdx ,

§3. AN

(where

lOS

ADDI'nONAL "MAX" INEQUALITY

* denotes the maximal function, and

¢ (x) = sup Ifk(x)l ), by the

maximal theorem.

k

We can now apply the generalized Riesz convexity theorem to conclude that T is bounded on Lp~q) if I < p S q S +oo. In particular, if I< p $2, then T is bounded on Lp(f 2 ), which is precisely the statement of the theorem! The case 2 S p < + oo follows by an obvious duality argument involving Lp(fq) -spaces.

Q.E.D.

REMARKS. The result of Theorem 8 does not hold when either p = 1 or p • "", and in fact the true order of growth of the bound Ap is O(p~) or p-+ ""• and O((p-1)-lh), as p-+ 1. This indicates that the theorem cannot be entirely trivial. The fact AP $ Ap~

as p -+ "" follows by an examination of the bounds

arising from the interpolation argument. To show that in fact Ap > Apy,_ in general, let E 1 ,E2 , ... ,Ek ... arise from the "dyadic interval" expecta· tions and set fk(x) = I if rk-t < X $ rk, fk(x) - 0) otherwise. Then

(IIfk(x)i 2 )~

=1, while

COROLLARY: If {fkl is any sequence of functions on (M, dx), then (l


.$ 2).

Proof: Carry out the proof of theorems, using the operator T (lfkJ)(x) •

{En(k, x)fk(x)l, where n( ·, ·) is an arbitrary (measurable) positive-integervalued function of (n, k). The corollary now follows by the argument we used to carry out the interpolation in the proof of the semigroup maximal theorem. (Note that we cannot use the duality argument here, since lfkl-+ lfk'l is non-linear, and T is non-self-adjoint.)

Q.E.D.

Interesting question: Is this true for 2 < p < +""? All our martingale inequalities are also valid for reverse martingales.

106

THE GENERAL LITI'LEWOOD·PALEY THEORY

Section 4. The Link Between Martingales and Semigroups Since (our) martingales depend on a discrete parameter n, and semigroups depend on a continuous time parameter t, it becomes expedient for us to discretize our semigroups. So we shall study the powers of an operator Q on Lp(M, dx) which satisfies the axioms

I Qfll p ~ II f I P

(I)

(II)

(1 $ p $ + oo)

•

Q is self-adjoint on L 2

(III)

Qf ~ 0 if f ~ 0.

(IV)

Ql

= 1.

THEOREM 9 (Rota): There is a huge measure space (0,{3 ,P), a col-

lectioo of sig~-fields ... sigma- field

S 1'n+l S 1' n s; ... S 1' 1 s; 1' 0 •

1'0 , all contained in f3, with the following

and another

properties:

,...

(1) The measure spaces (M,

lJl, dx) and (0,1'0 , P)

natural mapping i: 0 .... M. with Lp(M,

are isomorphic under a

The induced isomorphism of Lp(O,S: 0 , P)

lJl, dx) will also be denoted by

i .

J 0 , P):

Then Q2 n(i (f)) = i.(EE ,... nf), where E and E n are the conditional expectation operators for 1'0 and 1' n (respectively).

(2) Let f

f

Lp({l

Thus, the operator Q is associated with a reverse martingale, IEnl. From this result and the martingale maximal theorem, we can easily deduce the maximal theorem for semigroups, which satisfy (1), (II), (III), and (IV). For, let {Ttl be a semigroup satisfying our axioms. Then the operator Q = T 1/ 2 k+ 1 satisfies axioms (I)- (IV) above. The martingale maximal theorem and (2) above that

II supn Q 2n (if )(. )~ p

= IIi -l(supn Q 2n(i f

))II p

= ij supn i En( f (. np

,...

~

II E(supn En f )II p $ II supn En f I p

$

Apll

f

lip ,

107

§4. TiiE LINK BETWEEN MARTINGALES AND SEMIGROUPS

so that llsupnQ 2 nf(·)llp ~ Apllfllp for ffLP(M,dx). In other words, llsupn Tn/2kf( ·)lip ~ Apllf lip, with Ap independent of k. Letting k-+ oo, we conclude from the monotone convergence theorem that 1\supt>OTtf(·)\\p ~ Ap\lfl\p (recall that Ttf(x) is a continuous function of t f (O,+oo)). This deduction was not necessary at this stage-after all, we already knew a proof of the semigroup maximal theorem (which didn't even require axioms (III) and (IV).) But we can already surmise the power of the martingale theorems when combined with Theorem 9. Proof of Theorem 9:

(0, {3, P) is actually the result of an old construc-

tion from the theory of Markov processes. Imagine a particle located somewhere inside M, (say, at p0 ) at time t

= 0.

At time t = 1 the particle jumps to some other point p 1 of M, ac-

cording to some fixed probability distribution for p 1 . Having reached p1 , the particle forgets that it was ever at p0 • So at time t "'2, the particle jumps to a point p 2

M, and the probability distribution of p 2 depends only on p1 , not on Po • The process continues-at time t = n + 1 the particle jumps from Pn to Pn + 1 , having completely forgotten where it was at times f

0, 1, •.. ,n-1. There is a natural probability space (U, f3, P) for this random process. Suffice it here to define

n.

A point

(U E

n

should describe the complete

history of the peripatetic particle. So it is reasonable to set cu equal to the infinite sequence (p 0 , p1 , p 2 , ••• ). Thus, 0 consists of all possible sequences of points of M, i.e.' n = M X M X M X .... Now let us return to the case of an operator Q satisfying (I)- (IV) above, and try to use the above probabilistic ideas to construct an (0, {3, P). First of all, we agreed that

n = MX M X MX

• .. •

For the Borel field

the sigma- field generated by all sets of the form (*)

f3

we take

108

'niE GENERAL LITTLEWOOD-PALEY 'niEORY

where the Ai are measurable subsets of M. Note that the sets of the form (*)(so-called cylinder sets) already form a Boolean algebra. To start we aregoingtodefine the measure P. Let

s ..

A1

X

A2

X ••• X

AN

X

M

Start with the function XAN on M; then form

X

MX MX

Qx"N;

•••

then multiply by

XA

, to obtain XA • Q(xA ); apply Q again, to obtain N-1 N N -1 Q(XA • Q(xA )); multiply this by XA to obtain N-1 • N-2 XA • Q(xA • Q(xA )); apply Q again. Continuing this process, we N-2 N-1 N finally come to the function XA • Q(xA • Q( ••• (XA • Q(XA )) ••• ). 0 1 N-1 N Set

The reader may check that P is well-defined, non-negative, and finitely additive on the cylinder sets ((IV) is required to show that P is welldefined, since (A 1 x A2 x · •· x AN) x M x M x M x · · · and (A 1 x A2 x · · · x AN x M) x M x M x M x M x ·· · are the same cylinder set). It can be shown that P extends to a countably additive measure on fJ.

The proof is rather technical, so we omit it. The demanding reader may look in the paper of Doob, A Ratio Operator Limit Theorem [27]. Probabilistically, this corresponds to the situation explained at the beginning of the proof, where p0 is distributed according to the probability law Pr(p0

E

A) = fA dx,

and where a particle at position Pn "' x jumps to a position Pn +l cording to the pr~ability law Pr (pn+ 1 l A) Now define

Ml, and set

1n

10

"' (A 0

x M x M x Mx

---------

l

M ac-

= Q(XA)(x).

···I A 0

is a measurable subset of

n+l

= (M x M x M x ··· x M x S I S

Obviously · ·· .S:

E

,8, so that S .S: M x M x M x •··1.

1 n +1 .S: 1 n .S: ••• .S: 1 1 .S: 1 o = f3 -

§4. THE LINK BETWEEN MARTINGALES AND SEMIGROUPS

The mapping i:

n .. M defined

109

by i(xo, XJ' x2, ... ) "' Xo sets up an

isomorphism of measure spaces between (0, j'0 , P) and (M, JR,dx). Thus, part (1) of Theorem 9 is verified. In order to prove part (2), we make two claims: (a). If g:

n .. R

is such that g(lxo, XI' x2, .•• !), depends only on xn then

E(gXIx 0 , x1 , ••• })

=

Qng (x 0 ).

(b). If g: 0 .. R is such that g({x 0 , xl' x2 , ... !) depends only on x 0 (i.e., g is ~0 -measurable), then En(g)(lx 0 , xl' .•. }) = Qng (xn) . From these two claims, it is obvious that E En(i

-to = C 1(Q 2no.

which proves (2) of Theorem 9. Therefore, the proof of Theorem 9 is reduced to the task of checking (a) and (b). Proof of (a): We are given a c~ndidate, Qng(x 0 ). for the conditi~nal

expectation of g with respect to j'0 • Since Qng (x 0 ) is obviously j'0 measurable, it is enough to check that (*)

for S E j'0 , i.e., S = A 0 x M x M x M x ···. Both sides of(*) are equal to JA 0Qng(x)dx if g is the characteristic function of a subset of M, as follows from the definition of P. On the other hand, both sides of (*) are linear in g, and well-behaved under limit processes. So (*) is valid for all g. Note that so far we have not used the self-adjointness of Q. Proof of (b): As in (a), the problem reduces to showing that

(**) where S=MxMx ... xMxA xA 1 x .. ·xA xMxMxMx· .. n n+ n '

'--<-~ n

110

THE GENERAL LITTLEWOOD-PALEY THEORY

and g is the characteristic function of a subset of M. The left-hand-side of (**) is equal to

(*"'*)

fn

g (xo) XA · n

XA

n+l

· XA

n+2

· XA

N

= lg(x)Qn(XA · Q(xA •

M

n

dP(x)

· Q( ···) ... )dx n+ 1

whereas the right-hand-side of ("'*) equals

=

1

[Qng(x) . XA . Q(xA . Q( ... ) ... )]dx , n+l M n

since [ Qh(x)dx M

=

J

h(x)dx M

= XA

• Q(XA · Q( ···) ··· ), we obtain from (***) n n+l and (****) that the left-hand side of (**) is JM g (x) Qn¢ (x) dx, while the for any h. Setting ¢

right-hand side of (**) is fM Qng (x) ·

4> (x) dx

• These two quantities are

equal, by the self-adjointness of Q. The proof of (b) is complete. Q.E.D.

§S. THE LITTLEWOOD-PALEY INEQUALITIES

IN

GENERAL

111

Section 5. The Littlewood-Paley Inequalities in General

f

f

Recall that if {Ttl is a semigroup satisfying axioms (1)- (IV) and 72 Lp(M, dx), then g 1 (f)(x) f 000 t I(CI/ dt:) Ttf (x)J 2dt) . We are at last

=(

ready to prove our main result. THEOREM 10: If f

f

Lp(M, dx), then g 1(f)

II gl (f) I p

f

Lp(M, dx), and

.s Apll f II p

(l

< p < +oo).

Proof: Since we want to apply martingale theory, we shall have to deduce the desired inequality from a discrete analogue. The most natural candidate for a discrete analogue is the inequality (*)

(where the operator Q satisfied (I) - (IV)) and, in fact, it is easy to prove the theorem, given (*). Associated to {Q 2nl is a martingale {Enl on a huge measure space

(O,fJ, P); in accordance with Theorem 9. To facilitate passing from

{Q 2nl

to IEnl and back, let us adopt the notation that sn stands either for the operator Q2n, or for the operator En' depending on context. Write n

sn ,

! k=O

ak

where ak

=sk- sk-l •

and set

the Cesaro mean. We shall see shortly, that the most we can hope to deduce from martingale theory is (**)

niE GENERAL UTTLEWOOD-PALEY THEORY

112

In order to prove (*) from Theorem 9, we could try to show that

where f is defined on 0. Unfortunately, this inequality is false. Look at the simplest possible case, p = 2. Our inequality reduces to 2

00

I

S A2llflli ·

nii<En-En-1)£112

n=1 But this is preposterous! For the {(En-En_ 1)£1 are orthogonal, so that the most we could ever hope for is I;: 1 II (En- En_ 1) f IIi < + oo. But I;= 1 n II (En- En_ 1)£IIi could easily diverge. On the other hand, (*) holds for p = 2, as follows by the spectral theorem. In fact, if we write Q2 == fo 1 AdE (A), then Q2n = fo 1 ..\ndE (A), and (*) becomes

I

nii(Q2n_Q2n-2)£1122

n= 1

=

I

n i\-An-An-1)2(dE(A)f,f)

o

n= 1

which holds because

f i 1[

o

n(An_n-l)~(dE(A)f,f)

J

n= 1

:S A llflli since lin: 1n(An-An- 1)21 S A for A € [0, 1). Thus, we see that the analogy between semigroups and martingales is by no means perfect. Let us proceed with the proof of (**). We have the inequality II
;

§S. THE LITTLEWOOD·PALEYINEQUALITIES IN GENERAL

113

for this follows from Theorem 7 with {En! replaced by {E 2 nl. (Note that it is not true that Cik I ~kf 1 2 )~ K (In I Enf- En_ 1f 1 2 )~ ). Therefore,

2

to prove (**), we need only show that

un

=

-

n

s 0 + s 1 + ... + sn

=

~ (l - _k_ ) ak k= 0

n+l

n+l

To prove(***) we study un-un_ 1 • Write

+ where



denotes the largest power of 2 which is smaller than n.

Consider the inner sum,

I.~~+:j + 1

kak. If the "k" in the product k ak

were constant for each j, we could pull it out of the summation sign, and conclude that on - on_ 1 is essentially a sum of multiples of terms 2J+l I. k= 2 j+ 1 ak = ~j+ 1 (f). This would reduce(***) to a very easy computation. Since k is not constant, we resort to summation by parts:

(B)

THE GENERAL UTTLEWOOD-PALEY THEORY

114

by definition of Ee, Aj + 1 (f), and ak. This last equation is very encouraging-the first term on the right-hand side is precisely what I 2 k k < 2 j+1 k ak would be if k were constant (k = 2j+ 1); and since there are about 2j + 1 terms in the brackets, the sum in brackets should not be much larger than 2j+ 1 Aj+ 1 (£). We shall see in a moment that all these informal ideas can be made precise. But first.we need an analogue of equation (B), to handle the final sum in equation (A). The reader may easily check that n

I,

(C)

kak = n•En(A 1 +[log n](f))- [En_ 1(A 1 +[1og n](f)

k=+ 1

2

2

where j (k) is an appropriate integer-valued function of k. It is then a straightforward matter to use (D) together with Theorem 8

of Section 3 to obtain the inequality (***). The easy details are left to the reader.

§S.THE LITTLEWOOD·PALEY'INEQUALITIES IN GENERAL

115

The upshot of the last few paragraphs of estimates, is that (***) is actually an elementary consequence of some juggling with summation signs, and involves no deep ideas, except possibly Theorem 8. Where do we stand? We have proved that (1

(**)

< p < +oo)

where

and the {Ekl form a martingale or reverse martingale. Before we proceed to the proof of the Littlewood-Paley inequalities, let us prove the semigroup version of(**), which is: (1)

where {Ttl is our semigroup, and at(f)

=1/t J0 tTsfds.

It is enough to prove (1) with J0 "" replaced by JfM (0 < f < M < +oo)

and Ap independent of

f

and M. For obviously we could then let f-+ 0+,

M-+ +oo. In the interval [£, M], everything is real-analytic, so that we can

replace all integrals in the right-hand side of (1) by Riemann sums, and all derivatives by finite difference quotients. These changes produce errors which tend to zero as the Riemann sums approach integrals, etc. So when all is said and done, (1) reduces the discrete analogue (l
(1 ')

where

o 0

denotes (T 0 + TE + T 2£ + ... + Tnf)/n + 1.

But (1 ') is essentially trivial from (**) and Theorem 9. Let's plod through the proof. Theorem 9 immediately reduces (1 ') to the inequality

116

'niE GENERAL LITTLEWOOD-PALEY THEORY

n( i

(1 ")

\'h lp .$ Apllf/!p'

niE(un -un-1)1 2

n= 1

}

where the CTn are as in (**),and E is some other conditional expectation operator. On the other hand,

~t~. _.,£<•. _••_.),,) ~~p s ~c~. "'"·-··-·'')"lip . ~

simply because E has norm 1 on Lp(f-2). (The proof that LP<.£ 2 ) is the same as the proof that II Ell exercise for the reader.)

=1

II Ell

= 1 on

on Lp, and is left as an

Thus (1 ") follows from (**).

(1 ')is proved.

This completes the proof of inequality (1). As we have seen, (1) is the strongest relevant conclusion we can possibly hope to squeeze out of martingale theory. At the same time, it is not strong enough to prove the Littlewood-Paley inequalities. So we have to resort to our familiar method of proving a stronger theorem for L 2 , and then interpolating with (1). To rephrase (1) in a form amenable to interpolation, we make use of the fractional integrals and fractional averages which served us so well in the first proof of the semigroup maximal theorem. In that proof, as the reader no doubt recalls, we set

~

M (f}(t) =

['(a)

a

for a

f

C, Rea

> 0;

t

{ (t-s)a-l Tsfds 0

Jll

and we saw that Ma(f) could be continued analytically

into the entire complex line a

f

C.

We can rewrite (1) in the form

By an elementary argument we shall deduce from this its analogue where M1 is replaced by Ma, with Rea

> 1.

In fact since Iaf = Ia-l · I 1f, then

117

§s. THE LITTLEWOOD-PALEY INEQUALITIES IN GENERAL

Therefore

~-Ma

at

= -at-a-lla(t 2__M 1)+t-al

at

a-

+ Cala(

Write now ) Then

(!, ~l
1(tl.- M 1)-at-ala+l(.E-M 1) at

at

~ Ml).

at

= 1/['({3) fo 1 (1- s)f3- 1¢(ts)ds,

t

S

Re{3 > 0.

!, ~11>1 2 d: ,

Ap

with Af3 = 1/jr({3)J fJ{l-s)f3-tJds, which is at most O(e"IIm(f3)1), as 'Im ({3)J

-+ ""

in any fixed strip of finite width lying strictly in the right-

half plane of complex {3. Similarly

Combining this with the above and (1) gives,

Inequality (2) would be the Littlewood-Paley inequality if we could set a

=

0, for in that case Ma(f) = Ttf. Next, we claim that if p = 2, then inequality (2) is valid for a = -1, -2,

-3, ... ,-k, .... We shall give the proof for a= -1; the general case is no harder. M_ 1

(0 = t (a/at)Ttf, so that 2_ M

at - 1

(f)

=

l__(t l_Ttf)

at at

=

~ Ttf

at

+ t

£

at2

Tt

118

TilE GENERAL LITTLEWOOD-PALEY THEORY

Hence,

where

This shows that (2) holds for p = 2, a "" -1, since we already know from the proof of the semigroup maximal theorem that f ... gk(O is a bounded operator on L 2 • Having proved an inequality for a

= -1,- 2, ... , -

k, ... we can use the

same argument as before, to deduce

where Aa does not grow faster than O(e77 1Im(a)l), whenever a is restricted to a fixed strip of finite width, and Ilm (a)i ...

oo •

Formally, it follows from (2) and (3) that

for all a

f

C. In particular, as we have noted, the case a= 0 is precisely

the desired Littlewood-Paley inequality. The trouble is of course, that the operators

are non-linear, which means that we cannot apply the convexity theorem of Section 2, Chapter III in a totally simpleminded way. As in Chapter III, Sec-

§s. THE LITTLEWOOD-PALEY INEQUALITIES IN GENERAL

119

tion 3, the non-linearity is easy to overcome. Let ll>(t, x) be any measurable function on (O,oo)x(M,Ilt,dx) suchthat

(J0

00

tlll>(t,x)l 2 dt)Y.! $1 for

every point x. The operators T~: Lp ... Lp defined by T!(f)(x) = ..,

depend analytically on a

f

0

t

00

t[ll>(t,x).l._Ma(f)(x,t)]dt

at

C. Inequality (2) implies that II T;(OIIp $ Apallfllp

(1

< P < +oo)

for Rea> 1 and inequality (1) implies that 1IT<E
< p < +oo) for all a E C.

Taking the sup over all II> satisfying the defining condition, we obtain at last

Q.E.D.

120

THE GENERAL LITTLEWOOD-PALEY THEORY

Section 6. Dlmouement The following corollaries elaborate Theorem 10 and give an inkling of the applications of the Littlewood-Paley inequality. COROLLARY 1. For each k ~ l, llgk(flllp .$ Ap I flip (l

< p < +oo),

Proof: Proceed by induction on k. The case k .. l is what we have

just proved: We shall illustrate the induction step by considering k = 2; the general case is left to the reader. As we saw above,

a
--M

..

a at

2 t~ - T tf+ t a - T-x, at2

. a2 T-xt~ • 1.e., t 2 ~t

u

a


~t M_ 1 u

u

Therefore g2(0-

(~oot 3 1 ;t~ Tlfl 2dt)~

.$

(~ootift

+(

M_l
~ oot I ft T~ 12dt) Y.a

,

so that

by inductive hypothesis and the last line of the proof of Theorem 10. Q.E.D. CoROLLARY 2 (Converse inequality). llfllp .$ Ap11~ 1 (011p +

IIE0 (f)llp

< p < +oo), where E 0 is delined on L2 as the orthogonal projeclion onto the space of all funclions h such that Tth = h for t > 0. (1

Proof: Here, E 0 plays the role of the constant term of a Fourier series.

Note firstly that (*)

§6. DENOUEMENT

121

In fact, Ttf ... E 0(f) in L 2 as t ... +oo, for each f

f

L 2 • By our almost every-

where convergence theorems, E 0 (f) == limt ... 00Ttf ,::s supt > 0 ITtf I pointwise. (*) now follows from the maximal theorem.

The corollary is now a consequence of Theorem 10 combined with the standard "isometry" arguments, based on the familiar identity

(see Section 3 in Chapter II).

Q.E.D.

We can now formulate and prove a "Marcinkiewicz multiplier theorem" generalizing the Lie group version which appeared at the end of Chapter II. First of all, write Tt = fo 00 e-At dE(..\), as is possible by the spectral theorem. For each bounded function m(A) on (O,+oo), set Tmf = f0~m(..\)dE(..\)f. Thus T m is a bounded operator on L 2 • CoROLLARY 3. Suppose that m is of Laplace transform type, i.e.,

m(..\)

=,\ fo oo e-AtM(t)dt

(,\ > 0), where M(t) is a bounded frmction on

(O,+oo). Then T m is a bounded operator on all the spaces Lp (l

< p < +oo).

Proof: The arguments we used to prove the Lie group version of this

theorem in Chapter II apply to the present case, to prove that for f 1 1(T mf)

s Kg2(f)

E

Lp,

with K independent of f. Corollaries 1 and 2 now show

that IITmfllp S Apllfllp·

Q.E.D.

COROLLARY 4. Let A be the infinitessimal Aenerator of ITt!. (Thus

Tt "" etA.) Then (-A)it is a bounded operator on Lp (l < p < +oo) for each real t. Proof: ,\it is of ~aplace-transform type.

Q.E.D.

122

THE GENERAL LITTLEWOOD-PALEY THEORY

BIBLIOGRAPHICAL COMMENTS FOR CHAPTER IV

Section 1. For the theory of martingales see Doob [26], Chapter 7. The Marcinkiewicz interpolation can be found in Zygmund [20], Chapter XII. Section 2. See the remarks after Theorem 7. Section 3. Theorem 8 is new. Section 4. For Theorem 9 see Rota [30] and Doob [27].

CHAPTER V

FURTHER ILLUSTRATIONS In this final chapter we indicate some further illustrations of the theory, but our presentation is more in the spirit of Chapter II than Chapter III and Chapter IV.

Section 1. Lie groups We assume that G is a non-compact, connected, Lie group. We let Xl'X 2 , ... ,Xn be a basis for the (left-invariant) Lie algebra, considered as first-order differential operators on G. We set ~+ =

I. a lj.. x.x. 1 J

where laijl is any real symmetric positive definite matrix. (More specific choices of the {a. 1.1 will be made later.) Our first object is to consider the 1

heat-diffusion semigroup

T1 = et~

+

•

THEOREM. There exists a semigroup

I T~l,

which satisfies properties

(1), (II), (III) and (IV) of Chapter[[[, and such that

T~ = e~+ in the follow-

ing two related senses: (a)

(T! -l)f

as

for all sufficiently smooth f (b)

If f f Lp(G), 1 ~ p ~

oo,

then u(x, t)

dU (X, t) dt

= (T~ f)(x)

= A+ u (x, t ) Ll

f

C00 (G x R+), and

.

It should be noted that the operators T~ are left-invariant, that is

123

124

FURTHER ILLUSTRATIONS

La T~

= T~La,

where (LaO(x)

= f(a- 1x),

at G.

According to Hunt's paper (see [12, p. 279]), we can construct a probability semigroup T~ which satisfies (a) and (Ill) (T~ f ~ 0, if f ~ 0) and (IV) (T~ (1) = 1). By that same construction, symmetry (our property (II)) is then also guaranteed by the symmetry of ll+ (see [12, Section 7.3]). (I) is a consequence of (II), (III), and (IV), and the Riesz convexity theo-

rem. The fact that T + also satisfies conclusion (b) follows from the fact that

(T~f)(x) =

1

kt(y)f(xy)dy

G

where k!

t

kt

is a fundamental solution, and

L 1 (G)

Jet

l

C00 (G

X

R+), also

n L (G). (For these facts see Nelson [14].) 00

As a simple corollary of the above we also obtain the existence of the "Poisson semigroup" corresponding to the equation

a2u (x, t) at 2

+ fl+u(x, t)

=0

•

In fact, define P~ by pt 1 + = V"

~~ 0

e-u Tt 2 /4udu VTT +

We then claim that IP~l is a semigroup which satisfies (1), (II), (Ill), (IV), and instead of (b) of the above theorem, we have

a2 u at 2

+ ll +u

= 0,

where u(x, t)

= (P~ f)(x)

•

The details of this passage from T~ to P~ can be carried out as in the analogous case of compact groups treated in Section 2 of Chapter We now come to the g-function. For f ing two expressions (1)

t

n.

Lp (G) we consider the follow-

125

§1. LIE GROUPS

(2)

THEOREM 11. Let f

l

Lp{G),

1 < p < oo, and let g 1{f)(x) denote

either of the two expressions above. Then

This theorem is a direct consequence of Theorem 10 and its second corollary (Section 6 of the previous chapter), as soon as we verify that the projection E 0 is zero in this case. But suppose f l E 0(L 2(G)). Then T~ f = f for all t ~ 0, and so f E C 00 (G), and A+f = 0. Moreover since

kt

l

L 2 (G), for t

> 0 and f l L 2 (G),

it follows that f (x) = f0 ktCy) f (xy) dy,

vanishes at infinity. It may be assumed that f is real-valued. The above then shows that f attains its maximum and minimum values and thus must be constant and therefore zero in view of the local maximum principle of Hopf for the operator A+. Notice also-that T~f = f for all t > 0, if and only if P~ f

=f

all t, and therefore the result applies also the the semi·

t

group P+. The result for (2) can be extended when 1 < p

~

2 by taking into ac·

count the Xj derivatives. To do this define (.:\u)(x,t)

=

G at

+ A+u, and 1Vul 2

=

(*-)

2 +I aij(Xiu)(Xju) .

Let

Observe that g 1(f)(x)

~

g(f)(x).

THEOREM 12.

In proving this we follow closely the argument of Section 2 and 3 of

126

FURTHER ILLUSTRATIONS

Chapter 11. First, it suffices to consider the case where f ? 0, and f is

> 0,

C 00 and has compact support. Note that kt(x)

(see Nelson [14]). and

thus CT! f)(x) > 0, all (x, t) ( G x (0, oo) and hence (Pl f)(x) all (x, t)

f

= u (x, t) >

0,

G x (0, oo). Now it is immediate that

(A)

(See Lemma ~· Section 2 of Chapter II.) Also

by the general maximal theorem of Chapter III, or by the argument for the proof of Lemma 1 in Section 2, Chapter II. Finally,

f1

(C)

0

t(LiF)(x, t)sxdt=

G

1

F(x, O)dx

G

for appropriate F defined in G x [O,oo), and this class of F includes F(x, t)

=

(u (x, t))P. The proof of (C) requires a little bit of care.

We have to observe first that if f ( Lp(G), then [

ju(x,t)jPdx .... 0, as t

-+oo,

when l < p < "?,

G

This assertion (not valid when p

= l)

will be an immediate consequence

of the Lebesgue dominated convergence theorem, the maximal theorem, and the fact that lim u (x, t) = 0, for almost every x ,

t .... OC)

In proving the second assertion when p = 2, it suffices to consider a dense subset of f in L 2 (G). Now if P~ == f 0 00 e-AtdE(A), then we can write f = If- E(f) £1 + E(t )f and E (f )f -+ 0, as f -+ 0, since we already saw that E 0 (f) = 0. But when f 1 is of the form f- E(f) f, then II f 1 11 ~ e -it. Moreover

P!

§1. LIE GROUPS

127

Thus in view of the maximal theorem

This shows that lim u (x, t) = 0 almost everywhere, for f in a dense subt ->oo

set of L 2 (G), and therefore all f in L 2 (G). Finally, L 2 (G) n LP(G) is a dense subset of Lp(G) and therefore lim u(x, t) = 0, almost everywhere t-+oo for all f l LP(G), and finally J0 \u(x,y)\Pdx .... 0. Let us return to the proof of (C) for all F of the form (u (x, t))P. To establish this it will suffice, in view of what has just been done, to prove

1f

(C ')

N

0

t(Ll\F)(x, t)dt

=

G

i

G

F(x,O)dx-

f

F(x,N)dx .

G

Now if F, in addition to the smoothness it already has, also had as a function of x support in a fixed compact set of G, there would be no difficulty in verifying (C ') by the argument of integration by parts given in the proof of Lemma 3 (Section 2, Chapter II). To bring about this situation we construct a sequence { \P(X)¢k(x)\ < k X l G differential operators;

(i) sup

(ii) cpk(x) = 1, for x that Uk

~

l

for any polynomial P(X) of left-invariant

oo,

Uk, where Uk are open sets with the property

G;

(iii) ¢k+ 1 (x) ~ ¢k(x) ~ 0. An example of such a sequence can be obtained as follows. Let 17 ( t), 0

~

t < oo be a monotone Coo function in ( 0, oo ), such that 17 (t) = 1 for t

near zero and 17 vanishes outside a compact subset of t. Let d (x) denote the square of the distance from x

l

G to the group identity, measured by

any fixed smooth left-invariant Riemannian metric. Set cpk(x) = 77((d 2 (x))/k).

128

FURTHER ILLUSTRATIONS

Now with F(x, t)

= ¢k(x)(u(x, t))P

just indicated. We let k -+

co,

the identity (C ') holds, as we have

then the right side of (C ') clearly converges

to fa(u(x,O))Pdx- fa(u(x,N))Pdx. The left-hand side of (C') can be written as the sum of two integrals, whose integrands are respectively -t(&¢k)(u(x,t))P, and t¢k&(u)P. The first integral converges to zero since the 11 ¢k are zero inside Uk, Uk

-+

G; and 111 ¢kl

< A,

everywhere;

also uP(x, t) ·is integrable on G x [0, N]. The second integral converges monotonically to f0 N fa t ,& uPdx dt, since .\uP 2 0, and the ¢k converges monotonically to 1. This proves (C ') and therefore (C). Now that (A), (B), and (C) are established the rest of the proof of Theorem 12 is then the same as the corresponding argument given in the compact case (for 1

< p S 2)

in Chapter II, (see Section 3 of that chapter).

It is important to point out that the argument for p 2: 2 given in the

compact case cannot be extended in the present situation. This is because at that stage we would need to use the assertion that the Xj commute with

P~, which is the same as requiring that the Xj commute with 11+ =

I aij Xi Xj. For similar reaons some of the further applications given in Chapter II for compact groups do not have evident analogies in the case of general non-compact G, but there seem to be interesting possibilities if we make the assumption that G is semi-simple as we shall now see.

Section 2. Semi-simple

case

We now assume that G is a unimodular Lie group, K is a compact subgroup, and we consider the homogeneous space S = G/K. As usual LP(S, ds), (where ds is G invariant measure on G/K) is identifiable with the class of functions If Iff Lp(G), and f (gk) = f (g), k ( KJ. We also make a more specific choice of the left-invariant Laplacian

11+ "" I aij xi xj I by requiring that 11+ is also right-invariant under the action of K. More particularly if we write pk(f)(x) that 11+pkf

= p~+f,

= f (xk),

then we require

k f K, for all sufficiently smooth functions f on G.

We can obtain such a positive definite symmetric matrix laijl, by starting with any positive definite symmetric matrix la~j) I and performing the

129

§2. SEMI-8IMPLE CASE

appropriate integration with respect to the compact group K; (see the argument in Section 7 of Chapter I). When we have such a ~+ which is rightinvariant under K, then we denote by

~

its induced action on functions on

s. Let us denote by 0 the origin in S, that is, the point corresponding to the coset K. Then our non-unique choice of laijl corresponds to a choice of a positive definite quadratic form in the tangent space at 0, invariant under the action of K. For every such quadratic form we get a Riemannian metric on S, invariant under the action of G, and

~

then is the Laplace-

Beltrami operator for this metric on S (see the related problem at the end of Section 7, Chapter I). By the construction given in Section 1 above, the operator

~

leads to

semi groups which we now write as Tt and pt (instead of T~ and P~ ); since the latter semigroups are right-invariant under K, the former semigroups act on Lp(S, ds). It also follows that pt and Tt satisfy properties (1), (II), (III), and (IV), our fundamental properties for symmetric diffusion

semi-groups. We can write symbolically Tt = et~. and pt = e-t(-6.)¥.1 . In addition if U (x, t) ... (Ttf)(x), and u (x, t) = pt(f )(x), then au/at

~ u (x, t), and (a 2 u/at 2 ) + ~ u (x, t)

=

= 0.

There are now certain theorems for Lp(S, ds) which ate immediate consequences of the corresponding results for G (Theorems 11 and 12) in the previous section. We need not reformulate these theorems separately. We now make the assumption that S .. G/K is a symmetric space. If g is the Lie algebra, of G, and have that

!

= ~ + £. where

~

f. is the

respect to the Killing form. Let i~varia~t

Lie alg~bra

!

the sub-algebra corresponding to K, we

n~w

~rtho:onal c~mplement of ~ in

!

with

X1 , _x 2 , ••• , Xn be a basis of the right-

so that xl, ... , xr is a basis for ~ and

Xr+t'Xr+'2''"''Xr is a basis for£.. Then there exists two positive definite symmetric matrices {cijl. l

s i,

j ~ r, and lbkfl· r+ 1:$ k,

e~ n, so that

130

FURTHER ILLUSTRATIONS

c .. x.x. 1J

1 J

l~i, j~r

is not only right-invariant, but also left-invariant.

(~-

in effect is the

Casimir operatot: see Helgason [2, p. 451]). Let us say that a function f is zonal if f (k 1 xk 2 ) = f (x}, where k 1 , k2

f

K. The~e are exactly the functions on S

= G/K, which

are also invari-

ant under the a"ction of K on S. It is to be noted that if f is any smooth zonal function, then

_!

c .. x.x.

l~i, j~r

and therefore

~-f

1J

1 J

f:O,

is a Laplace-Beltrami operator of f, which is identical

with M for appropriate aij. We fix this choice of aij in what follows. Notice also that if

!

aij (Xif)(Xjf),

1_$i, j_$n

then if f is zonal we have .. I v £1 2 =~b . . 1J

o{.ocx.o 1 J

We can now state the version of Theorem 2 in Section 2, Chapter II, valid for all p, l

<

p

< oo,

THEOREM 13. Let f

= pt(f)(x).

Set

where

Then for l

< p < ""

f

but for zonal f.

Lp(S), and assume that f is zonal. Let u (x, t)

131

§2. SEMI.SIMPLE CASE

The theorem is proved as follows. Since g 1 ~ g, the inequality BpllfiJp ~ llg(f)llp follows directly from Theorem 11 in the previous section. In addition the case p ~ 2 of the inequality II g (f) /I p ~ Apll f I P' is contained in Theorem 12. It remains to consider the case p > 2 of the direct inequality. To do this we follow closely the argument given in Theorem 2 of Section 2, Chapter II, in particular "part II" of that proof. The basic step was to prove the inequality

~ (g (f)(x)) 2cp (x)dx G

=

1! 00

tj"VJ u (x, t)! 2¢> (x)dx dt

0

~A

1f 00

0

t!Vu(x,t)! 2¢>(x,t)dxdt

(A= 4),

G

if cp ~ 0, and cp (x, t) = pt(cp)(x). (Here cp is also zonal.)

In the compact case this can be proved, as is pointed out in Chapter II, because the Xi commute with pt. In the present case the Xi being rightinvariant commute with the pt, the latter being left-invariant, and of course as we have already remarked IV u (x, t)! 2 can be expressed in terms of the Xiu(x, t). The rest of the proof of the theorem is quite parallel with that of the compact case and may be left as an exercise to the interested reader. We now come to the analogue of the Riesz transforms (see e.g., Section 4 of Chapter II), in the present case of zonal functions for symmetric spaces. We prove first the inequality (*)

with A

independent of t. Fix a p

t

= t0 ,

and assume that to begin with -

t

f is C00 and has compact support. Then (XiP 0 )(£)

=P

t

*

0 (X/) and the,

left side of (*) is well-defined. The right side of(*) is of course defined on general principles in view of the fact that

t

-+

well~

Pt(f) is analytic

132

FURTHER ILLUSTRATIONS

in t (as a function with values in Lp) in a proper sector which contains the positive t semi-axis, when l < p < ""; see Theorem 2 of Chapter III. - to to t Now let fi = XiP (f) ""P (Xi(f)), and f 0 = (ap /atHOit=to. Bt Theorem 11 in the previous section

and because of Theorem 13 we have

Together this gives (**) which is (*) for smooth f with compact support. The passage of (*) and (**) to general f is then by a routine limiting argument. The definition of the Riesz transforms can be given symbolically as -'1-l f. R-(f) = X-(-A) 1 I In order to give a precise definition we proceed as follows. Consider the f0 of the form

I

wnere to fo = apt (f) at t=to

> 0 and f f L n L2 . ..

P

For these f 0 we define the Riesz transforms by Ri (f 0 ) .. - fi

=-

XiPt 0 (f)

in accordance with the inequality (*) or (**). Now purely formally since pt

= e-t(-1'\)'1-l

-'1-l = Xi(-&} fo

we have f 0 = -(-A)'n pt 0 (f), and therefore - fi

= XiPt 0 (f)

.

In order to show that these Ri are in fact well-defined or a dense subset of LP, we need to observe the following two simple facts: (i) The set of f 0 of the form

133

§2. SEMI-SIMPLE CASE

f 0 .. apt (f)l (for some to > 0, with f at t•to is dense in LP

E

Lp

n L2 ),

1 < p < oo. To see this, recall that for any

=h

lim P to h - pt 'h -oO t .... 00

t0

in both Lp and L 2 norm (when 1 < p < oc), as is shown in the proof of Theorem 12 of the previous section. Thus the set of f0 of the form f 0 =

n LP

with t' > t 0 , is dense in LP. Each such f 0 can be represented in the form

(Pto- pt ')h, h

l

L2

t

f 0 ..

aP (f) at

It =to

where f = -

j

t

~to

pt(h)dt

I

0

as an easy calculation verifies (ii) To see that the resulting Ri(f 0 ) is well-defined remark the following.

Suppose f 1 and f 2

E

Lp, and

fo .. apt -(fl)

at

Then Pt 1(f 1)

= Pt 2 (f 2).

I t=t 1

• -aPt (f2) j

at

t~t 2

,

This is because

ptj (f)

=-

lim t, .... ""

f

t,

t. J

apt (f)dt at

in the Lp norm, since pt '(f) ... 0, as t ' ... ""· However by the semigroup t t· t+t· property P (P J f) = P J (f) and therefore

from which our desired conclusion follows. The way the Ri(f) have been defined shows that by (*) (or (**)) we have

134

FURTHER ILLUSTRATIONS

for a dense linear subset in Lp' 1 < p < oo and hence the Ri have a unique bounded extension to all of Lp. We summarize this result and elaborate it somewhat as follows.

Suppose 1 < p < oo, and f is zonal f

THEOREM 14. (a)

BpJIIIIp .$ Ii IIR/OIIp $ ApJifiJp ·

(b)

When p = 2,

l

Lp. Then:

The proof of the inequality IJIRi(f)Jip _$ Apllfllp has been given above, and this shows in particular that it suffices to prove (b) for a class of f which are dense in L 2 • Start with f 0 which is C00 and has compact support and set f 1 = Pt 1(f 0 ), t 1 > 0, u(x,t) • Pt+t 1(f 0 ) = Pt(f 1). Apply the identity (C) (of the proof of Theorem 12), wtih F

= u2•

Then since

A(u 2 )

=

21 Wuj 2 , and if

f 0 is

zonal, so is f 1 and u (x, t), we have &u

2

a at

= 2[[__!!]

2

~ ~ + I b .. x.(u)X.(u)] -

..

1 ,J

1J 1

J

Now the identity

is valid for any one of our general semigroups, and follows easily by the spectral representation of p+, which can be written as

§2. SEMI-5IMPLE CASE

135

for appropriate E(.A). We have, in fact, already pointed out that in the present case E 0 ;;; 0. We therefore have

Hence

for all t > 0. This identity is (b) for f "' (apt;at)(f0 ), t > 0 and since this class of f is easily seen to be dense in L 2 the identity (b) is then fully proved. By polarization this identity yields, for f, g

f

fgds "'

s

f" s

~

L2

f

b lJ.. R.(f)R.(g)ds 1 J

and hence by Holder's inequality

where 1/p + 1/q

= 1,

wherever f ( L2 llf!lp =

n Lp,

sup gtL 2 nLq

I gil q:S 1

and g

1js

fgdsl

f

L2 n Lq. However,

136

§3. STURM•LIOUVILLE

Thus

II fliP

s sp- 1 :£ IIRi
I Ri (g) II q S Aq II gil q which we already know. Hence we have shown Bp I f II p S I II Ri(OJI p, whenever f c L 2 n LP and a final by the inequality ~

trivial limiting argument removes the restriction that f

E

L2 •

Section 3. Sturm-Liouville Let L be a Sturm -Liouville operator defined on an interval (a 1' a 2 ) of the line. Thus

L (f) • a(x) d 2 f + b(x) c.!!..+ c(x)f dx dx where it is assumed that a, b and c are continuous and a (x) > 0, while c (x)

S 0. Then we can find an appropriate measure q (x)dx on (a 1 , a 2 ) so

that with respect to L 2 (q(x)dx), Lis formally self-adjoint. With the appropriate boundary conditions the semigroup Tt - etL can be constructed and satisfies the general conditions (I), (II), and (III) of Chapter III. Moreover if c(x)

=0

then Tt also satisfies (IV) (see McKean, [13]).

These semigroups therefore give us interesting examples for which the results of Chapters III and IV apply. It will be the purpose of the concluding remarks that follow to follow up

the hint that part of the more refined analysis-carried out in Chapter II and in the present chapter in the context of groups-is most probably valid also in the setting of Sturm-Liouville expansions. We shall be suggesting only the statements and proofs of certain facts that are indicated by our experience with the theory for compact and semisimple groups and the results in [37] for other classical expansions. The reader who is wary of this kind of speculation need not pursue this matter further.

FURTHER ILLUSTRATIONS

137

Section 4. Heuristics It will be convenient to rewrite L in a more special form,

L (f)

= d 2f + a (x) ~ , dx 2

dx

(to which the general L may anyway be reduced by changes of variable). This L is formally self-adjoint with respect to the measure q (x)dx, where (q '(x))/(q (x)) ... a(x), i.e., q (x) • eSa(x)dx. In an important part of what follows it will be necessary to assume that a '(x) :;; 0. This is indeed the case for many interesting classical ex· pans ions. For example a (x} = 2A/x, then q (x)dx • -x 2 Adx, 0 < x < oo which corresponds to the Fourier-Bessel and ultra-spherical expansions (see [37]). Also if a(x)

= -2x,

q(x)dx"' e-x 2dx,

-oo

< x < oo which

gives rise to the Hermite expansion. We consider next the semigroup Tt

= etL, once we have

imposed appro-

priate boundary conditions, and as indicated before it satisfies all the conditions (I), (II), (III), and (IV) of a symmetric diffusion semigroup. We then consider the corresponding sub-ordinated semigroup pt .. e-t(-L)lh which can be expressed as

· ! V" vn 00

pt = _1

-u 2/ ~ Tt 4u du .

0

Then if u (x, t) "" (Ptf)(x), u satisfies the equation ~ u (x, t)

We write g(f)(x) =

and

II flip = (f

({•(1~;1' + ~~~ 1'H~

/f(x)JP.q (x)dx)liP,

The first conjectural theorem is the statement

•

= 0,

where

138

FURTHER ILLUSTRATIONS

H,l:

l
and conversely llfllp :5 Apllg(f)llp•

whenever f is orthogonal to the eigen-

function of L of eigenvalue zero, ('if such exists).

The proof should follow closely the pattern of Theorem 2 of Chapter II and Theorem 13 of the present chapter. To begin with, when p :5 2, the assumption a '(x) :5 0 seems unnecessary. We set down the following facts &uP .,. p(p-l)uP-2 [
(A')

if u > 0, u(x, t) = pt(f)(x) .

j1ts~p0 u(x,t)IP q(x)dx:5

(B')

(C')

f

00

Ap/lf(x)!Pq(x)dx

jt&F(x,t)q(x)dxdt = / F(x,O)q(x)dx- /F(x,oo)q(x)dx

0

and try to apply (C ') when F = (u(x, t))P. Only the argument involving (C ') would seem to require further analysis, since (B ') is a consequence of the general maximal theorem of Chapter III and (A') is merely straightforward differentiation. To pass to the case p ;::: 2

(h~re

we would require the condition

a '(x) :5 0) we would need the assertion that

f ooftt(au/ at 0

\'

+(aui)(x)q(x)dxdt ax

:5 A for ¢(x) .2: 0, with ¢(x,t)

~Jt (<~;/ + (~;) 2 ) ¢(x,t)q(x)dxdt

=

Pt(¢)(x). (A= 4.)

This can be obtained (at least formally) on the basis of a "sub-harmonic" property of (<1u/at) 2 + (du/ax) 2. In fact we claim that we always have

& r( au> 2 + o, L at

ax

-

if u satisfies Au

= o.

139

§4. HEURISTICS

Write U = du/dt, and V

= t1u/t1x.

~(V) = &(V) + a'(x)V = 0

Then clearly ~\u

. Next ti(U 2 )

= 0,

and also

= 2(U~ + uf) + 2u&_(u)

and

2(V~ + Vt) + 2V &_(V). Here Ux = t1U/t1x, Ut = t1U/t1t, etc. Together then ~\ (U 2 + V2) = positive + 2V ~\ v = positive -2a '(x) v 2

also Li(V 2) =

I

which is

~

0, if a '(x)

~

0.

The converse inequality Ap 1 11 f II p ~

II g (f) II p

for all f orthogonal to

the eigenfunction of eigenvalue zero is of course a consequence of the gen· eral result in Section 5 of Chapter IV, and the fact that

g~ (f) .::;

g (f).

Another converse inequality for f satisfying the same conditions is concontained in the following:

l
H,2:

where gx(f) • (J0 00 t (du/ax) 2 dt)'h,

u(x, t)

= Pt(f)(x).

In fact, it can be shown without difficulty, using (A') and the L 2 theorem for g 1 , that 411gx(011~ = llfll~ -IIE 0 (f)ll~ where E0 is the projection corresponding to the eigenfunction of eigenvalue 0. Thus, if E 0 (f) = 0, we get by polarization that

I

ff 1 q (x)dx =

!/<

t1P\f)(x}) ( apt (f 1 )(x)t dt q (x)dx) dX ax

and thus by Holder's inequality and the direct part of H, 1 we get H, 2. The next idea which merits some further examination is the possibility of studying the notion of the Hilbert transform (i.e., Riesz transform) in this context. To illustrate this idea, assume for the moment that L has a discrete spectrum: that is, suppose that ¢ 1 ,¢ 2 , ••. ,cpk, ... form a complete set of eigenfunctions with eigenvalues - Ai, ... , -A~, .... Assume therefore that L cf>k + A~ cf>k = 0, and

f cpk(x)cf>e(x) q (x)dx

= 8k,e •

Then a formal integration by parts shows that the functions

form an orthonormal set again, that is

140

FURTHER ILLUSTRATIONS

/ dcpk dcf>e q(x)dx = dx dx

.\~ Bk r '

The proposed Hilbert transform would provide the mapping l dcpk cpk(x) .. .\k dx

These observations can be put in a general setting by observing that if the cpk a!e eigenfunct~ons of L are eigenfunctions of

L=

=(d 2/dx 2) + a (x}(d/dx) then the

dcpk/dx

(d/dx 2 ) + a (x)(d/dx) + a '(x).

We recall the general assumption a '(x)

S _0 that we made earlier.

It

ensures the existence of a semigroup Tt '"' etl~ which satisfies our general conditions (1), (II), (Ill), (after appropriate boundary conditions have been supplied). Let now pt be the corresponding sub-ordinated semigroup corresponding to the differential equation & u (x, t)

~\u(x,t)=

a2u ax 2

=

+a(x)au +a'(x)u+a 2 u ... 0. 2 ax

at

We define

and we can expect that: l
H,3:

The indicated proof of this would follow the same pattern (A'), (B '), (C ') as outlined for the operator L, except that the identity for (A') would now read

if ~\u

= 0,

and u > 0.

141

§4. HEURISTICS

Observe that if we set

then ~\u

= 0,

and - (du/ dt)(x, 0) and (du/dx)(x, 0) are Hilbert transforms

in the indicated sense. The theorem that would seem to emerge from these considerations can be stated as follows: H, 4:

Let f

f

Lp(q (x)dx) and u (x, t)

II~~

(x,t)llp 5

Apll~:

= Pt(f)(x).

(x,t)llp,

Then

t>O,

l
with Ap independent of f or t. Since

on application of H, 2 shows that

Next

and therefore by H, 3,

Combining these two gives H, 4. There are also inequalities of this type for p 2= 2, but we shall not here pursue this matter further.

142

FURTHER ILLUSTRATIONS

BIBLIOGRAPHICAL COMMENTS FOR CHAPTER III

SectionS. The cases corresponding to Hermite and Laguerre polynomials have recently been treated by Muckenhoupt (38].

REFERENCES

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[17] R. E. Paley, 'Remarkable Series of Orthogonal Functions, I, Proc. London Math. Soc., 34 (1932), 241-79. [18] E. M. Stein, On the Functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc., 88 (1958), 430-466.

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The Spaces Lp With Mixed Norm,

Duke Math. Jour., 28 (1961), 301-324. [22] E. M. Stein, Interpolation of Linear Operators, Trans. Amer. Math. Soc., 83 (1956), 482-492. MARTINGALE AND MAXIMAL THEOREMS

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REFERENCES

[26]

J.

145

L. Doob, "Stochastic Processes," New York, 1952.

[27] _ _ , A Ratio Operator Limit Theorem, Zeit. Wahrscheinlich., 1 (1963), 288-294. [28] R. F. Gundy, A Decomposition for L 1 Bounded Martingales, Ann. Math. Statist., 39 (1968), 134-138. [29] R. E. Paley, A Proof of a Theorem on Averages, ~roc. London Math. Soc., 31 (1930), 289-300. [30] G. C. Rota, An "Alternieven de Verfahren" for General Positive Operators, Bull. Amer. Math. Soc., 68 (1962), 95-102. [31] E. M. Stein, On the Maximal Ergodic Theorem, Proc. Nat. Acad. Sci., 47 (1961), 1894-1897. ADDITIONAL REFERENCES

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J.

Marcinkiewicz, Sur les multiplicateurs des series de Fourier, Studia

Math., 8 (1939), 78-91. [35] F. Riesz and B. Sz-. Nagy, "Functional Analysis," New York, 1955. [36] E. C. Titchmarsh, "Eigenfunction Expansions Associated with Secondorder Differential Equations," Oxford, 1946. [37] B. Muckenhoupt and E. M. Stein, Classical Expansions and Their Relation to Conjugate Harmonic Functions, Trans. Amer. Math. Soc., 118 (1965), 17-92. [38] B. Muckenhoupt, Poisson Integrals for Hermite and Laguerre Expan.• sions, Trans. Amer. Math. Soc., 139 (1969), 231-242; Hermite Conjugate Expansions, ibid., 243-260. (39] A. P. Calderon, Intermediate Spaces, and Interpolation, Studia Math., 24, (1964), 113-190. (40] K. Yosida, "Functional Analysis," Berlin, 1965.

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