Perspectives in partial differential equations, harmonic analysis and applications: a volume in honor of Vladimir G. Maz'ya's 70th birthday

Perspectives in Partial Differential Equations, Harmonic Analysis and Applications Vladimir Gilelevich Maz'ya Proce...

Author: Mitrea | Dorina and Mitrea | Martus (editors)

7 downloads 616 Views 31MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Perspectives in Partial Differential Equations, Harmonic Analysis and Applications

Vladimir Gilelevich Maz'ya

Proceedings of Symposia in

PuRE MATHEMATics Volume 79

Perspectives in Partial Differential Equations, Harmonic Analysis and Applications A Volume

in Honor of

Vladimir G. Maz'ya's 70th Birthday Dorina Mitrea Martus Mitrea Editors

2000

Mathematics Subject Classification. Primary 01A50, 26D10, 31B15, 34L40, 35J25, 35Q53, 42B25, 46-D6, 46£35, 74Jl5. Photo on page ii courtesy of Tatyana Shaposhnikova

Library of Congress Cataloging-in-Publication Data Perspectives in partial differential equations, harmonic analysis, and applications : a volume in honor of V ladimir G. Maz'ya's 70th birthday / Dorina Mitrea, Marius Mitrea, editors. p. em. - (Proceedings of symposia in pure mathematics : v. 79) Includes bibliographical references. ISBN 978-0-8218-4424-3 (alk. paper) 1. Maz'ya, V. G. 2. Differential equations, Partial. 1965- II. Mitrea, Marius. QA377.P378

3. Harmonic analysis.

I. Mitrea, Dorina,

2008 2008030028

515'.353-dc22

Copying and reprinting. Material in this book may be reproduced by any means for edu cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permission(Dams. org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 2008 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America.

§

The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: I /vww. ams. org/ 10 9 8 7 6 5 4 3 2 1

13 12 11 10 09 08

Contents

On the scientific work of V.G. Maz'ya: a personalized account and

DORINA MITREA MARIUS MITREA Capacity, Carleson measures, boundary convergence, and exceptional sets NICOLA ARCOZZI , RICHARD ROCHBERG , and ERIC SAWYER

vii

1

On the absence of dynamical localization in higher dimensional random Schrodinger operators

JEAN BOURGAIN 21 Circulation integrals and critical Sobolev spaces: problems of optimal constants HAi' M BREZIS and JEAN VoN ScHAFTINGEN 33 Mutual absolute continuity of harmonic and surface measures for Hormander type operators and

LUCA CAPOGNA, NICOLA GAROFALO ,

DUY-MINH NHIEU

Soviet-Russian and Swedish mathematical contacts after the war. A personal account.

LARS GARDING

49 101

Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrodinger operators on bounded Lipschitz domains and

105 FRITZ GESZTESY MARIUS MITREA A local Tb Theorem for square functions STEVE HOFMANN 175 Partial differential equations, trigonometric series, and the concept of function around 1800: a brief story about Lagrange and Fourier 187 JEAN-PIERRE KAHANE Quantitative unique continuation, logarithmic convexity of Gaussian means and Hardy's uncertainty principle

CARLOS E. KENIG

Boundary Harnack inequalities for operators of p-Laplace type in Reifenberg flat domains and

JOHN L. LEWIS , NIKLAS LUNDSTROM , Waves on a steady stream with vorticity MARKUS LILLI and JOHN F . TOLAND v

KAJ NYSTROM

207 229 267

vi

CONTENTS

On analytic capacity of portions of continuum and a question of T. Murai FEDOR NAZAROV and ALEXANDER VOLBERG

279

The Christoffel-Darboux kernel BARRY SIMON

295

A Saint-Venant principle for Lipschitz cylinders MICHAEL E. TAYLOR

337

Wavelets in function spaces H ANS TRIEBEL

347

Weighted norm inequalities with positive and indefinite weights IGOR E. VERBITSKY

377

The mixed problem for harmonic functions in polyhedra of JR.3 MOISES VENOUZIOU and GREGORY C. VERCHOTA

407

On the scientific work of V.G. Maz'ya: a personalized account

Vladimir Gilelevich1 Maz'ya, one of the most distinguished analysts of our time, has recently celebrated his 70th birthday. This personal landmark is also a great opportunity to reflect upon the depth and scope of his vast, multi-faceted scientific work, as well as on its impact on contemporary mathematics. It is no easy task to re-introduce to the general public a persona of the caliber of Vladimir Maz'ya. Nonetheless, the narrative of his life is such an inspirational epic of triumph against adversity and seemingly insurmountable odds, of sheer perseverance and dazzling success, that such an endeavor is worth undertaking even while fully aware that the present abridged account will have severe inherent limitations. Simeon Poisson once famously said that Considering the sheer volume of his scientific work and scholarly activities, one might be tempted to regard Vladimir Maz'ya as the perfect embodiment of this credo. However, with his larger-than-life personality, boundless energy, strong opinions and keen interest in a diverse range of activities, Vladimir Maz'ya transcends such a cliche: he is a remarkable man by any reasonable measure. His life, however, cannot be separated from mathematics, regarded as a general human endeavor: much as his own destiny has been prefig ured by his deep affection for mathematics, so has Vladimir Maz'ya helped shape the mathematics of our time. Meanwhile, his views on mathematical ability are rooted in a brand of stoic pragmatism: he regards the latter not unlike the skill and sensitivity expected of a musician, or the stamina and endurance required of an athlete. In [6], I. Gohberg remarks: is

"life is good for only two things: dis covering mathematics and teaching mathematics".

"Whatever he writes aesthetic beautiful,feeling his love '. for art, music and literature seeming to feed his mathematical

I . Rough childhood. Vladimir Maz'ya was born on December 31, 1 937, in Leningrad (present day St. Petersburg) in the former USSR, roughly two years before World War II broke out in Europe. USSR was subsequently attacked and the capture of Leningrad was one of three strategic goals in Hitler's initial plans for Operation Barbarossa ( "Leningrad first, the Donetsk Basin second, Moscow third" ) , with the goal of "Celebrating New Year's Eve 1942 in the Tsar's Palaces." It is in this context that Vladimir Maz'ya's early life was marred by profound per sonal tragedy: his father was killed on the World War II front in December 1941, and all four of his grandparents perished during the subsequent siege of Leningrad, which lasted from September 9, 1941 to January 27, 1944. Vladimir was brought up by his mother, alone, who worked as a state accountant. They lived on her 1 patronymic after his father Hillel vii

viii

ON THE SCIENTIFIC WORK OF

V.G.

MAZ'YA: A PERSONALIZED ACCOUNT

meager salary in a cramped (nine square meter) room of a big communal apart ment. These days, it is perhaps difficult to imagine the hardship in which a young Vladimir was finding his feet, and yet he spoke of occasional glimmers in this deso late atmosphere. He once recounted a touching story about the lasting impression a children's botanical book he had received, about the fruits of the world, made on him: how the pictures he gazed upon over and over still vividly live in his memory, and how it took many long years before he had a chance to actually see and taste some of the fruits depicted there. Resolute and driven, Vladimir rose above these challenges. At the same time, his talent and ability were apparent from early on: he earned a gold medal in secondary school and, as a high-schooler, he was a frequent winner of city olympiads in mathematics and physics. II. The formative years.

While 17 years of age, Vladimir Maz'ya entered the Faculty of Mathematics and Mechanics (Mathmech) of Leningrad State University (LSU) as a student. His first publication, "On the criterion of de la Vallee-Poussin", was in ordinary differential equations and appeared in a rota-printed collection of student papers when he was in his third year of undergraduate studies. In the following year, while he was a fourth-year student, his article on the Dirichlet prob lem for second order elliptic equations was published in Doklady Akad. Nauk SSSR. Upon finishing his undergraduate studies at Mathmech-LSU, Vladimir Maz'ya se cured a position as a junior research fellow at the Research Institute of Mathematics and Mechanics of Leningrad State University. Two years later he successfully de fended his Ph.D. thesis on "Classes of sets and embedding theorems for function spaces". This remarkable piece of work was based on ideas emerging from his talks in Smirnov's seminar. In their reviews, the examiners noted that the level of qual ity and technical mastery far exceeded the standard requirements of the Higher Certification Commission for Ph.D. theses. Testament to the outstanding nature of his thesis, Vladimir Maz'ya was awarded the Leningrad Mathematical Society's prize for young scientists. Subsequently, Vladimir Maz'ya was a volunteer director of the Mathematical School for High School Students at Mathmech, an institution born out of his own initiative. Interestingly, Vladimir Maz'ya never had a formal scientific adviser, both for his diploma paper (master's thesis) , and for his Ph.D. thesis. Indeed, in each instance, he chose the problems considered in his work by himself. However, starting with his undergraduate years, he became acquainted with S.G. Mikhlin, and their relationship turned into a long-lasting friendship that had a great influence on the mathematical development of Vladimir Maz'ya. Ac cording to I. Gohberg, [6], "Maz'ya never was a formal student of Mikhlin, but Mikhlin was for him more than a teacher. Maz'ya had found the topics of his dis

sertations by himself, while Mikhlin taught him mathematical ethics and rules of writing, refereeing and reviewing."

III. Becoming established. During 1961-1986, Vladimir Maz'ya held a senior research fellow position at the Research Institute of Mathematics and Mechanics of LSU. Four years into that tenure, he defended his D.Sc. thesis, entitled "Dirichlet and Neumann problems in domains with non-regular boundaries", at Leningrad State University. From 1968 to 1978, he lectured at the Leningrad Shipbuilding Institute, where he became a professor in 1976. In 1986 he departed the university for the Leningrad Division of the Institute of Engineering Studies of the Academy

ON THE SCIENTIFIC WORK OF V.G. MAZ'YA: A PERSONALIZED ACCOUNT

ix

of Sciences of the USSR, where he created and headed the Laboratory of Mathe matical Models in Mechanics. At the same time, he also founded the influential Consultation Center in Mathematics for Engineers, serving as its head for several years. In 1990 Vladimir Maz'ya relocated to Sweden and became a professor at Linkoping University. At this stage in his career, in recognition of his fundamental contributions to the field of mathematics, Vladimir Maz'ya has become the recip ient of a series of distinguished awards in relatively quick succession. In 1 990 he received an honorary doctorate from the University of Rostock, Germany. In 1999 he was the recipient of the Humboldt Prize, and in 2000 was elected a corresponding member of the Royal Society of Edinburgh (Scotland's National Academy) . Two years later he became a full member of the Royal Swedish Academy of Sciences. In 2003 he received the Verdaguer Prize of the French Academy of Sciences, and in 2004 the Celsius Gold Medal of the Royal Society of Sciences at Uppsala. A number of international conferences in his honor have been organized during this period of time, such as the conference in Kyoto, Japan, in 1993, the conferences at the University of Rostock, Germany, and at Ecole Polytechnique, France, in 1998, and the conferences in Rome, Italy, and Stockholm, Sweden, in the summer of 2008. In 2002 Vladimir Maz'ya was an invited speaker at the International Congress of Mathematicians in Beijing, China. More recently, he has held appointments at the University of Liverpool, England, and at the Ohio State University, USA, while continuing to be a Professor Emeritus at Linkoping University, Sweden.

IV. The mathematical work. By any standards, Vladimir Maz'ya has been extraordinarily prolific, as his 50 years of research activities have culminated in about a couple dozen research monographs, and more than 450 articles, containing fundamental results and powerfully novel techniques. Besides being remarkably deep and innovative, his work is also incredibly diverse. Drawing upon several sources, most notably [1] , [2] , [5] and [9] , below we briefly survey some of the main topics covered by Vladimir Maz'ya's publications. The references labeled (Ma-X] refer to the list of books published by Vladimir Maz'ya, which is included following the current subsection. Boundary integral equations on non-smooth surfaces. One of the early significant contributions of Vladimir Maz'ya was his 1967 monograph [Ma-25] with Yuri D. Burago, where they developed a theory of boundary integral equations ( involving operators such as the harmonic single- and double-layer potentials ) in the space C0, of continuous functions, on irregular surfaces. The book contains two parts: the first of which concerns the higher-dimensional potential theory and the solutions of the boundary problems for regions with irregular boundaries, while the second part deals with spaces of functions whose derivatives are measures. This was happening around the time the Calderon-Zygmund program, one of its goals being a re-thinking of the finer aspects of Partial Differential Equations from the perspective of Harmonic Analysis, was becoming of age. In the early 60's, the solvability properties of elliptic multidimensional singular integral operators were well-understood, due to the fundamental contributions of people such as Tti comi, Mikhlin, Giraud, Calderon and Zygmund, and Gohberg, among others; but very little was known about the degenerate and /or non-elliptic case. Influenced by Mikhlin, Vladimir Maz'ya began in the mid 60's a life-long research program ( part of which has been a collaboration effort ) aimed at shedding light on this challenging o

x

0::'\

THE SCIENTIFIC WORK OF

V.G.

MAZ'YA: A PERSONALIZED ACCOUNT

and important problem. These innovative ideas did not get instantaneous recogni tion as a certain degree of skepticism has long accompanied efforts to understand non-smooth calculus. One well-known quotation attributed to H. Poincare, which typifies the aforementioned distrust, goes as follows: "Autrefois quand on invantait une fonction nouvelle, c'etait en vue de quelque but practique; aujourd'hui on les invente tout expres pour mettre en defaut les raisonnements de nos peres et on n'en tirera jamais que cela".

Such a point of view was by no means isolated. Even S.G. Mikhlin, years later, referring to the perspective of studying PDE's under minimal smoothness assumptions on the boundary, opined to the effect that "no mother would ever let her c hild play in such ravines".

The subject of analysis in non-smooth settings permeates through much of the work of Vladimir Maz'ya, who has had a most significant contribution in ensuring the eventual acceptance of this, nowadays fashionable, area of research. In collab oration with his Ph.D. student N.V. Grachev (1991) , Vladimir Maz'ya solved the classical problem of inverting the boundary integral operators naturally associated with the Dirichlet problem for the Laplacian, in the space C0, on a polyhedral sur face. Also, Maz'ya and A. A. Solov'ev were the first to consider (in 1990) boundary integral equations on a curve with cusps. Subsequently, they developed a logarith mic potential theory which is applicable to integral equations in elasticity theory in a plane domain with inward or outward peaks on the boundary (2001) . More recently, in collaboration with T. Shaposhnikova, Vladimir Maz'ya has studied the classical boundary integral equations of the harmonic potential theory on Lipschitz surfaces, and obtained higher fractional Sobolev regularity results for their solutions under optimal regularity conditions on the boundary. The method employed, going back to work of Maz'ya in the early SO's, consists of establishing well-posedness re sults for certain auxiliary boundary value and transmission problems for the Laplace equation in weighted Sobolev spaces. Counterexamples related to Hilbert's 19th and 20th problems. In his famous plenary address at the International Congress of Mathematicians in 1900, held at the Sorbonne, Paris, David Hilbert put forth a list of twenty-three open problems in mathematics, many of which turned out to be very influential for 20th century mathematics (strictly speaking, Hilbert presented ten of the problems: 1, 2, 6, 7, 8, 13, 16, 19, 21 and 22, at the conference, and the full list was published later). The 19th problem read: Are the solutions of regular problems in the cal culus of variations always analytic? Originally, Hilbert was referring to regular variational problems of first order in two-dimensional domains, but the issue of (local) regularity makes sense in higher dimensions and for higher-order problems as well. Hilbert's 19th and 20th problems, the latter asking "Is it not the case that o

every regular variational problem has a solution, provided certain assumptions on the boundary conditions are satisfied, and provided also, if need be, that the concept

have generated a large amount of attention and, in the second half of the 20th century, proofs were obtained in sufficient general ity. It was therefore natural to speculate that the conjectures continue to hold for higher-order variational problems. However, in 1968 Vladimir Maz'ya proved that this is not the case. In [8], Maz'ya constructed higher-order quasi-linear elliptic equations with analytic coefficients whose solutions are not smooth. Other counterexamples constructed in [8] (and, independently, by De Georgi [4]) concern the celebrated De Giorgi-Nash Holder regularity result for solutions of

of solution is suitably extended?'

ON THE SCIENTIFIC WORK OF V.G. MAZ'YA: A PERSONALIZED ACCOUNT

xi

the second order linear elliptic equations in divergence form with bounded measur able coefficients. Maz'ya showed that this property fails for higher-order equations which may admit variational solutions which are not locally bounded. The coun terexamples in [8] stimulated the development of the theory of partial regularity of solutions to nonlinear equations, i.e. , the study of regularity properties outside of a sufficiently small, exceptional set. The oblique derivative problem. The oblique derivative problem was first formulated by Poincare in his studies related to the theory of tides, and by the late 60's the two-dimensional setting was well-understood. At that time, much of the work in the multidimensional case has been restricted to the situation when the direction field of the derivatives is transversal to the boundary at each point, a condition which ensures that the ellipticity is nowhere violated. However, when the ellipticity degenerates, this problem turned out to be considerably more difficult and subtle. This case came under scrutiny in 60's when a series of papers were published in which the degenerate oblique derivative problem was considered in the scenario when the vector field is tangent to the boundary along a submanifold of codimension one, to which this vector is not tangent. This line of work received a big impetus when in 1970 Vladimir Maz'ya initiated a deep investigation of the problem in the case in which the boundary contains a nested family of subman ifolds r 1 :J r2 :J . . . :J r s with the property that the vector field is tangent to rk at points belonging to rk+ l , and is transversal to fs . By employing a new technique, Vladimir Maz'ya was able to prove in this setting the unique solvability of the problem in a formulation which includes an additional Dirichlet condition on the entry set of the vector field and allows the possibility of discontinuities of the solution at points of the exit set. Up to now, this is the only known result pertaining to the oblique derivative problem in the generic situation in the sense of V. Arnold, who has considered this problem as an illustration of his calculus of infinite co-dimensions ( see [3], §29 B ) . According to a hypothesis of Arnold, all submanifolds r l , . . . , rs, induce infinite dimensional kernels or co-kernels for the oblique derivative problem. Nonetheless, Maz'ya's striking theorem reveals that Arnold's hypothesis is inadequate, since it turns out that submanifolds of co dimension greater than one in the boundary are negligible, in the sense that they play the same type of role as removable singularities.

o

oBoundary-value problems in domains with piecew ise-smooth boundaries.

Vladimir Maz'ya has started working in this field at the beginning of the 1960's and from his early publications he was able to establish deep and unexpected results regarding second-order elliptic equations. For example, in studying selfadjointness conditions for the Laplace operator with zero Dirichlet data on contours of class C 1 (but not C2 ) , he discovered a surprising instability effect for the index under affine coordinate transformations. Following the emergence of Kondrat'ev's well-known 1 967 paper on elliptic boundary-value problems in domains with conic singulari ties, Vladimir Maz'ya began working actively in this field and, in collaboration with B .A. Plamenevskii, and later with V.A. Kozlov and J. Rossmann, has produced a string of papers which contain a fascinating theory of boundary-value problems in domains with piecewise smooth boundary, including regularity estimates, as ymptotic representations of solutions, well-posedness theorems, and methods for

xii

ON THE SCIENTIFIC WORK OF V.G. MAZ'YA: A PERSONALIZED ACCOUNT

computing the coefficients in the asymptotics of solutions near boundary singulari ties. The theory thus developed, together with important applications to problems arising in mechanics, engineering and mathematical physics, is presented in the monographs [Ma-7], [Ma-8], [Ma-15], and [Ma-16]. The aforementioned body of results complements the theory of elliptic bound ary value problems in Lipschitz domains, as initiated by A. Calderon, B. Dahlberg, E. Fabes, N. lliviere, M. Jodeit, C. Kenig, D. Jerison, J. Pipher, G. Verchota starting in the late 70's and early 80's. An authoritative account of the state of the art in this field, up to the mid 90's can be found in C. Kenig's book [7]. Compared with the latter, the former setting of domains with piecewise smooth boundaries allows for a wide range of non-Lipschitz domains. A simple example is offered by Maz'ya's "two-brick domain":

FIGURE

1

Indeed, a moment's reflection shows that near the point P, the boundary of the above domain is not the graph of any function (as it fails the vertical line test) even after applying a rigid motion. Most recently, progress in understanding such configurations from the Harmonic Analysis perspective has been recorded in [12], [13], [14]. Multipliers between spaces of differentiable functions. In the late 70's, Vladimir Maz'ya and Tatyana Shaposhnikova initiated a systematic study of mul tipliers in pairs of various spaces of differentiable functions. This resulted in their joint book [Ma-19], which for the time being, is the only monograph on this topic. The forthcoming book [Ma-l] by the same authors reports on the more recent progress in this area. The obvious motivation for a thorough investigation of prop erties of multipliers stems from the study of partial differential equations of the type

o

(1)

Lu

:=

L

0°(aa,{3(X)af3u) = f

in

n,

lo:l,lf31:S:m

in which the data and the solution belong to appropriate Sobolev spaces in the domain n c ffi.n. It is then of interest to understand how multiplication by the coefficients ao.,{3 transforms these classes of functions. A similar perspective comes from treating (1) via localization and flattening of the boundary of the domain n,

ON THE SCIENTIFIC WORK OF V.G. MAZ'YA: A PERSONALIZED ACCOUNT

xiii

for the purpose of transforming the original PDE into a problem in the upper half space JR.+.. In this scenario, the multiplier properties of the functions cp : JR_n-l -+ JR. which locally describe a n come into play. For example, in the case of the Poisson problem for the Laplacian with a Dirichlet boundary condition, i.e., (2)

{ .6.u = f

Tru = 0

in n, on a n,

such techniques allow for a sharp description of the analytical properties of n re quired for the implication (3) to hold (where 1 < p < oo is given) . One other route through which multipliers take center-stage in a natural fashion is when one considers PDE's on manifolds, in which case the transformational properties of (1) under changes of variables are of central focus. For general multipliers, Maz'ya and Shaposhnikova have established a wealth of basic results on the spectrum, traces and extensions, implicit functions, and two sided estimates for the essential norm. They have also identified various classes of mappings and classes of non-smooth manifolds on which these multiplier spaces are invariantly defined. In addition, a calculus of singular integral operators with symbols in the space of multipliers was developed. These efforts have been amply rewarded by the fact that such a theory permits for deep applications to elliptic boundary value problems in domains with non-smooth boundaries. o

Isoperimetric and integral inequalities, and theory of capacities.

While a fourth-year student at LSU, Vladimir Maz'ya made the remarkable dis covery that integral inequalities of Sobolev type are actually equivalent to certain isoperimetric and isocapacitary inequalities for subsets of the domain where a func tion is defined. Even today, Vladimir likes to recall that special moment of inspi ration, and he can artfully and fluidly reproduce the original calculations, to the delight of an interested interlocutor. These results, which eventually became part of his Ph.D. thesis, appeared in press in 1960-61 . This original approach enabled him to obtain sharp constants in the aforementioned integral inequalities. In particular, the sharp constant in Gagliardo's inequality (4)

= n - 1 v;;1 1n ,

proved to be equal to that in the classical isoperimetric inequality: Cn where Vn is the volume of the unit ball in JR_n (this was also found simultaneously and independently by G. Federer and W.H. Fleming) . More importantly, as Maz'ya himself emphasized in 1966, his proofs did not make use of any specific properties of the Euclidean space and, hence, could be carried over to the setting of Riemannian manifolds. An important inequality proved by Maz'ya (1964, 1972) , and which later be came known as the strong type capacitary inequality, allowed him to obtain capaci tary criteria for Sobolev-type estimates. In more recent papers (2005, 2006), he has also obtained some important generalizations of this inequality. He also discovered (2003) that embeddings in fractional Besov spaces, or Riesz potential spaces, are equivalent with the validity of a certain new type of isoperimetric inequalities. The 1 964, 1972 papers of Maz'ya, mentioned above, have motivated a thorough study

xiv

ON THE SCIENTIFIC WORK OF V.G. MAZ'YA: A PERSONALIZED ACCOUNT

of different aspects of the theory of Sobolev spaces, and have decisively influenced the development of this branch of mathematics. Currently, the methods in those papers are the driving force in the study of Sobolev spaces on metric spaces. The collection of results obtained up to 1985 are diligently presented in [Ma-20] , ar guably the most popular book authored by Vladimir Maz'ya (scheduled to appear in a new edition shortly). The systematic use of the notion of capacity of a set eventually became a recurrent theme of a sizable number of Maz'ya's papers. As early as 1963 he in troduced the polyharmonic capacity and successfully employed it in order to find optimal conditions for the well-posedness of the Dirichlet problem in the energy space for higher-order elliptic equations. At the beginning of 70's, V. Maz'ya and V.P. Khavin considered non-linear potentials and systematically studied their prop erties. Presently, the theory of non-linear potentials (naturally viewed as an exten sion of the classical linear theory) is a main-stream, active and fast-growing area of research, which has helped produce answers to many basic questions in the theory of functions, particularly for those concerning the nature of exceptional sets. By making essential use of his previ ously developed capacitary criteria, Vladimir Maz'ya was able to obtain (in 1962, 1964) , sharp conditions ensuring the validity of various spectral properties of the Schrodinger operator. More recently, in their masterful 2002 Acta Mathematica paper, Vladimir G. Maz'ya and Igor E. Verbitsky have identified the correct class of complex-valued potentials for which the Schrodinger operator -6. + V maps the energy space into its own dual. Subsequently, V. Maz'ya, V.A. Kondrat'ev and M.A. Shubin (2004) have proved necessary and sufficient conditions for the spectrum of the Schrodinger operator with a magnetic potential to be positive and discrete, thus generalizing the well-known work of A.M. Molchanov on this topic (who has treated the case when the magnetic field is absent) . In 2005, V. Maz'ya and M. Shubin succeeded in characterizing the sets which are negligible in Molchanov's criterion, thereby solving an long-standing open problem, originally posed by I.M. Gel'fand in 1953.

o Theory of the Schrodinger operator.

oBoundary behavior and maximum principles for elliptic and parabolic

One of the prevalent themes of research throughout Vladimir Maz'ya's career, is the issue of regularity of a boundary point in the sense of Wiener. As early as 1962, he has proved an estimate for the modulus of continuity of a harmonic function, formulated in terms of the Wiener integral which, in turn, has found important applications in the qualitative theory of linear and non-linear elliptic equations. Then in 1970 he formulated a condition for regularity, in the sense of Wiener, of a boundary point for a certain class of quasi-linear second-order elliptic operators, which includes the p-Laplacian. Conspicuously, all these years virtually nothing was known about the Wiener type regularity of a boundary point for higher-order equations. The breakthrough came in 2002 when Vladimir Maz'ya succeeded in generalizing the Wiener test to elliptic equations of arbitrary order. Subsequently, this fundamental result made the subject of Maz'ya's talk at the International Congress of Mathematicians in Beijing. In collaboration with G.I. Kresin, Vladimir Maz'ya has produced, in a series of papers starting around mid 80's, a necessary and sufficient condition formulated in

systems.

ON THE SCIENTIFIC WORK OF V.G. MAZ'YA: A PERSONALIZED ACCOUNT

xv

algebraic terms guaranteeing the validity of the classical maximum modulus prin ciple for second-order elliptic and parabolic systems. Next, in 1992, V. Maz'ya and J . Rossmann proved that the classical Miranda-Agmon maximum principle actu ally holds for any strongly elliptic operator of arbitrary order in a plane domain with a piece-wise smooth boundary, without peaks. While a similar result holds in the three-dimensional setting, in dimensions four and higher this principle fails for certain domains with conical vertices. For the polyharmonic (and biharmonic) equations in Lipschitz and C1 domains, this issue has been further investigated in [10] , [11]. Theory o f water waves. During his tenure at the Leningrad Shipbuilding In stitute, Vladimir Maz'ya became interested in the mathematical theory of linear surface waves and, in 1973, wrote two articles in collaboration with B.R. Vainberg, in which the basic boundary value problems of this theory are studied. Four years later, Vladimir Maz'ya was the first to obtain a rather general uniqueness condi tion for the problem of oscillations of a body fully immersed in a liquid, which was originally stated by F. John as far back as 1950. The papers produced by Vladimir Maz'ya and his collaborators on this topic eventually led to the monograph [Ma-6]. <>

Even from this brief review it is amply clear that Vladimir Maz'ya's work has an astonishing range and depth. However, he has left a lasting mark of original ity and technical virtuosity in many more other branches of mathematics, such as

estimates for general differential and pseudodifferential operators in a half-space, Sobolev perturbed spaces anddomains, asymptotic theory of elliptic boundary-value problems on sin gularly in which Maz'ya has developed a rather sophisticated theory, first in collaboration with S.V. Poborchi, then jointly with S.A. Nazarov and B. Plamenevskii, as well as V.A. Kozlov and A.B. Movchan, which makes the subject of [Ma-8], [Ma-9], and [Ma-11], respectively, numerical analysis (cf. [Ma-2] written with G. Schmidt) ; history of mathematics, an area in which he has co-authored with Tatyana Shaposhnikova a delightful and highly informative book about the life and work of J. Hadamard ([Ma-5] , [Ma-12]); asymptotic theory of solutions to differential equations withanalytic operatorfunctions coefficients [Ma-10] , written jointly with V.A. Kozlov; and estimates for with a bounded real part, described in the book [Ma3] , based on the joint research with G . Kresin. This list should also include pointwise an area in which he has co-authored with LV. Gel'man the monograph [Ma-21];

interpolation inequalities for derivatives, approximation by analytic and harmonic functions, degenerate elliptic pseudodifferential operators, uniqueness theorems for certain boundary value problems with data prescribed on only a portion of the boundary, characteristic Cauchy problems for hyperbolic equations, iterative pro cedures for solving ill-posed boundary value problems, etc. Always animated by large, important ideas, magnanimous in sharing his ex pertise with other, particularly younger, people, one can only wonder what other magnificent contributions Vladimir Maz'ya will make in the future; we wish him many more years ahead, in good health.

V. Books (co-)authored by Vladimir Maz'ya. [Ma-l]

[Ma-2]

Theory of Sobolev Multipliers with Applications to Differential and Integral Operators,

with T. Shaposhnikova, Grundlehren der Mathematischen Wissenschaften, vol. 337, Springer, 2008. Approximate Approximations, with G. Schmidt, American Mathematical Society, 2007.

BIBLIOGRAPHY

xvi

[Ma-3]

Sharp Real-Part Theorems. A Unified approach, with G. Kresin, Lecture Notes in Math ematics, No. 1903, Springer, 2007.

[Ma-4]

Imbedding and Extension Theorems for Functions in Non-Lipschitz Domains, with S. Poborchi, St-Petersburg University Publishers, 2007.

[Ma-5]

Jacques Hadamard, un Mathematicien Universe!, with T. Shaposhnikova, EDP Sciences, Paris, 2005 (revised and extended translation from English).

[Ma-6]

Linear Water Waves. A Mathematical Approach, with N. Kuznetsov and B. Vainberg, Cambridge University Press, 2002.

[Ma-7]

Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equa tions, with V. Kozlov and J. Rossmann, Mathematical Surveys and Monographs, Vol. 85, American Mathematical Society, 2000.

[Ma-8]

Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Do mains, Vol. 2, with S. Nazarov and B. Plamenevskij, Operator Theory. Advances and Applications, Vol. 112, Birkhiiuser, 2000.

[Ma-9]

Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Do mains, Vol. 1, with S. Nazarov and B. Plamenevskij, Operator Theory. Advances and Applications, Vol. 111, Birkhiiuser, 2000.

[Ma-10] Differential Equations with Operator Coefficients, with V . Kozlov, Springer Monographs in Mathematics, Springer-Verlag, 1999. [Ma-11] Asymptotic Analysis of F ields in Multistructures, with V. Kozlov and A. Movchan, Ox ford Science Publications, 1999 [Ma-12] Jacques Hadamard, a Universal Mathematician, with T. Shaposhnikova, American Math ematical Society and London Mathematical Society, 1998. [Ma-13] Differentiable Functions on Bad Domains, with S. Poborchi, World Scientific, 1997. [Ma-14] Theory of a Higher-order Sturm-Liouville Equation, with V. Kozlov, Springer-Verlag, Lecture Notes in Mathematics, 1997. [Ma-15] Elliptic Boundary Value Problems in Domains with Point Singularities, with V. Kozlov and J. Rossmann, American Mathematical Society, 1997. [Ma-16] Elliptic Boundary Value Problems, with N. Morozov, B. Plamenevskii, L. Stupyalis, American Mathematical Society Translations, Ser. 2, Vol. 123, 1984, AMS. [Ma-17] Encyclopaedia of Mathematical Sciences, Vol. 27, Analysis IV, Linear and Boundary Integral Equations, S.M. Nikol'skii (Eds.), Contributors: V.G. Maz'ya, S. Prossdorf, 233 pages, Springer-Verlag, 1991, V. G. Maz'ya: Boundary Integral Equations, pp. 127-222. [Ma-18] Encyclopaedia of Mathematical Sciences, Vol. 26, Analysis III, Spaces of Differentiable Functions, S.M. Nikol'skii (Ed.), Contributors: L.D. Kudryavtsev, V.G. Maz'ya, S.M. Nikol'skii, 218 pages, Springer-Verlag, 1990, V.G. Maz'ya: Classes of Domains, Measures and Capacities in the Theory of Differentiable Functions, pp. 141-211. [Ma-19] Theory of Multipliers in Spaces of Differentiable Functions, with T. Shaposhnikova, Pit man, 1985 (Russian version: Leningrad University Press, 1986). [Ma-20] Sobolev Spaces, Springer-Verlag, 1985 (Russian version:

Leningrad University Press,

1985). [Ma-21] Abschatzungen fur Differentialoperatoren in Halbraum, with I. Gelman, Berlin, Akademie Verlag, 1981; Birkhiiuser, 1982. [Ma-22] Zur Theorie Sobolewsche Raume, Series:

Teubner-Texte zur Mathematik BSB B. G.

Teubner Verlagsgesellschaft, Leipzig, 1981. [Ma-23] Einbettungssatze fiir Sobolewsche Raume, Teil 2, Series: Teubner-Texte zur Mathematik, Band 28, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980. [Ma-24] Einbettungssatze fiir Sobolewsche Raume, Teil 1, Series: Teubner-Texte zur Mathematik BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1979. [Ma-25] Potential Theory and Function Theory for Irregular Regions, with Yu. Burago, Seminars in Mathematics, Steklov Institute, Leningrad, Vol. 3, Consultants Bureau, New York, 1969 (Russian version: 1967).

Bibliography [1] M.S. Agranovich, Yu.D. Burago, V.P. Khavin, V.A. Kondratiev, V .P. Maslov, S.M. Nikol'skii, Yu.G. Reshetnyak,

M.A. Shubin,

B.R. Vainberg,

M.l. V ishik,

L.R. Volevich,

Vladimir

G.

BIBLIOGRAPHY

xvii

Maz'ya, On the occasion of his 65th birthday, Russian Journal of Mathematical Physics

10 (2003) , no. 3, 239-244. [2] M.S. Agranovich, Yu.D. Burago, B.R. Vainberg, M.I. Vishik, S.G. Gindikin, V.A. Kondrat'ev, V.P. Maslov, S.V. Poborchii, Yu.G. Reshetnyak, V.P. Khavin, M.A. Shubin, Vladimir Gilele vich Maz'ya (on his 70th birthday), Russian Math. Surveys 63:1 (2008), 189-196, Uspekhi Mat. Nauk 63:1 (2008), 183-189. [3] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Grundlehren der mathematischen Wissenschaften, Springer, 1983. [4] E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo el littico, Boll. Un. Mat. Ita!. 4 ( 1968), 135-137. [5] D. Eidus, A. Khvoles, G. Kresin, E. Merzbach, S. Prossdorf, T. Shaposhnikova, P. Sobolevskii, M. Solomiak, Mathemathical Work of Vladimir Maz'ya {on the occasion of his 60-th birth day), Functional Differential Equations, 4 (1997) , no. 1-2, 3-1 1 . [6] I. Gohberg, Vladimir Maz'ya: friend and mathematician. Recollections, i n J. Rossmann, P. Takac and G. Wildenhain (eds.), The Maz'ya Anniversary Collection, Birkhauser Verlag, Basel, 1999, pp. 1-5. [7] C. E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Prob lems, CBMS Regional Conference Series in Mathematics, Vol. 83, AMS, Providence, RI, 1994. [8] V. Maz'ya, Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients, Funkcional. Anal. i Prilozen., 2 ( 1968) no. 3, 53-57; English translation: Func tional Anal. Appl. 2 (1968) , 230-234. [9] The Maz'ya Anniversary Collection, edited by J. Rossmann, P. Takac, and G. Wildenhain, University of Rostock, Germany, Birkhauser Verlag, Switzerland, 1999. [10] J. Pipher and G.C. Verchota, Maximum principles for the polyharmonic equation on Lipschitz domains, Potential Anal. 4 ( 1 995), no. 6, 615-636. [11] J. Pipher and G.C. Verchota, A maximum principle for biharmonic functions in Lipschitz and C1 domains, Comment. Math. Helv., 68 ( 1993), no. 3, 385-414. [12] G.C. Verchota, The use of Rellich identities on certain nongraph boundaries, pp. 1 27-138 in "Harmonic Analysis and Boundary Value Problems", Contemp. Math., Vol. 277, Amer. Math. Soc., Providence, RI, 2001. [13] G.C. Verchota and A.L. Vogel, A multidirectional Dirichlet problem, J. Geom. Anal. 13 (2003), no. 3, 495-520. [14] G.C. Verchota and A.L. Vogel, The multidirectional Neumann problem in llt4, Math. Ann. 335 (2006), no. 3, 571-644.

Dorina Mitrea and Marius Mitrea Columbia, Missouri

Proceedings of Symposia in Pure Mathematics Volume 79, 2008

Capacity, Carleson Measures, Boundary Convergence, and Exceptional Sets N. Arcozzi, R. Rochberg, and E. Sawyer Dedicated to Prof. V. Maz'ya on the occasion of his 10th birthday. He has opened many doors.

ABSTRACT.

We study the relationships between a capacity theoretic descrip tion of Carleson measures for the Dirichlet space (as well as for a discrete model space, a function space on a dyadic tree) and a description based on a testing condition. Using those relationships we show how analysis in the model space can be used to establish boundary convergence results for both the model space and also the classical Dirichlet space.

1.

Introduction and Summary

There is a fundamental relation between the capacity of a set and energy in tegrals of probability measures supported on that set. If the capacity is small the energy integral will be large; in particular sets of capacity zero cannot support prob ability measures of finite energy. Here we develop similar ideas relating capacity to Carleson measures. We show that if a set has small capacity then any probability measure supported on it must have large Carleson embedding constant. In par ticular sets of capacity zero are exactly the simultaneous null set for all nontrivial Carleson measures. Functions having limited smoothness often exhibit attractive or convenient be havior at most points, the exceptional set being of capacity zero; that is, the good behavior holds henceforth . . Using the relationship between capacity and Carleson measures such a conclusion can be reformulated by saying the function exhibits the good behavior J.L- a.e. for every Carleson measure J.L. This can be useful because sometimes it is relatively easy, even tautological, to establish that a property holds J.l - a.e . . We will use this viewpoint to give a new approach to results related to boundary behavior of holomorphic and harmonic functions.

quasi-everywhere,

q.e

2000 Mathematics Subject Classification. 30A40, 46E35, 46E39. The first author partially supported by the COFIN project Analisi Armonica, funded by the Italian Minister for Research. The second author's work was supported by the National Science Foundation under Grant No. 0070642. The third author's work was supported by the National Science and Engineering Council of Canada.

2

ARCOZZI, ROCHBERG, AND SAWYER

The dyadic Dirichlet space is a Hilbert space of functions on a dyadic tree. In many ways that space models the classical Dirichlet space, the space of holomorphic functions, f, on the unit disk ][)) for which JD 1!' 1 2 < oo. In Sections 3 and 4 we present background material on the dyadic Dirichlet space and the associated theory of Carleson measures. In Section 5 we present our new results on Carleson measures for the dyadic Dirichlet space. Those include the use of Carleson measures to measure capacity and a direct proof of the equivalence of the measure theoretic and capacity theoretic criteria for a Carleson embedding. In Section 6 we show how those results can be used to study boundary behavior and exceptional sets. Roughly, the idea is to work with the tree geometry to construct a majorant of the variation of the function being studied. If this majorant is in a discrete function space, X, then the boundary set on which the majorant is infinite must be a null set for every X -Carleson measure. We then appeal to results of Section 4 to recast this as a statement about the capacity of the exceptional set. The discrete case is a model case in which the proofs are relatively straight forward and the geometric issues we wish to highlight are particularly clear; and in this paper we will focus almost exclusively on that model case. However the ideas we present are very general and we believe they can be extended to the study of smooth functions whenever there is control of local oscillation. The details of extending these techniques to classes of smooth functions, and to function spaces that are not Hilbert spaces, requires some further ideas and technical machinery. In Section 7 we briefly discuss the types of extensions we have in mind and our approach to them; however we postpone a detailed presentation of that work to a later paper [ARSp] . 2.

Embedding Maps

2 . 1 . Capacity Conditions. Here we record some results which go back to classic work by Maz'ya. For proofs and references we refer to [AH] and [AE] . For a function f defined on JR.2 we define its Riesz potential of order one by -.6.) - l/ 2 f. We define the associated (p 2) capacity of a compact E c JR.2 hf ( by =

=

Cap(E)

=

=

�

Cap1 (E) 2

inf { 12 g2 dx : h g � 1 on E , g � 0, g inf { 12 I 'Vu l 2 dx : u � 1 on E, u

E

E

£ 2 (JR.2 ) }

C0 } .

( Below we will tacitly assume all sets discussed are capacitable.) The basic trace theorem in this context is:

THEOREM 2 . 1 . Suppose w is a nonnegative Borel measure on JR.2 . The following are equivalent: (1) There is a constant C so that Vu E C0

ll u ll£2 (dw ) � C II 'Vu ll£2 (dx)

(2) There is a constant C so that Vf

E

£2 ( dx)

·

llh f l l £2 cdwJ � c I I J I I £ 2 (dx) .

CARLESON MEASURES AND EXCEPTIONAL SETS

3

(3) There is a constant C so that for all compact E JR2 w(E) C Cap (E) . c

::;

There are similar results for functions defined in regions of JR2 and for spaces of holomorphic functions. In particular we recall the result of Stegenga [St] for D, the Dirichlet space of holomorphic functions on the unit disk !DJ;

{f

l f l i, l f(O)I2 + L 1 !' 12 dV < oo} . Recall that functions in V have boundary values a.e. on 11'. For any E !DJ we define the associated interval Iz in 11', the boundary of !DJ, to be the interval with center j l l and length 2 ( 1 - l z l ) . For any open set 0 contained in we define the tent over 0, T( 0 ) by Iz C 0} . T ( O ) = {z For such an 0 we define its D-capacity by Capv ( O ) = inf { ll f ll i, : Re f � 1 on 0 } . In particular, for an interval I T we have the estimate 1 Capv ( I ) "' l log I I I I THEOREM 2.2. [St] The following are equivalent for a positive measure w sup ported on ( 1 ) There is a constant C so that Vf E V (2.1) llfll£2 cdw ) c llfllv . (2) There is a constant C so that for any open 0 C (2.2) w ( T ( O ) ) ::; C Capv ( O ) . D=

E Hol (!DJ)

:

=

z

z

1I'

z

E

!DJ :

c

ii)) :

::;

1I'

2.2. Testing conditions. One approach to results such as the previous two is to find a good collection of testing functions and evaluate the desired inequality, for instance on those functions. The inequalities obtained are of course necessary in order to have ( 2.1 hold for all functions and, in the favorable cases, those conditions are also sufficient. A famous example where this approach succeeds is the following. Let be the Hardy space, the subspace of consisting of functions, whose negative Fourier coefficients vanish. Such a function, extends to a holomorphic function in !DJ which we also denote by

(2.1),

)

2 L2 (11', dB) f, f, H f. THEOREM 2.3. (Carleson) The following are equivalent for a positive measure w supported on ( 1 ) There is a constant C so that f E H2 (11') (2.3) llfll£2cdw ) ::; c llfiiPc'll' , diJ) (2) There is a constant C so that for any open interval I C (2.4) w(T(I)) C I I I . H2 is a Hilbert space with reproducing kernels; the functions kz (ei0) ( 1 .zei0) -1 are the kernel functions. Testing (2.3) on kz gives the estimate (2. 4 ) for the interval Iz. ii)) :

V

::;

1I'

=

ARCOZZI, ROCHBERG, AND SAWYER

4

A digression on terminology: The issues in the previous theorems are certainly similar to each other in spirit. However the questions arose in different contexts and are discussed using different words. The result in Theorem 2.1 is referred to as a On the other hand the measures in Theorem 2.3 are referred to as ( ) and the results in Theorems 2.2 and 2.3 are described as for the respective spaces. (Actually there is a lack of consensus on terminology here. Some call a measure a " Carleson measure" if it satisfies a measure theoretic condition in the style of (2.4) ; the measure of a region of simple shape is controlled by a function of the size of that region. Other people use the name to denote measures for which conclu sions such as (2. 1 ) and (2.3) hold; there is a continuous embedding of the space being studied into a Lebesgue space associated with the measure. We use the second style and call the measures of The orem 2.2 Carleson measures. Also we call the embeddings whose continuity is insured by (2.1) or (2.3) Carleson embeddings and re fer to the norms of those embeddings as the embedding constants of the measures or as the Carleson measure norms of the measures.)

trace theorem. Carleson measures for the Hardy space characterizing the Carleson measures

Although the testing philosophy is a guide to the correct conjecture in this case, that is not always so. Similar ideas for the space V would lead to speculation that it was sufficient to have (2.1) for all intervals rather than all open sets. However that is not the correct answer and Stegenga provides the appropriate examples. Yet there is a slightly more sophisticated testing scheme which can be used effectively to study Carleson measures on the Dirichlet space. The idea, which goes back to work by Kerman and Sawyer [KS] , is to test the embedding operator and also test its adjoint. One main theme of this paper is the relationship between results such as The orems 2 . 1 and 2.2 formulated in terms of capacity and the more measure theoretic results suggested by the testing scheme just described. For more about capacity in general, and in particular the fact that set func tionals which appear to be quite different can be used to measure capacity, we refer to [AH] , [AE] , [V] , and [KV] .

I,

3 . The Dyadic Dirichlet Space We will work with functions on the dyadic tree T which we now define. T is a rooted, directed, loopless graph. It contains a root vertex, which is connected by two edges to vertices Or and 01 (its right and left children) . Every other element a is connected to three vertices; one, its parent, a:- , is on the path connecting a to o, the other two are its children, O:r and 0:1 . For a:, (3 E T we denote by [a:, (3] the set of vertices on the geodesic path from a to (3. We let d(a, (3) denote the length of that path, d( a, (3) = I [a:, (3] I - 1 , and we will abbreviate ( a) to ( a). For a:, (3 E T we write (3 -< a if (3 E [o, a:] and we define the successor set of (3 to be a t (3 } 8 ((3)

o,

d o,

= {a: :

d

CARLESON MEASURES AND EXCEPTIONAL SETS

5

f a function D I I* f if a = o Df (a) = { f (��o� f(a+ ) otherwise, If (a) = L f (J) , I* f (a) = L f (J) . We have I D = D I = Id, the identity operator, and I* is the adjoint of I with respect to the inner product on l2 (T) . We envision T as a subset of ][)) with at the origin and with the 2n vertices in {aand ofd( radius ) = n} located at equally spaced points on the circle centered at the origin 1 - 2-n . With this in mind we define the dyadic Dirichlet space, 'Dd, to be the space of functions F on T for which I F I i,d = I F (o)i2 + Lr I DFI 2 < oo. Informally we think of T as sitting in ][)) and of holomorphic functions F E 'D as being modeled by their restrictions to T. Oversimplifying a bit we have these restriction are in 'Dd and there is an informal correspondence between the operators J; and (1 - i z l 2 ) d� acting on holomorphic functions and the operators I and D operating on functions on the tree. This point of view is developed in detail in

We will work with several operators defined on functions on T. For by on T we define functions J, J, and

-yE[O ,n]

")' ES(n)

:

o

a

[Ar] , [AR] , [ARSl], and [ARS2] . In this context we will associate to any measure, p, defined on ][)) a measure f..LT which is defined on T by setting equal to the f..L measure of the set of z E ][)) which are hyperbolically closer to (thought of as sitting in than to any other point of T (with any convenient choice in cases of ties). We will also be interested in functions and measures defined on the (abstract) boundary of T, &T, which we now introduce. A point of aT is an equivalence class of geodesics in T each infinite in one direction. The geodesics r and f' are equivalent exactly if their symmetric difference, (r u f') \ (f n f') is finite. A convenient representative for a class is the unique geodesic in the class starting at the root. For each E T let be the set of T in aT with the property that every representative geodesic for T intersects The topology of aT is defined by declaring the sets to be a basis for the open neighborhoods. We set T T u aT. Note that any representative geodesic r for a point of &T, when viewed as a subset of ][)) , is a sequence that converges to a point of 1!'. Hence when we think of T as a subset of ][)) it is also convenient to think of aT as identified with 1!' with its usual topology. This lets us identify open intervals of 1!' with open sets &T. Using this identification we will regard measures defined on 1!' as also defined on aT and vice versa. (The pairing of aT with 1!' has a countable set of ambiguous points, but these cause no difficulty in the discussion which follows.) Finally, if a function F on T has a limit along a geodesic r then we extend it by continuity to the corresponding point of &T. In particular if has finite support then extends to the entire boundary. We define capacity for subsets E of T by

J.Lar (a)

D)

,

a

=

aS(a) aS(a)

S(a) .

f

If

(3.1)

6

ARCOZZI, ROCHBERG, AND SAWYER

Alternatively, there is a straightforward variation which we will use later Capr(E) = inf{ I!I*J.LII �2 : J.L a positive measure on E,

2':

Il*J.Li e 1 } . (3.2) The quantity I I* J.LII �2 is sometimes called the energy integral of J.L and denoted

£ (J.L) .

'Dd and 'D Here is the characterization of Carleson measures for 'Dd and 'D from [ARSl] which complements Stegenga's description of Carleson measures for the Dirichlet 4. Carleson Measures for

space. (Actually the final statement of the theorem, a capacitary condition in the style of Stegenga, is not in [ARSl] but can be included, for instance, by virtue of Theorem 5.6 below.) We will also be interested in the radial variation of functions in and we now introduce a function which measures that. For in we define

'D

f 'D

THEOREM 4. 1 . Let J.L be a positive Borel measure on Then, the following conditions are equivalent: a Carleson measure for 'D; i.e. (2. 1) holds. (1)(2) J.LJ.L issatisfies stant C so athatCarleson fV E 'Dinequality for the radial variation. There is a con k I V ( !)( ) l 2 dJ.L :s; C l f llb · (3) J.LT is a Carleson measure for 'Dd. There is a constant C so that Vf E Dd ( 4.1) L I F ( a) l 2 J.Lr ( a) c I F I ;d . (4) J.LT satisfies the tree condition. There is a constant C so that \fa E T (4.2) (5) J.LT satisfies a tree capacity condition. There is a constant C so that for all sets E Uj8S(xj ) in 8T, (4.3) J.Lr (uj S(xj )) C Capr ( Uj8S(xj ) ) . The condition (4.1) states that the " integration map" I is bounded from l2(T) into l 2 (T, J.LT) . We could analyze that condition by testing it on simple functions as follows. Pick a E T and set f = d( a )-1X(o. a]· If (4. 1 ) holds then we find (I* J.Lr )(a) J.L (S (a)) :s; I If ii Z2 (T,fL) C I I JI I ;d d (a) - 1 . The condition obtained, d ( a) (S (a)) C, is equivalent to the condition (4.3) restricted to the case where { Xj } has only one element. As was true in Stegenga's theorem that condition is necessary for the boundedness of I but not sufficient. However if I is bounded then so is its adjoint, I*, mapping 1 2 (T, J.Lr) to l 2 (T). if».

z

:s;

o.ET

=

:s;

=

:s;

J.L

:s;

=

CARLESON MEASURES AND EXCEPTIONAL SETS

Condition (4.2) tests the boundedness of weakened version of the statement that

7

I* on the functions Xs(a) ; (4.2) is a slightly

Let 11 be a positive, Borel measure on T. We denote the best constant in the testing condition (4.2) by

[ M [ cM(T) ; I M I [ cM(T) = aEsupT I* (II** ll11)(a2)(a) .

(4.4)

By the previous theorem this quantity is comparable to the norm of the Carleson embedding on VJ. 5. Capacity and Carleson Measures on Trees. 5 . 1 . Computing Capacity Using Carleson Measures. The ordinary def inition of capacity for a subset E of T is that in We now give an alternative way of computing capacity for subsets of T using Carleson measures. To minimize the notational burden we give the proof below for E c which is the case of primary interest. The extension to the general case is straightforward.

(3.1).

fJT

THEOREM 5.1. Let E C T be compact. Then, f..l ( E) ( 5. 1 ) Capr (E) = sup (!')CE [ M [ cM(T) Before proceeding with the proof we note that the quantities which appear in the definition [ M [ c M(T) are in fact related to quantities which show up in the study of capacity. From the definitions we have that I* 11 (a) = f..l ( S(a)). If we multiply out the square in I* ( I* 11 )2 ( ) we find the comparable quantity I* (I*f..l) 2 (a) "' L a d(a,(3 /\ {31)f..l(f3)f..l (f3') . Supp

a

{3,{3' 'r

Thus, completely informally, the ratio in (4.2) is comparing the mass of the measure f..lX S( a ) to a type of energy integral of the same measure.

a T a

S(a)

P ROOF. We start with inequality [ � ] . For fixed E we denote by Ta = the subtree of T having root a and we add a subscript to the corresponding tree objects: is the capacity of subsets of Ta , Ea is the energy in Ta , Wa is the extremal measure in the definition of capacity, and so on. Let Ea = E n Ta . The extremal measure Wa and the function 'Pa = satisfy

Capa

I�w

We claim that Wa is a rescaling of the extremal measure for E in f', to Ea :

(5.2)

w, restricted (5 . 3) Here, a- 1 is the predecessor of in T. In fact, Wa mm1m1zes Ea ( f..l ) over all measures such that Iai�f..l (�) � 1 on Ea (with the possible exception of a set having null-capacity) . On the other hand, we claim that w[ E" minimizes Ea (v) among all measures on Ea such that Ia � v (� ) � 1 - II*w (a - 1 ) p. p . on Ea . a

11

z;

i

ARCOZZI, ROCHBERG, AND SAWYER

8

Suppose this is not the case. Then there exists a measure � E Ea and

Iai;v(O � 1 - II*w(a- 1 ) for

p.p.

Eo: (v) = L (I * v) 2 <

L (I* w) 2

Ta

Consider the functions T:

=

v on Eo: such that

Eo: (w iEJ ·

Ta

I*w in T and I;v = I*v in Ta . Define a new function 'lj; on 'lj; (x) =

{ l*w(x) I*v(x)

if x E To; , if x E T \ To: .

We have l'lj;(�) hence ll'l/J II i2

�

�

1 q.e. on E,

Cap(E) . On the other hand, ll'l/J IIi2 = Ea (v) + [£ (w) - Ea (w) ] < £ (w) = Cap(E),

and we have reached a contradiction. The measure

A

=

1-

w iEa Il*w(a -1 ) '

Ea (J.J-) over the set of the measures p.then, p. By� minimizes in Ea , hence A = Wa· The claim is proved. the homogeneity of the energy,

J.J-

such that

II* J.J-(�) �

1 for

Ea (w iEJ = ( 1 - Il* w(a-1)) 2 Ea (wa) Il* w(a-1)) 2 wa(Ea) = II* w(a-1))w(Ea)· =

(1 (1 -

As a consequence,

:t�=)

"' L..J x >a (I*w)P' pl -p' = £ ( I ) (1 - V(w)(a-l))P'-1 = l*w

(5.4)

o w(E) Cap(E) = w(E) = llwllcM .

with equality if and only if a ll w llcM = and

1

We now prove J.J-. Then,

:::; 1 ,

= (we set the default value II*w(o- 1 ) = 0). Hence,

[�] . By definition of II · llcM , £ ( J.J- ) :::; IIJ.J-IIcMJ.J- ( E) for all measures J.J-(E) 2 J.J-(E) < J.J-(E) < = (J.J-) M £ IIJ.J-IIc J.�.( E )

as wished.

( t:(IJ.) )

Cap(E ),

D

Carleson measures.5.2. E has capacity zero if and only if it is a null set for all C OROLLARY

9

CARLESON MEASURES AND EXCEPTIONAL SETS

4 5

4

5.2. On a Question of Maz'ya. Conditions and of Theorem are, by that theorem, equivalent to each other. Some time ago Prof. Maz'ya asked if we could give a direct proof of that equivalence, one not relying on the fact that both conditions characterize the same class of Carleson measures. We do that now. In our proof we will use the fact that if a measure satisfies condition then so does any smaller positive measure. That fact follows from knowing that condition characterizes a class of Carleson measures. However, in the spirit of this section, we also give a proof of that fact which is relatively elementary and which does not involve the theory of Carleson measures. We begin with the monotonicity. For a measure on let =

4

4

11 T, aJ.L (I*11)2. THEOREM 5.3. Let 11 be a measure on T and let >. be a measurable function on T, 0 � >. � 1 . If I*aJ.L � I*/1 on T, then I*a>.J.L � 2 I*( .AJ1) . 5. 4 . If v � f1 and f1, v are measures on T, then ll v ll cM(T) � ·

C OROLLARY

2ll f1 llcM(T) ·

P ROO F . By rescaling, it suffices to verify the conclusion at the root. We use a simple argument based on distribution functions. Let

I*(.AJ1)(y) MJ.L >.(a) = omax �y�a I *J1 (Y ) be the discrete maximal function we used in [ARS3]. If necessary, we can extend the definition to a E &T in the obvious way. Then, r aAJ.L (o) � [ I*I�>-;��) r (l* J1 ( a))2 � L [M J.L.A (a)] 2 aJ.L (a ) aET = 2 1 1 taJ.L(( E T : MJ.L.A (() > t)dt. Now, {( E T : MJ.L>.(() > t} US(aj ) is the disjoint union in T (by the definition of the maximal function, we do not need to consider the closure of S(aj) in T) . Then, taJ.L (( E T : M J.L.A (() > t) Lj taJ.L (S(aj)) � 2:tl* J1 (aj ) j � L J * ( .AJ1) (aj) j I* � ( .AJ1 ) ( ). Inserting this estimate in the previous one, we have I* a>.J.L (o) � 2 · I* aJ.L (o). =

=

=

0

0

We now give a direct proof that the tree condition is equivalent to the capacitary condition, Theorem below. That the capacitary condition implies the tree

5.6

10

ARCOZZI, ROCHBERG, AND SAWYER

condition was noted in [KS] and earlier as Theorem 4 in [Ad] . We proceed to the opposite implication. We need an estimate for measures supported in T. L EMMA

5.5. We have

Capr (S(E))

� 4 Capr (E),

for E = U8S(aj), where S(E) = US(aj)· Note that Capr (S(E)) P ROOF .

Let

=

Capr ( { aj } ). Obviously, Capr(S(E)) � Capr (E).

cp be the extremal function for Capr(E) :

We show that it is near extremal for S(E). The function cp can be recovered from the equilibrium measure, J.L is clearly constant on each 8S(aj): there is rj > 0 such that Now, for all ( E 8S(aj ) , 1 - Icp(aj)

= Icp(() - Icp(a) = I: cp([J) =

=

-

cp = I* J.L, and

i3E[oj ,(] rj I: T d(/3) [oj ,(] rjT d( oj ) cp(aj)·

Hence, 1 cp(aj ) = I cp(aj) . Note that cp, the extremal function, is monotone increasing with respect to the partial ordering in thus cp(aj) � 1 / (aj)· Hence,

T,

This means that

d

2cp is an admissible function for E: D

5.6. Let be a positive, Borel measure on T. Then J.L satisfies I* [I* J.L] 2 (a) sup (5. 5 ) � CI (J.L). oET I* J.L(a) if and only if satisfies, for all sets E = Uj8S(aj) in 8T, J.L ( UjS(aj) ) � C2 (1-L) Capr(Uj8S(aj ) ) . (5.6) Moreover, C2 (J.L) ,....., CI (J.L) . T HEOREM

J.L

J.L

CARLESON MEASURES AND EXCEPTIONAL SETS

(5.5)

Suppose that f.t satisfies with C1 (t-t) U S (o:j ) · Then, f.tE = t-t l s ( E ) :::; f.t satisfies ll t-tE II cM(T) :::; P ROOF.

Capr ( S ( E ))

=

sup sup(v)�S(E)

11

1. Recall that S ( E )

=

=

2 by Theorem 5.3. Hence,

S(E)) v( ll v llcM(T)

t-t(S(E)) - li t-tE iicM(T) > t-t(S(E)) >

- 2 .

i.e., t-t ( S ( E )) < 2 Capr ( S ( E )) < 8 Capr ( E ) , by Lemma opposite inequality is already known.

5.5.

As mentioned, the 0

6. Boundary Behavior and Exceptional Sets In this section we give a number of results about boundary behavior and ex ceptional sets for the dyadic Dirichlet space. In several cases we show that certain behavior occurs with an exceptional set that is a null set for a class of Carleson measures. Then by Corollary or a variation of that corollary, we conclude that the possible exceptional set has capacity zero. The results we present now are discrete analogs of established results about boundary convergence of smooth functions and about the associated exceptional sets. The literature on those problems is extensive. We offer [A] , [AC] , [Ca] , [DB] , [GP] , [Ki] , [M] , [NRS] , [NS] , [Tw] , and [W] as recent representatives as well as places where the reader can get more information. In particular the versions of our next few results for harmonic functions are in [Tw] . A main theme here is to show that when appropriate oscillation estimates are available then there is a unified approach to such results. In particular this approach highlights the basic geometry of the tree model, or, what is roughly the same thing, the geometry of a Whitney covering of the domain.

5. 2 ,

6.1. Boundary Values. It is not clear at first glance that functions in Vd must have boundary values on a large subset of oT. We now establish that with an argument whose basic form is at the core of the later discussions. For each positive integer n let Xn be the characteristic function of the set {a: E T : a:) :::; n} . Given F E Vd , set, for each n

d(

F�

I ( I D F I · Xn ) · It is immediate that the sequence of functions { F; } is increasing in n, that each function extends by continuity to all of OT, and that each function is in Vd and has norm at most I I F I . By monotonicity the extended limit lim F; = F* is defined everywhere. It then follows using Fatou's lemma that I F* 1 2 has finite integral with respect to any Carleson measure f.t for Vd . Hence F* is finite f.t - a.e . . With F* in hand as a majorant for the variation of F along each geodesic it is easy to see that F also has boundary limits f.t - a.e. Hence, by Corollary F has finite boundary limits q.e. =

5. 2 ,

12

ARCOZZI, ROCHBERG, AND SAWYER

6.2. Beurling's Theorem. The prototypical result in this area is Beurling's theorem that any E V has radial boundary values He did this by showing that the boundary values of the radial variation, were finite and noting that at such points must have radial boundary values. In fact the argument in the previous paragraph is essentially a complete proof of a discrete analog of Beurling's theorem. For F E Vd we measure its radial variation by =

f

V(f)(eiq.e.t )

f

q. e . Vr(F)(a) I(I DFI ) (a:).

6.1 (Discrete Beurling's Theorem) . For F E vd, Capr ({r E 8T : Vr(F)(r) +oo }) 0. Hence Capr ( {r E 8T : lim F(a:) does not exist}) = 0. Start with F E vd. Hence DF E l 2 (T) , and hence I D F I E l 2 ( T) , and thus Vr(F) E Vd . This insures that the set on which Vr(F) = oo is a null set for any Vd Carleson measure and hence, by Corollary 5. 2 , has vanishing capacity. Finally, one easily checks that if Vr(F)(r) < oo then limaa F(a:) exists. T HEOREM

=

=

aEr

P ROOF.

0

6.3. Algebraic Approach Regions. We continue to focus on Vd and now consider boundary limits through more general approach regions. For any subset S c T we define the boundary limit through S of a function defined on T by

F

limF(f)) = S

lim

{3E S d(/3) ---> oo

F(f)).

At the level of metaphor, convergence along the geodesics C T is similar to both radial convergence in the disk and to non-tangential convergence through a narrow wedge with apex at the boundary point corresponding to The analog of non tangential convergence with wider wedges is obtained by also including points that are at most a fixed distance, from f. For E T, E N let

r

r.

k, a: n {a: + n} = {(f) E T : f) >- a:, d(a:,f)) n)} and we define r 1 (k) = U {a: + k} . 6.2. Fix k 2:: 1. For F E Vd, F has f1 (k) limits q.e.; that is Capr ({ f E 8T : lim F(a:) does not exist}) = O. For F E Vd we define DF ( 1 ) by DF< 1 l (a:) = I DF(a) l + L I DF (f)) l . Because DF E l 2 (T) we also have DF< 1 l E l2 (T) , the reason is that each value DF(J)I shows up(las part of DF( 1 l(J) and as part of at most one other DF(ll(J'). IHence G = I(DF l) E Vd and thus G has boundary values q.e. Suppose now that rG(r). is a geodesic representing a point in the boundary at which G has boundary limit We claim that it follows that F has a boundary limit through the approach region r 1 ( k). We are assuming (6.1) limG(a:) - G(f) 0, =

aEr

T HEOREM

rl (k)

P ROOF .

/3E{a+k}

r

=

13

CARLESON MEASURES AND EXCEPTIONAL SETS

aa. inLetr andk belet thej3, {3'direct be two points of r ( k) which are further from the ancestor of a of order k. Now we have that aj F (/3) - F(/3' )1 S j F(/3) - F(a) j + I F(a) - F(/3')1 L I D F( 'y) l + L I D F('y') l "IE[o:, ,6'] "tE[o:,,6) k (6.2) 4 (G(r) G(a- )) . To see this last inequality we concentrate on the first sum. The geodesic segment [a, /3] has two regions, the first consisting of 1's in r, the second part consisting of the 1"s not in r. In the first case j D F b) l S DG('y ). In the second case set {3('-y') = 1'- k , the ancestor of 1' of order k. We have {3('-y') E [ a - k , a] and thus I D F('y') l DG(/3('-y' )). Thus the sum of the I DFI along [a, /3] is dominated by twice the sum of the DC along [a- k , a]. Finally, because G is increasing we get (6.2). Furthermore (6. 1) insures that the right hand side of (6.2) will tend to zero as a tends to the boundary; and hence F has the desired limit through r ( k). We now consider larger regions. Fix an integer k and set r2 (k) = o:Eur {a + kd(a)} . We will consider limits over the sets f 2 (k). To do this we start with F and construct a majorant for its variation. Define DF(2l (a) = I DF(a)l + kL- 1 max {I DF(/3)1 : j3 E {a + kd(a) + j}} IfF E Dd then DF E l 2 (T). We now claim that DF< 2l E l2 (T). Each DF<2l(a) is a sum of k + 1 terms taken from the square summable sequence {DF(a)} so we only need show that there is an upper bound on how many times an individual element of that latter sequence is used in this construction. However, in fact, no term is used more than twice. The term I D F(/3) 1 does appear in DF< 2l(j3). The only other time that term can be used as a summand is in D F ( 2) (a) for the unique a which satisfies the two conditions a E [a, /3] and (k + 1) d(a) S d(/3) < (k + 2) d(a). This observation together with an application of the Cauchy-Schwarz inequality to the sum used in defining DF< 2 l(a) shows that DF (2) E l 2 (T). A minor modification of the argument we used before now shows that F con verges along f 2 (k). The only required change is where before we backed up from a to an a- k now we have to back up from a to a' E [a, a] with (k + 1 ) d(a') > d(a). 6.3. Fix k 2:: 1 . For F E Dd , F has f2 (k) limits q.e.; that is Capr ({f E 8T : lim F(a) does not exist}) = 0. Now pick root than

1

s;

s;

s;

1

0

j =O

T HEORE M

r2 (k)

Again at the level of metaphor we can describe the geometry of the regions which correspond to these types of approach. We give the description in the upper half plane with the positive imaginary axis as the geodesic of interest. The geodesic convergence in the tree corresponds to convergence in the Stoltz region y > The thickened geodesics correspond to the wider, but still non-tangential, approach regions y > The regions correspond to regions which are tangent to the boundary with the tangency being of finite order; roughly,

2- kfl1x(k)l .

f2 (k)

lx j . r2 (k)

ARCOZZI, ROCHBERG, AND SAWYER

14

y

corresponds to the region > l xlk+l . Results on tangential convergence and the size of the associated exceptional sets go back to Kinney [Ki] and more general versions are in [Tw] .

6.4. Beyond Algebraic Approach Regions. The previous result can be extended to regions which are tangent of infinite order to the boundary but at a cost; the convergence will be quasi-everywhere but now quasi-everywhere with respect to a The capacities will be those associated with the spaces the Hilbert space of F functions on T for which

Vd,o,

different capacity.

(6.3) The approach regions of interest are these. For 0 T which defines an element of aT we set

r3 (c)

=

< c :::; 1 and r a geodesic in

u { a + [2cd (a)] } . a Er

(Hereafter we will regard the nearest integer brackets as implicit and will not write them.) We begin by a straightforward modification of the argument which gave Theo and construct a majorant for its variation. Define rem 6.3. We start with F E

Vd

j= 2· 2ed(c.) (6.4) DF(3l (a) IDF(a) l + L max {IDF(;J) I : ;J E {a + j}} = ed(c.) j 2 If F E Vd then DF E 2 (T) , but now it need not hold that DF(3) E 2 ( T) . For each let {;J(a) j } be the vertices of T which appear on the right hand side of (6.4); that is, and the selected elements where max IDF(;J )I is attained. Thus DF(3 l(a) = L IDF(;J(a) j ) l . j =

l

0!

l

0!

Hence

I DF(3l (a) l 2 :::; (Lj IDF(;J(a)j ) l 2 ) · (number of j 's) :::; C2cd(a) (L IDF(;J(a)j ) l 2 ) .

j Hence the sequence of numbers { 2-cd(a ) j DF(3l (a) \ 2 } is summable because, again, no vertex shows up as a ;J(a) j more than a few times. Thus DF(3) E ( T, 2-cd (<>l). We now use the same arguments as before. Set G J(DF(3l ); G will have finite =

l2

radial limits along every geodesic r with the possible exception of a set which is a null set for every Carleson measure for the space Also as in the previous proof, any boundary point r at which J(DF(3l )(r) < oo will be a boundary point where we have good convergence of F; in this case the good convergence meaning convergence over r (c) . 3 The description of the Carleson measures for these spaces is given in [AR] . Here is the description for 0 < c < 1.

Vd,o·

6.4. Suppose 0 < c :::; 1 . Let be a positive Borel measure on T. Then, the following are equivalent: T HEOREM

p,

15

CARLESON MEASURES AND EXCEPTIONAL SETS

is a Carleson measure for Vd,c: i i.e. there is a constant C so that Vf E vd,E (6.5) aET (2) satisfies the E - tree condition. There is a constant C so that Va E T (6.6) (1)

p,

p,

Hence we have

THEOREM 6.5.exception Fix E, 0 of< ca <set1 .which For FisE aVnull d , F has f3 ( c ) limits for all f E aT with the possible set for every measure which satisfies the condition (6.6). p,

In this case the approach regions have infinite order tangency, in fact subexpo nential contact. The Euclidean analogs of these regions shaped like the part of the upper halfplane where > exp ( Finally if a variant of Theorem 5.1 is available i n this context the result can be reformulated as q.e. convergence with respect to the appropriate capacity. In fact such a theorem does holds; its statement and proof are similar to the c = 0 case considered earlier; details will be in [ARSp]

- lx l -c).

y

6.5. The Result of Nagel, Rudin, and Shapiro. It is a result of Nagel, Rudin, and Shapiro [NRS] that, with a possible exceptional set of Lebesgue mea sure zero, functions in the Dirichlet space approach their radial boundary values through approach regions of full exponential contact; that is, with the shape of the set > exp ( Further work in that direction is in [NS] , [DB], and [Tw] . It would be interesting to know if an analogous result holds for Vd and we leave that as a question. The proof just given shows that any Vd has limits along regions f3 ( l ) with an exceptional set that is a null set for all the measures which satisfy with c = 1. However the full boundary is, in fact, such a set. The quickest way to see that is to note G = with = d(a) -1 has boundary values identically +oo. We could consider the subspace of Vd ,1 consisting of martingales; the Carleson measures for that subspace have null sets which are exactly the sets of Lebesgue, for that see [AR] . However it is not clear how to use that result in this context. One difference between the two cases is that the proofs for harmonic functions make systematic use of the reconstruction of the interior values of functions from the boundary values; in contrast the values of an Vd on T are not determined by the boundary values.

y

- lx l - 1 ).

FE

Ig

(6.6)

g(a)

FE

6.6. Boundary Convergence for a BMO-type Space. The Carleson mea sures for V are the positive measures on the disk which satisfy the equivalent con ditions of Theorem 4. There is an interesting subspace X of V consisting of those V which generate Carleson measures in the following sense:

fE

{ f E V : l f '(z) l2 dxdy is a Carleson measure for V } . The discrete analog is Xd = { F E vd : = I DFI 2 is a Carleson measure for vd } . X=

/1F

16

ARCOZZI, ROCHBERG, AND SAWYER

As discussed briefly in [AR] , the space X has a relation to V similar to the relation the space B M 0 has to the Hardy space In fact one of the characterizations of functions in BMO is that E BMO if and only if (1 is a Carleson measure for the Hardy space. The functions in BMO are both smaller than and smoother than generic functions in Similarly functions X are smaller than generic functions in V. Specifically if E V then for some small c f it holds that exp(c f has integrable boundary values; however E X insures that the boundary values of exp (exp(cJ are integrable. Similar results also hold for the model spaces on trees, for all this see [LL) . Here we obtain a different result, but one in the same spirit, the functions in xd have nicer properties than the generic elements of vd . For comparison recall that Theorem 6.1 states that

H2.

f

l f'(z) l 2

H2. f

lfl2)

l z l 2 )dxdy

f

lf l ) )

(6.7) VF E Vd lim F(a) " � DF(a) exists for quasi-every r. Suppose now that F E xd is fixed J.l(Ct) /lF(a) I D F(a) l 2 . Recall that the tree condition for is that there is a C so that for all a I* (l* J.1) 2 (a) :::; Cl* J.i (a). 6.6. Suppose is a Carleson measure vd then L J.l(S(a)) < 00 for quasi-every r. Equivalently j log+ l z _1 ei l dJ.i (z) < oo for quasi-every () . In particular, if F E Xd then L (I* (DF) 2 )(a) < 00 for quasi-every r. The argument proving Theorem 6.1 applies to Ih for any square sum mabie function h defined on T.The tree condition evaluated at the origin insures that h(a) l * J.i ( a) 11(S(a)) is such a sequence. That gives the first statement. The second follows from the first by estimating how often each value J.l ( f3) occurs in the sum. By writing out all the terms in ( J * (DF) 2 )(a) and then discarding those corresponding to vertices not on r we obtain a weaker, but more transparent, a:Er

=

a:Er

=

=

11

T HEOREM

/1

a:Er

e

a: E r

P ROOF.

=

=

D

corollary. C OROLLARY

If F E Xd then L d(a)I D FI 2 (a) < 00 for quasi-every r.

6.7.

a:Er

These results describe radial convergence and, as with the results for functions in V, they can be extended to larger convergence regions for both

II* (I DF I 2 )(f3)and 1 + lim j log I 1 - wz I dJ.i (z) .

lim (3

w

17

CARLESON MEASURES AND EXCEPTIONAL SETS

7. Possible Extensions

Here we briefly and very informally discuss how some of these ideas will be taken further in [ARSp] .

T

7.1. Other Function Spaces on Trees. Various function spaces on have been studied both on their own and as discrete models for spaces of smooth functions such as Besov spaces. This view is developed among other places in [Ar] , [AR] , [ARSl], [ARS2] where in addition to study is also made of various weighted spaces. The arguments of the previous sections adapt directly to show that such functions converge to boundary values through various approach regions with exceptional sets that are null sets for classes of Carleson measures. One way to get further insight is to develop geometric characterizations of the relevant classes of Carleson measures. For the spaces mentioned that is done in the earlier work by the authors. To go to results involving capacity we need a result such as Corollary 5.2. For the function space described in (6.3) the proof we gave in the case a = 0 continues to work with straightforward changes. However for p # 2, for instance for the dyadic Besov spaces of [AR] , one needs to work with the nonlinear potential theory appropriate for spaces and the arguments are more complicated. That work will be presented in [ARSp]

l2(T)

lP(T)

lP

7.2. Holomorphic, Harmonic, and other Smooth Functions. Results such as those we described for model spaces such as Vd can be used to obtain results for spaces of smooth functions. Suppose for instance that we want to derive a version of Beurling's theorem [Beu] .

For all j E V {eit f(rei0 ) fails to exist) 0. First select and fix f E V. Also select R so large that the hyperbolic disks of radius R centered at points of the tree, { D( R) E T} is a cover for and so that for all E T we have { C D( R). For each E T we measure the } local oscillation of f by Osc(a) Osc(a, f) = sup { l f (z) - f(z')l : z, z' E D(a. R)} . Straightforward considerations of the geometry of the placement of T in show that the disks { D( 5R) T} have the property that there is an M so that no point is in more than M disks. This insures that { Osc ( )} E l2(T) because T HEOREM 7 . 1

(Beurling, 1940) . Capv( lim :

a

a

-

=

r

a,

, an at

IDI

: a

a,

a

=

a,

][JI

: a E

,

function theoretic estimates yield

Osc(a ? :S

a

C j l f' l2 dxdy. D (a,5R)

The finite overlap of the disks and the definition V insures that the integrals on the right can be summed. Theorem 6.1 then insures that limr I* Osc(a) is finite except for a set of f of 'Dr capacity zero. This insures that has a limit along the path connecting the vertices in and that in turn is enough to insure that also has a limit along the radius which terminates at the point of the circle corresponding to the boundary element determined by r.

r

f

f

18

ARCOZZI, ROCHBERG, AND SAWYER

This outline gives convergence off of an exceptional set is of Vr capacity zero rather than V capacity zero. We hope to return to the general question of the relationship between null sets for discrete capacities and for continuous capacities. In this particular case however, the capacities associated with Vr and with V, the two collections of null sets are known to agree, as is shown by Benjamini and Peres [BP].

A limitation of the proof we outlined is that it established the existence of boundary limits rather than finiteness of the variation functional How ever that was just for convenience of presentation. A slightly more elaborate def inition of oscillation together with a similar argument, but using the fact that functions in Vr have nontangential limits, would establish the variation result. We would like to emphasize that large parts of the previous argument do not involve holomorphy at all. If one has local oscillation estimates and knows that the oscillation numbers live in a space X, for instance X could be a weighted then the argument shows that limits exist along with an exceptional set of that is a null set for all the Carleson measures for X. So, for instance, these arguments can certainly be used with harmonic functions or holomorphic functions of several variables. Also, there are other, rather different types of function spaces such as A-harmonic functions and monotone Sobolev functions where such oscillation estimates are available; see for instance [KMV] , [MV] . These types of variations have not been explored

V(f)(ei0).

rlP(T),

r

7.3. Final Questions. One of the themes in the study of boundary value results for, say, harmonic functions is consideration of whether the description of the exceptional sets is sharp. That is also a natural question in this context but we haven't considered it. We conclude by mentioning two areas where we do not know if the approach we have been describing can be used but the possibility is intriguing. First, the study of radial variation for bounded holomorphic (or harmonic, p- harmonic, etc.) functions on the unit disk (or ball, tree, etc.) is a very active research area. The indications so far are that the results there are deeper than and different from the results for, for instance, general Hardy or Besov spaces. The paper [CFPR] includes some general discussion of the area and references. Second, although we considered various types of approach regions for bound ary convergence, they were all of the same sort, a geodesic with a symmetrical enveloping shell which, in the Euclidean sense, narrowed as the region approached the boundary. These are all versions of having boundary limits along a collection of paths, the geometry of the envelope controlling the type of paths. However one can also consider convergence to boundary values through a collection of sets which is (in some appropriate sense approximately) translation invariant and contains no paths. This theme has a long history, recently it shows up in the alternative ap proach to the results of Nagel, Rudin, and Shapiro given by Nagel and Stein [NSJ. In this more general context it also makes sense to look for descriptions of approach regions that are optimal in various senses. These topics are treated fully by DiBiasi in [D B] . Particularly interesting to us is that a substantial part of the work there proceeds through analysis on model spaces defined on dyadic trees.

r

CARLESON MEASURES AND EXCEPTIONAL SETS

19

References D. Adams, On the existence of capacitary strong type estimates in Rn , Ark. Mat. 14 ( 1976), no. 1, 125-140. [AHJ D. Adams, L. Hedberg, Function spaces and potential theory, Grundlehren der Mathema tischen Wissenschaften, 314, Springer-Verlag, Berlin, 1996. [ACJ P. Ahern, W. Cohn, Weighted maximal functions and derivatives of invariant Poisson integrals of potentials, Pacific J. Math. 163 ( 1994) , no. 1, 1-16. H. Aikawa, Capacity and Hausdorff content of certain enlarged sets, Mem. Fac. Sci. Eng. [A] Shimane Univ. Ser. B Math. Sci. 30 ( 1997), 1-2 1 . [AE] H. Aikawa, M. Essen, Potential theory- selected topics, Lecture Notes in Mathematics, vol. 1633. Springer-Verlag, Berlin, 1996. [Ar] N. Arcozzi, Carleson measures for the analytic Besov spaces: the upper triangle case, J. Inequal. Pure Appl. Math. 6 (2005), no. 1 , Article 60, [AR] N. Arcozzi, R. Rochberg, Topics in dyadic Dirichlet spaces, New York J. Math. 10 (2004), 45-67 [ARS1] N. Arcozzi, R. Rochberg, E. Sawyer, Carleson measures for analytic Besov spaces, Revista Math. Iberoamericana 18 (2002) , 443-510. [ARS2] N. Arcozzi, R. Rochberg, E. Sawyer, Carleson measures and interpolating sequences for Besov spaces on complex balls, Mem. Amer. Math. Soc. , Vol. 182, 2006, no. 859, vi+163 pp. [ARS3] N. Arcozzi, R. Rochberg, E. Sawyer, The Characterization of Carleson measures for analytic Besov spaces: a simple proof, Complex and Harmonic Analysis, A. Carbery, P. Duren, D. Khavinson, A. Sistakis, Eds. , Destech Pub!. 2007, 167-178. [ARSp] N. Arcozzi, R. Rochberg, E. Sawyer, Carleson Measures, Capacity, and Exceptional Sets, in preparation. [BP] I. Benjamini, Itai; Y. Peres, Random walks on a tree and capacity in the interval, Ann. Inst. H. Poincare Probab. Statist. 28 (1992), no. 4, 557-592. [Beu] A. Beurling, Ensembles exceptionnels, (French) Acta Math. 72 (1940) . 1-13. [CFPR] A. Canton, J. L. Fernandez, D. Pestana, J. M. Rodriguez, On harmonic functions on trees, Potential Anal. 15 (2001), no. 3, 199-244. [Ca] L. Carleson, Selected problems on exceptional sets, Van Nostrand Mathematical Studies, No. 13 D. Van Nostrand Co. , Inc., Princeton, N.J.-Toronto, Ont.-London 1967. [DB] F. Di Biase, Fatou type theorems. Maximal functions and approach regions, Progress in Mathematics vol. 147 Birkhauser Boston, Inc., Boston, MA, 1998. [GP] D. Girela, J. A. Pelaez, Boundary behaviour of analytic functions in spaces of Dirichlet type, J. Inequal. Appl. 2006, Art. ID 92795, 12 pp. [KV] N. Kalton, I. Verbitsky. Nonlinear equations and weighted norm inequalities, Trans. Amer. Math. Soc. 351 (1999) , no. 9, 3441-3497. [KS] R. Kerman, E.Sawyer, The trace inequality and eigenvalue estimates for Schrodinger op erators, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 4, 207-228. [Ki] J. Kinney, Tangential limits of functions of the class S"" , Proc. Amer. Math. Soc. 14 ( 1963), 68-70. [KMV] P. Koskela, J. Manfredi, E. Villamor, Regularity theory and traces of A-harmonic func tions, Trans. Amer. Math. Soc. 348 ( 1996), no. 2, 755-766. [LL] Y. Lin, Thesis, Washington University, in preparation, 2008. [MV] J. Manfredi, E. Villamor, Traces of monotone Sobolev functions, J. Geom. Anal. 6 ( 1996), no. 3, 433-444 ( 1997). [MJ Y. Mizuta, Existence of tangential limits for -harmonic functions on half spaces, Potential Anal. 25 (2006), no. 1, 29-36. [NRS] A. Nagel, W. Rudin, J. Shapiro, Tangential boundary behavior of functions in Dirichlet type spaces, Ann. of Math. (2) 116 ( 1982) , no. 2, 331-360. [NSJ A. Nagel, E. Stein, On certain maximal functions and approach regions, Adv. in Math. 54 ( 1984) , no. 1, 83-106. [St] D. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math. 24 (1980) , no. 1, 1 13-139. [Tw] J. Twomey, Tangential boundary behaviour of harmonic and holomorphic functions, J. London Math. Soc. (2) 65 (2002), no. 1, 68-84. [Ad]

ARCOZZI, ROCHBERG, AND SAWYER

20

[V] [W]

I. Verbitsky, Nonlinear potentials and trace inequalities, The Maz'ya anniversary collection, Vol. 2 ( Rostock, 1998), 323-343, Oper. Theory Adv. Appl. , 1 10, Birkhauser, Basel, 1999. D. Walsh, Radial variation of functions in Besov spaces, Publ. Mat. 50 (2006), no. 2, 371-399.

( Arcozzi) DIPARTIMENTO DO MATEMATICA, UNIVERSITA DI BOLOGNA, 40127 BOLOGNA, ITALY E-mail address:

( Rochberg) 631 30, U.S.A.

arcozzi�Ddm . unibo . it

DEPARTMENT OF MATHEMATICS, WASHINGTON UNIVERSITY, ST.

E-mail address:

4K1 , CANANDA

E-mail address:

MO

rr�Dmath . wustl . edu

(Sawyer) DEPARTMENT OF MATHEMATICS & STATISTICS,

0NTAIRO, LSS

LOUIS,

Saw6453CDNtDao l . com

MCMASTER UNIVERSITY, HAMILTON,

Proceedings of Symposia in Pure Mathematics Volume 79, 2008

On the absence of dynamical localization in higher dimensional random Schrodinger operators Jean Bourgain Dedicated to V. G. Maz'ya

ABSTRACT. It is shown that dynamical localization fails in random Schrodinger operators with a slowly decaying potential in dimension at least 5, exhibiting a higher dimensional phenomenon. The method is perturbative and uses suitable renormalization of the Bohr series.

§ 1 . Introduction

In what follows, we consider lattice Schrodinger operators on zd , of the form

(1.1)

A is the lattice Laplacian on zd , i.e. ln 1 - n� I + + lnd - nd l 1 (1. 2) A(n,n ) { 01 ifotherwise

where

'

· · ·

=

=

and Vw is a random potential, which we take of the form

(1.3)

or

(1.3' ) with { Wn I n E zd } independent Bernoulli or normalized Gaussian random variables and {vn l n 7J.,d } , {wn l n 7J.,d } decaying deterministic sequences. More precisely, (1 .4) Vn = ( l n l 1)"' for some a: :::>: 0 and Wn will satisfy (1.5) E

E

"'

+

2000 Mathematics Subject Classification.

Primary 4606, 5206. ©2008 American Mathematical Society

21

22

JEAN BOURGAIN

Let us start with the well-known Anderson model (of transport in inhomogeneous media) where = K-Wn (o: = 0, Wn = 0) with coupling constant K > 0. In the !-dimensional case (d = 1), the spectral theory of Hw has been extensively studied and is well-understood. Rewriting the equation 'ljJ = E'lj; as

Vw(n)

H

(1.6)

the main tool is the transfer matrix formulation of

(1.6)

(1.7) where

(1.8) and the Lyapounov exponent

1

(1.9)

IE log ii MN (E) II · L(E) = Nlim --> oo N For all K > 0, Hw satisfies almost surely Anderson localization (AL) (pure point spectrum with exponentially decaying eigenfunctions) and dynamical localization (DL) . This last property relates to the associated Schrodinger group ( eitH ) t E IR and means that for any given exponent A > 0

(1.10)

ln l

whenever � decays rapidly for --+ oo . Recall that (DL) implies pure point spectrum but not conversely. Despite quite precise and widely believed conjectures, the higher dimensional case > 1 is still poorly understood. The belief is that for d = 2, the spectrum re mains pure point with localized eigenfunctions (but for small K a different behaviour of the localization length at the edge and in the bulk of the spectrum) and for d � 3 and small K, there is a component of absolutely continuous spectrum. But at this time, we only dispose of the Frohlich-Spencer multiscale analysis, which enables us to produce point spectrum (also ( AL) and (DL)) for large K and at the edge of the spectrum. We do not know of a method to produce continuous spectrum. The only rigorous result distinguishing d = 1 and d > 1 is due to [ESY] and relates to localization length for small "' · For d = 1 , the behaviour is � for "" --+ 0 (i.e. the reciprocal of the Lyapounov exponent) and [ESY] establishes in = 3 localization lengths at least ,JH o > 0 some constant, in the bulk of the spectrum. Let us now turn to (1.1) with decaying random potential. For d = 1 and using again transfer matrix methods, it is shown in [DSS] that if then = (see [DSS] , Theorem 1.1). • I f 0 < o: < � ' Hw satisfies a.s. (AL) and (DL). In fact the eigenfunctions 'ljJE have a decay rate

d

"'

d Vw(n) wn ! n ! -"',

,

(1.11) ••

If o:

1'1/Je (n) l < C(E) exp{ -c' l n l 1 -2"'}.

> � , Hw has pure a.c.-spectrum, a. s.

23

DYNAMICAL LOCALIZATION

The situation a = � is more complicated and we do not discuss it here (see [DSS] and later papers) . Our purpose is to point out a higher dimensional phenomenon (we need in fact) regarding (DL) with a random decaying potential. We will sketch a proof of the following (1. 1) THEOREM.

d :2: 5

Let d :2: 5 and Hw be given by with (1.12) Vw(n) ( l n l + 1 ) Wn + Wn (n E d ) where a of> size 0 is arbitrary, is sufficiently small and Wn is some deterministic potential �

=

Z

"'

�

(1.13)

Then, with high probability in w, Hw fails (DL). More precisely sup I n l A IeitH (0, n) I oo if A > d. ( 1.14) tEIR =

nEZd

For d = 1, this behaviour is indeed impossible if a < � ·

Remarks.

(1). In what follows, we take for simplicity {wn } to be Bernoulli, but we could take other symmetric i.i.d random variables satisfying suitable moment estimates (we do not intend to specify). (2). The deterministic potential { wn } is used when renormalizing the Bohr ex pansion. It does not play an essential role in the statement of the Theorem and likely can be removed (this would require replacing the free Laplacian .b. by .b. + w with a deterministic smooth decaying potential with decay (1.13)) . (3). The proof of the Theorem relies heavily on [B] and is in fact a corollary of this result. In [B] , we consider = .b. + with as in (1.12) and show that at a specific energy (the lower edge of the spectrum of .b.), has an extended state and the Green's function of behaves like the free Green 's function i.e. w

E Hw Vw G(E)Vw Hw Hw G0 (E), ( 1.1 5) I G(E) (n,n') l "' l n - �'l d_2 . Statement (1.14) is then easily deduced from ( 1.1 5) . The analysis in [B] is actually only carried out for a > � but may be generalized

to arbitrary a > 0 (it requires considering more terms in the multi-linear expansion of We will recall the main ideas below. (4). [B] was inspired by an unpublished preprint of T. Spencer and W. Wang [S - W] on Lipschitz trails. Note that for in the bulk of the spectrum, l n - n' l ---r (with an essential phase factor), while at the edge we have l n -;' l d The latter is square inte grable for which explains our assumption. The renormalization for in the bulk is quite different and much harder cf. [ESY] ).

G(E)).

I Go(E)(n, n') l 1 E_, I Go(E- )(n, n') l d :2: 5, E "'

d

E

1

"'

2 •

(

24

JEAN BOURGAIN

(5) . It is conceivable that an analogue of [B) and the above Theorem remains valid with a = 0 (i.e. no decay). This would require a different strategy however, as infinite multilinear Bohr extension in is unlikely to be tractable (even after renormalization). (6). Instead of lattice models, one could also consider a model on !Rn, of the form Hw = -� I: EZd Vw (n)rp(· - n) where � is the usual Laplacian, Vw n a random potential as above and rp( ) a localizing bumpfunction on JRd . The remainder of the paper is organized as follows. We will first briefly sketch the argument from [B) leading to (1. 15). We then deduce (1.14), which is a rather elementary spectral consideration (and may well be known) . an

K-

+

x

§2. The Green's function estimate

(i) Redefine the Laplacian by subtracting 2d from the lattice Laplacian, thus d cos 2'1r�j = l � l 2 0 ( 1�14) - .i (� ) = 2d - 2 (2. 1 )

c�= j=l

) c + and consider the Green's function G0 = ( -� io)- 1 at 0-energy. Hence +

(2.2)

Go (n, n1) =

(ii) Write

I

e - 21l'i (n -n' H - �(0 �

(2.3)

'

rv

1

I n - n l d'

2.

H = -� + V

with � as above and where V = Vw

(2.4)

+w

with V(n) = Vw (n) =

(2.5)

(1

(wn Bernoulli) ln l ) a Wn

K,

+

and Wn deterministic satisfying (1.13). The role of w will become clear later on. Denote G the Green's function of H at 0-energy. The basic idea is to write a Bohr series (with finitely many terms) for G and make probabilistic estimates on the terms. Thus iterating the resolvent identity (for the time we let V = V, dropping the w-potential) G = Go - GVGo

(2.6)

we obtain (2 . 7) G = Go - GoVGo + GoVGoVGo - GoVGoVGoVGo +

I

I

· ·

·

± GVGo

· ·

·

VGo.

Our aim is to bound Go VGo ll , Go VGo VGo ll , . . . using the randomness of V = Vw and probabilistic considerations. In an ideal situation of square cancellations, the

25

DYNAMICAL LOCALIZATION

estimate iGo (n, n') i ;S l n- , 1a 2 with l n iL2 square-summable for d 2: 5), together with some decay of Vn , would easily imply bounds of the form

; (

i ( Go VGo . . . VGo) (n, n' ) i

(2.8)

«

K/

in - n' i d - 2

and also (2.9)

i (VG o V . . . VG o) (n, n ' ) i

/'i,s in - n' id - 2 min (ini sa , i n' is<>)

«

with 8 the number of V-factors in (2.8), (2.9). Letting s > � and taking in (2.7) an expansion of length 3, one derives that G is a perturbation of Go , i G (n, n' ) - Go (n, n' ) i <

( 2 . 10)

O(K)

. in - n' i d- 2

The difficulty with the terms in (2.7) is that the V-factors are not independent however and the Wn -factors may cancel out. We need therefore to proceed with more care when iterating the resolvent identity and certain renormalization is needed, requiring the additional deterministic potential in (2.4). iii As the V-factors are not independent, we will need an appropriate proba bilistic estimate which we describe next. Recall that {wn in E zd } are independent Bernoulli variables this is not essen tial for what follows however . Considering a 8-tuple (n1 , . . . , n8 ) , we say that there is 'cancellation' if (2.11) Note that this property is invariant under translation. Say that (n1 , . . . , n8 ) is 'admissible' if for any segment 1 ::; 81 < 8 2 ::; 8, the sub-complex (n81 , n81 + l , . . . , n82 ) does not cancel. Use the notation ,ns to indicate summation restricted to admissible 8tuples. The interest of this notion is clear from the following estimate. LEMMA 1 . For 8 2: 2 "L * Wn, . . . Wns a��' a�l,ln2 . . . w

()

(

)

E�, ,

I

(2.12)

n1 ,

...

...

, n8

a�s},n' t�

:S Cs [ n L

t ,··· ,ns

ia�0},n,

· ·

·

a�},n' i 2

J 1 12

·

a�Pn 2: 0. Since in the summation no (n 1 , . . . , n8 ) cancels, there is some index ns ' which is not repeated or repeated an odd number of times. Specifying a subset I of {1, . . , 8 of odd size at most 28 possibilities , we consider now 8-tuples of the form ( v ( l l m v(2) m v (3 ) m . . . ) Proof. We may clearly assume

E*

}

.

'

'

(

'

'

)

'

where m E zd appears on the /-places and v (l l , v(2l , . . are admissible complexes indexed by sub-intervals of { 1 , . . . , 8 determined by I.

}

.

26

JEAN BOURGAIN

Thus, enlarging the I: * -sum, (which we may by the positivity assumption) , it follows that

L* wn 1

(n , , ... ,n, 1 ) v{l) II

·

·

·

wn an(o)n 1 !

S]

·

· ·

an(s t ) m S]

l

J

(2.13)

where, for fixed m , We used here that if (n 1 , . . , n8 ) is admissible, then so is (n8 " n51 + 1 , . . , n8 J for all 1 ::; 8 1 ::; 82 ::; Enlargement of the original sum I: * enables thus to get the product structure in (2. 15) . The number of factors is at least 2. Thus the preceding and a standard decoupling argument implies .

.

8.

(2. 14) Next, by Holder's inequality and moment-equivalence, we get (2.15) Proceeding by induction on 8, we obtain thus

�

[ m�J . , ,L.n. ,

Ja;::� ,

a}:.�l,m J'

]

]

l am,n81 (s ,+ I)+2 · · · an82 (s2),m 1 2 1 < (2.12) . � ] n s1 + 2 , . .. ,n8 2 This proves Lemma 1 . Expressions considered in Lemma 1 appear when writing out matrix elements of products (A(0) Vw A( 1 ) Vw A ( sl ) (n , n' ) where the A (s ' ) are matrices and Vw a random potential Vw (n) = Wn Vn · We use the notation (2.16) to indicate that, when writing out the matrix product as a sum over multi-indices, we do restrict the sum to the admissible multi indices.

[

"'

· · ·

· ·

·

1

2

27

DYNAMICAL LOCALIZATION

Lemma 1 then implies that (2.17 ) (iv) Returning to (2.3), (2.4), write G = Go - GV"Go = Go - GVGo - GwGo

and iterating once G = Go - Go VGo + GVGo VGo - GowGo + GwGo VGo.

(2.18)

In order to replace GVG0 VG0 by ( GVG0 VG0 ) * , we need to remove aGv2G0, where a = Go(O, 0)

(2.19) since Hence, form (2.18)

(2.20) G = Go - GoVGo + ( GVGo VGo ) * - G( w - av2) Go + GwGo VGo . Continuing the iteration, write (GVGoVGo ) *

=

(GoVGoVGo)* - GV(GoVGoVGo) * - Gw(GoVGoVGo ) *

= (GoVGoVGo)* - (GVGoVGoVGo)* - aGv2GoVGo + a2Gv2VG0 - Gw(GoVGoVGo) *

and substituting in (2.20 ) (2.21)

G = Go - GoVGo + (GoVGoVGo) * - (GVGoVGoVGo ) *

+

a2Gv2VGo

- G(w - av2) Go + G(w - av2) GoVGo - Gw(GoVGoVGo ) * .

Next ( 2.22 )

(GVGoVGo VGo)* = (GoVGoVGoVGo)* -GV(Go VGoVGo VGo)* -Gw(Go VGo VGo VGo)*

and GV(GoVGoVGoVGo)* =

(2.23)

(GVGoVGoVGoVGo) * + aGv2(GoVGoVGo) * - a2 (Gv2VG0VG0 ) *

where W is the non-diagonal operator (2.24)

W ( n ' n' ) =

{

v;;v;,Go (n, n')3

0

if n -:/- n' otherwise.

+

GWG0

28

JEAN BOURGAIN

Substituting (2.22), (2.23) in (2.21) and expanding the a2Gv2VG0-term gives G =Go - GoVGo + (GoVGoVGo) * - (GoVGoVGoVGo)* + a2Gov2VGo + (GVGoVGoVGoVGo)* - a2 (Gv2VGoVGo) * - a2 (GVGov2VGo) * - G(w - av2 - a3 v4) Go + G(w - av2) GoVGo - G(w - av2) (GoVGoVGo)*

(2.25)

+ Gw(GoVGoVGoVGo ) * - a2GwGov2VG0 + GWGo.

We need a more appropriate expression for W. Write (2.26) where M is the convolution operator (2.27)

M ( n, n')

=

{�

o (n, n')3

if n :f n' otherwise

and the operator P has matrix elements bounded by (2 . 28) l p ( n, n' ) I < - "'4 1 1 n l -2a - I n' 1 -2 "" 1 2 I n - n' l - 3 (d - 2 ) < l n - n ' ld+"'(l n l 2+a +l n' l 2+"') "'4

since d ;::: s (and assuming < � ). The convolution operator M is symmetric and even in each variable with be haviour M (n) "' l nl - 3 (d-2 ) . Hence o:

(2.29)

a� M E L� for lo:l

<

3(d - 2)

and (2.30) Start by subtracting M (O) from M(�), introducing an additional potential M (O)v4 in (2.26). We factor then (2.31) where the convolution operator satisfies in particular (2.32) This brings W in the form (2.33)

29

DYNAMICAL LOCALIZATION

Note that 1:100 = - I and GD. = -I + Gil. Substitution in (2.25) gives G

=

�

(1 - Mv4) Go - GoVGo + (GoVGoVGo) * - (GoVGoVGoVGo) * + a2 Gov2 VGo

(2.34)

- a2 (GVG0Vv 2 Go) * - a 2 (Gv 2 VGoVGo) * - G(w - av2 - a3 v4 - M (O)v4)G0

+

� GVM1 v4Go + (GVGoVGoVGoVGo) *

+ G(w - av 2 ) (GoVGo)

(2.3 5) - G(w - av2)(GoVGoVGo) * + Gw(GoVGoVGoVGo) *

(2.36) 1

+

� GwM1v4Go - a2 GwGov2VGo

- 2 cv 4 M1 + GPGo.

In continuing the process, we only expand G further in the (2.34) terms. In general, when replacing GV ( ) * by ( GV ( ) ) * , there will be additional terms of the form GW( · ) * where the operator W appears from cancellations. These operators W need to be processed further. Observe first that, dismissing the v factors, we certainly have an estimate ·

-

·

· ·

·

· ·

Wn-

�

I W (n , n' ) l < In - n l3(d - 2 )

(2.37)

which is obtained by removing the repeated variables iteratively, using the estimate n

Our aim is to bring W in the form (2.33). The first step is to move the v-factors outside the product in order to remain with a convolution operator. Decompose as follows Vn

_1_ l nl <>

(2.39)

=

=

a n'ln'. (nl 2-+<>n') O (ln n' l 2 ( l n l - 2 -a 1 - a n' (n - n') + 0 ( l n - n' l 2 + <> ) . l n' l 2 + a In' I<> l n ' 1 2 +<>

_1

_

ln' l <>

_

+

_

+

l n' l - 2 - <> ) )

This decomposition is applied to each v-factor (n' being the left matrix element). '12+" All contributions with at least one I1:�2+" or at least two n1'�C� , 2��' ) are absorbed by P satisfying (2.40) IP(n' , n" ) l < Kin' - n " l - 3d+ 8 +<> ( 1 n' r 2 - <> + ln" l - 2 - <> ) . Remain the contributions (2.41)

v�, M(n' - n" )

and (2.42)

n' (nI - nII ) vS n' ln' l 2 K ·

30

JEAN BOURGAIN

where M satisfies (2.29), (2.30) and K is odd (1Rd-valued) , satisfying 7 IK (n) l < l n l -3d + .

(2.43)

Symmetrizing the operator W, the preceding gives 1 1 W - v8M + - Mv 8 + P' (2.44) 2 2 with P' still satisfying (2.40). Treating M as before, we obtain with the same notation as earlier =

GW(Go

(2.45)

· ·

·

) * = M (O)Gv8 (GoV . . . ) *

� Gvs Mt (V . . ) *

(2.46)

-

(2.47)

1 s - 2, M1 v ( GoV . . . ) *

(2.48)

+

.

� GVM1v8 (GoV . . . )* + � GwMt V8 (GoV . . . ) *

(2.4 9)

+ GP ( . . . ) * .

(2.50)

Term (2.45) is added to (2.35), (2.46) to (2.34) or (2.36); (2.47) , (2.48) to (2.34); (2.49) to (2.35); (2.50) to (2.36). Note that by (2.17) (2.51)

Ew [ I (GoVGo . . . ) * (n, n' ) I J ,____, j V -factors

«

��;J i n - n' l - d+ 2 ( 1 n l

1\

l n' l ) - min(j a , d- 2 )

with large probability. We carry out the preceding iteration s times, with s s ( a ) large enough. Denoting the sum of the terms in (2.34) that do not involve G, by G0 + A, A = Aw , it follows from (2.51) that =

(2.52) Write the remainder terms in (2.34) as GB, B

=

Bw

where

(2.53)

d 2:

by our choice of s and since 5. The potential is chosen to make (2.35) vanish up to sufficiently high order to capture the remainder by a term of the form GB with B satisfying (2.53). Finally, the term (2.36) are of the form GB1 , GB2 with w

as

(2.54) and, cf. (2.28), (2.40) (2.55)

' I Bz ( n , n') l < In - n ld- 2 (ln' l - 2 - a + ln l - 2 -a) . ' /\;

DYNAMICAL LOCALIZATION

31

Summarizing, we obtain (2.56) G = Go + A + G(B + B1 ) hence (2.57) G = (Go + A) (I - B') - 1 with IB'(n, n') l < �>:(In - n'l -d-1 + I n - n' l - d+2 ( 1 nl - 2 -a + ln'l - 2 -a )) . (2.58) The inverse (I - B') -1 may be estimated by a Neumann series and (2.58) . One verifies that (2.59) (I - B') -1 = I + B" with B" again satisfying the estimate (2.58) . It follows that IG( n, n') - Go (n, n')l < _ (2.60) + In 'l d- 2 In _ n' ld�41n' l 2 + a and by symmetry considerations

�

(2.61)

IG(n, n') - Go (n, n') l < I n

��id-2

which is (2.10). From the preceding we may also derive a bound (2.62) IG( O + ic) (n, n') l ;S I n - n' l -d+ 2 .

§3. Diffusion for the Schrodinger group

It remains to derive (1.14) from the Green 's function behaviour (3.1)

�

I G( O) (n, n') l "' I n - ' ld- 2

at E = 0. Write from the spectral theorem (3.2)

where (3.3)

Im G(>-. + io) ( O , x)

=

Im(H - )... - io) -1 (0, x).

Denote P" the Poisson kernel on IR (thus Fe: (t) e- e: Jtl) and Qe: the conjugate Poisson kernel. From (3.2) IeitH (0, x) l ee: ltl i (JLo,x Pe:r' (t ) l . (3.4) Now assume dynamical localization in the form (3.5) sup IeitH (0, x) I « lxi - A for lxl ---+ oo t for some exponent A > 0. We will show that A :::; =

=

*

d.

32

JEAN BOURGAIN It

follows from (3.4) , (3.5) that

(3.6) 1 ( 1-lo ,x Pc ) "' (t)l < Ce-c lti l x i - A for some constant C. Next, we have from the resolvent identity *

G(O + ic ) implying by (3. 1) , (2.62) (3.7)

=

G( O) - icG(O )G(O + ic)

[ G( O + ic ) (O, x) - G(O) (O, x) [ � c L [ x

_

y

Again using (3. 1), this gives [ G( O + ic) ( O, x) [ "'

(3.8)

�� d-2

_ for [ x [ [ x l! 2

«

Estimate

I Y[!- 2 � l x l:- 4 ·

�-

[ G(O + ic ) (O , x) [ :::; [[tLo ,x Pc [[oo + 11 1-lo,x Qc [[oo :::; ( [[P� ll 2 + IIQ� 1 [ 2 ) [ 1 �-Lo ,x P� ll 2 *

*

*

< Cc -1 [ x i - A

(3.9 ) invoking (3.6) . Combining (3.8), (3.9) , it follows indeed that A

:::; d.

References [B] . J. Bourgain, Random lattice Schrodinger operators with decaying potential: Some higher dimensional phenomena, Springer LNM. [DSS] . F. Delyon, B. Simon, B. Souillard, Prom power pure point to continuous spectrum in disordered systems, Ann. Inst. H. Poincare, Vol. 42, n3 (1985) , 283-309. [ESY] . L. Erdos, M. Salmhofer, H.T. Yau, Quantum diffusion for the Anderson model in the scaling limit, Ann. Henri Poincare 8 (2007) , no. 4, 553-555. [S-W] . T. Spencer, W-M. Wang, Lipschitz tails, (unpublished ms) . INSTITUTE FOR ADVANCED STUDY, PRINCETON, NJ 08540 E-mail address:

bourgainil)ias . edu

Proceedings of Symposia in Pure Mathematics Volume 79, 2008

Circulation integrals and critical Sobolev spaces: problems of optimal constants Ha!m Brezis and Jean Van Schaftingen Dedicated to Vladimir Maz'ya on the occasion of his 70th birthday, with high esteem and friendship

ABSTRACT. We study various questions related to the best constants in the following inequalities established in [1, 2, 3] ;

and

I[ l h.n

cp ·

fl

:S:

Cn !I\7'PIJLn l f l ,

Cn i i \7'PIILn ll il ll , where r is a closed curve in R n, cp E c;:_o (Rn ; Rn) and i1 is a bounded measure on Rn with values into Rn such that div i1 0. In 2d the answers are rather
· il l

::::

=

complete and closely related to the best constants for Sobolev and isoperimetric inequalities.

1 . Estimates for curves and divergence-free vector fields

W1 •n (Rn)

In general, functions in the Sobolev space do not need be bounded continuous functions. However, it was shown recently that in circulation integrals, such functions behave as if they were bounded continuous functions:

THEOREM 1 (Bourgain, Brezis and Mironescu [4]). There exists C > 0 de pending only on n such1j5 EthatC�(Rn; if r is Rn), a closed rectifiable oriented curve whose tangent vector is f, for every (1.1) 1991 Mathematics Subject Classification. 26010, 35J25, 46E35, 49Q15, 51M16, 58C35. Key words and phrases. Circulation integrals, optimal constants, Sobolev inequalities, isoperimetric inequalities, divergence-free vector fields. The second author (JVS) was partially supported by the Fonds de Ia Recherche Scientifique FNRS (Belgium) and by the Fonds Special de Recherche (Universite catholique de Louvain) . He acknowledges the hospitality of the Laboratoire J.-L. Lions, Universite de Paris VI and of the Rutgers University Mathematics Department, at which part of this research was carried. @2008 American Mathematical Society

33

HA IM BREZIS AND JEAN VAN SCHAFTINGEN

34

Here and in the sequel, the subscript in C_;x'(Rn; Rn) means compact support. The original proof of Theorem 1 relied on a Littlewood-Paley decomposition; an elementary proof was given by the second author In two-dimensional space, this estimate follows easily from the classical isoperi metric theorem. This is not anymore the case in higher dimensions, but the in equality still retains some isoperimetric flavour. The aim of this paper is to present various problems concerning optimal con stants in Theorem 1. A first family of problems consists in fixing the curve r and finding the optimal constant Ar and the optimal vector field rp. Such a vector field always exists. In two dimensions, the value of the associated optimal constant can even be given in terms of the area enclosed by r. The next problem consists in optimizing Ar over all possible curves i.e., study ing A = supr Ar. In R2 , one merely needs to maximize the area under a constraint on the perimeter; the classical isoperimetric theorem then yields a circle as the optimal curve. Similar results in higher dimensions would be desirable. However, none of the two-dimensional arguments work. The question whether the optimal constant is attained is even open. As explained by Bourgain and Brezis the optimal constant A in Theorem 1 is the same as the optimal constant in THEOREM 2 (Bourgain and Brezis C>0 E C_;x' (R n; R n), c

[18].

[2], a divergence-free bounded measure, then[2for, 3]).everyThererp exists

such that if i1 is

(1.2)

[3],

Theorem 2 also has a direct proof by Littlewood-Paley decompostion and an elementary proof The latter inequality provides thus a relaxed problem for the optimal constants, which could be more tractable. This paper is organized as follows. In section 2, we explain how for a given curve, one can find an optimal vector field, how this leads to the optimal problem, how the problem of the optimal constant for divergence-free measures is the same as the problem for closed curves. We end this sections by some problems related to the confinement of the inequalities to domains. In section 3, we show that for a given divergence-free vector measure, one can find an optimal vector field. We then raise various questions concerning the optimal constant in the inequality.

[17].

2. The two-dimensional case 2.1. Curves in 2d. Throughout this section, we assume that r is a simple, closed, smooth, oriented curve with tangent vector f. The optimal constant for the

inequality of Theorem 1 is given by Ar

=

sup

{fr rp f : rp ·

E

C� (R2 ; R2 ) and

l2 IY'$I 2 1 } , $

where 1 · 1 denotes the Euclidean norm (of a vector or a matrix). By Theorem 1 , this can be rewritten as Ar = sup rp · [ : rp E H1 (R2 ; R2 ) and IY'$\ 2 $ 1 , 2 where

{fr

l

}

CIRCULATION INTEGRALS AND CRITICAL SOBOLEV SPACES

35

is the completion of C� (R2; R2) equipped with the norm IIY'$IIL2 . Note that there are two possible definitions for fr $ f when $ E H1 (R2; R2) . The first one is in the sense of traces; secondly, starting from one can define fr $ f abstractly via density. Both definitions coincide. Since the space H1 (R2 ; R2) modulo constants is a Hilbert space, one has ·

(1.1),

·

2.1. There exists a unique $ E H1 (R2; R2) (modulo constants) such that (2.1) (2.2) The vector field $ satisfies the equation, 1 Lr (2.3) -!::::4 = .i_ Ar 1i (2.4) div $ = 0 (2.5) $ E (U'0 C) (R2; R2 ) , where 1i1Lf denotes the ! -dimensional Hausdorff measure restricted to r. Note that the additional property (2. 5 ) allows to give a classical meaning to fr $ · f. 2.1. Since H1 (R2; R2) is a Hilbert space, there exists a unique $ such that l $ · t = Ar and P ROPOSITION

n

P ROOF OF P ROPOSITION

The vector field $ satisfies the Euler equation -A.D.$ = fH 1 Lr for some Lagrange multiplier ).. E R. Obviously ).. = Ar . In particular, div $ satisfies - D.. div $ = 0 Since div $ E L2 (R2), one has necessarily div $ = 0. Finally, since -D.$ is a divergence-free bounded measure, the boundedness and the continuity of $ follow from the results of Bourgain and Brezis (see Remark and Theorem 0

.

[2,

3]).

5] [3,

The constant Ar can be determined explicitely: P ROPOSITION If f = 8'£, Ar = 1'£1 ! P ROOF . To fix the ideas, we shall assume that r is positively oriented. Apply ing successively Stokes' theorem and the Cauchy-Schwarz inequality, we obtain ! $ . [= 1V' 1\ $12 \7 1\ $ '5. 1 '£ 1 1 / 2 2 ! ! I2 , '5. 1'£1 1 1 2 1\ $12 + ldiv $12 = 1'£1 1 1 2 IY'$ IY' 2 2

2. 2 .

then

·

(2.6)

l

h

(l_

(l_

)

)

(l_

)

HAI M BREZIS AND JEAN VAN SCHAFTINGEN

36

where \1 1\ 1{5 = fhcpz - 82 'P 1 · This proves that Ar :::; 1�1 ! . In order to obtain the equality, choose 1{5 E H 1 (R2; R2) to be the solution of \l /\ 1{5 = 1 in � , \1 1\ 1{5 = 0 in R2 \ � , div 1{5 = 0 in R2 . 0 By construction, 1{5 realizes the equality in (2.6); hence, Ar = 1�1 ! .

{

The main result is P ROPOSITION 2.3. (2.7)

One has

Moreover, equality holds in (2.7) if, and only if, r is a circle.

Writing r = {)�, we combine Proposition 2.2 together with the clas sical isoperimetric inequality, in order to obtain Ar = 1 � 1 ! :::; 1 . 2 Moreover, one has equality if and only if r = a� is a circle. 0 2.2. A relaxed problem. Let i1 be a bounded R2-valued vector measure such that div i1 = 0 in the sense of distributions, and set P ROOF .

}rr

Ajt = sup {

L2 1{5 · i1 : 1{5 E C� (R2 ; R2) and L2 I"V012 :::; 1 } .

A particular case is when r is as in Section 2.1 and the measure is i1 = t1i1 Lr; one has then Ai1 = Ar. By Theorem 2, the integral JR2 1{5 i1 is well-defined when 1{5 E H 1 (R2; R2) as an extension of a linear functional1. One can thus relax the supremum to ·

Ai1 = sup

{L2 1{J · i1 : I{J E H 1 (R2 ; R2 ) and L2 I"VI{JI 2 ::; 1 } .

As previously, one can show that if i1 =/= 0, Ajt is attained by a vector field 1{5 E ( H 1 n L00) (R2; R2) that satisfies (2.8)

so that in particular div 1{5 = 0. By the result of Bourgain and Brezis mentioned above, 1{5 is a bounded continuous vector field. One can also compute explicitely the value Ai1. Recall that by the Sobolev Nirenberg embedding, if i1 is divergence-free, it can be written as i1 = "V..L(, for some unique ( E L2 (R2), where "V..L( = (82 (, -81(). P ROPOSITION 2.4. ( i1

Let and be as above, then

1In fact, one can even show using Theorem 2 that null sets for the H 1 capacity in R2 are null sets for the variation li11 of j1, so that every H 1 - quasicontinuous function is j1-measurable. In particular, the integral makes sense for every <ji E (H1 n L"")(R2 ; R2 ).

CIRCULATION INTEGRALS AND CRITICAL SOBOLEV SPACES

PROOF. One has, by the Cauchy-Schwarz inequality, for every t.jJ E

jRr 2 0 . i1 }Rr 2 'Vl.c 0 }Rr 2 (('V t.jJ) (L2 1 ( 1 2 ) ! (L2 1 'V 01 2 ) ! � ( L 1 ( 1 2 )! ( L I 'V 01 2 ldiv 01 2 ) ! 2 2 (L2 1 c 1 2) ! (L2 I 'V$1 2 ) ! , =

37

C�(R2 ; R2 ),

(\

=

�

(\

+

(\

=

whence one obtains an upper bound

For the equality, observe that the solution t.jJ E

H1(R2 ; R2 ) to

{'VI\t.{J= ( , = div t.jJ

0,

achieves equality in the above inequality.

0

PROPOSITION 2.5. One has l i11 (2.9) A - < y'7r 2 , where l l i1 1 sup {L t.jJ i1 'Vx E R2, l t.{J( x ) l � 1 } . 2 Moreover, equality holds in (2.9) if, and only if, 1-'

·

=

i1

-

:

= A'Vl.XB(x,r)

for some A E R, x E R2 and r > 0.

PROOF. By the optimal Sobolev-Nirenberg inequality, one has

I C I P
0

=

(2.7)

(2.9).

f'H1Lr Jr l

2.5

it is clear that implies The main interest of Proposition is that the relaxation to Lipschitz curves (having possibly self-intersections), or to divergence free measures, does not introduce new maximizers. There is a slight improvement of Proposition

2.5

HA IM BREZIS AND JEAN VAN SCHAFTINGEN

38

For every $ E H1 (R2 ; R2 ) , r

0, x E R2 and r > 0. C OROLLARY

2.7.

<5:

Given $ E H1 (R2 ; R2 ) , let -If solve V' A = V' A $ , div '!j! = O . One then has

{

P ROOF.

�

( r 1v�1 2 ) � 2 ft JR2 }R2r

If equality holds, one has by Proposition 2.5, = AY'j_XB(x,r ) so that, by (2.8), - f:l.'!jl = Vf-l = VAV' j_ XB(x, r ) · On the other hand, V' j_ (V' J\ $) = V' j_ (V' J\ �) -!::.� since div -If = 0. Therefore, V' j_ (V' A $) = VA V'j_XB(x,r) 0 and consequently V' J\ = VAXB(x , r) · Confinement to domains. Let n c R2 be a smooth, bounded, simply connected domain with normal vector field ii. We will work with the class of measures on 0 satisfying div = 0 in n and [1 · ii = 0 on 80. This class is defined as Mff (O, R2 ) = il E C(O; R2 ) * E 0 1 (0) , V'( · = 0 .

j1

>

-

=

2.3.

$

j1

{

:

(2 · 12)

k j1 }

j1 E Mff(O, R2 ) (just take ((x) = Xi, = 1, 2 in the 2.8. For every $ E H1 (0) C(O) and j1 E Mff (O, R2 )

Note that fn. il = 0 for every definition). P ROPOSITION

V(

f

i

n

- < ll ll II" A -11 r.p ln . f-l 2 ft r.p V(n) · -

-

il

v

holds in the nontrivial cases if j1 = At1t1 LaB(x, r) with aB(x , r) C 0 and Equality

0.

39

CIRCULATION INTEGRALS AND CRITICAL SOBOLEV SPACES P ROOF .

Let

-¢ E H 1 (R2 ; R2 ) satisfy

{

�

V' (\ = V' (\ cp in n ' V' 1\ 'ljJ = 0 in R2 \ n , in R2 , div -¢ = 0

and define the measure ij E M (R2 ) by

{ J· ii = ln( J . p ,

}R2

for every

J E Cc(R2 ; R2 ) . One has, by Corollary 2 . 7,

( cp . i1

ln

=

r -¢ . ij "5o 11 m 1

(r

2 y'Jr JR2 1

JR2

)

v (\ -¢l 2 ! =

I PI I ( r v

2 y'Jr ln 1

(\

The equality cases follow again from the conclusion of Corollary 2. 7 . REMARK

)

tPI 2 !

. 0

2.9. If 0 is not simply connected, the inequality

in cp · i1 -5o Clli111 11V

A

tPI!L2(n)

cannot be true. Indeed, assume that 0 belongs to a bounded connected component of R2 \ n . Take r c n to be any closed curve of index 1 with respect to 0, and set = t'H 1 Lr and cp(x) One then has

= ��2 •

j1

in cp j1 = 21r ' 0

while

0

in n . From (2.12) , we also have

(20 13) since

f

Jn

II V'

A

� �< 'P · _ p,

I I ili i II " 'P�11 L2(fl)

V2if

v

tPIIv
j1 E M�(D, R2 ) , let (2. 14) Ap,n = sup {£ cp · j1 : cp E H 1 (R2 ; R2 ) and fo 1 V'cpl2 1 } ; of course the supremum in (2.14) is achieved by some unique cp E H 1 (D; R2 ) modulo constants; moreover, cp E (100 n C) ( D ; R2 ) by Theorem 2 in [5] (when j1 is an 1 1 function - the case of measures is similar). Thus we have for every j1 E M� (D, R2 ) For every

"5o

lli1 11 AM-,n < V2if . _

Set

(2.15)

A n = sup { Ap, n

:

j1 E M� (D, R2 ) and ll i111

"5o

1} .

HAIM BREZIS AND JEAN VAN SCHAFTINGEN

40

PROPOSITION 2.1 0 . One has

1 An 2 vn <

1 ..j'5;; .

::=:::

Moreover, if 0 is a disc, then An = Jh- and supremum in (2.15) is achieved. Let j1 >..tH1 L8B(xo, r) with 8B(xo , r) C 0 and { (x(x -- xoo )j_) j_ ifif lxx -- xoxo l > rr ., (X ) = l i X P ROOF.

= _

:S:

Jx - xo J2

t.p

r2

One then has

in $ . i1 = k2 $ . i1 �� (fa/v$1 2 ) > �� (in l \7 $1 2) � , so that An > 2},r . The inequality An k follows immediately from (2. 13) . Finally, assume 0 is a disc; without loss of generality, 0 = B(O , 1 ) . One sets then �

=

:::;

r = 8B(O, 1), j1 = tH1 Lf and $(x)

=

xj_ ; immediate computations give

in $ . i1 = 271" ll flll

=

'

2 71" '

in l \7$12 = 271" .

0

We have no clue about the dependence of An on 0 and whether the supremum in (2.15) is achieved. The only information we have is PROPOSITION 2.11. Assume that An = k and An is achieved. Then n is a

disc.

There are two extreme scenarios: S CENARIO 1 . An = k only when n is a disc . S CENARIO 2. An = k for every domain n R2 . P ROBLEM 1 . Decide between Scenario 1 , Scenario 2 and intermediate scenarios . PROBLEM 2. Is it true that for every domain n, An is achieved? By Proposition 2.11, a positive answer to Problem 2 would lead to Scenario 1. This scenario would be reminiscent of the situation of the balls who have the worst best Sobolev inequalities [12). There is a variant of Proposition 2.8 where the boundary condition fl· ii 0 is replaced by the condition that $ should vanish on 80 . Set M#(O , R2 ) = {fl E C(O; R2 )* : V( E C� (O), in \7( - j1 = 0 } · PROPOSITION 2.12. For every j1 E M # (O ; R2 ) and for every $ E H6(0; R2 ) c

=

C(O;

R2), one has

(2.16)

n

CIRCULATION INTEGRALS AND CRITICAL SOBOLEV SPACES

41

for every $ H6(0) where (2.17) Sn = sup { J l u i! L2 (n) : u BV(n) , j \7 uj J � 1 and k u = 0 } , and II \lulcannot l denotes the total mass of the measure \7u. Moreover the constant Sn in (2.16) be improved. Inequality (2. 1 6) is established as above, see also Theorem 2.1 in [5]. For the last statement, assume that for every j1 M#(n; R2) and for every tj5 E

E

P ROOF .

E

H6(n; R2) C (O; R2), one has

E

n

In tP i1 � AI! ill! (fn l \7 A $1 2 ) � , for some constant A. We claim that for every u BV(O) with fn u = 0, we have l u i ! L2 � AJI \7u l ! . Indeed, set j1 \7.Lu, and choose any function tj5 H6(n)nC (O) such that \7 A$= u 0 in n [1, Theorem 3]. 3. Is the supremum in (2.17) achieved by some u E BV(O)? Or equivalently, does equality hold in (2.16) in the nontrivial cases? The problem has been treated on the sphere [19] and on the unit ball [1 1]. For a general domain n Rn, with n � 3 and when BV(O) and 12(0) are replaced by the n 2 spaces H (0) and Ln-2 (0), an affirmative answer has been given [10, Proposition 1.2]. 2.13. As is well known, there is no universal bound on Sn, even when replacing the constraint ll\7u ll � 1 by the constraint J l \7ul ! v � 1. This is related to the eigenvalue problem for the Laplacian with Neumann boundary condition. In ·

E

E

=

P ROBLEM

c

1

REMARK

the similar inequality

inf l ! u - c i L �' � Snll \7u l u , the best constant Sn is proportional to a relative isoperimetric constant of n [14, Theorem 3.2.3 and § 6.1.7]. A consequence of Proposition 2.12 is the inequality k tP i1 � Sn I ili (fn l \7$12 ) � , for every j1 M#(n; R2) and for every tj5 H6(n; R2) n C (O; R2), since fn 1 \7 $12 � fn l \7$12 . By analogy with the above, for j1 M#(f2, R2), set (2.18) A�,n = sup { k $ i1 $ E H6(R2 ; R2) and k l \7 $1 2 � 1 } ; and (2.19) An = sup{ A�,n i1 M# (n, R2) and lli111 � 1 } , so that n

cER

·

E

E

(\

E

·

:

E

HAIM BREZIS AND JEAN VAN SCHAFTINGEN The supremum in (2.18) is uniquely achieved since M#(fl, R2) H-1 (0; R2) , and the maximizer is bounded and continuous [5]. In general, we do not expect having A� So. Indeed, the maximizing vector fields cp in Proposition 2.12 need not be divergence-free. One has PROPOSITION 2.14. There exists a > 0 such that for every domain n R2' A� ;:::: a . PROOF. Simply take some compactly supported divergence-free measure [1 E f 42

c

=

c

M (O ; R2) , and some compactly supported vector field E C.;"' (R2; R2) JR2 f [1 :/:- 0. By translation and dilation, one has that

such that

·

A'0 >

-

JR2 f . i1

0

llilll llV'fllv

This raises the question PROBLEM 4. Compute info A� and info So. Are they achieved? In [19, Question 4. 1 ], the question was asked whether info So = SB (O, l) · Remember that An does not have an upper bound independent of the geometry. If we allow n to be multiply connected, A� has no upper bound. On the other hand, we do not know whether A� has an upper bound independently of the geometry for simply connected domains. PROBLEM 5. Does one have sup{A� : n R2 is a simply connected domain} < oo? c

3. Higher dimensions

3.1. Inequalities for curves. Throughout this section r C Rn is a simple, closed, rectifiable curve. The optimal constant in Theorem 2 is Ar = sup {[ r.p . r :

such that

1r

The vector field

=

vector field. This is an interesting open problem:

CIRCULATION INTEGRALS AND CRITICAL SOBOLEV SPACES

43

PROBLEM 6.

Does one have rp E (L00 n C) (an; an). More generally, let i1 E M (an; an) be a bounded measure such that div i1 = 0. From [2, 3] , we know that i1 E (W1·ncan; an) ) * and hence there exists a unique $ E W1·n can; an) modulo

constants that solves

n - :E ai ( IY'
=

;1 .

Does one have $ E (L00 n C)(an; an)? In two dimensions, Proposition 2. 2 gives the exact value of Ar in terms of :E , I I the area of the surface spanned by r. This is not anymore the case in higher dimensions. Let us examine what happens when r an is planar, i.e., if r II where II an is a (two-dimensional) plane. Recall that the trace on of a function u E W1 ·n(an) belongs to \V!. ,n (II) with c

c

II

c

u

ll l llw� ·n (TI ) ::::; C llull w l,n (Rn ) ·

Conversely, given any

g

n

E w!.,n(II),

there is an extension u to an such that

llull w l,n(Rn ) ::::; C ll9llw � ·n (ll )

(one proceeds for example by successive harmonic extensions). As a consequence, we have, when r II c an, Ar c:::: sup { £ $ · t : rp E C� (II; a2 ) and ll$llw* · n( ll) ::::; 1 } , (3. 1) Since W1·2 (a2 ; a2 ) w!.,n (a2 ; a2 ) [15, Theorem 2.2.3], we see, by Proposi tion 2. 2 , that c

c

(3.2)

I :E I �

::::; CA r ,

for every planar curve r an, where :E II is the surface spanned by r. This leads to the problem PROBLEM 7. Let r an be a simple, closed, rectifiable curve, and let I :E I be the area of an area-minizing surface spanned by r. Does (3. 2 ) hold? On the other hand, one cannot find an upper bound on Ar in terms of I :E I : c

c

c

3.2. There exists a sequence of planar surfaces :Ek and planar curvesPROPOSITION r k a:Ek such that, as k - oo, Ark -> oo while In view of (3.1), the conclusion follows from LEMMA 3.3. If p 2 and s 2/p, then there exists a sequence of planar surfaces :Ek and planar curves rk a:E k . and a sequence of vector fields rpk E ;:" (a2 ; a2 ) such that, as k -> oo, =

>

C

=

=

r
lrk

HAIM BREZIS AND JEAN VAN SCHAFTINGEN

44

while and ll
·

c

R2

such that

where t2 = f. €2 . Set and define

One has

rk = Tk (r) , �k = Tk ( � ) , ;jk = k - 8 €2 ('1/J r;; 1 ) . 1 '1/Jk tk = k -s 1 '1/J t2 0

�

rk

·

�

On the other hand, one has

1

r

----) 00 .

and

D

Let r C Rn be a simple closed rectifiable curve. Recall that (see [2, 18])

£ $ f :::; Cs,p lf i ii
lfl 2 C l � l ! ,

where 1 � 1 is the area of an area-minimizing surface spanned by r. question is whether for every $ E W8·P(Rn ; Rn ) ,

A natural

(3.3) with 0 < s < n and sp = n. From Lemma 3. 3 , we deduce that (3. 3 ) fails when p > 2. We ask the question whether (3. 3 ) holds when p = 2 and s = � : PROBLEM 8. Does (3.3) holds when p = 2 and s = �?

The final question concerning the inequality for curves is finding the optimal constant among all curves: PROBLEM 9 . When n 2 3, is A = sup{ Ar r Rn is a closed rectifiable curve } attained? By which curve? :

c

CIRCULATION INTEGRALS AND CRITICAL SOBOLEV SPACES The answer to Problem 9 is open even when r is a planar curve of Rn, n 2 3. There are numerous variants of Problem 9. In particular, one can define Ar = sup {£ $ · f $ E C�(Rn ; Rn ) and Ln I V' !fi n :S 1 } , or one could work with different norms on W1·n (Rn ; Rn), or even on W8·P(Rn ; Rn), with 0 < s < n and sp n . 45

1\

:

=

3.2. Inequalities for measures. As in two dimensions, we can also consider the relaxed problem with measures. When j1 is a finite vector measure such that div j1 0, define AJl = sup {L2 $ j1 $ E C� (Rn ; Rn ) and I Y'!fi! Ln :S 1 } . As explained in [2] and in Remark 2.6, the optimal constants in Theorems 1 and 2 are the same, i.e. =

·

:

PROPOSITION 3.4. One has A = sup { AJl j1 is a measure, div j1 = 0 and ll f111 :S 1 } . In view of Proposition 3. 4 , Problem 9 can be relaxed to PROBLEM 10. Is the supremum in Proposition 3.4 attained? By what measure? The advantage of the formulation of Problem 10 versus Problem 9, is that while :

rmeasures. was taken among closed curves, j1 is taken in the vector space of divergence-free One could then hope that some kind of concentration-compactness could

provide the existence of an optimizer. The divergence-free condition however is quite rigid for this kind of approach. In two dimensions, the maximizing measures are integrals along circles. In higher dimensions, we ask PROBLEM 11. Let j1 be a maximizing measure in Proposition 3.4 (assuming that the supremum is achived). Is j1 an integral along a curve? A partial answer is given by

PROPOSITION 3.5. If j1 achieves the supremum of Proposition 3.4, then j1 is an n n extremal point of the unit ball in M# (R ; R ) . Proposition 3.5 means that maximizing measures are atomic, i.e., they do not have any nontrivial decomposition preserving the mass into divergence-free mea sures. One might be tempted to claim that atomic divergence-free measures are circulation integrals. However, as explained by Smirnov [16] , there are divergence free atomic measurekthat are not circulation integrals: Consider for k 2 2 a k dimensional torus T and a constant vector fieldk iJ on Tk such that the equation = iJ does not have periodic solutions. If 4> T Rn maps Tk on 8 and iJ on J, one has that j1 defined by }rRn $ · f1 = Jre $ · -& is atomic but is clearly not a circulation integral. x

:

�

HAIM BREZIS AND JEAN VAN SCHAFTINGEN PROOF OF PROPOSITION 3 . 5 . Since one has clearly !Iilii = 1, assume by con tradiction that il = >.il1 + (1 >.)ilz, where E (0, 1), il1 , ilz E M# (Rn ; Rn ), ll ili ll = 1 and ili =f. il. Let tp E W1• n (Rn; Rn ) such that IIVtpiiLn = 1 and

46

)..

-

r

}Rn Because tp is a maximizer for At7,

tp . il

=

A[.i .

Therefore, tp cannot be a maximizer for AJ7, and r il < A[.i, }Rn tp . i

whence A = A[.i = r tp . il }R n

=

)..

r

}Rn

which is a contradiction.

tp . il1

+

,

(1 - >.) r tp . il2 }Rn < >.Af.i,

+

(1

-

>.)Af.i,

:::;

A, 0

References =

1. J. Bourgain and H. Brezis, On the equation div Y f and application to control of phases, J. Amer. Math. Soc. 16 (2003), no. 2, 393-426 (electronic). 2. , New estimates for the Laplacian, the div-curl, and related Hodge systems, C. R. Math. Acad. Sci. Paris 338 (2004) , no. 7, 539-543. 3. , New estimates for elliptic equations and Hodge type systems, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 2, 277-315. 4. J . Bourgain, H. Brezis, and Mironescu, H 112 maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation, Pub!. Math. Inst. Hautes Etudes Sci. (2004) , no. 99, 1-115. 5. H . Brezis and J. Van Schaftingen, Boundary estimates for elliptic systems with L 1 -data, Calc. Var. Partial Differential Equations 30 (2007) , no. 3, 369-388. 6. Andrea Cianchi, A quantitative Sobolev inequality in BV, J. FU.nct. Anal. 237 (2006), no. 2, 466-481 . 7 . H. Federer, Geometric measure theory, Springer-Verlag, New York, 1969. 8. H. Federer and W.H. Fleming, Normal and integral currents, (1960) . 9. N. Fusco, F. Maggi, and A. Pratelli, The sharp quantitative Sobolev inequality for functions of bounded variation, J. FU.nct. Anal. 244 (2007) , no. 1 , 315-341 . 10. P . Giriio and T. Weth, The shape of extremal functions for Poincare-Sobolev-type inequalities in a ball, J. Funct. Anal. 237 (2006 ) , no. 1, 194-223. 1 1 . M. Leckband, On the existence of extremals for the Sobolev inequality on the ball Bn for functions with mean value zero, preprint. 12. F. Maggi and Cedric V., Balls have the worst best Sobolev inequalities, J. Geom. Anal. 1 5 (2005), no. 1 , 83-121. 13. V.G. Maz'ya, Classes of domains and imbedding theorems for function spaces., Sov. Math., Dokl. 1 (1960) , 882-885. 14. , Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. 15. T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, de Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co. , Berlin, 1996. 16. S . K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows, Algebra i Analiz 5 (1993) , no. 4, 206-238. 17. J. Van Schaftingen, Estimates for U -vector fields, C.R. Math. 339 (2004), no. 3, 181-186. ___

___

P.

___

CIRCULATION INTEGRALS AND CRITICAL SOBOLEV SPACES 18.

47

___

, A simple proof of an inequality of Bourgain, Brezis and Mironescu, C.R.Math. 338 (2004), no. 1, 23-26. 19. M. Zhu, Sharp Poincare-Sobolev inequalities and the shortest length of simple closed geodesics on a topological two sphere, Commun. Contemp. Math. 6 (2004), no. 5, 781-792.

Added in proof. We have been informed by N. Fusco that he has obtained jointly with A. Pratelli partial answers to our Problems 3 and 4. More precisely, info Sn SB(O, l ) and if an is Lipschitz , the supremum in (2.17) is achieved by the characteristic function of a subdomain (paper in preparation). =

DEPARTMENT OF MATHEMATICS - HILL CENTER, RUTGERS, THE STATE UNIVERSITY OF NEW JERSEY, l l O FRELINGHUYSEN RD. , PISCATAWAY, NJ 08854-8019 E-mail address:

brezis(gmath . rutgers . edu

DEPARTMENT DE MATHEMATIQUE, UNIVERSITE CATHOLIQUE DE LOUVAIN, CHEMIN DU CY CLOTRON 2, 1348 LOUVAIN-LA-NEUVE, BELGIUM E-mail address:

Jean . VanSchaft ingen!guclouvain. be

Proceedings of Symposia in Pure Mathematics Volume 79, 2008

Mutual absolute continuity of harmonic and surface measures for Hormander type operators Luca Capogna, Nicola Garofalo, and Duy-Minh Nhieu Dedicated to Professor Maz'ya, on his 70th birthday

ABSTRACT. In this paper, we consider the Sub-Laplacian £. which consists of a sum of squares of smooth vector fields satisfying the Hi:irmander finite rank condition. We study the Dirichlet problem associated with this operator on domains that satisfy certain geometric conditions. For such domains, several key results are established. These results consist of 1 ) A reversed Holder in equality for the Poisson kernel 2) The £.-Harmonic measure and the surface measure (as well as the H-Perimeter measure) are mutually absolutely con tinuous 3) A representation (hence solvability of the Dirichlet problem) for solutions to the Dirichlet problem.

1. Introduction In this paper we study the Dirichlet problem for the sub-Laplacian associ ated with a system = of coo real vector fields in �n satisfying Hormander's finite rank condition

X {X1 , ... , Xm } rank Lie[X1 , ... , Xm] n. ( 1.1) Throughout this paper n 2 3, and Xj denotes the formal adjoint of Xj . The sub-Laplacian associated with X is defined by m (1.2 ) £u jL=l x;xju . A distributional solution of £u = 0 is called £-harmonic. Hormander's hypoel lipticity theorem [H] guarantees that every £-harmonic function is coo , hence it is a classical solution of £u = 0. We consider a bounded open set D C �n , and study =

=

2000 Mathematics Subject Classification. 31C05, 35C15, 65N99. First author supported in part by NSF Career grant DMS-0134318. Second author supported in part by NSF Grant DMS-07010001. @2008 American Mathematical Society

49

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU

50

{

the Dirichlet problem ( 1 .3)

£u = in D , u = ¢ on aD .

0

Using Bony's maximum principle [B] one can show that for any ¢ E C(aD) there exists a unique Perron-Wiener-Brelot solution Hf to (1.3). We focus on the boundary regularity of the solution. In particular, we identify a class of domains, which are referred to as AD Px domains for which we prove the mutual absolute continuity of the £-harmonic measure dw x and of the so-called horizontal perimeter measure d(Jx = Px (D; ) on aD. The latter constitutes the appropriate replacement for the standard surface measure on aD and plays a central role in sub-Riemannian geometry. Moreover, we show that a reverse Holder inequality holds for a suitable Poisson kernel which is naturally associated with the system X. As a consequence of such reverse Holder inequality we then derive the solvability of ( 1 .3) for boundary data ¢ E LP(aD, d(Jx) , for ::; oo. If instead the domain D belongs to the smaller class (J - ADPx 1 < introduced in Definition below, we prove that £-harmonic measure is mutually absolutely continuous with respect to the standard surface measure, and we are able to solve the Dirichlet problem for for boundary data ¢ E LP(aD, d(J) , for 1 p ::; 00 . The connection between harmonic and surface measure is a central question in the study of boundary value problems for second order partial differential equations. As it is well-known a basic result of Brelot allows to solve the Dirichlet problem for the standard Laplacian when the boundary datum is in £ 1 with respect to the harmonic measure. However, since the latter is difficult to pin down, it becomes important to know for what domains one can solve the Dirichlet problem when the boundary data are in some LP space with respect to the ordinary surface measure d(J. In his ground-breaking paper [Dal] Dahlberg was able to settle the long standing conjecture that in a Lipschitz domain in .!Rn harmonic measure for the Laplacian and Hausdorff measure Hn- l restricted to the boundary are mutually absolutely continuous. One should also see the sequel paper [Da2] where the mutual absolute continuity was obtained as a consequence of the reverse Holder inequality for the kernel function k = dwjd(J . For C 1 domains Dahlberg's result was also independently proved by Fabes, Jodeit and Riviere [FJR] by the method of layer potentials. The results in this paper should be considered as a subelliptic counterpart of Dahlberg's results in [Dal] , [Da2] . There are however four aspects which substan tially differ from the analysis of the ordinary Laplacian, and they are all connected with the presence of the so-called characteristic points on the boundary. In order to describe these aspects we recall that given a C 1 domain D c .!Rn' a point Xo E aD is called for the system X = {X1 , . . . , Xm} if indicating with N( xo) a normal vector to aD in Xo one has

(admissible for the Dirichlet problem), ·

p

8.10

(1.2)

<

1977

characteristic

The characteristic set of D, hereafter denoted by � = � D , x , is the collection of all characteristic points of aD. It is a closed subset of aD, and it is compact if D is bounded. We next introduce the most important prototype of a sub-Riemannian space: the Heisenberg group IHin. This is the stratified nilpotent Lie group of step

51

MUTUAL ABSOLUTE CONTINUITY

= (z Z1 , two whose underlying manifold is en X JR with group law o . . . , Yn ), and we identify z = � /m(z · z')). If X = l . . . Xn , = E en with the vector E JR 2n, then in the real coordinates E JR2n+ l a basis for the Lie algebra of left-invariant vector fields on IH!n is given by the vector fields

t'-

(x ,

(1 •4)

X· = J

(x, y) 8 y 8' 8 j 2 8t

-

x

,

- 2 -

) y (yl , Xn+j

=

In view of the commutation relations

[Xj , Xn + k]

=

8jk

(z, t) (z't') + t + x iy (x,y,t) + � 8 , j 1, ... , n, + 8yj 2 8t Xj -

-

=

8 8t , j, k = 1, ... ,n ,

the system X = {X1 , ... , X2 } generates the Lie algebra of IHin . The real part of n the Kahn-Spencer sub-Laplacian on IH!n is given by

(1.5)

2n Co = l: Xi2 = j= l

�z

n lzl2 Du + Dt(l: xiDYi - YiDxJ . +4 j= l

This remarkable operator plays an ubiquitous role in several branches of math ematics and of the applied sciences. We stress that .Co fails to be elliptic at every point. Concerning the distinctions mentioned above we note:

(1.3)

1) Differently from the classical case, in the subelliptic Dirichlet problem the Euclidean smoothness of the ground domain is of no significance from the stand point of the intrinsic geometry near the characteristic set I:. In this geometry even a domain with real analytic boundary looks like a cuspidal domain near one of its characteristic points. Since bounded domains typically have non-empty character istic set it follows that the notion of "Lipschitz domain" is not as important as in the Euclidean setting, and one has to abandon it in favor of a more general one based on purely metrical properties, see [CGl] . With these comments in mind, in this paper we will assume that the domain D in be NTA x with respect to the Carnot-Caratheodory distance associated with the system X, see Definition below) and c= . The former property allows us to use some fundamental results developed in [CGl] , whereas the smoothness assumption permits to use tools from calculus away from the characteristic set. In this con nection we mention that the c= hypothesis guarantees, in view of the results in [KNl] , that the Green function for and singularity at a given point in D is smooth up to the boundary away from I:, see Theorem below. Another striking phenomenon is that in the subelliptic Dirichlet problem nonnegative £-harmonic functions which vanish on a portion of the boundary can do so at very different rates. The dual aspect of this phenomenon is that nonnegative £-harmonic functions which blow up at the boundary (such as for instance the Poisson kernel) have very different rates of blow-up depending on whether the limit point is characteristic or not, see [GV] . This is in sharp contrast with the classical setting. It is well-known [G] that in a C 1 • 1 domain all nonnegative harmonic functions ( or solutions to more general elliptic and parabolic equations) vanishing on a portion of the boundary must vanish exactly like the distance to the boundary itself. This fails miserably in the subelliptic setting because of characteristic points on the boundary. For instance, in IH!n the so-called gauge ball

(non-tangentially

(1.3)

accessible

8.1

(1.3)

2)

B

=

{ (z,

3.12

t) E IH!n l l z l4 + 16t2 < 1}

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU

52

is a real analytic domain with two isolated characteristic points p ± = ±t ) . With .C0 defined by the function + t is a nonnegative .C0-harmonic function in B which along the t-axis vanishes at the (characteristic) boundary point -t) as the square of the Carnot-Caratheodory distance to p - . On the pother hand, the function = x1 + 1 is a nonnegative .C0-harmonic function in B which along the x 1 -axis vanishes at the (non-characteristic) boundary point P1 = ( e 1 ) where = E JR.2n , like the distance to P1 . Thus, there is not one single rate of vanishing for nonnegative .C0-harmonic functions in smooth domains in lH!n ! Despite this negative phenomenon in [CGl] two of us proved that in a Ax domain all nonnegative £-harmonic functions vanishing on a portion of the boundary (characteristic or not) must do so at the same rate. This result, known as the plays a fundamental role in the present paper. Returning to the above example of the gauge ball B C lH!n, the comparison theorem which vanish in a implies in particular that all nonnegative solutions of -t), must vanish non-tangentially boundary neighborhood of the point plike the square of the distance to the boundary (and not linearly like in the classical case) ! 3 ) The third aspect which we want t o emphasize is closely connected with the discussion in 1) and leads us to introduce the third main assumption in the present paper paper. In [Jl] , [J2] D. Jerison studied the Dirichlet problem (1 .3) near characteristic points for .C0. He proved in [Jl] that for a coo domain D c lH!n if the datum ¢ belongs to a Folland-Stein Holder class ri3 , then Hf? is in r" (D) , for some o: depending on (3 and on the domain D. It was also shown in [Jl] that, given 1 ) there exists M = M(o:) > for which the real analytic domain any o: E

(1.5),

=

(0,

-

,

0

,

(0,

u(z, t) t =

u(z, t) e1 (1,0, ... ,0)

NT

comparison theorem,

=

(0,

(0,

[0u 0 =

0

u

u 0

admits a .C0-harmonic function such that = on 80-M and which belongs exactly to the Holder class r"" (in the sense that it is not any smoother) in any neighborhood of the (characteristic) boundary point = Once again, this example shows that, despite the (Euclidean) smoothness of the domain and of the boundary datum, near a characteristic point the domain appears quite non-smooth with respect to the intrinsic geometry of the vector fields X1 , . . . , Xz n · In fact, since the paraboloid n is a scale invariant region with respect to the non-isotropic group the smooth domain OM should be thought of as a non dilations � convex cone from the point of view of the intrinsic geometry of .C0 (for a discussion of Jerison's example see section This suggests that by imposing a condition similar to the classical Poincare tangent outer sphere [P] one should be able to rule out Jerison's negative example and possibly control the intrinsic gradient X G of the Green function near the characteristic set. This intuition was proved successful in the papers [LUI], [CGNl], which were respectively concerned with the Heisenberg group and with Carnot groups of Heisenberg type. In this paper we generalize this idea and prove the boundedness of the Poisson kernel in a neighborhood of the boundary under the hypothesis that the domain D in (1 .3) satisfy what we call a X It is worth emphasizing that the X-balls in our definition are not metric balls, but instead they are the (smooth) level sets of the fundamental solution of the sub-Laplacian .C. The metric balls are not smooth (see [CGl]) and therefore it would not be possible to have a notion of tangency based on these sets.

e (0, 0).

(z, t) (Az, A2t),

4).

tangent outer -ball condition.

MUTUAL ABSOLUTE CONTINUITY 4) In Dahlberg's mentioned theorem on the mutual absolute continuity between harmonic and surface measure in a Lipschitz domain D C IRn there is one important property which, although confined to the background, plays a central role. If we denote by Hn- li &D the surface measure on the boundary, then there exists constants (3 > 0 depending on n and on the Lipschitz character of D such that a rn- l :::; CJ(aD n B(x,r )) :::; (J rn- 1 , for any aD and any r > 0. A property like this is referred to as the 1-Ahlfors regularity of CJ, and thanks to it surface measure is the natural measure on aD. Things are quite different in the subelliptic Dirichlet problem. Consider in fact the gauge ball B as in 2), with its two (isolated) characteristic points p ± (0, ±�) of aB. Simple calculations show that denoting by B(P±,r) a gauge ball centered at one of the points p± with radius r, then one has for small r > 0 (1.6) CJ(aB B(P±' r)) rQ - 2 ' where Q 2n + 2 is the so-called homogeneous of lH!n relative to the 2t) associateddimension non-isotropic dilations (z, t) (.A. z , .A. with the grading of the Lie algebra of lH!n. The latter equation shows that at the characteristic points p± surface measure becomes quite singular and it does not scale correctly with re spect to the non-isotropic group dilations. The appropriate "surface measure" in sub-Riemannian geometry is instead the so-called horizontal perimeter Px(D; ) introduced in [CDG2] which on surface metric balls is defined in the following way CJx (aD n Bd(x, r)) d;j Px(D; Bd(x, r)) . To motivate such appropriateness we recall that it was proved in [DGNl], [DGN2] that for every C2 bounded domain D JH[n one has for every aD and every 0 < r < R0(D) a rQ - 1 :::; CJx(aD n Bd(x,r)) :::; (J rQ- l . 53

CJ =

a,

X E

=

n

�

=

--->

·

c

XE

Now it was also shown in these papers that the inequality in the right-hand side alone suffices to establish the existence of the traces of Sobolev functions on the boundary. Remarkably, as we prove in Theorem 1.3 below, such a one-sided Ahlfors property also suffices to establish the mutual absolute continuity of £ harmonic and horizontal perimeter measure. Such property will constitute the last basic assumption of our results, to which we finally turn. We need to introduce the relevant class of domains.

DEFINITION 1 . 1 . Given a system X = {X1 , .. . , Xm } of smooth vector fields satisfying ( 1 . 1), we say that a connected bounded open set D C IRn is admissible for the Dirichlet problem ( 1 .3) with respect to the system X, or simply ADPx, if: iii)) DD iiss non-tangentially of class coo ; accessible (NTA x) with respect to the Carnot-Caratheodory metriciii) associated to the system {X1 J . . . , Xm } (see Definition 8. 1); DThesatisfies a uniform tangent outerisXupper -ball condition (see Definition 6. 2}; iv) horizontal perimeter measure 1-Ahl f ors regular. This means exist A, Ro > 0 depending on X and D such that for every aD and 0that< rthere < Ro one has X E

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU The constants appearing in iv) and in Definitions 6.2 and 8.1 will be referred to as the ADPx-parameters of D. We introduce next a central character in this play, the subelliptic Poisson kernel of D. In fact, we define two such functions, GD (x,y) G(y,x) indicate the each one playing a different role. Let G(x,y) Green function for the sub-Laplacian (1. 2 ) and for an ADPx domain D 1 . By Hormander's theorem (H] and the results in (KNl] , see Theorem 3.12 below, for any fixed X E D the function y ---+ G(x, y) is up to the boundary in a suitably small neighborhood of any non-characteristic point Yo E &D. Let v(y) indicate the outer unit normal in y E &D. At every point y E &D we denote by Nx (y) the vector defined by (< v(y), X1 (y) >, ... , < v(y), Xm (Y) >) Nx (y) We also set W(y) INX (y) l J< v(y), X1 (y) > 2 + ... < v(y) , Xm (Y) > 2 . We note explicitly that it was proved in [CDG2] that on &D dax W da . x Denoting with I: the characteristic set of D, we remark that the vector N (y) if and only if y E I: . For y E &D \ I: we define the horizontal Gauss map at y by 0letting NxX (y) v x (y) IN (y) l DEFINITION 1.2. Given a bounded open set D JRn, for every (x,y) E D ( &D \ I:) we define the subelliptic Poisson kernels as follows x P(x, y) < XG(x, y), Nx (y) > , K(x, y) P(x,y) W(y) < XG(x, y), v (y) > . We emphasize here that the reason for which in the definition of P(x, y) and K(x, y) we restrict y to &D \ I: is that, as we have explained in 3) above (see also section 4), the horizontal gradient XG(x, y) may not be defined at points of I: . Since as we have observed the function W vanishes on I:, it should be clear that the function K(x, y) is more singular then P(x, y) at the characteristic points. However, such additional singularity is balanced by the fact that the density W of the measure ax with respect to surface measure vanishes at the characteristic points. As a consequence, K(x, y) is the appropriate subelliptic Poisson kernel with respect to the intrinsic measure ax , whereas P ( x, y) is more naturally attached to the "wrong measure" Hereafter, for x E &D it will be convenient to indicate with b. (x , r ) &D Bd(x, r ) , the boundary metric ball centered at x with radius r > 0. The first main result in this paper is contained the following theorem. THEOREM 1.3.existLetpositive D C JRn be a ADPx domain. For every p > 1 and any E D there fixed x , depending 1 parameters, such thatconstants the ADPx for EC,&DR1and 0 < r < Ron1 p,oneM,hasR , x1 , and on ;; , a K(x 1 , y)dax (y) . K(x y) (y) d ::; C a ( b.�Xo r) ) }{ 1 ) { x P � ( ax (b. r)) } , x 54

=

=

c=

=

·

=

=

+

=

=

c=

c

x

=

=

=

a.

=

o

X0

1

Xo,

fl.( x 0 ,r)

fl. ( x 0 ,r)

1In [B] it was proved that any bounded open set admits a Green function

n

MUTUAL ABSOLUTE CONTINUITY Moreover, the measures and dax are mutually absolutely continuous. By combining Theorem 1.3 with the results if [CGl] we can solve the Dirichlet problem for boundary data in LP with respect to the perimeter measure dax . To state the relevant results we need to introduce a definition. Given D as in Theorem 1.3, for any y E 8D and a > 0 a nontangential region at y is defined by r,(y) {x E D I d(x, y) ( 1 + a)d(x , 8D) } . Given a function u E C(D) , the a-nontangential maximal function of u at y is defined by N,(u)(y) sup l u (x)l . THEOREM 1 .4. Let D C JRn be a ADPx domain. For every p > 1 there exists a constant C > 0 depending on D, X and p such that if f E LP(8D, dax ), then Hf (x) { J(y) K(x, y) dax (y) , lav and IIN"'( Hf ) IILv(aD,dax) :=:; C IIJIILv(aD,dax) Furthermore, Hf converges nontangentially ax -a.e. to f on 8D. 55

dw x 1

:::;

=

=

xH, ( y )

=

·

Theorems 1.3 and 1.4 constitute appropriate sub-elliptic versions of Dahlberg's mentioned results in [Dal] , [Da2] . These theorems generalize those in [CGN2] relative to Carnot groups of Heisenberg type. We mention at this point that, as we prove in Theorem 8.3 below, for any C 1 • 1 domain D C JRn which is Ax the horizontal perimeter measure is lower 1-Ahlfors (this is a basic consequence of the isoperimetric inequality in [GNl] ) . Combining this result with the assumption is in Definition 1 . 1 , we conclude that for any ADPx domain the measure 1-Ahlfors. In particular, is also doubling, see Corollary 8.4. This information plays a crucial role in the proof of Theorem 1 .4. On the other hand, even if the ordinary surface measure is the "wrong one" in the subelliptic Dirichlet problem, it would still be highly desirable to know if there exist situations in which ( 1.3) can be solved for boundary data in some with respect to To address this question in Definition 8.10 we introduce the class of ADPx domains. The latter differs from that of ADPx domains for the fact that the assumption is replaced by the following assumption on there exist B, Ro > 0 depending on X and D such that for every X0 8D and 0 < < R0 one has

iv)

NT ax

ax

a

da. aa: r E

iv)

LP balanced-degeneracy

As we have previously observed surface measure becomes singular near a char acteristic point. On the other hand, the angle function W vanishes, thus balancing the singularities of ADPx-domains we obtain the following two re For sults which respectively establish the mutual absolute continuity of £-harmonic and surface measure and the solvability of the Dirichlet problem with data in D

LP(8 da). ,

a. a da,

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU THEOREM 1.5. Let D C !R.n be a 0' - ADPx domain. Fix x 1 E D . For every p0' ADPx 1 there exist positive constants C, R 1 , depending on p, M, Ra, x 1 , and on the - parameters, such that for every y E oD and 0 < r < R1 one has 56

>

Moreover, the measures dwx and dO' are mutually absolutely continuous. We mention explicitly that a basic consequence of Theorem 1.5 is that the standard surface measure on the boundary of a 0' - ADPx domain is doubling. D JR.n be a 0' - ADPx domain. For every p 1 there existsTHEOREM a constant1 .6C. Let0 depending on D, X and p such that if f E LP (oD, dO'), then Hf (x) { f (y) P(x , y) dO' (y ) , lew and II Na (Hf ) II LP(8D,da) :S C l l f i i LP (8D,da) · Furthermore, Hf converges nontangentially 0'-a.e. to f on oD. C

>

>

=

Concerning Theorems 1 .3, 1 .4, 1.5 and 1 .6 we mention that large classes of domains to which they apply were found in [CGN2], but one should also see [LUl] for domains satisfying assumption in Definition 1 . 1 . The discussion of these examples is taken up in section 9 . In closing we briefly describe the organization o f the paper. In section we collect some known results on Carnot-Caratheodory metrics which are needed in the paper. In section 3 we discuss some known results on the subelliptic Dirichlet problem which constitute the potential theoretic backbone of the paper. In section we discuss Jerison's mentioned example. Section 5 is devoted to proving some new interior a priori estimates of Cauchy Schauder type. Such estimates are obtained by means of a family of subelliptic mollifiers which were introduced by Danielli and two of us in [CDGl], see also [CDG2] . The main results are Theorems 5 . 1 , 5.5, and Corollary 5.3. We feel that, besides being instrumental to the present paper, these results will prove quite useful in future research on the subject. In section 6 we use the interior estimates in Theorem 5 . 1 to prove that if a domain satisfies a uniform outer tangent X-ball condition, then the horizontal gra dient of the Green function is bounded up to the boundary, hence, in particular, near I;, see Theorem 6.6. The proof of such result rests in an essential way on the linear growth estimate provided by Theorem 6.3. Another crucial ingredient is Lemma 6.1 which allows a delicate control of some ad-hoc subelliptic barriers. In the final part of the section we show that, by requesting the uniform outer X-ball condition only in a neighborhood of the characteristic set I;, we are still able to obtain the boundedness of the horizontal gradient of up to the characteristic set, although we now loose the uniformity in the estimates, see Theorem 6.9, 6.10 and Corollary 6 . 1 1 . In section we establish a Poisson type representation formula for domains which satisfy the uniform outer X-ball condition in a neighborhood of the charac teristic set. This result generalizes a similar Poisson type formula in the Heisenberg group JH[n obtained by Lanconelli and Uguzzoni in [LUl] , and extended in [CGN2]

iii)

2

4

G

G

7

MUTUAL ABSOLUTE CONTINUITY to Carnot groups of Heisenberg type. If generically the Green function of a smooth domain had bounded horizontal gradient up to the characteristic set, then such 57

Poisson formula would follow in an elementary way from integration by parts. As we previously stressed, however, things are not so simple and the boundedness of XC fails in general near the characteristic set. However, when D C Rn satisfies the uniform outer X-ball condition in a neighborhood of the characteristic set, then combining Theorem 6.6 with the estimate K(x, y) :::; IXG (x, y) l

'

X E D, y E aD ,

see (7.7) , we prove the boundedness of the Poisson kernel y -+ K (x, y) on aD. The main result in section 7 is Theorem 7.10. This representation formula with the estimates of the Green function in sections and 6 lead to a priori estimates in for the solution to (1 .3) when the datum ¢ E C(aD). Solvability of ( 1 . 3 ) with data in Lebesgue classes requires, however, a much deeper analysis. The first observation is that the outer ball condition alone does not guarantee the development of a rich potential theory. For instance, it may not be possible to find: a) Good nontangential regions of approach to the boundary from within the domain; b) Appropriate interior Harnack chains of nontangential balls. This is where the basic results on NTAx domains from [CGl] enter the picture. In the opening of section 8 we recall the definition of NTAx-domain along with those results from [CGl] which constitute the foundations of the present study. Using these results we establish Theorem 8.9. The remaining part of the section is devoted to proving Theorems 1.3, 1 .4, and 1 .6. Finally, section 9 is devoted to the discussion of examples of ADPx and a ADPx domains and of some open problems.

5

LP

1.5

2. Preliminaries

n�

In Rn , with 3, we consider a system X = {X1 , . . . , Xm } of coo vector fields satisfying Hi::irmander's finite rank condition ( 1 . 1 ) . A piecewise C 1 curve [0 , T] -+ Rn is called [FP] if whenever exists one has for every � E ]Rn

'Y :

sub-unitary

'Y'(t)

m

< 'Y' (t) , � >2 :::; j=l L < Xj ('Y (t) ) , � > 2 . We note explicitly that the above inequality forces "(1 ( t) to belong to the span of {X1 ( . . . , Xm ( ( ) ) } . The sub-unit length of is by definition 8 ( ) = T. l 'Y 'Y(t)), 'Y t 'Y Given x, y E !Rn , denote by Sn (x, y) the collection of all sub-unitary 'Y : [O, T] -+ n which join x to y . The accessibility theorem of Chow and Rashevsky, [RaJ , [Ch] , states that, given a connected open set r2 c !Rn , for every x, y E n there exists 'Y E Sn (x, y) . As a consequence, if we pose dn ( x, y)

=

inf { l s ( 'Y ) I 'Y E Sn ( x, y) } ,

Carnot-Carathiodory distance dJRn (x,on y).n, associated It is clear

we obtain a distance on n, called the with the system X . When n = Rn , we write d(x, y) instead of

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU that d(x, y) ::::; dn(x, y), x, y E n, for every connected open set n Rn. In [NSWJ it was proved that for every connected n Rn there exist C, € > 0 such that (2.1) C lx - y l :::=; dn(x, y) ::::; C- 1 Ix - yl • , x,y E O. This gives d(x,y) ::::; c - 1 l x - yl • , x,y E 0, and therefore is continuous. 58

c

CC

It is easy to see that also the continuity of the opposite inclusion holds [GNl] , hence the metric and the Euclidean topology are compatible. For and > we let < } . The basic properties of these balls were established by Nagel, Stein and Wainger in their seminal paper [NSW] . Denote by Y1 , . . , Yi the collection of the Xj 's and of those commutators which are needed to generate A formal "degree" is assigned to each namely the corresponding order of the commutator. If is a n-tuple of integers, following [NSWJ we let ) and = 2:7= 1 = det ( Yi, , . . . , Yi J . The is defined by

x E Rn r 0,

Bd(x, r) {y E Rn I d(x, y) r . Rn. ::::; }i, I = (i1, ... , i ), 1 n ::::; l d( deg(}i1 , ij J) Nagel-Stein-Wainger polynomial a1(x) A(x,r) L l ai (x) l rd(I), (2.2) r > 0. For a given bounded open set U Rn, we let (2.3) Q sup {d(I) I la i (x) l =J 0, x E U} , Q(x) inf {d(J) l lai (x) l =J 0}, and notice that ::::; Q(x) ::::; Q. The numbers Q and Q(x) are respectively called the local homogeneous dimension of U and the homogeneous dimension at x with =

=

I

c

=

=

n

respect to the system X .

C

THEOREM 2.1 ( [NSW] ) . For every bounded set U Rn, there exist constants C, Ro > 0 such that, for any x E U, and 0 < r ::::; R0, (2.4) c A(x, r) ::::; I Bd(x, r) l ::::; c - l A(x, r). As a consequence, one has with C1 2Q (2.5) for every x E U and 0 < r :::=; R0• The numbers C1 , Ro in (2. 5 ) will be referred to as the characteristic local pa rameters of U. Because of (2.2), if we let E(x,r) = A(xr� r) , (2.6) then the function r ----. E(x, r) is strictly increasing. We denote by F(x, ) the inverse function of E(x, ) , so that F(x, E(x, r)) r. Let r(x, y) = r(y, x) be the positive fundamental solution of the sub-Laplacian =

·

·

=

.c

and consider its level sets

=

m

l: x;xj , j=l

O(x, r) The following definition plays a key role in this paper.

MUTUAL ABSOLUTE CONTINUITY DEFINITION 2.2. For every x E JRn, and r > 0, the set B (x , r) { y E JRn I r (x, y) > E(�, rJ will be called the X -ball, centered at x with radius r.

59

=

We note explicitly that

B(x,r) D.(x,E(x,r)),

and that

=

D.(x, r) = B (x, F(x, r)) .

One of the main geometric properties of the X-balls, is that they are equiv alent to the Carnot-Caratheodory balls. To see this, we recall the following im portant result, established independently in [NSW] , [SC] . Hereafter, the no tation Xu = (X1 u , ... , Xmu) indicates the sub-gradient of a function u , whereas IXu! = (2::;: 1 (Xj u ) 2 ) ! will denote its length.

THEOREM 2.3. Given a bounded set U JRn, there exists R0, depending on U and X, such that for x E U, 0 < d(x,y) s; R0, one has for s E N {0}, and for someonconstant C C(U, X , s) > 0 d(x, y) 2- s IX11 Xh ··· X1sr (x , y) l (2.7) c - 1 I Bd (x, d(x, y) ) ' l 2 d(x,y) r(x , y) � C I Bd(x, d(x , y ) ) l Inallowed the first to actinequality on eitherinx(2.or7),y. one has ]i E {1, ... , m} for i 1 , .. . , s, and X}i is C

U

=

::;_

=

(2. 5 ), a(2.>7),1 , itdepending is now easy to recognize that, given a bounded set such that on U and

I n view of UC there exists

JRn ,

X,

(2.8) for x E U, 0 < r s; R0 • We observe that, as a consequence of (2. 4 ), and of (2.7), one has 1 ) (2. 9) C d(x, y) s; F (x , s; C - 1 d(x , y), x r( , y) for all x E U, 0 < d(x, y) s; R0• We observe that for a Carnot group G of step k, if Vk is a V1 stratification of the Lie algebra of G, then one has A(x,r) = canst r Q , for every x E G and every r > 0, with Q 2::�=1 j dimVj , the homogeneous dimension of the group G. In this case Q ( x ) Q. In the sequel the following properties of a Carnot-Caratheodory space will be g =

EB . . . EB

=

=

useful.

PROPOSITION 2.4. (JR", d) is locally compact. Furthermore, for any bounded set U C lRn there exists Ro R0(U) > 0 such that the closed balls B(x0, R) , with X0 E U and 0 < R < R0, are compact. =

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-lv!INH NHIEU REMARK 2.5. Compactness of balls of large radii may fail in general, see [GNl] . However, there are important cases in which Proposition 2.4 holds glob in the sense that one can take U to coincide with the whole ambient space and = oo . One example is that of Carnot groups. Another interesting case is that Raally, when the vector fields Xj have coefficients which are globally Lipschitz, see [GNl] , [GN2] . Henceforth, for any given bounded set U C JRn we will always assume that the local parameter Ra has been chosen so to accommodate Proposition 2.4. 60

3. The Dirichlet problem

X = {X11 , ..p, X::;moo} ofwec=denote vector fields in !Rn (1.1),{f LP(D) X fD CLP(D),j by £1· P (D) the 1, ... , m} endowed with its natural m = + l lfl l c1·P(D) l lf i i LP (D) L 1 1 Xj fi i £P (D ) j =l The local space Cz��(D) has the usual meaning, whereas for 1 p < oo the space £6•P (D) is defined as the closure of C0(D) in the norm of £ 1 ·P (D). A function u Cz��(D) is called harmonic in D if for any ¢; E C0(D) one has jD f xjuXj ¢; dx = 0 , j =l i.e., a harmonic function is a weak solution to the equation Cu = 2:;':, 1 XJ Xj u = 0. By Hi:irmander's hypoellipticity theorem [H] , if u is harmonic in D, then u lRn , and a function ¢; c=(D). Given a bounded open set D £1 •2 (D), the Dirichlet problem consists in finding u cf� �(D) such that { uCu =¢; 0E £6'2 (D)in .D (3. 1 ) By adapting classical arguments, see for instance [GT] , one can show that there exists a unique solution u £ 1 • 2 (D) to (3.1). If we assume, in addition, that ¢; C(D), in general we cannot say that the function u takes up the boundary value ¢; with continuity. A Wiener type criterion for sub-Laplacians was proved in [NS]. Subsequently, using the Wiener series in [NS] , Citti obtained in [Ci] an estimate of the modulus of continuity at the boundary of the solution of (3.1). In [D] an integral Wiener type estimate at the boundary was established for a general class In what follows, given a system satisfying and an open set JRn , for Banach space E I j E norm

.

::;

=

·

::;

E

E

c

E

E

-

E

E

of quasilinear equations having p-growth in the sub-gradient. Since such estimate is particularly convenient for the applications, we next state it for the special case of linear equations.

p=2 THEOREM 3.1. Let ¢; E £1• 2 (D) n C(D). Consider the solution u to (3.1). There = C(X) > 0, and Ra = Ra(D, X) > 0, such that given X0 8D, and 0 < r <existR
::;

r

MUTUAL ABSOLUTE CONTINUITY In Theorem 3. 1 , given a condenser (K, 0), we have denoted by capx (K, 0) its Dirichlet capacity with respect to the subelliptic energy £x (u) = fn 1 X u l 2 dx associated with the system X = {X1 , ... , Xm } · For the relevant properties of such capacity we refer the reader to [D] , [CDG4] . A point X0 E aD is called regular if, for any ¢ E £ 1 • 2 (D) n C(D), one has (3.2) lim u(x) ¢(x0) If every Xo E aD is regular, we say that D is regular. Similarly to the clas sical case, in the study of the Dirichlet problem an important notion is that of generalized, or Perron-Wiener-Brelot (PWB) solution to (3.1). For operators of Hormander type the construction of a PWB solution was carried in the pioneering 61

=

x .-.... x o

•

work of Bony [B] , where the author also proved that sub-Laplacians satisfy an el liptic type strong maximum principle. We state next one of the main results in [B] in a form which is suitable for our purposes.

2 . Letharmonic D C !Rn befunction a connected, boundedsolves open(1.set,3) andin the¢ E sense C(aD).of ThereTHEOREM exists a 3.unique H{j which Perron-Wiener-Brelot. Moreover, H{j satisfies (3.3) supD I H/? 1 sup 1 ¢1 · oD Theorem 3. 2 allows to define the harmonic measure for D evaluated at x E D as the unique probability measure on aD such that for every ¢ E C(aD) H{j(x) = la{ D ¢(y) dwx (y), x E D. A uniform Harnack inequality was established, independently, by several au thors, see [X] , [CGL] , [L] : If u is £-harmonic in D C !Rn and non-negative then there exists C, a > 0 such that for each ball B ( x, ar) C D one has sup u C inf u. (3.4) and Using such Harnack principle one sees that for any x, y E D, the measures are mutually absolutely continuous. For the basic properties of the harmonic :S:

dw x

B(x,r)

:::;

B ( x ,r)

dw x

dwY

measure we refer the reader to the paper [CGl] . Here, it is important to recall that, thanks to the results in [B] , [CGl] , the following result of Brelot type holds.

1 (aD, dwx ), for THEOREM 3. 3 . A function ¢ is resol u tive i f and only i f ¢ E L one (and therefore for all) x E D. The following definition is particularly important for its potential-theoretic im plications. In the sequel, given a condenser (K, 0), we denote by cap(K, 0) the sub-elliptic capacity of K with respect to 0, see [D] . DEFINITION 3. 4 . An open set D c !Rn is called thin at Xo E aD, if capx (Dc n Bd(x0,r), Bd(Xa,2r)) > 0. . . (3.5) lIm lnf capx (Bd(X0, r), Bd(Xo, 2r)) 3. 5 . If a problem. bounded open set D c !Rn is thin at Xo E aD, then Xo is regularTHEOREM for the Dirichlet r--+0

62

LUCA CAPOGNA , NICOLA GAROFALO, AND DUY-MINH NHIEU Proof. If D is thin at X0 &D, then 1 [ capx (Dc n Bd(x0, t), Bd(Xa, 2t)) ] dt capx (Bd(xa, t),Bd(x0, 2t)) t E

-

R

=

o

oo

.

to Theorem 3 . 1 , the divergence of the above integral implies for 0 < r < Thanks R/3 osc {u, D n Bd(x0,r)} :::; osc {¢,&D n Bd(X0,2R)} . Letting R ----> 0 we infer the regularity of X0•

0

A useful, and frequently used, sufficient condition for regularity is provided by the following definition.

An open set n �n is said to have positive density at Xo J O n Bd(xa,r)l > 0. . . lIm mf j Bd(Xo, r) l PROPOSITION 3. 7. If De has positive density at X0, then D is thin at X0•

DEFINITION 3.6.

&n,

c

if one has

E

r-+0

Proof. We recall the Poincare inequality

In I

(diam(0))2 In IX¢1 2 dx , valid for any bounded open set n �n , and any

0. From the latter, we obtain capx (Dc Bd(Xa, r), Bd(Xa, 2r)) > c I DC n Bd(Xa, r)l (3.6) capx (Bd(Xa, r), Bd(xa, 2r)) r 2 capx (Bd(X0, r), Bd(Xa, 2r)) :::;

C

E

c

=

n

-

-

---'=:-----'--�'--

Now the capacitary estimates in [D] , [CDG3] give

capx (Bd(X0,r),Bd(X0,2r)) c- l rQ - 2 , C(O, X) > 0. Using these estimates in (3.6) we find capx (Dc n Bd(xa, r ), Bd(Xa, 2r)) -> C* IDe Bd(xa, r)l , I Bd(xa, r) I capx (Bd(xa, r), Bd(xa, 2r)) where C* C* (0, X ) > 0. The latter inequality proves that if De has positive density at X0, then D is thin at the same point.

C rQ -2 for some constant C

:::;

:::;

=

n

=

0

A basic example of a class of regular domains for the Dirichlet problem is provided by the (Euclidean) C1 • 1 domains in a Carnot group of step r = It was proved in [CGl] that such domains possess a scale invariant region of non-tangential approach at every boundary point, hence they satisfy the positive density condition in Proposition 3.7. Thus, in particular, every such domain is regular for the Dirichlet problem for any fixed sub-Laplacian on the group. Another important example is provided by the non-tangentially accessible domains (NTA domains, henceforth) studied in [CGl] . Such domains constitute a generalization of those introduced by Jerison and Kenig in the Euclidean setting [JK] , see Section 8.

2.

MUTUAL ABSOLUTE CONTINUITY 63 DEFINITION 3.8. Let D !Rn be a bounded open set. For 0 < a 1, the class r�·0 (D) is defined as the collection of all f E C(D) £=(D), such that l f(x) - f(y) l < oo. sup x,yED , xfy d ( X, Y ) 0 We endow r�·"' (D) with the norm l f(x) f(y) l sup (x,y) 0 x,yED,xfy The meaning of the symbol r?��( D ) is the obvious one, that is, f E r ?��( D ) if, for every w D, one has f E r�·0(w). !Rn denotes a bounded closed set, by f r�·0 (F) we mean that f coincides on the set F with a function g E r�· 0 (D), where D is a bounded open set containing F. The Lipschitz class r�· 1 (D) has a special interest, due to its connection with the Sobolev space .C 1 • 00 (D). In fact, we C

::;

n

d

E

CC

-

If F C

have the following theorem of Rademacher-Stepanov type, established in [GNl] , which will be needed in the proof of Lemma 6.1.

THEOREM 3.9. (i) Given a bounded open set U C !Rn,1 there exist Ro R0(U, X) > 0, and C = C(U, X) > 0, such that if f E .C · = (Bd(x0, 3R)), with and 0R),< soR as< R0, then fforcaneverybe modified on a R)set of dx-measure zero in x0Bd = UBd(xo, to satisfy x, y E Bd(x0, l f (x) - f(y) l :S C d(x, Y) llfll.cr . oo(Bd(x0,3R)) · If,inequality furthermore, f replace E c=(Bd(x0, 3R)), then in the right-hand side of the previous one can the beterman l open lf l l .c' . oo (Bd(x0 , 3R)) with I I X fi i Loo (Bd (xa , 3R)) · (ii) Vice-versa, let D !Rn set such that SUPx ,yE D d(x, y) < oo . If 1 1 f r�· (D), then f E .C •00 (D). note explicitly that part ( i) of Theorem 3. 9 asserts that every function f .C 1 ·=We(Bd(x0,3R)) has a representative which is Lipschitz continuous in Bd(xo,R) with respect to the metric d, i.e., continuing to denote with f such representative, one has f E r0• 1 (Bd(xo,R)). Part (ii) was also obtained independently in [FSS]. The following result was established in [D] . 3.10. Let D !Rn be a bounded open set which is thin at every x0 THEOREM fJD. If ¢ E r0·i3 (D), for some j3 E (0, 1), then there exists a E (0, 1), with a = a ( D , X , /3) such that IHf (x ) - Hf (y) l sup d(x, y) 0 < x,yED,xfy Given a bounded open set D C !Rn, consider the positive Green function = G(y,x) for C and D, constructed in [B] . For every fixed x E D, one Gcan(x,y)represent G(x, · ) as follows (3.7) G(x, ) r(x, ) hx , where hx Hf.(x, .) . Since, by Hormander's hypo-ellipticity theorem, r(x, · ) E c=(JRn \ {x}), we conclude that, if D is thin at every X0 8D, then there exists a E (0, 1) such that, for every E > 0, one has (3.8) G(x, · ) E r�·0( D \ B(x0,E)) . =

E

C

E

E

C

E

,

00 ·

·

=

·

=

-

E

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU

64

We close this section with recalling an important consequence of the results of Kohn and Nirenberg [KNl] ( see Theorem 4), and of Derridj [Del] , [De2] , about smoothness in the Dirichlet problem at non-characteristic points. We recall the following definition.

1 domain D !Rn, a point x0 aD is called DEFINITIONfor3.11. GivenXa =C{X characteristic the system 1 , ... , Xm } iffor j = 1, . . . one has < Xj(X0), N (xo) > = 0 , where N(x0) indicates a normal vector to aD at x0• We indicate with = :Ev, x the collection of all characteristic points. The set is a closed subset of aD. THEOREM 3.12. Let D !Rn be a COCJ domain which is regular for (1.3). Consider the point harmonic withan¢ openCOCJ(8D). If X0 V ofaDX0 issucha non characteristic for£,function then thereH!{,exists neighborhood that H!{ C00(D n V ) . REMARK 3.13.3. 12Wefailsstressin that, as weat characteristic indicated in thepoints. introduction, theit fails concluso sion of Theorem general In fact, D andproblem the boundary are realthatanalytic, incompletel generaly thethatsoleven utionif ofthethedomain Dirichlet H!{ maydatum be not¢better Holder continuous up to the boundary, see Theorem 3. 10. An example of such negative phenomenon indedicated the Heisenberg groupa related lHin was constructed by Jerison in [J 1] . The next section is to it. For example concerning the heat equation see [KN2] . C

E

,m

:E

:E

C

E

E

E

4. The example of D. Jerison Consider the Heisenberg group ( discussed in the introduction) with its left invariant generators .4 of its Lie algebra. Recall that lH!n is equipped with the non-isotropic dilations

(1 )

6\(z, t) = (>.z, >.2t) , whose infinitesimal generator is given by the vector field n (xi-a + Yi a . ) + 2 -a . =L ·i=l ax, 8y, at We say that a function u lH!n IR is homogeneous of degree a lR if for every (z, t) lH!n and every >. > 0 one has u(o>.(z, t)) = u(z, t) . One easily checks that if u C 1 (lH!n) then u is homogeneous of degree a if and only if Zu = a u . We also consider the vector field n ( a a) e = L xi - - Yi- , (4.1) i= l ayi axi which is the infinitesimal generator of the one-parameter group of transformations Ro lH!n lHin, () IR, given by Ro(z, t) = (e z, t), z = X + iy en . z

.

:

E

--+

E

E

-->

>."'

E

:

-

i()

E

MUTUAL ABSOLUTE CONTINUITY n

65

Notice that when 1, then in the z-plane Ro is simply a counterclockwise rotation of angle B, and in such case in the standard polar coordinates (r, B) in
a . ae In the sequel we will tacitly identify z = x E IR2n, and so l z l = Jlxl 2 I Y I2. We note explicitly that in the real coordinates t) the real part of the Kohn-Spencer sub-Laplacian ( 1.5) on IH!n is given by 2n a z l 2 a2 Co L xi2 �z l � !:It e . u ut i=l

e

=

+ iy (x,y) �

+

(x, y,

+ -4 + It is easy to see that if u has cylindrical symmetry, i.e. , if =

=

u(z, t)

then

=

eu

Consider the gauge in IH!n

N

=

N(z, t)

=

f( l z l , t) ,

=

0.

(lzl4

+ 16t2 ) 1 14 .

The following formula follows from an explicit calculation

'¢ d� IY'H N l 2 �: �HN Q� 1 where Q 2n + 2 is the so-called homogeneous dimension associated with the non-isotropic dilations {.>O· As a consequence of (4.2), if f o N for some function f : [0, ) IR, then one has the beautiful formula (4.3) Since f(t) t 2- Q satisfies the ode in the right-hand side of (4. 3 ) one can show that a fundamental solution of - 0 with pole at the group identity e (0, 0) IH!n (4.2)

=

=

=

u =

=

C

is given by

(4.4)

oo

f(z, t)

0

=

=

CQ , (z, t) =f. N(z, t) Q -2

___,

E

e,

where CQ > needs to be appropriately chosen. The following example due to D. Jerison [Jl] shows that, even when the domain and the boundary data are real analytic, in general the solution to the subellip tic Dirichlet problem (1.3) may not be any better than f0 ·" near a characteristic boundary point. Consider the domain

M E IR . Since nM is scale invariant with respect to { J>. h>o we might think of nM as the analogue of a (M 2:: or a (M < Introduce the variable

convex cone T

=

concave cone

0),

T(z, t)

=

4t N2

(z, t) =f e .

0).

66

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU It is clear that T is homogeneous of degree zero and therefore ZT 0 . Moreover, with 8 as in (4.1) , one easily checks that =

87

=

0.

T = 'Y} are constituted by the t-axis { 1, t - 4 J1 - "(2 iz l 2 ' if !'YI < 1. Furthermore, the function T takes the constant value T v'l +4M16M2 ' on anM. We now consider a function of the form (4.5) v v(z,t) Na u(T) , where the number a > 0 will be appropriately chosen later on. One has the following result whose verification we leave to the reader. PROPOSITION 4.1. For any a > 0 one has £0v 4¢Na- 2 { (1 - T2 )u11 (T) - � Tu1(T) + a(o:+4Q - 2) u(T) } 41/!Na-2 { (1 - T2 )u"(T) - (n + 1)Tu1(T) + a(a ; 2n) u(T) }· Proposition 4.1 we can now construct a positive harmonic function in nM Using which vanishes on the boundary (this function is a Green function with pole It is important to observe the level sets when 'Y = and by the paraboloids 'Y

=

=

=

=

at an interior point).

E

PROPOSITION 4.2. For any a (0, 1) there exists a number M M(a) < 0 such that the nonconvex cone nM admits a positive sol u tion of £0v 0 of the form (4.5) which vanishes on f)0,M · Proof. From Proposition 4.1 we see that if v of the form (4. 5 ) has to solve the equation £0v = 0, then the function u must be a solution of the Jacobi equation (4.6) (1 - T2 )u11(T) - (n + 1)Tu1(T) + a(a 4+ 2n) U(T) 0 . 1} is degenerate and corresponds to the As we have observed the level { T t-axis {z 0}. One solution of (4. 6 ) which is smooth as T -> 1 (remember, the t-axis is inside nM and thus we want our function v to be smooth around the t-axis since by hypoellipticity has to be in C00(0,M)) is the hypergeometric function n +-1 ; 1 - T ) 9a (T) F (- a2 , n + 2o: ; 2 -2When 0 < o: < 2 one can varify that 9a (1) 1 , and that 9a(T) -> -oo as T -> - 1+ . =

=

=

=

=

V

=

=

MUTUAL ABSOLUTE CONTINUITY Therefore, 9a has a zero Ta . One can check (see Erdelyi, Magnus, Oberhettinger and Tricomi, vol. l , p . l lO (14)), that as o + , then Ta - 1 + . We infer that for > 0 sufficiently close to 0 there exists - 1 < Ta < 0 such that 9a (Ta ) = 0 · If we choose Ta < 0 , M = M(o:) = � 1 - r,; then it is clear that On 8D,M we have T Ta , and therefore the function of the form (6.10), with u(r) = 9 of being harmonic and nonnegative in n,proof.and furthermore ona (r),anMhaswethehaveproperty that v = NOtga (Ta ) 0. This completes the 67

0: �

�

o:

V

=

=

0

a (z, t)ga (r) v = N , a ee Ero8D,M(fiM), D,M.

1),

Since o: belongs the interval (0, then it is clear that belongs at most to the Folland-Stein Holder class but is not any better than metrically Holder in any neighborhood of = (0, 0). What produces this negative phenomenon is the fact that the point is characteristic for

5. Subelliptic interior Schauder estimates

In this section we establish some basic interior Schauder type estimates that, besides from playing an important role in the sequel, also have an obvious indepen dent interest. Such estimates are tailored on the intrinsic geometry of the system . . . , Xm} , and are obtained by means of a family of sub-elliptic mollifiers which were introduced in [CDGl] , see also [CDG2] . For convenience, we state the relevant results in terms of the X-balls introduced in Definition 2.2, but we stress that, thanks to ( 2.8 ) , we could have as well employed the metric balls Since in this paper our focus is on .C-harmonic functions, we do not explicitly treat the non-homogeneous equation .C = with a non-zero right-hand side. Estimates for solutions of the latter equation can, however, be obtained by relatively simple modifications of the arguments in the homogeneous case. The following is the main result in this section.

X = {X1 ,

B(x, r) u f

Bd(x, r).

5 . 1 . Let D C !Rn be a bounded open set and suppose that u is har monicTHEOREM in D. There exists Ro > 0, depending on D and X, such that for every x E D and 0 < r ::=; Ro for which B(x,r) C D, one has for any s E N I Xh Xh· · ·Xjs u(x) l --;rc �ax l u i , for some constant C = C(D, X, s) > 0. In the above estimate, for every i = 1, . . . , s, the index ji runs in the set {1, . . . , m} . 5.2. We emphasize that Theorem 5. 1 cannot be established similarly tooremitsREMARK classical ancestor for harmonic functions, one usesof atheharmonic mean-value the coupled with the trivial observation that anywhere derivative function istionsharmonic. In the present are no longer harmonic!non-commutative setting, derivatives of harmonic func ::=;

B ( x ,r)

A useful consequence of Theorem 5 . 1 is the following.

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU C OROLLARY 5.3. Let D !Rn be a bounded, open set and suppose that u is a non-negative harmonic function in D. There exists Ro > 0, depending on D and X,anysuch that for given s N any x D and 0 < r � Ro for which B(x, 2r) D, one has for for some C = C(D X, s) > 0 . Proof. Since u 2: 0 , we immediately obtain the result from Theorem 5.1 and from the Harnack inequality (3.4) . 68

C

E

E

C

,

0

To prove Theorem 5 . 1 , we use the family of sub-elliptic mollifiers introduced in [CDGI] , see also [CDG2] . Choose a nonnegative function E Cg"(IR) , with We C [1, 2] , and such that = and let = define the kernel

f 1 f(R-1 s). fR(s) R1-) 1Xyr(x,y)l2 KR (X, y) fR (f(x, y) f(x, y)2 u Lfoc (!Rn), following [CDGl] we define the subelliptic mollGiven ifier ofaufunction by JR u(x) = fa,. u(y) KR(x, y) dy, R > 0. ( 5.1 ) We note that for any fixed x !Rn, (5.2 ) supp KR(x, ) f!(x, 2R) \ f!(x, R). One of the important features of JR u is expressed by the following theorem. THEOREM 5.4. Let D C !Rn be open and suppose that u is harmonic in D. There exists Ro > 0 , depending on D and X, such that for any x E D, and every 0 < R Ro for which D(x, 2 R) C D, one has u(x) JR u(x). Proof. Let u and f! (x, R) be in the statement of the theorem. We obtain for 'ljJ C00 (D) and 0 < t :S R, see [CGL] , ( 5.3) 2 1 '1/J (x) = 18fl(x,t) '1/J (y) 1 XIDyf(x,y)i f(X, Y)I dHn-l (Y) + lfl(x,t) £'1/J(y) [r(x,y) - t] dy. Taking 'ljJ = u in ( 5.3) , we find i2 ( 5.4 ) u(x) Jan( { x ,t) u(y) 1 XI Dy f(x,y) f(x, y) i dHn_I (y). We are now going to use ( 5.4) to complete the proof. The idea is to start from the definition of JR u( x), and then use Federer co-area formula [Fe] . One finds y) i 2 dHn l(Y)l dt. JR u(x) = loo fR (t) [l u(y) IXI Dyr(x, r (x, y)I The previous equality, ( 5.4 ) , and the fact that fiR !R (s)ds 1, imply the con clusion. JIR f(s)ds 1,

supp f

E

=

E ·

C

:S

=

as

E

=

0

8fl(x,t)

=

0

MUTUAL ABSOLUTE CONTINUITY

69

The essence of our main a priori estimate is contained in the following theorem. JR.n .

THEOREM 5.5. Fix a bounded set U C There exists a constant Ro > 0, depending only on U and on the system X , such that for any u E Lfoc (JR.n ), x E U, 0 < R :::; R0, and s E N one has for some C C(U, X , s) > 0, 1 1 l u(y) l dy. IXj, Xh · · ·Xi, JR u(x) l :::; R F( R)2 + s !1(x,R) c

=

X,

Proof. We first consider the case s = 1 . From (2.7) , and from the support property (5.2) of KR (x ) we can differentiate under the integral sign in (5. 1 ) , to obtain ,

·

,

u(x) l :::; jrB(x,2R) i u(y) i iXx KR (x , y)i dy. By the definition of KR(x, y) it is easy to recognize that the components of its sub-gradient XxKR ( x , y) are estimated as follows R - 2 I Xr ( x, y) l 3 r ( x , y)- 4 IXj KR ( x y)i < + C R - 1 r(x, y) - 2 L IXi Xkr ( x, y) i i Xkr ( x, y) i k=1 3 y) + R-1 I X r(x, l r (x, y)- 3 = Ih{x , y) + Ik ( x, y) + Ik {x, y) . IX JR

c

,

m

c

To control the three terms in the right-hand side of the above inequality, we use the size estimates (2.7) , along with the observation that, due to the fact that on the support of KR ( , · ) one has 1 1 < r ( x, :::; R ' 2R then Theorem 2.3, and (2.9), give for all x U, 0 < R :::; Ra, and D ( x, 2R) \ D (x, R) c < d( x, y) < c - 1 . (5.5) - F (x, R) -

x

y) E

Using (2.7) , (5.5) , one obtains that for i = 1 , 2, 3 .

yE

c

sup l lk (x, Y) l :::; RF(x, R) yE!1(x,2R)\!1(x,R)

3

for any x E U, provided that 0 < R :::; R0• This completes the proof in the case s = 1. The case 2 is handled recursively by similar considerations based on Theorem 2.3, and we omit details. It may be helpful for the interested reader to note that Theorem 2.3 implies

s�

1Xi, Xi2 ·· ·Xj, r ( x , y ) l < C d( x, y ) - s r ( x , y ) ,

so that by (5.5) one obtains (5.6)

sup IXh X12 . . .Xj, r (x, y)i yE!l(x,2R)\!1(x,R)

<

We are finally in a position to prove Theorem 5.1.

c

RF ( X, R

)s .

0

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU Proof of Theorem 5 . 1 . We observe explicitly that the assumption states that with R E(x,r)/2, then O(x,2R) B(x,r) C D. By Theorem 5.4, and by (5.6), we find 70

=

=

_max J uj . To complete the proof we only need to observe that Q(x, R) B(x, r), and that, thanks to Theorem 2.3, (2.9), one has C J B(x, r)J+s < c- 1 < r8 - RF(x, R)2 r8 0 REMARK 5.6. We observe explicitly that when is a Carnot group with X . . . , Xm being of the horizontal layer of its Lie algebra, then the constant C in Theorema fixed5. 1 basis and Corollary 5. 3 can be taken independent of the open set D. f!(x ,R) =

G

1,

6. Lipschitz boundary estimates for the Green function In this section we establish some basic estimates at the boundary for the Green function associated to a sub-Laplacian, when the relevant domain possesses an appropriate analogue of the outer tangent sphere condition introduced by Poincare in his famous paper [P] . Analyzing the domain nM in Remark 3.13 one recognizes that Jerison's negative example fails to possess a tangent outer gauge sphere at its characteristic point. We thus conjectured that by imposing such condition one should be able to establish the boundedness near the boundary of the horizontal gradient of the Green function (see for instance [G] for the classical case of elliptic or parabolic operators) . This intuition has proved correct. In their paper [LUl] Lanconelli and U guzzoni have proved the boundedness of the Poisson kernel for a domain satisfying the outer sphere condition in the Heisenberg group, whereas in [CGN2] a similar result was successfully combined with those in [CGl] to obtain a complete solution of the Dirichlet problem for a large class of domains in groups of Heisenberg type. The objective of this section is to generalize the cited results in [LUl] and [CGN2] to the Poisson kernel associated with an operator of Hi:irmander type. Namely, lRn X =

ifwith D C respectis a tobounded domain satisfying an intrinsic unif(1.1), orm outer sphere condition a system {X , ... sati s fying and having , X } m 1 Green function G (x, y) G ( x, y), if we fix the singularity at an interior point x -> JXG(x1, x)J,Thewhichexactis well defined for x E D\ {xl }, xbelongs 1 E D, tothenL00theinfunction a neighborhood of oD. statements are contained in Corollaries 6.7 and 6 . 1 1 . =

D

We emphasize that, in view o f Theorem 3 . 1 2 , the main novelty o f this result lies in that we do allow the boundary point to be characteristic. As it will be clear from the analysis below, the passage from the group setting to the case of general sub-Laplacians involves overcoming various non-trivial obstacles. Our first task is to obtain a growth estimate at the boundary for harmonic functions which vanish on a distinguished portion of the latter. We show that any such function grows at most linearly with respect to the Carnot-Caratheodory distance associated to the system The proof of this result ultimately relies on

X.

MUTUAL ABSOLUTE CONTINUITY

71

delicate estimates of a suitable barrier whose construction is inspired to that given by Poincare [P] , see also [G] . We begin with a lemma which plays a crucial role in the sequel. The function = denotes the positive fundamental solution of the sub-Laplacian associated with the system see Section

r(x, y) r(y, x) X, 2. bounded exist R0,C > 0, depending on U andLEMMA X, such6.1.thatForforanyevery X0 EsetU, Uand Rn, x, y there E Rn \ Bd(x0, r), one has r d(x,y ). jr (x0,x) - r(x0,y) j S C j Bd(xo,r) j two cases: (i) d(x, y) > Or; (ii) d(x, y) S Or. Here, () E Proof. (0, 1) is toWebedistinguish suitably chosen. Case (i) is easy. Using (2. 7 ) we find j r (x0,x) - r(x0,y) j r(x0,X) + r(x0,y) d(xo, x)2 + d(x0, y) 2 C [ j Bd(X0, d(xo, x)) j j Bd (X0, d(xo, y) ) j ] { 1 1 } - C 1 < C r d(x,y) · + - C E (X0,r) () j Bd(Xo,r) j E (X0,d(X0,X )) E (X0,d(X0,y) ) We next consider case (ii) , and let p = d(x, y) S er. Let be a sub-unitary curve joining x to y with length l (! ) S p + p/16. The existence of such a curve d is guaranteed by the definition of d(x, y). Consider the function g(P) ;j d(x, P) d(y, P). By the continuity of g {I} R, and by the intermediate value theorem, we can find P E {!} such that d(x, P) = d(y, P). For such point P, we must have d(x, P) d(y, P) S 43 p. (6.1) If (6.1) were not true, we would in fact have 4P3 + 34P < d(x,P) + d(y, P) S ls (!) S p + 16p , which is a contradiction. From (6.1) we conclude that x, y E Bd (P, 3p/4 ) . Moreover, d(P, X0) d(x, X0) - d(x, P) r - �P ( 1 - �()) r. C

$

< <

<

-

1

s

:

�

=

:?:

:?:

:?:

We claim that

(6.2)

� · In fact, let z E Bd(P, �p), then d(z ' x0 ) -> d(P' x0) - d(z , P) -> ( 1 -34 {}) r - 94 () r = (1 - -43 {} - -94 B) r = -r2 This proves (6. 2 ). The above considerations allow to apply Theorem 3.9, which, keeping in mind that r(x0, ) E C00(Bd(P, �p) ) , presently gives (6.3) j r (xo,x) - r(x0,y) j S C p sup jX r(x0,�) j . Using (2.7) we obtain for � E Bd(P, �p) 1 j Xr(xo, �)I S C d(x0,�) E(xo,d(xo,�)) '

provided that we take () =

-

-

·

eEBd (P, £ p)

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU where --> E(x0, t) is the function introduced in (2.6). Since by (6.2) we have d(x0,�) ;::: r/2, the latter estimate, combined with the increasingness of E(x0,·), leads t o the conclusion 1 r) sup I X f(xo, � ) I ::::; C ) r Xo, E( eE Bd ( P,�p Inserting this inequality in (6.3), and observing that rE(!o , r) ::::; C I Bd(�o,r)l , we find lf (xo, x) - r(xo , y) l ::::; c I Bd(Xo,r r) l d(x, y). 72

t

This completes the proof of the lemma.

0 The following definition plays a crucial role in the subsequent development.

DEFINITION 6.2. A domain D C !Rn is said to possess an outer X-ball tangent at Xo E aD if for some r > 0 there exists a X -ball B(x 1 , r) such that: (6.4) X0 aB(x1,r), B(x1 ,r) n D We say that D possesses the uniform outer X-ball if one can find Ro > 0 such that for every Xo aD, and any 0 < r < Ro, there exists a X -ball B(x1 , r) for which (6.4) holds. Some comments are in order. First, it should be clear from (2.8) that the existence of an outer X-ball tangent at Xo E aD implies that D is thin at Xo ( the reverse implication is not necessarily true) . Therefore, thanks to Theorem 3.5, X0 is regular for the Dirichlet problem. Secondly, when X = { a�, , . . . , a�n } , then the distance d(x, y) is just the ordinary Euclidean distance l x - Yl · In such case, Definition 6.2 coincides with the notion introduced by Poincare in his classical paper [P] . In this setting a X-ball is just a standard Euclidean ball, then every cu domain and every convex domain possess the uniform outer X-ball condition. When we abandon the Euclidean setting, the construction of examples is technically =

E

0.

E

much more involved and we discuss them in the last section of this paper. We are now ready to state the first key boundary estimate.

THEOREM 6.3. Let D !Rn be a connected open set, and suppose that for some rdepending > 0, D has an outer X-ball B(x 1 ,r) tangent at Xo aD . There exists c > 0, D and on X , such that if ¢ E C(aD) , ¢ 0 in B(x 1 , 2r)naD, then we haveonlyforonevery xED I H�(x)l ::::; C d(x,r Xo) max aD 1 ¢ 1 . Proof. Without loss of generality we assume max l ¢ 1 1. Following the idea in [P] we introduce the function ,x) 1 X E D, 1 ,r)-1 1- -E(xr(x,12r)f(x) - E(xE(x1 , r)(6.5) 1 ' where X f(x1 , x) denotes the positive fundamental solution of £, with singularity at x 1 , and t --> E(x 1 , t) is defined as in (2 .6 ) . Clearly, f is £-harmonic in !Rn \ { x } . Since r(x 1 , · ) ::::; E(x 1 ,r) - 1 outside B(x 1 , r), we see that f ;::: 0 in !R n \ B(x 1 ,r), c

E

=

aD

=

_

-t

l

MUTUAL ABSOLUTE CONTINUITY hence in particular in D. Moreover, f 1 on 8B(x1 , 2r) D, whereas f 2: 1 in (JR.n \ B(x 1 , 2r)) D . By Theorem 3.2 we infer I H.f (x) l � f(x) for every x D. The proof will be completed if we show that (6.6) J(x) � C d(x,x0) r , for every x E D. Consider the function h(t) E(x 1 , t) - 1 . We have for 0 < s < t < R0, h(s) - h(t) (t - s) E(E'(xX1,T1 , T))2 , for some s < T < t . Using the increasingness of the function r ----* rE(x 1 , r), which follows from that of E(x 1 , ) and the crucial estimate ' (x 1 , r) -< c-1 C -< rEE(xt,r) which is readily obtained from the definition of A(x 1 ,r) in (2.2), we find (6.7) C tEt(-X 1s,t) � h(s) - h(t) � C-1 s E(t -x1s, s) . in mind the definition (6.5) of j, from (6.7), and from the fact that E(xKeeping 1 , ) is doubling, we obtain f(x) � C E(x 1 ,r) {f(xt,Xo) - f(x 1 ,x)}, where we have used the hypothesis that X0 E 8B(x 1 , r). The proof of (6.6 ) will be achieved if we show that for x JR.n \ B(x 1 , r) 73

n

=

n

E

=

=

·

,

'

·

E

In view of (2.8), the latter inequality follows immediately from Lemma 6.1. This completes the proof.

0

G(x, y)

Let D C JR.n be a domain. Consider the positive Green function asso ciated to and D. From Theorem 3.2 and from the estimates (2.7) one easily sees that there exists a positive constant such that for every ED

C

CD x, y 2 ' 0 < - G(x, y) � CD I Bd(x,d(x,y) d(x, y))l

(6.8)

x, y E D. Our next task is to obtain more refined estimates for G. the uniform outer X-ball condi tion.THEOREM There exists6.4.a Suppose constant that C DC(X,JR.nD) satisfy > 0 such that d(x,y) d(y, 8D) G( x,y) <- C I Bd(x,d(x,y) )i for each x, y D, with x =j:. y.

for each

C

=

E

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU Proof. Consider a > 1 as in (2.8), and let Ro be the constant in Definition 6. 2 of uniform outer X-ball condition. The estimate that we want to prove is immediate if one of the points is away from the boundary. In fact, if either d(y, aD) ad(�� , or d(y, aD) R0, then the conclusion follows from (6.8). We may thus assume that a d(y, aD) < a(d(x,a +y)3) , and d(y, aD) < R0• (6.9) We now choose aRo ) . r - ( 2ad(x,y) , (a + 3) 2 One easily verifies from (6. 9 ) that ad(y, aD) < 2r. Let Xo be the point in aD such that d(y,aD) d(y,x0) and consider the outer X-ball B(x 1 ,rfa) tangent to the boundary of D in X0• We claim that y E D n B(x1 , (a + 3)r). To see this observe that by (2. 8 ) X0 E B(x 1 , � ) Bd(x 1 , r), and therefore a + 2 < -a + 3 r. d(y, Xl) :::; d(y, X0) + d(x0, X1 ) d(y, aD) + d(x0, Xl ) :::; --r a a This shows y E Bd(x 1 , a - 1 (a + 3)r). Another application of (2. 8 ) implies the claim. Next, the triangle inequality gives d(x,x1 ) d(x, y) - d(x1 , y) > d(x, y) - a-+a-3 r > d(x, y)(1 - 2a21 ), and consequently x E !Rn \ Bd(x 1 , ( 1 - 2�2 )d(x, y)). On the other hand (2. 8 ) implies IRn \Bd(xl , (1 - _2_2 )d(x, y)) IRn \B(xl , a� (1 - 2a1 2 )d(x, y)) IRn\B(x l , (a+3)r), the last inclusion being true since a > 1. now consider the Perron-Wiener-Brelot solution v to the Dirichlet problem .CvsuchWethat 0 in B(x 1 , (a+3)r)nD, with boundary datum a function ¢ E C(a(B(x 1 , (a+ 3)r) n D)), 0 :::; ¢ :::; 1, ¢ 1 on aB(x 1 , (3+a)r)nD, and ¢ = 0 onweaDnB(x 1 , (1+a)r). can only say that We observe in passing that, thanks to the assumptions on v is continuous up to the boundary in that portion of a(B(x 1 , (a + 3)r) n D) that is common to aD. However such continuity is not needed to implement Theorem 3.outer 2 and.C-ball deduce that 0 1. We observe that D n B(x1 , (a+ 3)r) satisfies the condition at the point Xo E aD. Applying Theorem 6.3 one infers for every y E D n B(x 1 , (a+ 3)r) (6.10) jv (y) j :::; C d(y,aD) r . Let CD be as in (6. 8 ) and define w(z) Ci/ E(x, {3d(x, y))G(x, z), where {3 (1Observe - 1a ). Since X tt B(x l , (a + 3)r), then .Cw 0 in B(x l , (a + 3)r) n D. - b that 2 if z E aB(x 1 , (a + 3)r), then d(x, z) d(x, xl ) - d(z , x1 ) (1 - 2a21 ) - (a+ 3)r {3d(x, y), 74

?:

?:

_

.

mtn

=

C

=

?:

C

�

C

=

=

:::;

D,

v

:::;

=

=

=

?:

?:

?:

MUTUAL ABSOLUTE CONTINUITY r

75

r E (x, r) , (a+ w :::; Ci/ E(x,w(y)d(x,:::;z))G(x, z) :::; &(B(x 1 v(y) DnB(x1 , (a + 3)r). 3)r) n D).

from our choice of and (3. Consequently, in view of the monotonicity of ---> and (6.8) , we have that 1 on By Theorem 3.2 one concludes that The estimate in of established above, along with (2. 1 ) , completes the proof. 0

v

in a Carnot group, by exploiting G(y,50]x)that= G(x, y), one can actually improve G(x, y) :::; C d(x, y) -Qd(x, &D)d(y, &D) , x, y E D , x =I y ,

It was observed in [LU2, Theorem the symmetry of the Green function the estimate in Theorem 6.4 as follows

where Q represents the homogeneous dimension of the group. An analogous im provement can be obtained in the more general setting of this paper. To see this, note that the symmetry of and the estimate in Theorem 6.4 give for every

(6 . 1 1 )

G G(y, x) - G(x, y) <- C I Bd(x,d(x,y) d(x, y)) l d(y, &D) '

x, y E D

_

C 0

where > is the constant in the statement of Theorem 6.4. We now argue exactly as in the case in which (6.9) holds in the proof of Theorem 6.4, except that we now define

d(x, y)) l G(z,x) , z E B(x1 , (a + 3)r) n D . w(z) = C- 1 d(x,&D) _ 1 1 Bd(x,d(x,y)

Using (6. 1 1) instead of (6.8) we reach the conclusion that

w(z) :::; 1 , for every z E &(B(x1 , (a+ 3)r) n D) . Since .Cw = 0 in B(x 1 , (a + 3)r) n D, by Theorem 3.2 we conclude as before that w(y) :::; v(y) in D n B(x1 , (a + 3)r). Combining this estimate with (6.1 0) we have proved the following result. the uniform outer X -ball con dition.COROLLARY There exists6.5.a Suppose constant that C = C(X,�nD) satisfy > 0 such that &D)d(y, &D) ' G(x, y) < C d(x,I Bd(x, d(x, y))i for each x, y E D, with x =I y. D

C

We now turn to estimating the horizontal gradient of the Green function up to the boundary. The next result plays a central role in the rest of the paper.

There orm outer existsTHEOREM a constant6.6.C =Assume C(X, D)the>uni0 fsuch that X -ball condition for D �n . d(x,y) I XG(x,y) i :::; C I Bd(x,d(x,y)) i' for each x, y E D, with x =I y. Proof. Let Ra b e as in Definition 6.2. Fix x, y E D and choose 0 < r < Ra such that x rf:. Bd(y, ar) D. Applying Corollary 5. 3 and ( 2 . 8 ) to G(x, ·) we obtain for every z E B(y, r) IXG(x, z) i :::; -cr G(x, z). C

C

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU d If d(y,aD) � 2ad(x,y), we choose r = min ( ( y2�D), lf ) and then the latter inequality implies the conclusion via Theorem 6.4. If d(y, aD) > 2ad(x, y), then r(x, ) - hx , we use (2.7) to bound j X rj , and, keeping in mind that G(x, ·) d with r min ( ( y2�D ), If), we apply Corollary 5.3 and the maximum principle to 76

·

=

obtain

=

I X hx (Y) i � -cr hx (Y) -cr hy (x) � -cr sup r(y, w) -cr r(y, z) for some z E aD. On the other hand, one has d(x, y) d(y,2aaD) d(y,2az) so that using ( 2. 7) one more time 1 z)) - c E(y,2ad(x,y)) 1 - c r(y, x) - c I Bd(x,d(x,y))i d(x, y) 2 . r(y,z) - c E(y,d(y, Replacing this inequality in the estimate for I X hx ( Y)I we reach the desired conclusion. =

wE8D

<

<

<

<

<

<

D

COROLLARY 6.7. If D C JR.n satisfies the uniform outer X -ball condition, then for any E D and every open neighborhood U of aD, such that ¢_ U, one has G(x0, ) E £ 1 • 00 (U). Moreover, its £ 1 • 00 (U) norm depends on D,X and U but it is independent of Localizing the hypothesis. It is interesting to note that one can still prove that G(x , ) E £ 1 • 00 (U) under the weaker hypothesis that the uniform outer X-ball condition be satisfied only in a neighborhood of the characteristic set of D. In this case, however, the uniform estimates in will be lost. We devote the last part of this section to the proof of this result. Let I: I:v aD denote the compact set of all characteristic points. DEFINITION 6.8. Let D be a C 1 domain. We say that D possesses the uniform outer X-ball in a neighborhood of I: iffor a given choice of an open set V containing I:, one can find such (6.4) that forholds.everyMore Q V n aD and 0 < r there exists > 0which atheX-ball B(x ,r) for in general, we say that D possesses 1 uniform outer X -ball along the set V n aD if one can find Ro > 0 such that for every E V n aD and 0 r < there exists a X -ball B(x1 , r) for which (6.4) holds. X0

X0

·

X0 •

0

·

X0

=

Ro

X0

E

<

c

<

Ro

Ro

Our first step consists in proving " localized" versions of Theorems 6.4 and 6.6.

THEOREM 6.9. Let D C JR.n be a domain that is regular for the Dirichlet prob lem. LetX -ball P E aD and assume that for some <:: > 0 the set D possesses the uniform outer along Bd( P, 2<::) n aD. There exists a constant C = C(X, D) > 0 such that G(x,y) :::; C I Bd(d(x,y) ,d(x , y)) i d(y,aD) for each y E Bd(P, <::) n D, and x E D, with =f. y. x

x

MUTUAL ABSOLUTE CONTINUITY Proof. The proof follows closely the one of Theorem 6. 4 and we will adopt the same notation as in that proof. Let x0 be the point in 8D closest to y . In order to apply the arguments in the proof of Theorem 6.4 we need to show that the set D has an outer .C-ball B(x 1 ,rja) at X0 for every 0 < r < Ro . Given our hypothesis it suffices to show that x0 E Bd(P, 2E) n 8D . Observe that d(y, x0) :::; d(y, P) < E, and consequently d(P,x0) < 2E. Since D has an outer X-ball B(x1,rja) at X0 for every 0 < r < Ro , then so does the subset B(x 1 , (a + 3)r) n D. The rest of the proof is a word by word repetition of the one for Theorem 6. 4 . THEOREM 6.10. Let D JRn be a domain that is regular for the Dirichlet problem. Let P E 8D and assume that for some E > 0 the set D possesses the formthatouter X -ball along Bd(P, 2E) n 8D . There exists a constant C = C(X, D) > 0unisuch I XG(x , y) l :::; C I Bd(x,d(x,y) d(x, y)) l , for each y E Bd(P, � E) n D, and x E D, with x =f. y. Proof. In the proof of Theorem 6.6 there is only one point where the outer X-ball condition is used. Consider y E Bd(P, � E) n D and assume that d(y, 8D) :::; d(y:D) and observe that if z E B(y,r ) then d(z, y) < ar :::; E/2. d(x,y). Choose2r = Consequently d( z , P) :::; d(z, y) + d(y, P) :::; E, and we can apply Theorem 6. 9 to the function G(x, z) concluding the proof in the same way as before. COROLLARY 6.11. Let D JRn be a C00 domain. If D satisfies the uniform outer X -ball condition in a neighborhood V of �' then for any X0 E· ) l Dl.cLooand(u) ev ery open neighborhood U of 8D, such that X0 ¢. U, one has IIG(x0, :::; C(x0,D, V, U, X). Proof. Observe that D is regular for the Dirichlet problem. The regularity away from the characteristic set follows by Theorem 3.12 and the regularity in a neighborhood of � is a consequence of the uniform outer X-ball condition and of the cited results in [Ci] , [D] , [NS] and [CDG3] . Denote by V the neighborhood of � where the uniform outer X-ball condition holds. In view of the compactness of � , we have that W = UP E E B(P, 2E) V, for some E > 0. We will consider also the set A = U P EE B(P, � E) W. In view of Theorem 3.12, we have that G(x0, ) E C00(1J \ {A U {x0 }}). In particular, G(x0, · ) is smooth in U \ A. This implies the estimate I I G (x0, ·) l l c'""(U\A) Co = Co(x0, D, V, X). To complete the proof of the corollary we consider y E A and observe that there must be a P E � such that y E B(P, �E). Denote by Q the homogeneous dimension associated to the system X in a neighborhood of D. In view of Theorem 6.10 we have that l - Q :::; cl , with cl depending only on X, D and At this Ipoint XG(xo,we y)choose Xo) i :::; Cd(y, C(x0, D, V, U, X) = min{ C0 , Cl }, and the proof is concluded. 0 77

0

c

0

C

c

·

C

:::;

u.

7. The subelliptic Poisson kernel and a representation formula for £-harmonic functions In this section we establish a basic Poisson type representation formula for smooth domains that satisfy the outer X-ball condition in a neighborhood of the characteristic set. This results generalizes an analogous representation formula

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU

78

for the Heisenberg group lH!n obtained by Lanconelli and Uguzzoni in [LUl] and extended in [CGN2] to groups of Heisenberg type. Consider a domain which is regular for the Dirichlet problem. For a fixed point E we respectively denote by and = the fundamental solution of and the = Green function for D and c with pole at Recall that = where h is the unique £-harmonic function with boundary values We also note that due to the assumption that be regular, h are continuous in any relatively compact subdomain of We next consider a coo domain n c 11 c containing the point For any E we obtain from the divergence theorem

r(x) r(x, Xo)

G(x) G(x, Xo),Xo · G, D\ {x0}. D Xo. u, v C00(D)

D X0 D C G(x). r(x)-h(x),, r D

k [u Cv - v Cu] dx

(7.1)

v

[H] dC! s X /-+ r(xo, x) C00(D \ k {x{xo} r(xo, x) x0,sk } Ek F(xo , s; 1 ), F(xo, · ) B(Ek ) B(xo, Ek ) B(xo, Ek ) E(xo, X0• aB(Ek ) v g�/::�:l v(x) G(x), \ B(Ek ), CG

where denotes the outer unit normal and the surface measure on 8!1. By Hormander's hypoellipticity theorem the function is in oo such that the sets ) . By Sard's theorem there exists a sequence are c= manifolds. Since by (2.7) the fundamental solution E JRn I = we can assume without restriction that such manifolds are has a singularity at strictly contained in n. Set = where is the inverse function c n ) introduced in section two. The sets of = c are a sequence of smooth X-balls shrinking to the point We note explicitly that the outer unit normal on is = - 1 1• Applying (7. 1) with = and !1 replaced by !t,k = !1 where one has = 0, we find ·

=

+

f i =l lao{ [u XjG - G Xju] < Xi, v > dC! { [G Xiu - u XjG] < Xj, v > dC! . jf=l laB(,k)

Again the divergence theorem gives

(7.2)

MUTUAL ABSOLUTE CONTINUITY Using (7.2) , and the fact that G r - h, we find

79

=

(7.3)

f J=l la{ B(< k) [G Xju - u XJ G] < X1,v > da

j=fl 18B( da - jf=l 18B( da

+

0

Using (5.3) we find

lxr� 2 da u(xo) - J{B(
r

=

I B( Ek ) l < c 2 E(xo, Ek ) - Ek ,

letting (7.4)

k -+ oo, so that Ek 0, we conclude from (7.2 ) , (7.3) , u(x0) � lan [G Xju - u XjG] < Xj, v > da In G .Cu dx . -+

+

=

To summarize what we have found we introduce the following definition.

l , at every E NITION 7. 1 . Given a bounded open set n of class C point y E an we let ( < v(y), X1 (y) > , . . . , < v(y), Xm ( Y ) > ) , Nx (y) where v(y) is the outer unit normal to n in y. We also set W(y) I Nx (y) l L j=l < v(y),XJ (Y) >2 • lfy E an \ I:, we set vx (y) INNXx (y)(y) l One has lvx (y)l 1 for every y E aD \ I: . We note explicitly from Definitions 3 . 1 1 and 7.1 that one has for the charac teristic set I: of n I: {y E an 1 W(y) o} . D FI

c

n

c

]Rn

=

m

=

=

=

=

=

=

Using the quantities introduced in this definition we can express (7.4) in the following suggestive way.

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU PROPOSITION 7.2. Let D C JRn be a bounded open2 set with (positive) Green function and consider a C domain n 0 D. For any u E CG00of(D)theandsub-Laplacian every x E 0(1one.2) has u(x) lao{ G(x, y) < Xu(y), Nx (y) > da(y) - lao{ u(y) < XG(x, y), Nx (y) > da(y) + in G(x, y) Cu(y) dy . If moreover Cu=O in D, then u(x) lao{ G(x, y) < Xu(y), Nx (y) > da(y) - lao{ u(y) < XG(x, y), Nx (y) > da(y) . In particular, the latter equality gives for every E n < XG(x, y), Nx (y) > da(y) 1 . { lao REMARK 7.3. If u E c=(D), then we can weaken the hypothesis on n and require only 0 D rather than 0 D. 80

c

c

=

X

-

=

c

c

We consider next a coo domain D C ]Rn satisfying the uniform outer X-ball condition in a neighborhood of I;. Our purpose is to pass from the interior repre sentation formula in Proposition 7.2 to one on the boundary of aD. The presence of characteristic points becomes important now. The following result due to Derridj [Del, Theorem 1 will be important in the sequel.

]

THEOREM 7.4. Let D C ]Rn be a coo domain. If I; denotes its characteristic set, then a('E) 0. =

We now define two functions on D x (aD \ I;) which play a central role in the results of this paper. They constitutes subelliptic versions of the Poisson kernel is the Poisson kernel from classical potential theory. The former function The latter for D and the sub-Laplacian (1.2) with respect to surface measure is instead the Poisson kernel with respect to the perimeter measure This comment will be clear after we prove Theorem 7.10 below.

P(x, y)

a. ax. K(x, y) Poisson kernels) . With the notation of Definition 7.1, DEFINITION for every (x,7.5y) E(Subelliptic D (aD \ E) we let P(x, y) < XG(x, y), Nx (y) > (7.5) We also define (7.6) K(x, y) P(x, < XG(x, y), x (y) > . W (y)y) extend the definition of P and K to all D fJD by letting P(x, y) K(x, y) 0 forWe-a.any x E D andy E E. According to Theorem 7. 4 the extended functions coincide a e. with the kernels in (7.5) , (7.6) . It is important to note that if we fix x E D, then in view of Theorem 3.12 the functions y P(x, y) and y K(x, y) are coo up to fJD \ I;, The following estimates, which follow immediately from (7.5) and (7.6) , will play an important role in the sequel. For (x, y) E D (aD \ E) we have (7.7) P(x, y) ::; W (y) IXG(x,y) i , K(x,y) ::; I XG(x, y)i . x

=

-

=

=

-

v

x

-->

-->

x

=

=

MUTUAL ABSOLUTE CONTINUITY

81

We now introduce a new measure on aD by letting

(7.8)

dax

W da . We observe that since we are assuming that D coo the density W is smooth and bounded on aD and therefore implies that dax « In view of this observation Theorem implies that also ax (E) =

E (7. 8 ) da. 7.4 0. REMARKmeasure 7.6. PxWe(Dmention explicitly that the measure dax in (7.8) is the X perimeter (following concentrated on &D. To explain this point we recall that for; ) any open setDenGiorgi) ]Rn Varx (xv ; O ) , (7.9) Px (D; O) where Varx indicates the sub-Riemannian X -variation introduced in [CDG2] and also developed in [GNl] . Given a bounded C2 domain D C ffi.n one obtains from [CDG2] that =

·

c

=

Px (D; O) (7.10) ( W da . lavno From (7.10) one concludes that for every y E aD and every r > 0 Px (D ; Bd(y, r)) (7.11) ax(&D Bd(y, r)) which explains the remark.and geometry The measure ax Px (D; ) on aD plays a perva sive role in the analysis of sub-Riemannian spaces, andinitsgeometric intrinsic properties have many deep implications both in subelliptic pde ' s and measure theory. For an account of some of these aspects we refer the reader to [DGN2] . PROPOSITION 7. 7. Let D C JRn be a bounded coo domain satisfying the uniform outer we haveX -ball condition in a neighborhood of its characteristic set E . For every x E D f P(x, y)da(y) 1 lav ( K(x, y)dax (Y ) . lav Proof. We x E D and recall that E is a compact set. In view of Theorem 7.n 4 weOkcan D,choose an exhaustion of D with a family of coo connected open sets with n k / D as k _...., oo, such that &n k r� ur�, with rk &D\E, k rk / aD, a(r0 _...., o. B y Proposition 7.2 (and the remark following it ) we obtain for every k E N (7.12) XG(x, y), Nx (y) > da (y) ( - 1 lank x x f . f lar� XG(x, y), N (y) > da(y) + lar� XG(x, y), N (y) > da (y) We now pass to the limit as k oo in the above integrals. Using Corollary 6.11 and a(r%} 0, we infer that x (y) > da(y) 0 . XG(x, y), lim ( N -+ k oo Jar� =

n

,

=

·

=

=

=

fix

c

c

=

=

c

=

<

<

<

_....,

_....,

<

=

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU Theorem 3.12, Corollary 6.11, and the fact that rl / aD, allow to use domi nated convergence and obtain lim f XG(x, y), Nx (y) > da(y) f XG(x, y), Nx (y) > da(y) . 82

k-+oo

lor�

<

=

lav

<

In conclusion, we have found

{ XG(x, y), Nx (y) > da(y) , - 1 lav which, in view of (7.5), proves the first identity. To establish the second identity we return to (7.12), which in view of (7.6), (7. 8 ) we can rewrite as follows { XG(x, y), Nx (y) > da(y) 1 - lor� { < XG(x, y), vx (y) > dax(y) - lor� { XG(x, y), Nx (y) > da(y) . { K(x, y) dax(y) - lor� lark Since as we have observed dax da, in view of the second estimate K (x, y) ::::; (7.7),respect we can again use Theorem 3.12, Corollary 6.11 and dominated Iconvergence XG(x,y) l in(with to ax ) to conclude that lim f K(x, y) dax (Y ) { K(x, y) dax(Y) . lav laq =

<

<

=

<

«

k-+oo

=

This completes the proof.

D

THEOREM 7.8 . Let D satisfy the assumptions in Proposition 7. 7. If ¢ C00(8D) assumes a single constant value in a neighborhood of then Hf .C1 •00 (D). Furthermore, iffor ¢ E C(8D) we have Hf .C1•00 (D), then { K(x, y) ¢(y) dax (Y) , D. { P(x, y) ¢(y) da(y) lav Hf(x) lav :E,

E

=

E E

x E

=

Proof. We start with the proof of the regularity result. Let ¢ be as in the first part of the statement. We mention explicitly that, by definition, ¢ is 000 in a neighborhood of Denote by U a neighborhood of I: in which the function ¢ is constant and along which the domain D satisfies the uniform outer X-ball con dition. As in the proof of Corollary we can assume that U = U P E E t: ) , for some E = t:(U, If we denote by Ro the constant involved in the defi nition of the uniform outer X-ball (see Definition then we can always select . In a smaller constant so that E = R0 (here > is the constant from view of Proposition we can assume without loss of generality that ¢ vanishes l ¢1 = in a neighborhood of I: and We want to show that the horizontal

aD. 6.11, Bd(P, X) > 0. 6.8), 1 (2.8) ) 2a a 7.7 max 1. aD gradient of Hf is in L00 in such neighborhood. By Theorem 3.12 the conclusion R0, where R0 is as in Hf E .C 1 • 00 (D) will follow. Fix X0 I:, and 0 Definition 6. 2 . Theorem 6. 3 implies (7.13) E

<

r

<

MUTUAL ABSOLUTE CONTINUITY for every y E D. Let now x E B(x0,r/2) D and consider the metric ball Bd(x,a - 1 T) C B(x,T), see (2. 8 ), where : Corollary 5.3 implies (7.14) I XH¢ (x) l :::; d(x,CaD) H

T =

n d(x D )

D

.

D

=

=

<

H;j (x)

G(x ,

=

< X( H;j) ( y),

H;j (y) < XG(x, y) ,

>

(y)

>

D

x

D >

x

:::;



n

=

0:

<

>

:=

:=

>

:::;

B(x0 ,3r/2)n8D

0

84

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU

X E D. For every a-measurable E c aD we set vx (E) l K(x, y) dax (y) . According to Proposition 7.9, dvx defines a Borel measure on aD. Using The orems 7.4 and 7.8 we can now establish the main result of this section. THEOREM 7. 10. Let D c �n be a domain possessing the uniform outer Xwe-ball condition in a neighborhood of the characteristic have wx = vx , i.e., for every ¢ E C(aD) one has set :E. For every x E D, { ¢(y) K(x, y) dax(Y) = laD { ¢(y) P(x, y) da(y) , x E D . H!j(x) = laD x Ineveryparticular, (x, y) E Ddw aDis absolutely one has continuous with respect to dax and da, and for dwx (y) = K(x, y), dwx (y) P(x, y) . ( 7.16) dax da Proof. We begin with proving ( 7. 16) . Let F c aD be a Borel set. If F = aD then the result follows from Proposition 7.7. We now consider the case when the inclusion F c aD is strict. Choose E > 0. Since both K(x, y) and W(y) are bounded, there exists open sets E, , F, c aD such that F c F, c F, E" and vx (E, \ F) < E/2. Theorem 7.4x guarantees the existence of open sets :E" U, such that :E c :E, c :E, c U, and v (U,) < E/2. We now choose a function ¢ E Ccgc(aD) and 0 :::; ¢ :::; 1 with ¢ = 1 on U€ and vx (supp¢) < iE. We have wx (U,) = 1 dwx (y) � 1aD ¢(y)dwx (y) H!j (x) ( 7. 17) (by Theorem 7.8) = { cp (y) K(x, y)dax(y) :::; v x (supp ¢) < �E 4 laD Let now 'l/!0, '�PI E C;:o(aD) such that 0 :::; 'l/Jo , 'l/! 1 :::; 1 and 'l/!o = l in aD \ U, , 'l/!o = O in 'l/!1 = l in F, 'l/!1 = 0 in aD \ E€ . One has wx (F) < wx (U,) + wx (F \ U,) (by (7. 1 7) ) < �E + la 'l/Jo ( Y ) 'l/Jl ( Y ) dwx (y) 3 E + H!jDo,h (x) (by Theorem 7.8) 4 � E + laD 'l/!o (Y)'l/!1 (y)K(x, y) dax(y) :::; � E + vx (E, ) 3 7 4 E + vx (F) + vx ( EE \ F) < vx (F) + 4 E . Since E > 0 is arbitrary, we infer that wx(F) :::; v x (F). If we repeat the same argument with E€ \ F playing the role of the set F, we can prove w x (E, \ F) :::; vx (E€ \ F). This allows to exchange the role of wx and vx in the computations above and conclude v x (F) :::; wx (F). We now fix

=

c=

X

=

c

=

U,

•

:E"

MUTUAL ABSOLUTE CONTINUITY

85

E C(aD)(7.16). Hf(x) laD { ¢(y)dwx (y). (7.18) On the other hand (7. 1 6) yields dwx(y) K(x, y)dux (y). If we substitute the latter in (7.18) we reach the conclusion.

To complete the proof of the theorem we now use From the definition of harmonic measure we know that for each ¢ and x E D we have =

=

0

8. Reverse HOlder inequalities for the Poisson kernel This section is devoted to proving the main results of this paper, namely The orems and In the course of the proofs we will need some basic results about domains from the paper [CGl]. We begin by recalling the relevant definitions.

1.3 , 1.4NT , 1.5A 1.6. x DEFINITION 8.1. We say that a connected, bounded open set D C :!Rn is a non-tangentially (NTAx domain, accessible hereafter) domain if there with exist respect M, r0 >to0 thefor system which: X {X1, .. ., Xm } (i) (Interior corkscrew condition) For any Xo E aD and r :::; ro there exists Ar(Xo) E D such that II d(Ar(Xa), Xo) :::; r and d(Ar(xa), aD) > II (This implies that Bd(Ar(x0), 2�) is (3M, X)-nontangential.) (ii) (Exterior corkscrew condition) De :!Rn \ D satisfies property (i). (iii) (Harnack chain condition) There exists C(M) > 0 such that for any > 0 and x, ay Harnack E D suchchain that d(x, aD) >to yd(y,whose aD) length > and d(x, y) C , there exists joining depends on C but not x on We note the following lemma which will prove useful in the sequel and which follows directly from Definition 8.1. LEMMA 8.2. Let D C :!Rn be NTA x domain, then there exist constants C, R 1 depending 0 r R1ononethehasNTAx parameters of D such that for every y E aD and every c I Bd(y,r) l :::; min {I D n Bd(y , r) I , I D c n Bd(y , r) l } :::; c-l I B d(y,r) l . has positive density at every 3.boundary point Inandparticular, every NTAforx domain therefore it is regular the Dirichlet problem (see Definition 6, Proposition 3. 7, and Theorem 3. 5). In the sequel, for y E aD and r > 0 we denote by fl (y, r) aD n Bd(y, r) the surface metric ball centered at y with radius r. We next prove a basic non degeneracy property of the horizontal perimeter measure dux in (7. 8 ). THEOREM 8.3. Let D C :!Rn be a NTA x domain of class C2 , then there exist C* , R1 > 0 depending on D, X and on the NTAx parameters of D such that for every y E aD and every 0 < r R1 ux ( fl (y, r)) � C* I Bd(y,r r) l . In particular, ux is lower 1-Ahlfors according to [DGN2] and ux(fl (y,r)) > 0. =

0

<

=

E,

E.

<

<

=

<

E,

<

E

E

86

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU

Bd(y, r)

Proof. According to (I) in Theorem 1 . 15 in [GNl] every metric ball is a PSx (Poincare-Sobolev) domain with respect to the system We can thus apply the isoperimetric inequality Theorem 1 . 18 in [GNl] to infer the existence of R 1 > such that for every E aD and every < < R1

X.

0 y 0 r Bd(y,r) Px (D Bd (y r) ) , . mm{! D n Bd(y,r) ! , ! D n Bd (y,r) ! } 9zl :S Ciso diam ! Bd(y, r)! Q' where Q is the homogeneous dimension of a fixed bounded set U containing D. On the other hand, every NT Ax domain is a PSx domain. We can thus combine the c

1

;

,

latter inequality with (7.1 1 ) and Lemma 8.2 to finally obtain

ax (ll (y, r)) > C* ! Bd(y,r r) !

This proves the theorem.

D

8.4. Let D c JRn be a NTAx domain of class C2 satisfying the upperCOROLLARY 1-Ahlfors assumption in iv) of Definition 1 . 1 . Then the measure ax is 1fors, in the sense that there exist A , R1 > 0 depending on the NTA x parameters ofAhl D and on A > 0 in iv), such that for every y E aD, and every 0 < r < R1 , one has (8. 1)

Inandparticular, the measure ax is doubling, i.e. , there exists C > 0 depending on A on the constant C1 in (2.5), such that C ax (tl(y, r)) . (8.2) ax ( ll ( y, 2r ) ) for every y E aD and 0 < r < Proof. According to Theorem 8.3 the measure ax is lower 1-Ahlfors. Since by Definition 1 . 1 it is also upper 1-Ahlfors, the conclusion ( 8 1 ) follows. From iv)the oflatter and the doubling condition (2.5) for the metric balls, we reach the desired R1 .

::;

.

conclusion (8.2).

D

The following results from [CGl] play a fundamental role in this paper. THEOREM 8.5. Let D C JRn be a NTAx domain with relative parameters M, r0• There C > 0, depending only on X and on the NTAx parameters of D, Mexistsanda constant such that for every Xo E aD one has W Ar (x o ) (6.(x0, r)) 2: C . THEOREM 8.6 (Doubling condition for .C-harmonic measure) . Consider a NTAx domain ]Rn with relative parameters M, To . Let Xo E aD and r To· There exist C >D0,c depending on X, M and r0, such that wx(tl(xo, 2r)) ::; cw x (tl(xo, r)) for any x E D \ Bd(X0, Mr). To ,

::;

87

MUTUAL ABSOLUTE CONTINUITY

THEOREM 8.7 ( Comparison theorem) . Let D C ]Rn be a X - NTA domain with relativefunctions parameters r0. Let E {)D and 0 on aDr \ �(x0, ii · If u , v are £ harmonic in D,M,which vanishX0 continuously 2r), then for every x E D \ Bd(x0, Mr) one has u (x) 0 _ 1 u(A,. (xo)) u(A,. (xo)) C v(A,. (xo)) - v(x) v(A,.(x0)) for some constant C > 0 depending only on X, M and r0• For any y E an and a > 0 a nontangential region at y is defined by ra ( Y) = {x E n I d(x , y) :::; (1 + a)d(x , en)} . Given a function u the cx-nontangential maximal function of u at y E aD is defined by Na (u)(y) = xEf',sup(y) i u (x) l . THEOREM 8.8. Let D ]Rn be a NTA x domain. Given a point x 1 E D, let f E L1 (8D, dwx1 ) and define u(x) laD { f(y)dwx (y) , x E D . Then, u is £-harmonic in D, and: ( i) Na (u)(y) :::; CMwxl (f)(y), y E aD ; (ii) u converges non-tangentially a. e. ( dwx1 ) to f. Theorem 8. 7 has the following important consequence. THEOREM 8.9. Let D ]Rn be a ADPx domain, and let K(·, ·) be the Pois son in (7.6). There exists r1 > 0, depending on M and r0 , and Kernel a constantdefined C = C(X, M, r0, R0) > 0, such that given X0 E aD, for every X E D \ Bd(X0, Mr) and every 0 r r 1 one can find Ex 0 ,x , r C �(xo, r), with <Jx (Ex0, x ,r ) = 0, for which K(x,y) C K(Ar(X0),y) wx (�(x0, r)) for every y E �(X0, r) \ Ex0,x,r · Proof. Let Xo E aD. For each y E �(xo, r) and 0 s r/2 set <

<

<

<

C

=

c

<

<

:::;

<

u v

<

D and vanish continuously on aD \ wx (�(y , s ) ) w A r(x o) (�(y , s)) C (8.3) wx (�(x0, r/2)) wAr (xa ) (�(X0, r/2)) for every x E D \ B(x0, Mr). Applying (8.3) we thus find wx (�( y , s )) wA r(x o) (�( y , s)) (8.4) C wx(�(X0, r/2)) wAr (xo ) (�(X0, r/2)) The functions and are £-harmonic in Theorem 8.7 gives

�(x0,

2r).

<

<

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU

88

Upon dividing by s )) in (8.4) ( observe that in view of Theorem 8.3 the ux measure of any surface ball s ) is strictly positive) , one concludes

ux (�(y,

(8.5)

�(y, wx (�(y, s)) < C wAr (xo) (�(y, s )) wx (�(x0, r/2)) ux(�(y, s)) ux(�(y, s )) wAr (xa ) (�(x0, r/2))

Using Theorem 8.5 in the right-hand side of (8.5) we conclude

(8.6)

( )

We now observe that 8 . 2 in Corollary 8.4 allows to obtain a Vitali covering theorem and differentiate the measure with respect to the horizontal perimeter wx � yy,,ss measure ux . This means that for ux-a.e. E r ) the limit lim ax exists and equals �;: we obtain for ux-a.e.

wx

y �(x0,

s---+ O

i i ��

(y). This being said, passing to the limit as s ----+ o+ in (8.6) y E �(xo, r)

w Since by (7.16) in Theorem 7. 10 we know that d;x = = dax we have reached the desired conclusion. We observe in passing that the exceptional set here depends on and on but this fact will be incon sequential to us since we plan to integrate with respect to ux the above inequality on the surface ball r) . d x

K(Ar (x0), y),

x

(y) K(x, y),

d Ar ( xo )

(y)

Ar (x0),

�(x0,

D

We now turn to the

Xa

Xr Rr d(x 1 ,x0) MR1 . �(x0,

Proof of Theorem 1.3. We fix p > 1, E aD and E D. Let be the minimum of the constants appearing in Definitions 6.2, 8.1, and in Theorem 8.9. Moreover, the constant should be chosen so small that > Let 0
R1 . Ar (x0) R1 (

X0,

89

MUTUAL ABSOLUTE CONTINUITY Now we have (by ( 7 . 16 ))

:::;

C

(wxa'x

(D.(xo, r))P-1 (D.(xo , r ) )

1 x 1 ( ll (

o ,r )

ll(xo,r)

j XG( Ar (x0) , y) jP-1 dwx' (y) d(Ar (xo), y) jBd ( Ar ( Xo) , d (Ar (Xo) , y) ) j dwx ' (y)

)

)

1

;;

(by Theorem 6.6)

)p- l 1

;;

)

dwx' (y) f;

(by iv)

(by (S . 7))

in Definition

1.1)

This concludes the proof of the reverse Holder inequality. Regarding absolute continuity, we already know from (7.16) that dw x 1 is absolutely continuous with respect to dax . To prove that is absolutely continuous with respect to dwx1 we only need to observe that the reverse Holder inequality for established above and the doubling property for from (8.2) in Corollary 8.4 allow us to invoke Lemma 5 from [ CF] .

dax dax

K

D

We next establish a reverse Holder inequality for the kernel

P(x, y) defined in

(7.5) . The main trust of this result is that, under a certain balanced-degeneracy

8D,

assumption on the surface measure a of it implies the mutual absolute con tinuity of £-harmonic measure and surface measure. Given the fact that, as we have explained in the introduction, surface measure is not the natural measure in the subelliptic Dirichlet problem, being able to isolate a condition which guaran tees such mutual absolute continuity has some evident important consequences. To state the main result we modify the class of domains in Definition 1.1. Specifically, we pose the following

ADPx DEFINITION 8.10. Given a system X , ... ,Xrn} ofCsmooth vector fields {X1 open (1.1), we say that a connected bounded set D JRn is a-admissible satisfying for the Dirichlet problem (1. 3 ), or simply a - ADPx , if: i) DD iiss non-tangentially of class 000; accessible (NTA ) with respect to the Carnot-Caratheodory ii) metric associated to the system {X1 , ... ,Xrn} x(see Definition 8.1); Definition 6.2); iii) D satisfies a uniform tangent outer X-ball condition (see =

90

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU

and iv)0 0 depending on X and D such that for every Xo E aD (yEL\.(x0 max,r) W(y) ) a(�(xo, r)) � B I Bd(Xo,r r) i . We note that Definitions 1 . 1 and 8.10 differ only in part iv). Also, ( 7.8 ) gives max,r) W(y) ) a(�(x0, r)) . ax (�(x0, r)) 1L\.(x0,r) W(y)da(y) � (yEL\.(xo =

This observation shows that

a - ADPx c ADPx .

a

The reason for which we have referred to the new assumption on as a balanced-degeneracy condition is that, as we have seen in the introduction the measure badly degenerates on the characteristic set E. On the other hand, the vanishes on E, thus balancing such degeneracy. angle function

a W

P(x1,

Proof of Theorem 1.5. The relevant reverse Holder inequality for ·) is proved starting from the second identity x 1 = in (7.16 ) and then argu ing in a similar fashion as in the proof of Theorem 1 .3 but using the non-degeneracy estimate in of Definition 8.10 instead of the upper 1-Ahlfors assumption in Def inition 1 . 1 . We leave the details to the interested reader. 0

iv)

dw P(x 1 , ·)da

A consequence of Theorem 1 .5 and of Theorem 8.6 is the following result. We stress that such result would be trivial if the surface balls would just be the ordinary Euclidean ones, but this is not the case here. Our surface balls are the metric ones. Another comment is that away from the characteristic set the next result would be already contained in those in [MM].

�(y, r)

THEOREM 8 . 1 1 . Let D C !Rn be a a - ADPx domain. There exist C, R1 > 0 depending on the a - ADPx parameters of D such that for every y E aD and 0 < r < R1 , a(�(y, 2r)) � C a(�(y, r)) . Proof. Applying Theorem 1.5 with p = 2 , we find 1 { P(x1, y)2 da(y) � ( (�rXo, r)) J{L\. (xo ,r) P(x 1 , y) da(y) ) (�( Xo, r )) JL\.(xo ,r) x1(�(xo,r)) ) 2 = C ( wa(�(x0, r)) a

(]"

This gives

2

9I

MUTUAL ABSOLUTE CONTINUITY

0 Our final goal in this section is to study the Dirichlet problem for sub-Laplacians when the boundary data are in LP with respect to either the measure or the surface measure We thus turn to the

ax a. Proof of Theorem 1.4. The first step in the proof consists of showing that functions f E LP(8D, dax) are resolutive for the Dirichlet problem (1. 3 ). In view of Theorem 3. 3 it is enough to show that f E £ I (8D, dwx 1 ) for some fixed x i E D. This follows from (7.16) and Proposition 7. 9 , based on the following estimates { l f (y) l dwx1 (Y) laD { l f (y)I K (x i ,y) dax(y) laD 1 1 ' ::; (� l f (y)I P dax ( Y) ) (� K(x i , y) P dax (y) ) D 1 D ::; C (� l f (y)I P dax(y) ) D This shows that LP(8D, dax) C LI(8D, dw x1 ) and therefore, in view of Theo rem 3. 3 , for each f E LP(8D, dax ) the generalized solution solution Hf exists and it is represented by Hf (x ) { f(y) dwx (y) . laD At this point we invoke Theorem 8. 8 and obtain for every y E 8D (8.8) Na (Hf ) (y) ::; C Mwx1 (f)(y) . Moreover, Hf converges non-tangentially dwx1-a. e . to f. By virtue of Theo rems 1.3 and 1. 5 , we also have that Hf converges dax-a.e. to f. To conclude the proof, we need to show that there exists a constant C depending on 1 < < D and X such that IINa (Hf ) IILP (aD,d<7x) ::; CII JIILP(aD,d<7x) ' for every f E LP(8D, dax ) . In order to accomplish this we start by proving the following intermediate estimate (8.9) 11 Mwx 1 (f) l b (8D,d x ) ::; CIIJII LP (8D,de7x)> 1 < Since > 1, choose f3 so that 0 < f3 < and fix E D as in Theorem 1. 3 . From (7.16) and the reverse Holder inequality in Theorem 1. 3 we have 1 r f(y)dwX1 (y) Wx1 (�(xa, )) 16.(x0,r) 1 73 73' ' 13 ::; wx1 (��x0> )) (1{6.(x0,r) l f (y) l dax(y)) (1{6.(x0,r) K(xi ,y)13 dax (Y) ) ax (�(xa, ))!J' ( 1 1 K(X I , Y) dax (Y) ) II I II £!3(6.(x0,r) ,d<7x) ::; C Wx 1 ( ( )) ax ( ( )) 6.(x0,r) 1 73 13 =C( ax (�t ) 16.(x0,r) l f (y) l dax(y) ) =

r;

i7

r;

.

=

p

P ::;

<7

p

p

r

1

r

r

A u X0, r

Xa, r

A u Xa, r

XI

00

·

oo,

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU If we now y aD and take the supremum on both sides of the latter inequality by integrating on every surface ball � (x0, r) containing y, we obtain (8.10) By the doubling condition (8 . 2) in Corollary 8. 4 we know that the space (sults 8D d( x, y ) d x ) is a space of homogeneous type. This allows us to use the re in [CWJ and invoke the continuity in £P(8D, d x ) of the Hardy-Littlewood maximal function obtaining 11 Mw"1 f lliP(oD , dax ) I Max ( l f l13 ) � lliP (8D ,dax ) r Max ( l fl13 ) � dax :::; c r l f i P d x c ll fl liP(8D,dax ) ) lav lav which proves (8. 9 ). The conclusion of the theorem follows at once from (8. 8 ) and (8.9). 92

fix

,

,

E

a

a

<

C

=

a

=

D

Finally, we give the

Proof of Theorem 1.6. If the domain D is a a - ADPx-domain, instead of a ADPx-domain, then using Theorem instead of Theorem 1 .3 we can establish the solvability of the Dirichlet problem for boundary data in with respect to the standard surface measure. Since the proof of the following result is similar to that of Theorem ( except that one needs to use the second identity dwx 1 = P ( x 1 , ) a in and also Theorem we leave the details to the interested reader.

1.5

(7.16) 1.4

£P

·

8.11),

d

D

9 . A survey of examples and some open problems In the study of boundary value problems for sub-Laplacians one faces two type of difficulties. On one side there is the elusive nature of the underlying sub Riemannian geometry which makes most of the classical results hard to establish. On the other hand, any new result requires a detailed analysis of geometrically significant examples, without which the result itself would be devoid of meaning. This task is not easy, the difficulties being mostly related to the presence of char acteristic points. In this perspective it becomes important to provide examples of ADPx-domains. In this section we recall some of the pertinent results from recent literature.

Examples of NTAx domains. In the classical setting Lipschitz and even B M 01 domains are NTA domains [JK] . In a Carnot-Caratheodory space it is considerably harder to produce examples of such domains, due to the presence of characteristic points on the boundary. In [CG 1 J it was proved that in a Carnot group of step two every C 1 • 1 domain with cylindrical symmetry at characteristic points is NTAx . In particular, the pseudo-balls in the natural gauge of such groups are NTAx . This result was subsequently generalized by Monti and Morbidelli [MM] . C 1 •1

9.1. with In a respect Carnottogroup ofCarnot-Caratheodory step r 2 every bounded (Euclidean) domain is the metric associated to a system X of generators of the Lie algebra. THEOREM NTAx

=

93

MUTUAL ABSOLUTE CONTINUITY

Examples of domains satisfying the uniform outer X-ball property. The following result provides a general class of domains satisfying the uniform X ball condition, see [LUI] and [CGN2] . We recall the following definition from [CGN2] . Given a Carnot group
A 0 and x0 E aD there exists a X -ball that (6.4) is satisfied. From this it follows that every bounded convex subsetthe ofgauge

=

C

We mention explicitly that, thanks to the results in [K] , in every group of the Heisenberg type with an orthogonal system X of generators of g = EB fundamental solution of the sub-Laplacian associated with X is given by

V1 V2 ,

r(x, y) N(x-C(
where Q

=

=

is the non-isotropic Kaplan's gauge. Kaplan's formula for the fundamental solution shows, in particular, that in a group of Heisenberg type the X-balls coincide with the gauge pseudo-balls (incidentally, in this setting the gauge defines an actual distance, see [ Cy] As a consequence of this fact, the exterior X-balls in Theorem 9.2 can be constructed explicitly by finding the coordinates of their center through the solution of a linear system and a second order equation.

).

Ahlfors type estimates for the perimeter measure. Recall that if C JRn is a standard or even a Lipschitz domain, then there exist positive constants a:, {3 and Ro depending only on and on the Lipschitz character of such that for every E and every < < Ro one has (9.1) a n- l � = � {3 n - l .

D

C1 , Xo aD,

D, 0 nr a(aDn B(x0,r)) P(D;B(x0,r)) r r Here, we have denoted by P(D, B(x0, r)) the perimeter of D in B(x0, r) accord ing to De Giorgi. Estimates such as (9.1) are referred to as the 1-Ahlfors property

of surface measure. They play a pervasive role in Euclidean analysis especially in connection with geometric measure theory and its applications to the study of boundary value problems. In what follows we recall some basic regularity results for the X-perimeter measure which generalize (9. 1 ) and play a central role in the applications of our results. We have mentioned in the introduction that from the standpoint of the Carnot-Caratheodory geometry, Euclidean smoothness of a do main is of no significance. Even for coo domains one should not, therefore, expect 1-Ahlfors regularity in general, see [CG2] for various examples. For this reason we introduce the notion of type of a boundary point, and recall a result showing that if a domain possesses such property, then the corresponding X-perimeter satisfies Ahlfors regularity properties with respect to the metric balls. Given a system of coo vector fields X = {X1 , . . . , Xm} satisfying con sider a bounded We say that domain c JRn with an outer normal a point E is � 2 if either there exists E {1, such that

1 C X0 aD of type D

j0

N.... , m}

(1.1),

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH N HIEU

94

( i.e. , is non-characteristic, see Definition 3. 1 1 ) , or there < Xjjx0) , exist indices 0 , j0 E } such that < [Xio , XjJ (x0) , N(x0) We say that is of type � 2 if every point E is of type � 2. We stress that when the system has rank � 2, then every C1 domain is automatically of type � 2. An important instance is given by a Carnot group of step = 2. In such a group, every bounded C1 domain is of type � 2. The following theorem is from [CGl] . THEOREM 9.3. CI , I c JR.n . E � 2 = = x0, < <

Ni(xo) >i={1,0 ... , mX0 D X0 {)D r

>i= 0.

r

a bounded domain D For every point 0, aD ofcontinuously type Consider there exist A Ro(D, Xodepending A(D, Xo) > 0 and Ro on such that for any 0 r Ro one has Xo) > (9 . 2) O'X(�(xo, r)) � (yE�(maxx0 ,r) W(y)) O'(�(x0, r)) � A I Bd(xo,r r) l . The same conclusion holds if {)D is real analytic in a neighborhood of x0, regardless of Xo · C2 domain, then for every point X0 E 8D of type � 2 there of theIf Dtypeis a bounded exist that A forA(Dany, X0)0 <> r0 0, depending continuously on X0, such (9.3) O'X(�(Xo, r)) A - I I Bd(Xo,r r) l . We mention that in Carnot groups of step r 2 the upper 1-Ahlfors estimate (9.2) was first proved in [DGNl] , whereas for vector fields of rank r 2 the lower estimate (9.3) was first established in [DGN2 . In the setting of Hormander =

�

=

=

] vector fields, upper Ahlfors estimates for the surface measure 0' away from the characteristic set were first established in [MM2] . As a consequence of Theorem 9.3 we obtain the following

COROLLARY 9 . 4. Let X = { XI , . . . , Xm} be a set of C00 vector fields in JR.n satisfying Hormander's condition with rank two, i.e. such that span{Xl , . . . , Xm , [XI , X2] , . . . . , [Xm-l , Xm] } = JR.n , atmeasure every O'Xpoint.is aFor1-Ahlevery bounded CI , I domain D lR.n the horizontal perimeter fors measure. Moreover the stronger estimate (9.2) holds. c

As a consequence of the results listed above we obtain the following theorem or even the stronger which provides a large class of domains satisfying the property. 0' -

ADPx THEOREM 9.5. Let be a Carnot group of Heisenberg type and denote by X = { XI , . . . , Xm } a set of generators of its Lie algebra. Every coo convex bounded domain C ADP is a ADP and also a ADPx domain. In particular, the gauge balls in D are x andx also ADPx domains. To conclude our review of Ahlfors type estimates, we bring up an interesting connection between 1-Ahlfors regularity of the X-perimeter O'X and the Dirichlet problem for the sub-Laplacian, see [CG2] : THEOREM 9.6. Let D be a bounded domain in a Carnot group If the perime ter measure O'X is 1-Ahl f ors regular, then every x0 E 8D is regular for the Dirichlet problem. ADPx CG

CG

CG

0' -

0' -

G.

This result, in conjunction with a class of examples for non-regular domain constructed in [HH] yields the following

MUTUAL ABSOLUTE CONTINUITY 2:

95

2:

COROLLARY 9. 7 . If r 3 and m 1 3, or m 1 = 2 and r 2: 4, then there exist Carnot groups IG of step r E N, withisdimnot Vl-Ahl and bounded, c= domains 1 = fmors1 , regular. D C IG, whose perimeter measure ax

Beyond Heisenberg type groups. The above overview shows that, so far, the known examples of ADPx domains are relative to group of Heisenberg type. What happens beyond such groups? For instance, what can be said even for general Carnot groups of step two? One of the difficulties here is to find examples of domains satisfying the outer tangent X-ball condition. The explicit construction in Theorem above rests on the special structure of a group of Heisenberg type, and an extension to more general groups appears difficult due to the fact that, in a general group, the X-balls are not explicitly known and they may be quite different from the gauge balls. In this connection it would be desirable to replace the uniform outer X-ball condition with a uniform outer gauge pseudo-ball condition (clearly the two definitions agree for groups of Heisenberg type) . It would be quite interesting to know whether for general Carnot groups a uniform outer gauge pseudo-ball condition would suffice to establish the boundedness of the horizontal gradient of the Green function near the characteristic set (this question is open even for Carnot groups of step two which are not of Heisenberg type!) . Concerning the question of examples we have the following special results.

9.2

9.8. Let IG be a Carnot group and denote by g its Lie algebra. We say that a family F of smooth open subsets of g is a T- family if it satisfies {i ) For any F E F, the manifold 8F is diffeomorphic to the unit sphere in the Lie algebra. {ii) The family F is left-invariant, i.e. for any x E IG and F E F we have log(x exp(F)) E F. If D C is a smooth subset and F is a T-family, then we say that D is tangent tohyperplanes F if for every x E 8D there exists F E F such that x E 8F and the tangent to 8F and 8D at x are identical, i.e. Tx8F = Tx 8D. 9.9. Let g be the Lie algebra of a Carnot group of odd dimension. If DTHEOREM C is a smooth open set and F is a T -family, then D is tangent to F. Proof. In order to avoid using exp and log maps for all x, y E g we will denote by xy the algebra element log( exp x exp y). We will assume that g is endowed with a Euclidean metric, so that notions of orthogonality and projections can be used. Fix X0 E 8D and choose any element F E F. We will show that there exists z E g such that the left-translation zF is tangent to 8D at x 0 . Let n = dim(IG) = dim(g) be odd, and denote by sn-1 the unit (Euclidean) sphere of dimension n - 1. Define the map N 8F sn- 1 as follows: For each point X E 8F set b = xx;;- 1 D and observe that this is a smooth open set with X E ab 8F. Set N(x) = the outer unit normal to the boundary of the translated set 8D at the point x. This amounts to left-translating the point X0 to the point x and considering the unit normal to the translated domain at that point. The smoothness of D and of the group structure of IG implies that N is a smooth vector field in 8F . In order to prove the theorem we need to show that for some point x E 8F the vector N(x) is orthogonal to Tx 8F. In fact in that case the set F would be tangent to the translated set b at the point X0, and its left translation X0x -1 F could be chosen as DEFINITION

g

g

:

--->

n

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU the element of F tangent to D at the point Recall that left translation, being a diffeomorphism, preserves the property of being tangent. The conclusion comes from the fact that there cannot be any smooth tangent non vanishing vector field on oF since it is diffeomorphic to sn - 1 . Consequently the vector fields obtained by projecting N(x) on Tx oF must vanish for some point E oF. 0 COROLLARY 9.10.C Letbe a smooth be a Carnot group ofIfF stepistwoaTwithfamily, odd-dimensional Lie algebra and D convex subset. composed of - 1 -then for any x E oD convex subsets, and invariant by the transformation x x there exists F E F such that F and E oF. 96

X0•

X

G

g

g

c nc,

-->

X

Proof. In a Carnot group of step two the left translation map is affine and hence preserves convexity. The same holds for the inverse map. Consequently at any boundary point x0 there will be a convex manifold tangent to Being convex as well then and must either be on the same side at or lay at different sides of the common tangent plane By translating to the origin and considering either or we can pick the manifold lying on the 0 opposite side of and hence disjoint from it.

D x0.

D D

E oD

FEF D F Tx00D. F F- 1

x0

Choosing appropriate T-families of convex sets we can now prove our two main results concerning the uniform outer gauge pesudo-ball and -ball conditions.

X COROLLARY 9.11. aLetconvexbe seta Carnot group of step twoE oD with and odd-dimensional Lie algebra Given D C for every every r >the0 there exists a gauge pseudo-ball B(x ,r) which is tangent to oD in x0 from 1 Furthermore, every bounded convex set in outside, i.e. , such that (6. 4 ) is satisfied. satisfies the uniform outer gauge pseudo-ball condition. Proof. If D is smooth then the proof follows from the immediate observation that the gauge balls are convex sets in the Lie algebra and are diffeomorphic to sn- 1 (see for instance [F2] ) . For non-smooth convex domains D, we consider E oD and a new domain fJ obtained as the half space including D and with boundary Tx00D. Since fJ is a smooth convex domain then we can apply to it the previous theorem and find an outer tangent gauge ball at the point with radius r > 0. Clearly this ball will also be tangent to the original domain D at and will be 0 contained entirely in the complement of D. COROLLARY 9.12.everyLet x Ebe!Rna and Carnot group of step small two withthe Xodd-dimensional Lie algebra If for for r sufficiently B(x, r) - 1 , r) B(x, r) -1 then every bounded convex set-inballssatisfies are convex, and B(x the uniform outer X -ball condition. Proof. We need only to show that the family of balls B(x, r) form a T-family. In [DG2] it is shown that X -balls are starlike with respect to the family of homo geneous dilations in the Carnot group. In particular, one has the estimate (vT(·, x), Z ) > 0 on oB ( , r) where we have denoted by Z the generator of the homogeneous di lations. This inequality, coupled with Hormander's hypoellipticity result, implies that oB ( , r) is a smooth manifold, while the starlike property immediately implies that oB (x , r) is diffeomorphic to the unit ball. G

g.

G,

X0

G

X0

X0

X0,

G

g.

=

G

x

x

0

MUTUAL ABSOLUTE CONTINUITY

97

We recall from the classical paper of Folland [F2] that in a Carnot group the fundamental solution of the sub-Laplacian is a function r(x, y) = r(y - 1 x ) and r( x- 1 ) = rt (x) , where rt is the fundamental solution of the transpose of the sub Laplacian .C. However, a sub-Laplacian on a Carnot group is self-adjoint, hence the group gauge, if we assume £ * = -.C and r(x) = r(x - 1). Let us denote by that for all x, y G one has r(xy - 1 ) = r(yx- 1 ) (this happens for instance if r(x) = r(llxll)), and set B(x, = { y l r(y - 1 x ) > then

11 · 1 1 E r) c} 1 1 1 B(x, r)- = { y - 1 r(x - y) > c} = B(x - 1 , r) .

We conclude by explicitly noting that a serious obstruction to extending the previous results to Carnot groups of higher step consists in the fact that, unlike in the step two case, the group left-translation may not preserve the convexity of the sets. Beyond linear equations. Another interesting direction of investigation for the subelliptic Dirichlet problem is provided by the study of solutions to the non linear equations which arise in connection with the case p -1- 2 of the Folland-Stein Sobolev embedding. In this direction a first step has been recently taken in [GNg] where, among other results, Theorem 6.4 has been extended to the Green function of the nonlinear equation

2n £v u = .l: Xj (IXul v- 2 Xj u) =

(9.4)

j=1

in the Heisenberg group IHln . Here is the relevant result.

0,

THEOREM 9.13. Let D C be a bounded domain satisfying the uniform outer X -ball condition. Given 1 p ::; Q, let G denote the Green function associated with (9.4) and D. Denote by g = (z,t),g1 = (z1, t1) E (i) If 1 p Q there exists a constant C = C(G, D, p) > 0 such that ( d( ) ) 1 /(p- 1) d(g1,&D) ' g,g1 E D ' g1 -1- g . GD,p(g1, g) ::; c I B (g , �(:, g l ))l (ii) If = Q, then there exists C = C(G, D) > 0 such that ) d(g1,0D) G , (g1 , g) C log ( diam(D) d(g , g1 ) d(g,g1) , g ,g1 E D , g1 -=f- g . IHln

<

<

D ,p

IH!n .

<

I

p

Dp

::;

One might naturally wonder about results such as Theorem 6.6 in this setting. However, before addressing this question one has to understand the fundamental open question of the interior local bounds of the horizontal gradient of a solution to For recent progress in this direction see the paper [MZZ] .

(9.4).

References [Bel] A. Belta'iche, The tangent space in sub-Riemannian geometry. Sub-Riemannian geometry, Progr. Math., 144 (1996), Birkhiiuser, 1-78. [B] J. M. Bony, Principe du maximum, inegalite de Harnack et unicite du probleme de Cauchy pour les operateurs elliptique degeneres, Ann. Inst. Fourier, Grenoble, 1, 119 (1969) , 277-304. [CFMS] L. Caffarelli, E. Fabes, S. Mortola & S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana J. Math, 4, 30 (1981), 621-640. [CDG1] L . Capogna, D. Danielli & N. Garofalo, Subelliptic mollifiers and a characterization of Rellich and Poincare domains, Rend. Sem. Mat. Univ. Pol. Torino, 4, 54 ( 1993), 361-386. [CDG2] , The geometric Sobolev embedding for vector fields and the isoperimetric inequal ity, Comm. Anal. Geom. 2 ( 1994) , no. 2, 203-215. __

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU

98

___

[CDG3] , Subelliptic mollifiers and a basic pointwise estimate of Poincare type, Math. Zeitschrift, 226 (1997) , 147-154. [CDG4] , Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, American J. of Math., 118 (1997) , 1 153- 1 196. [CGl] L. Capogna & N. Garofalo, Boundary behavior of nonegative solutions of subelliptic equa tions in NTA domains for Carnot- Caratheodory metrics, Journal of Fourier Anal. and Appl. , 4 4 (1995) . [CG2] , Ahlfors type estimates for perimeter measures in Carnot- Carathodory spaces, J. Geom. Anal. 16 (2006), no. 3, 455-497. [CGNl] L. Capogna, N. Garofalo & D. M. Nhieu, A version of a theorem of Dahlberg for the subelliptic Dirichlet problem, Math. Res. Letters, 5 (1998) , 541-549. [CGN2] , Properties of harmonic measures in the Dirichlet problem for nilpotent Lie groups of Heisenberg type, Amer. J. Math. 124, vol 2, (2002) 273-306. [Ch] W.L. Chow, Uber System von linearen partiellen Differentialgleichungen erster Ordnug, Math. Ann., 117 ( 1939), 98-105. [Ci] G. Citti, Wiener estimates at boundary points for Hormander's operators, Boll. U.M.I., 2B (1988), 667-681. [CGL] G. Citti, N. Garofalo & E. Lanconelli, Harnack 's inequality for sum of squares of vector fields plus a potential, Amer. J. Math. , 3,115 ( 1993) , 699-734. [CF] R. Coifman & C. Fefferman, weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. [CW] R. Coifman & G. Weiss, Analyse harmonique non-commutative sur certains espaces homo genes, Springer-Verlag, ( 1971 ) . [CGr] L. Corwin and F. P. Greenleaf Representations of nilpotent Lie groups and their applica tions, Part 1: basic theory and examples, Cambridge Studies in Advanced Mathematics 18, Cambridge University Press, Cambridge ( 1990). [CDKR] M. Cowling, A. H. Dooley, A. Koninyi and F. Ricci H-type groups and Jwasawa decom positions, Adv. in Math. , 87 ( 1991) , 1-41 . [Cy] J. Cygan, Subadditivity of homogeneous norms on certain nilpotent Lie groups, Proc. Amer. Math. Soc., 83 (1981), 69-70. [Dal] B. E. J. Dahlberg, Estimates of harmonic measure, Arch. Rat. Mech. An., 65 (1977) , 272-288. [Da2] , On the Poisson integral for Lipschitz and C 1 -domains, Studia Math. , 66 ( 1979), 13-24. [D] D. Danielli, Regularity at the boundary for solutions of nonlinear subelliptic equations, Indi ana J. of Math. , 1, 44 (1995), 269-286. [DG] D. Danielli & N. Garofalo, Geometric properties of solutions to subelliptic equations in nilpotent Lie groups, Lect. Notes in Pure and Appl. Math. , "Reaction Diffusion Systems" , Trieste, October 1995, Ed. G. Caristi Invernizzi, E. Mitidieri, Marcel Dekker, 194 (1998) , 89-105. [DG2] D.Danielli & N. Garofalo, Green functions in nonlinear potential theory in Carnot groups and the geometry of their level sets, preprint 2000. [DGNl] D. Danielli, N. Garofalo & D. M. Nhieu, Trace inequalities for Carnot-Caratheodory spaces and applications, Ann. Sc. Norm. Sup. Pisa, Cl. Sci. (4) , 2, 27 ( 1998), 195-252. [DGN2] , Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Caratheodory spaces, Mem. Amer. Math. Soc. 182 (2006), no. 857, x+1 19 pp. [DGN3] , Sub-Riemannian calculus on hypersurfaces in Carnot groups, Adv. Math. 215 (2007), no. 1, 292-378. [Del] M. Derridj , Un probleme aux limites pour une classe d 'operateurs du second ordre hypoel liptiques, Ann. Inst. Fourier, Grenoble, 21, 4 ( 1971) , 99-148. [De2] , Sur un theoreme de traces, Ann. Inst. Fourier, Grenoble, 22, 2 (1972) , 73-83. [Diaz] R. L. Diaz, Boundary regularity of a canonical solution of the Eib problem, Duke Math. J . , 1 , 64 ( 1991 ) , 149-193. [FJR] E. B. Fabes, M. Jodeit, Jr. & N. Riviere, Potential techniques for boundary value problems on C1 -domains, Acta Math. 141 (1978), 165-186. [Fe] H. Federer, Geometric measure theory, Springer-Verlag, ( 1969 ) . ___

___

___

___

___

___

___

MUTUAL ABSOLUTE CONTINUITY

99

[FP] C. Fefferman & D.H. Phong, Subelliptic eigenvalue problems, Proceedings of the Conference in Harmonic Analysis in Honor of A. Zygmund, Wadsworth Math. Ser., Belmont, CA, ( 1981) , 530-606. [FSC] C. Fefferman & A. Sanchez-Calle, Fundamental solutions for second order subelliptic op erators, Ann. Math. , 124 ( 1986), 247-272. [F1] G. Folland, A fundamental solution for a subelliptic operator, 19 ( 1973), 373-376. [F2] , Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Math. , 13 (1975) , 161-207. [FS] G.B. Folland & E.M. Stein, Hardy Spaces on Homogeneous Groups, Princeton Univ. Press. , (1982). [FSS] B. Franchi, R. Serapioni & F. Serra Cassano, Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields, Boll. Un. Mat. Ita!. B (7) 11 ( 1997) , no. 1, 83- 1 17. [G] N. Garofalo, Second order parabolic equations in nonvariational form: Boundary Harnack principle and comparison theorems for nonegative solutions, Ann. Mat. Pura Appl. (4) 138 (1984) , 267-296. [GN1] N. Garofalo & D.M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot- Caratheodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. , 49 (1996), 10811 144. [GN2] , Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot- Caratheodory spaces, J . d'Analyse Math., 74 (1998), 67-97. [GNg] N. Garofalo & Nguyen C. P., Boundary estimates for p-harmonic functions in the Heisen berg group, preprint, 2007. [GV] N. Garofalo & D. Vassilev, Regularity near the characteristic set in the non-linear Dirich let problem and conformal geometry of sub-Laplacians on Carnot groups, Math. Ann. 318 (2000) , 453-516. [GT] D. Gilbarg & N. S. Trudinger, " Elliptic Partial Differential Equations of Second Order" , 2nd edition, rev. third printing, Springer Verlag, Berlin, Heidelberg ,1998. [HH] W. Hansen & H. Huber, The Dirichlet problem for sub-Laplacians on nilpotent groups Geometric criteria for regularity, Math. Ann., 246 (1984) , 537-547. [H] H. Hiirmander, Hypoelliptic second-order differential equations, Acta Math., 119 ( 1967), 147-171. [HW1] R. R. Hunt & R. L. Wheeden, On the boundary values of harmonic functions, Trans. Amer. Math. Soc., 32 ( 1968), 307-322. [HW2] , Positive harmonic functions of Lipschitz domains, Trans. Amer. Math. Soc., 41 ( 1970), 507-527. [J1] D. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, Parts I and II, J. Funct. Analysis, 43 (1981 ) , 97-142. [J2] , Boundary regularity in the Dirichlet problem for Db on CR manifolds, Comm. Pure Appl. Math. , 36 ( 1983), 143-181 . , The Poincare inequality for vector fields satisfying Hormander's condition, Duke [J3] Math. J . , 53 ( 1986), 503-523. [JK] D. Jerison & C. E. Kenig, Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. Math., 46 1982, 80-147. [JK1] , An identity with applications to harmonic measure, Bull. Amer. Math. Soc., 2, 2 ( 1980), 447-451 . [K] A . Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated b y composition of quadratic forms, Trans. Amer. Math. Soc . , 258 (1980) , 147-153. [Ke] C. E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, Amer. Math. Soc., CBMS 83, 1994. [KN1] J. J. Kohn & L. Nirenberg, Non-coercive boundary value problems, Comm. Pure and Appl. Math. , 18, 18 ( 1965), 443-492. [KN2] , Degenerate elliptic-parabolic equations of second order, Comm. Pure Appl. Math. 20 1967, 797-872. [Ko] A. Koninyi, Kelvin transform and harmonic polynomials on the Heisenberg group, Adv. Math. 56 ( 1985), 28-38. [LU1] E. Lanconelli & F. Uguzzoni, On the Poisson kernel for the Kahn Laplacian, Rend. Mat. Appl. (7) 17 ( 1997) , no. 4, 659-677. ---

---

___

---

___

---

___

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU

1 00 ___

, Degree theory for VMO maps on metric spaces and applications to Hrmander operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), no. 3, 569-601. [L] G. Lu, On Harnack 's inequality for a class of strongly degenerate Schrodinger operators formed by vector fields, Diff. Int. Equations, 1 ( 1994) , no. 1 , 73-100. [MZZ] G. Mingione, A. Zatorska-Goldstein & X. Zhong, Gradient regularity for elliptic equations in the Heisenberg group, preprint, 2007. [MM] R. Monti & D. Morbidelli, Regular domains in homogeneous groups, Trans. Amer. Math. [LU2]

Soc., 357 (2005), no. 8, 2975-301 1 . [MM2] R . Monti & D. Morbidelli, Trace theorems for vector fields, Math. Z., 239 (2002), no. 4 , 747-776. [NSW] A. Nagel, E.M. Stein & S. Wainger, Balls and metrics defined by vector fields I: basic properties, Acta Math. 155 (1985) , 103-147. [NS] P. Negrini & V. Scornazzani, Wiener criterion for a class of degenerate elliptic operators, J. Diff. Eq. , 166 ( 1987), 151-167. [P] H. Poincare, Sur les equations aux derivees partielles de la physique mathematique, Amer. J. of Math. , 12 ( 1890), 2 11-294. Rashevsky, Any two points of a totally nonholonomic space may be connected by an [RaJ P. admissible line, Uch. Zap. Ped. Inst. im. Liebknechta, Ser. Phys. Math., (Russian) 2 (1938), 83-94. [RS] L. P. Rothschild & E. M. Stein, Hypoelliptic differential operators and nilpotent groups. Acta Math. 137 (1976) , 247-320. [SC] A. Sanchez Calle, Fundamental solutions and geometry of sum of squares of vector fields, Inv. Math., 78 (1984) , 143-160. [St] E.M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory In tegrals, Princeton Univ. Press., (1993) . [V] V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Grad. Texts in Math., vol. 102, Springer-Verlag, (1984) . [X] C-J, Xu, On Harnack 's inequality for second-order degenerate elliptic operators. Chinese Ann. Math. Ser. A 10 (1989), no. 3, 359-365.

K.

-

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ARKANSAS, FAYETTEVILLE,

E-mail address, Luca Capogna:

DEPARTMENT OF MATHEMATICS, PURDUE UNIVERSITY, WEST LAFAYETTE

E-mail address, Nicola Garofalo:

IN 47907-1968

garofalo
DEPARTMENT OF MATHEMATICS, SAN DIEGO CHRISTIAN COLLEGE, CAJON

AR 72701

lcapogna
CA 92019

E-mail address, Duy-Minh Nhieu:

dnhieu
2 100

GREENFIELD DR, EL

Proceedings of Symposia in Pure Mathematics Volume 79, 2008

Soviet-Russian and Swedish mathematical contacts after the war. A personal account. Lars Garding Dedicated to Prof. V. Maz'ya on the occasion of his 70th birthday.

After the second world war ended it still took a long time for normal commu nications to be restored but when the change came it could be abrupt. The subject of this note is my personal experience of contacts between Soviet and later Russian mathematicians from the beginning of the 1950's. It all started in 1946-47 when I spent a year at Princeton University USA on a stipend from the Swedish-American Foundation. I soon came into contact with a group of young American mathematicians sharing their time between the university and the Institute of Advanced Study. A running subject of conversation among them was the theory of normed rings, a subject blending algebra and analysis and later becoming a corner stone of harmonic analysis. The start was a paper in the thirties by the great Soviet mathematician Israel Gelfand. The chief ideologue of our group at the Institute was my friend Irving Segal. He told me that he even had wanted to go to the Soviet Union but his request got a negative answer from lin's office. My chief interest at the the time was hyperbolic differential operators and one day Irving told me that he had seen an interesting hyperbolic paper by one Ivan Georgievich -Petrovsky in the main Soviet periodical the Matematitjeski Sbornik. I spent half the night with this paper without understanding its mixture of analysis and algebraic geometry. The subject was lacunas, regions where the fundamental solution of a hyperbolic operator unexpectedly vanishes. Lacunas in a special but interesting case had been the subject of a paper of mine. Later, in the middle sixties, Michael Atiyah and Roul Bott helped me with the topology to make a complete extension of Petrovsky's paper issued with a dedication to him in his native language. In Princeton I had met a biologist who had started learning Russian seduced by some interesting paper. I decided to do the same. After this encounter, on the boat home to Sweden I found a prospective teacher who, unfortunately, lived in Stockholm and my home town was Lund in the south. Coming home I started studying Russian with a Russian emigree who had been a lawyer at the tsarist high court. My first success was being able to read Petrovsky's Russian summary of

sfa,

2000 Mathematics Subject Classification. Primary 01A60.

101

1 02

LARS CARDING

his lacuna paper. I was not alone with my Russian teacher. In the late 1940's the political situation and the reputation of the many classical Russian writers made it interesting to study Russian and many did. The Soviet mathematical world was opened up to me with a big baW'fll the form of a meeting in Moscow in the summer of 1956 to which the Russian mathematical society had invited a number of foreign mathematicians and their wives. We lived in one of the big hotels and every national group had its own interpreter. Our stay coincided with the twentieth Soviet party congress of which we knew nothing until the papers one day carried Chrustjev's speech about the cult of the personality of the Stalin era. When we asked out interpreters about the sense of this speech they could only give us a literal translation. Arriving in Moscow, my wife and I were met by M.I. Vishik and a woman student. Vishik and I had common interests, we became friends and he later was a frequent visitor to Lund. My once imagined encounter took place when we were invited to dinner by Petrovsky together with Gelfand and his wife. Gelfand spoke then no English and very little German which hampered out conversation. Future Russian friends were Olga Ladyzhenskaja and Olga Olejnik. Others were S. Sobolev and the Georgian mathematician Vekua. With time my wife and I were to make many mathematical visits to the Soviet Union, both to Siberia and the South. After the fall of the Soviet Union, mine and the Swede's relations with Russian mathematicians underwent a drastic change. Our former hosts now appeared as emigrants looking for better economic conditions in the West. Some of them could find positions at a Swedish university. Long ago there were besides two Institutes of Technology only two of them, one in Uppsala and one in Lund but from 1950 on many new ones were created. Their somewhat hasty appearances made them in dire need of a competent scientific work force. The university of Linkoping profited from the arrival of Vladimir Maz'ya a specialist in nonlinear differential equations and the Institute of Technology adopted Ari Laptev who had had some political difficulties in Leningrad. The university in Blekinge that specialized in computing received competence in analysis with the arrival of Nail N. Ibrahimov whom I had met in Akademgorodok, a Siberian academic town. By now Vladimir Maz'ya retains a connection with Linkoping and has position at the University of Liverpool and Laptev is employed by London University. I had sparse but cordial relations with Maz'ya. He was the opponent of a Lund thesis and I visited him and his family in Linkping. With his wife Shaposhnikova he wrote a book on the French mathematician Hadamard and Shaposhnikova trans lated my dialogue between God and von Neumann into Russian. As he himself told Vladimir was worried about his ability to support his family, wife and mother in-law, after his Swedish retirement. As things have turned out, his worries were unsubstantiated. The Russian mathematicians I have mentioned were my friends. They are only a small part of the many reputable Russian mathematicians who left Russia after the fall of the Soviet State to get jobs in the West. At the same time Russian mathematics seem to regain its previous strength as evidenced by Perelman's proof of the Poincare conjecture. The upheavals after the Second World War reflected in

SOVIET-RUSSIAN AND SWEDISH MATHEMATICAL CONTACTS

1 03

the text above are now being replaced by a world of free communication and travel recalling the conditions of the nineteenth century. LUNDS UNIVERSITET MATEMATISKA !NSTITUTIONEN

E-mail address:

lars . garding
Box

1 18 221 00 LUND

Proceedings of Symposia in Pure Mathematics Volume 79, 2008

Generalized Robin Boundary Conditions, Robin-to-Dirichlet Maps, and Krein-Type Resolvent Formulas for Schrodinger Operators on Bounded Lipschitz Domains Fritz Gesztesy and Marius Mitrea Dedicated with great pleasure to Vladimir Maz'ya on the occasion of his 70th birthday.

ABSTRACT. We investigate generalized Robin boundary conditions, Robin-to Dirichlet maps, and Krein-type resolvent formulas for Schrodinger operators on bounded Lipschitz domains in IRn , n � 2. We also discuss the case of bounded C 1 ·r -domains, ( 1 /2) < r < 1 .

1 . Introduction This paper is a continuation of the earlier papers [43] and [46] , where we studied general, not necessarily self-adjoint, Schrodinger operators on C1· r -domains n c JR.n, n E N, n � 2, with compact boundaries an, (1/2) < r < 1 (including unbounded domains, i.e., exterior domains) with Dirichlet and Neumann boundary conditions on an. Our results also applied to convex domains n and to domains satisfying a uniform exterior ball condition. In addition, a careful discussion of locally singular potentials with close to optimal local behavior of was provided in [43] and [46] . In this paper we push the envelope in a different direction: Rather than discussing potentials with close to optimal local behavior, we will assume that E L00 (n; and hence essentially replace it by zero nearly everywhere in this paper. On the other hand, instead of treating Dirichlet and Neumann boundary conditions at an, we now consider generalized Robin and again Dirichlet boundary conditions, but under minimal smoothness conditions on the domain n, that is, we now consider Lipschitz domains n. Additionally, to reduce some technicalities, we will assume that n is bounded throughout this paper. Occasionally we also discuss the case of bounded C1• r-domains, ( 1/2) < r < 1 . The principal new result in

V

V

V

dnx)

2000 Mathematics Subject Classification. Primary: 35110, 35125, 35Q40; Secondary: 35P05, 47 A10, 47F05. Key words and phrases. Multi-dimensional SchrOdinger operators, bounded Lipschitz do mains, Robin-to-Dirichlet and Dirichlet-to-Neumann maps. Based upon work partially supported by the US National Science Foundation under Grant Nos. DMS-0400639 and FRG-0456306. @2008 American Mathematical Society

105

F. GESZTESY AND M. MITREA

1 06

this paper is a derivation of Krein-type resolvent formulas for Schrodinger opera tors on bounded Lipschitz domains n in connection with the case of Dirichlet and generalized Robin boundary conditions on an. In Section 2 we provide a detailed discussion of self-adjoint Laplacians with generalized Robin (and Dirichlet) boundary conditions on an. In Section 3 we then treat generalized Robin and Dirichlet boundary value problems and introduce associated Robin-to-Dirichlet and Dirichlet-to-Robin maps. Section 4 contains the principal new results of this paper; it is devoted to Krein-type resolvent formulas connecting Dirichlet and generalized Robin Laplacians with the help of the Robin to-Dirichlet map. Appendix A collects useful material on Sobolev spaces and trace maps for C1 ·r and Lipschitz domains. Appendix B summarizes pertinent facts on sesquilinear forms and their associated linear operators. Estimates on the funda mental solution of the Helmholtz equation in JRn, n � 2, are recalled in Appendix C. Finally, certain results on Calder6n-Zygmund theory on Lipschitz surfaces of fundamental relevance to the material in the main body of this paper are presented in Appendix D. While we formulate and prove all results in this paper for self-adjoint gener alized Robin Laplacians and Dirichlet Laplacians, we emphasize that all results in this paper immediately extend to closed Schrodinger operators He,n = -6.e,n + dom (He , n ) = dom ( - 6.e,n ) in L 2 (n; dnx) for (not necessarily real-valued) poten tials satisfying E L 00 (n; dnx), by consistently replacing -6. by -6. + etc. More generally, all results extend directly to Kato--Rellich bounded potentials relative to -6.e,n with bound less than one. Next, we briefly list most of the notational conventions used throughout this paper. Let 1i be a separable complex Hilbert space, ( , ) rt the scalar product in 1i (linear in the second factor) , and the identity operator in 1i. Next, let T be a linear operator mapping (a subspace of) a Banach space into another, with dom(T) and ran(T) denoting the domain and range of T. The spectrum (resp. , essential spectrum) of a closed linear operator in 1i will be denoted by a ( · ) (resp., O"ess ( ) ) . The Banach spaces of bounded and compact linear operators in 1i are denoted by B(1i) and B00 (1i), respectively. Similarly, B(1i1 , 1i2 ) and B00 (1i1 , 1i2 ) will be used for bounded and compact operators between two Hilbert spaces 1{ 1 and 1i2 . Moreover, X1 '--+ X2 denotes the continuous embedding of the Banach space X1 into the Banach space X2 . Throughout this manuscript, if denotes a Banach space, denotes the of continuous conjugate linear functionals on that is, the of (rather than the usual dual space of continuous linear functionals on X) . This avoids the well-known awkward distinction between adjoint operators in Banach and Hilbert spaces (cf. , e.g., the pertinent discussion in [37, p. 3-4] ) . Finally, a notational comment: For obvious reasons i n connection with quantum mechanical applications, we will, with a slight abuse of notation, dub -6. (rather than 6.) as the "Laplacian" in this paper.

V

V, V, V

V

Irt

·

·

·

space X* conjugateadjoint dual space X

X

X,

2. Laplace Operators with Generalized Robin Boundary Conditions In this section we primarily focus on various properties of general Laplacians - 6.e , n in L 2 (n; dnx) including Dirichlet, -6.D , n, and Neumann, - 6. N, n , Lapla cians, generalized Robin-type Laplacians, and Laplacians corresponding to classical

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 1 07 Robin boundary conditions associated with open sets 0 lRn, n E N, n 2, intro duced in Hypothesis 2 . 1 below. We start with introducing our assumptions on the set 0 and the boundary operator e which subsequently will be employed in defining the boundary condition on 80: is an open, HYPOTHESIS 2 . 1 . Let n E N, n bounded, nonempty Lipschitz domain. 2, and assume that 0 lRn We refer to Appendix A for more details on Lipschitz domains. For simplicity of notation we will denote the identity operators in L2(0; dnx) and L2(80; an - 1 w) by In. and Ian., respectively. Also, we refer to Appendix A for 2':

c

2':

c

our notation in connection with Sobolev spaces.

HYPOTHESIS 2.2. Assume Hypothesis 2.1 and suppose that ae is a closed 1 w) with domain H112(80) H112(80), bounded sesquilinear forme EinlR L2(80; dnin-particular, from below by (hence, is symmetric). eelan. c the self-adjoint operator in L2 1(80; dn- 1 w) aeuniquely associatedDenote with aeby(cf. (B.27)) and by(B.32) e E B ( H112(80), H- 12(80) ) the extension of as discussed in (B.26) and . Thus one has (2.1) (!, e g ) 1 /2 = (g, f ) 1 /2 ' f, g E H 1 /2 (80). (2.2) (!, E) f ) 1/2 ce ll f lli2(an.;dn-lw) ' f E H 112 (80). X

8 ):

8

8

2':

Here the sesquilinear form

antilinear in the first, linear in the second factor) , denotes the duality pairing between and

(

H8(80)

(2.4) such that

(f,g) s = l{an. dn- 1 w(0 f(�)g(�), f E H8 (80), g E L2 (80; dn- 1 w) H -s (80), s E [0, 1] , (2.5) and dn- 1 w denotes the surface measure on 80. Hypothesis 2 . 1 on 0 is used throughout this paper. Similarly, Hypothesis 2.2 <---+

is assumed whenever the boundary operator e is involved. ( Later in this section, and the next, we will occasionally strengthen our hypotheses. ) We introduce the boundary trace operator "'/b ( the Dirichlet trace ) by

0 C(O)- --+ C(80), 0 = u l a . (2.6) Then there exists a bounded, linear operator (cf. , e.g. , [69, Theorem 3.38] ) , H8 (0) ----. Hs - (1 /2) (80) <---+ L2 (80; dn-1 w), 1 / 2 s 3/2, (2.7) H312 (0) --+ H 1 -"'(80) L2 (80; dn-1 w), E E (0, 1), 'YD :

"'fD U

n.

'YD

<

'YD :

'YD :

<---+

<

F. GESZTESY AND M_ MITREA

108

whose action is compatible with that of �/b. That is, the two Dirichlet trace oper ators coincide on the intersection of their domains. Moreover, we recall that �w :

H8 (fl)

____.

Hs- (l/2l (afl) is onto for 1/2 <

s <

(2. 8 )

3/2.

\Vhile, in the class of bounded Lipsehitz subdomains in IRn , the end-point cases s = 1/2 and s = 3/2 of /D E B(H•(fl), H-'-(1 /2l (afl)) fail, we nonetheless have /D E B(H(3/2) + t: (fl), H1 (afl)) ,

(2 .9 )

c > 0.

See Lemma A.4 for a proof. Below we augment this with the following result : LEMMA 2.3. Assume Hypothesis 2.1. Then for each s > -3/2, the restriction to boundary operator (2.6) extends to a linear operator

/D

:

{ u E H112 (fl) I Llu E H8 (fl) }

---->

L2( 8fl ; dn-1w ) ,

( 2.1 0)

is compatible with (2.7), and is bo·unded when {u E H112(fl) I L\u E H8(fl) } is equipped with the natural graph norm u f---> lluiiHI/2 ( f!) + IIL\.uiiH• (Il) · In addition, this operator has a linear, bounded right-inverse (hence, in particular, it is onto) . Furthermore, for each s > -3/2, the restriction to boundary operator (2.6) also extends to a linear operator (2.11) which is again compatible with (2. 7) , and is bounded when { u E H312 (fl) I L\u E H l+• (fl) } is equipped with the natural graph norm u f---> l l u 1 1 Hs t2 ( f!) + II L\.u ii H l+•(ll) · Once again, this operator has a linear, bounded right-inverse ( hence, in particular, it is onto) . PROOF. For each s E IR set H{:,_ (fl) = { u E H" (fl) I L\u = 0 in Sl} and observe that this is a closed subspace of H8(0.). In particular, H{:,_ (fl) is a Banach space when equipped with the norm inherited from H" (fl). Next we recall the nontan gential maximal operator M defined in (D.9). According to [39] , or Corollary 5. 7 in [51], one has

H�2 (fl)

=

{ u harmonic in n I M(u)

E

£2(80.; dn- lw) }

(2.12)

and u f---> IIM(u) ll £2 (8l!;dn- lw) is an equivalent norm on H;{2 (fl ) . To continue, fix some K- > 0 and set d(y) dist (y, 8fl) for y E n. According to [28), the nontangential trace operator =

Crn.t. u) ( x) =

(2. 13)

lim u(y) 03y-x lx-y l < (l+ K) d( y)

is then well-defined when considered as a mapping "Yu.L.

:

{ u harmonic in !1 1 M (u)

E

L2 (afl; � - 1 w) }

____.

L2 (afl; dn -1 w) .

(2. 14)

Furthermore, the operator (2.13), (2. 14) is bounded. Granted these results, for a fixed s > - 3/2 we may then attempt to define (2.15) by setting (2. 16)

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS

1 09

A moment's reflection shows that, in order to establish that the mapping (2.15), (2.16) is well-defined, it suffices to prove that (2. 1 7) in the case when n is a bounded Lipschitz domain which is star-like with respect to the origin in �n (cf. (A.6)) . Assuming that this is the case, pick u E H (1 /2l+s (D) for some E > 0, and for each E (0, 1 ) set Ut(x) = u ( tx ) , X E D. We claim that

t

(2. 18) To justify this, it suffices to prove that this is the case when u E coo (0) as the result in its full generality then follows from a standard density argument. However, for every u E C00 (0) one trivially has Ut __... u as t __... 1 in H 1 (D), hence in H(1 / 2 ) +s (n) . Having disposed of (2.18), we may then conclude that "fDUt --> "fDU in L 2 (aD; dn- lw) as t --. 1. Since for each E (0, 1) we have Ut E C(O) , it follows that "fDUt = 1fjy ut = Ut lan . Thus, altogether,

t

t -->

(2. 19) 1. Ut l an --> "fDU in L 2 (an; dn -lw) as On the other hand, for almost every X E an, and every t E (0, 1), we have that 1 , and lx - Yl :=::; ( 1 + ) dist ( y , an) for y = tx belongs to n , converges to X as some sufficiently large = ��:(D) > 0 (independent of x and This implies that

t

11:

__...

t).

��:

Ut(X) --> ('Yn.t. u) (x) pointwise, for a.e. X E an, as t --> 1 .

(2.20)

Combining (2.19), (2.20) we therefore conclude that the functions 'Yn.t . u , "fDU E L2 (aD; dn- 1 w) coincide pointwise a.e. on an. This proves (2.17) and finishes the justification of the fact that the mapping (2.15), (2.16) is well-defined. Granted (2.15), (2. 16) is well-defined, it is implicit in its own definition that the mapping (2.15), (2.16) is also bounded when we equip H;{ 2 (D) + H5+ 2 (D) with the canonical norm (2.21) JJ u JJ H,;; 2 (!1 ) + ll v i i H•+2 (!1) · w��t uE H,;; 2 (!1 ) , vE H•+2(!1) The same type of argument as above (i.e., restricting attention to pieces of W 1-+

n which are star-like Lipschitz domains, and using dilations with respect to the respective center of star-likeness) shows the following: If w E C (O) can be de composed as u + v with u E H;!2 (D) and v E H8+2 (D) for some s > -3/2, then w l an = /n.t.U "(vv. In other words, the action of the trace operator '1v in (2.15), (2.16) is compatible with that of (2.6). This completes the study of the nature and properties of '1v in (2.15), (2.16). Consider next the claim made about (2.10). As regards its boundedness and the fact that this acts in a compatible fashion with (2.7) , it suffices to prove that

+

(2.22) continuously. To see this, pick u E H11 2 (D) such that t:. u E H8(D) and extend (cf. t:.u to a compactly supported distribution w E H8 (�n). Next, set

[87])

v ( x)

=

r � y En (X - y )w(y ) , X E n, }JF. n

(2.23)

110

F.

GESZTESY AND M. MITREA

where

E

n

(X ) -

_

{

},. ln(l x l ) , 1 ( 2 -n)wn -l l x l 2 n

n = 2, n 2: 3,

'

(2.24)

is the standard fundamental solution for the Laplacian in JRn (cf. (C.1) for z = 0). Here Wn-l = 2nn/2 /f(n/2) (f( · ) the Gamma function, cf. [1, Sect. 6.1]) represents the area of the unit sphere in !R.n. Then v E Hs+2 (�) and .6-v = .6-u in 0. As a consequence, the function w = u - v is harmonic and belongs to H112(0), that is, u = w + v with w E H)j2(0), v E H8+2 (0) . Furthermore, the estimate

sn-l

(2.25)

for some C = C(D, s) > 0 is implicit in the above construction. Thus, the inclusion (2.22) is well-defined and continuous, so that the claims ahout the boundedness of (2.10), as well as the fact that this acts in a compatible fashion with (2. 7) , follow from this and the fact that 1v in (2. 15) , (2. 16) is well-defined and bounded. As far as the existence of a linear, bounded, right-inverse is concerned, it suffices to point out (2. 12) and recall that the mapping (2. 14) is onto (cf. [28] ) . \Ve now digress momentarily for the purpose of developing an integration by parts formula which will play a significant role shortly. First, if 0 is a bounded star like Lipschitz domain in JRn and G is a vector field with components in H)j 2 (D) + H8+2(�), s > -3/2, such that div(G) E L1 (0), then =

ln{ dxn div(C) lau{ d"- 1w

v

· 1v G .

(2.26)

Indeed, if as before Gt (x) = G(tx), x E D, t E (0, 1), then (2.2 7) div(C1) t(div(C))t in the sense of distributions in n. Writing (2.26) for Gt in place of G, with 0 < t < 1, and then passing to the limit t -+ 1 yields the desired result . As a corollary of (2.26) and (2.22), we also have that (2.26) holds if n is a bounded star-like Lipschitz domain in !Rn and G is a vector field with components in {u E H112(�) l 6u E H·'(D)}, s > -3/2, such that div(C) E L1 (D). Since the latter space is a module over C0 (JRn) and any LipHchitz domain is locally star-like, a simple argument based Oil a smooth partition of unity shows that the star-likeness condition on 0 can be eliminated. More precisely, Hypothesis 2.1, (2.28) G E { u E H11 2 (D) i 6u E H"(O)} n , s > -3/2, ===> (2.26) holds. =

}

div(G) E L1 (0; dnx )

Moving on, consider the operator (2. 1 1 ) . To get started, we fix s > -3/2 and ::u,;,;ume that the function u E H31 2 (0) is such that 6u E H 1 +8 (0). Then, by the . second line in (2.7),

/DU E H1 -"(0D.) for every

c

> 0.

(2.29)

To continue, we recall the discussion (results and notation) in the paragraph containing (A.l l )-(A.16) in Appendix A. For every j, k E { 1 , . . . , n}, we now claim that (2.30)

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS Since the functions Oju, /A u belong to the space {w E H 1 1 2 (0) may then conclude from (2.30) and (2.10) that

o('YD u) OTj,k

111

J tl.w E H8(0) } , we

E L 2 (80 ; dn-lw),

(2.31)

and, in addition, (2.32) for every j, k E { 1 , ... , n }. In concert with (2.32) and (2.29), the characterization in (A.16) then entails that /D U E H 1 (80) and II !D u llH l (an) ::; C( J i u iiH3/2(!1 ) IJCl.uJI H l +s (n )) · In summary, the proof of the claims made about (2. 1 1) is finished, modulo establishing (2.30) . To deal with (2.30) , let '¢ E C0(JR.n) and fix j, k E { 1 , ... , n}. Consider next the vector fields Fj,k = ( o, . . . , o , u k'¢ , 0 . . . , 0, -u j'¢ , 0 ... , o) , (2.33)

+

Gj,k

a

=

a

( o, ... , 0, '¢8ku, O ... , o, -'¢8ju, O ... , o) ,

with the nonzero components on the j-th and k-th slots. Then Fj,k, Gj,k have components in the space { u E H 1 12 (0) J tl.u E H8 (0)} with > -3/2 and satisfy div ( Fj,k )

=

s

- di ( Gj,k ) = (ojuok'¢ - OkUOj'¢) E L2 (0; dn x),

v

(2.34)

in the sense of distributions. Also,

v '1D (Fj,k) ('YD u) ( vkoj 'lj; - lljOk'¢) , v '1D (Gj,k) = 'l/J ( vk !D (OjU) - llj/D (oku)) . ·

=

(2.35)

·

Hence, using (2.28), we obtain

{

lan

dn-lw ('YD u) ( vkoj'¢ - vjok'¢) = =

{

lan

dn- 1 w v '1D (Fj,k) ·

In dnx div(Fj,k)

=-

{

lan

=

-

In dnx div(Gj,k)

dn -l w 'l/J (vk/D ( Oju) - Vj !D (oku)) .

(2.36)

This justifies (2.30) and shows that the operator (2. 1 1 ) is well-defined and bounded. Clearly, this acts in a compatible fashion with (2.7) and (2.10). To finish the proof of Lemma 2.3, there remains to show that this operator also has a bounded, linear, right-inverse. This, however, is a consequence of the well-posedness of the boundary value problem u

E H 31 2 (0),

Cl.u = 0 in 0, !D (u) = E H 1 (80),

f

a result which appears in [101] . Next, we introduce the operator

/N

(2.37)

D

(the strong Neumann trace) by

/N = v " /D '\1 : Hs +l (O) -+ L2 (80; dn- 1 w),

1/2

< s < 3/2,

(2.38)

where v denotes the outward pointing normal unit vector to 80. It follows from (2.7) that IN is also a bounded operator. We seek to extend the action of the

112

F. GESZTESY AND M. MITREA

Neumann trace operator (2.38) to other (related) settings. To set the stage, assume Hypothesis 2.1 and recall that the inclusion (2.39) t : H8 (D.) '--7 (H1 (D) ) * , s > -1/2, is well-defined and bounded. We then introduce the weak Neumann trace operator :YN : { u E H1 (D) I �u E H 8 (D) } ___. H-1 12 (80.) , s > -1/2, (2.40) as follows: Given u E H1 (0.) with �u E H8 (0.) for some s > -1/2, we set (with L as in (2.39)) ( ¢, :YNuh;z

=

l dnx V7

( x )

·

V7u(x) + Hl ( f!j ( , L (�u) )(Hl (rl ) ) • ,

(2.41)

for all ¢ E H112 (80.) and E H1 (0.) such that 1o = ¢. We note that this definition is independent of the particular extension of ¢ , and that :YN is a bounded extension of the Neumann trace operator IN defined in (2.38). The end-point case s = 1/2 of (2.38) is discussed separately below. LEMMA 2.4. Assume Hypothesis 2.1. Then the Neumann trar-e operator (2.38) also extends to (2.42) :YN : { u E H312 (0.) I �u E L2 ( D ; dnx) } ___. L2 (8r!; dn-1w) in a bounded fashion when the space { u E H312 (0.) I �u E L2 (0.; dnx) } is equipped with the natural graph norm u f--l lluiiH3/ 2( r!) + ll�ull ucn ; dn x) · This extension is compatible with (2.40) and has a linear, bounded, right-inverse (hence, as a conse quence, it is onto) . Moreover, the Neumann trace operator (2.38) further extends to :YN : { u E H112 (r!) I �u E L2 (0. ; dnx ) } ___. H- 1 (80.) (2.43) in a bounded fashion when the space { u E H112 (0.) I �u E L2(r!; dnx) } is equipped with the natural graph norm u ,..... llui1 H l/ 2 (f!) + ll�uiiP ( rl ;d"x) · Once again, this extension is compatible with (2.40) and has a linear, bounded, right-inverse (thus, in particular, it is onto) . PROOF.

Fix 'l/J E c=(fi). Applying (2.28) to the vector field G = "1f;V7u yields

r dn-lw 1f v · -w('lu) = r dxn V7lf; · V7u + r dxn 1f; �u. Jn Jn �n

(2.44)

f dn-1w "¢ v · !o ('lu) = f dxn 'l · Vu + f dxn � �u. lao ln ln

(2.45)

Consider now ¢ E H112 (80.) and E H1 (0.) such that 1o = ¢. Since c= (n) '--7 H 1 (0.) is dense, it is possible to select a sequence 'l/Jj E c=(f!), j E N, such that and 'l/;j l ao ___. ¢ '1/Jj ___. 4> in H 1 (r!) as j ___. oo. This entails V'l/;j ___. 'l in L2 (!1; dnx) · in H 1 12( 8!1) as j ___. oo. Writing (2.44) for 'l/Ji in place of '1/) and passing to the limit j ___. oo then yields This shows that the Neumann trace of u in the sense of (2.40), (2.41) is actually v · !o ('lu). In addition, II:YNull £2 (8!1; dn -1w) = ll v · !o ( V7u ) IIL2 (ofl;dn-lw) ::; llfn ( Vu) II£2(Di1;dn- lw)n ::; C(IIV7 ui 1Hl/2 (f!)" II �( Vu) IIH-l(f!)n.)

+

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS C ( ll\7ui !Htl2(n)n + ll\7(6-u) IIH-t (n)n) :::; C ( ll u iiH3/2(!1) + ll 6.u ii L2(!1;dnx)) ,

113

=

(2.46)

where we have used the boundedness of the Dirichlet trace operator in (2. 10) with s = - 1 . This shows that, in the context of (2.42) , the Neumann trace operator

;:;Nu

= v

· 'Yv (\7u)

(2.47)

has is well-defined, linear, bounded and is compatible with (2.40) . The fact that this has a linear, bounded, right-inverse is a consequence of the well-posedness result in Theorem 3.2, proved later. As far as (2.43) is concerned, let us temporarily introduce

;yn { u E H 112 (0.) l 6.u E L 2 (0.; dnx) } :

by setting

____.

H - 1 (80.)

=

(H1 (80.) )

*,

(2.48)

(2.49) (¢, ;:;n u h (;:;N( IP ) , -yv u)o + ( IP , 6.u)L2(!1;dnx) - (6. 1P , u)p(n;dnx) > for all ¢ E H1 (80.), where IP E H312 (0.) is such that 'Yvlfl = ¢ and 6.1P E L2(0.; dnx). That such a IP can be found (with the additional properties that the dependence ¢ IP linear, and that IP satisfies a natural estimate) is a consequence of the fact =

f---+

that the mapping (2. 1 1) has a linear, bounded, right-inverse. Let us also note that the first pairing in the right hand-side of (2.49) is meaningful, thanks to the first part of Lemma 2.3 and what we have established in connection with (2.42). We now wish to show that the definition (2.49) is independent of the particular choice of IP. For this purpose, we recall the following useful approximation result: (2.50) where the latter space is equipped with the natural graph norm u f---+ ll u ii H • ( O ) + ll 6.u iiL2(!1;dnx) · When s = 1 this appears as Lemma 1 .5.3.9 on p. 60 of [48] , and the extension to s < 2 has been worked out, along similar lines, in [26]. Returning to the task ast hand, by linearity and density is suffices to show that (2.51) whenever IP E H312 (0.) is such that 'Yvlfl = 0, 6.1P E L2 (0.; dn x) , and u E c= (fi") . Note, however, that by (2.41) with the roles of IP and u reversed we have

(;yN( IP) , -yvu)o = /{ dnx \71P(x) · \7u(x) + (6.1P, u )L2(!1 ;dnx) > .n

(2.52)

so matters are reduce to showing that

in �X \71P(x) · \7u(x)

=

- ( IP , 6.u)L2(!1;dnx) ·

(2.53)

Nonetheless, this is a consequence of Green's formula (2.28) written for the vector field G = �\7u (which has the property that 'YvG = 0). In summary, the operator (2.48), (2.49) is well-defined, linear and bounded. Next, we will show that this operator is compatible with (2.40), (2.4 1 ) . After re-denoting ;y by ;:;N, then this becomes the extension of the weak Neumann trace n operator, considered in (2.43). To this end, assume that u E H1 (0.) has 6.u E L 2 (0.; dn x). Our goal is to show that (2.54)

F.

114

GESZTESY AND M. MITREA

for every ¢ E H1 (80.) or, equivalently,

for

E

ln cf'x 'V(x)

·

'Vu(x)

=

(7N(iP ) , �fDu)o - ( A

(2.55)

H312 (0.) such that A il> E L2(0.; d"x). However,

(7N(), J'nu)o

=

(7N() , J'nu) 1;2

=

k d"x 'V(x)

·

\i'u(x)

+ (A, n)vcn;d"x)•

(2.56)

where the first equality is a consequence of what we have proved about the operator (2.42) , and the second follows from (2.41) with the roles of u and reversed. This justifies (2.55) and finishes the proof of the lemma. 0 For future purposes, we shall need yet another extension of the concept of Neumann trace. This requires some preparations (throughout, Hypothesis 2.1 is enforced). First, we recall that, as is well-known (sec, e.g., [51]) , one has the natural identification (2.57) Note that the latter is a closed subspace of H-1 (1Rn). In particular, if Rn u uln denotes the operator of restriction to n (considered in the sense of distributions), then (2.58) is well-defined, linear and bounded. Furthermore, the composition of Rn in (2.58) with t from (2.39) is the natural inclusion of H� (n) into H- 1 (D.). Next, given z E C, set Wz (D. ) = { (u, J) E H1 (0.) x (H1 (0.) ) * I (-A - z)u = /In in D'(D.) } , (2.59) =

equipped with the norm inherited from H1 (0.) x (H1 (0.) ) * . We then denote by 7.11( : Wz (D.) ----. H-1 12 (80.) (2.60) the ultra weak Neumann trace operator defined by (¢ , 7N (u, /) /!;2 =

ln dnx V'

(x)u(x) - H' (!1J (, f) cH'.(fl))· · ·

- z

(u, f) E Wz (0.),

(2.61)

for all ¢ E H112 (80.) and E H1 (D.) such that /'niP = ¢. Once again, this definition is independent of the particular extension of ¢. Also, as was the case of the Dirichlet trace, the ultra. weak Neumann trace operator (2.60), (2.61) is onto (this is a corollary of Theorem 4.5) . For additional details we refer to equations (A.28)-(A.30) . The relationship between the ultra weak Neumann trace operator (2.60) , (2.61) and the weak Neumann trace operator (2.40), (2.41) can be described as follows: Given s > - 1/2 and z E C, denote by (2.62) Jz : {u E H1 (D.) I Au E H ' (D.) } -t Wz (fl) the injection (2 . 63) ]z (u) = (u , t ( -Au - zu) ), u E H1 (0.) , Au E H8 (D.),

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 115 where is in ( 2.39) . Then ( 2. 64 ) Thus, from this perspective, -;:yN can also be regarded a bounded extension of the Neumann trace operator IN defined in ( 2. 38 ) . Moving on, we now wish to discuss generalized Robin Laplacian:,; in Lip:,;chitz as

�

as

subdomains of �n. Before initiating this discussion in earnest, however, we formu late and prove the following useful result:

Hypothesis 2.1. Then for every c > 0 there exists a > 0 ({3(t:2.5. {3(10)LEMMA ) Assume ( 1/t:)) such that 0 ll fvu lli2(8!1;dn-lw ) ciiVu lli2(!1; dnx ) n + {3(t:) llu111, 2(0 ; dnx) for all E H 1 (0). ( 2.65 ) PROOF. Since n is a bounded Lipschitz domain, there exists an h E CO' (�n)n 40]) with real-valued components and > 0 such that (cf., [48, Lemma 1 5. 1 . 9 ( 2. 66) ( h)cn � a.e. on an. =

dO

�

U

K

.

v .

Thus,

l l !vull i2(8!1;dn-lw)

,

p.

/'\,

ddx � ( in d x

� r o n lw ( h) cn l {vul 2 "' la = � r n div (l u l 2 h) , "' lo = n (V I u l 2 , ) c n lu l 2 div �

v .

(h) ). u E H1 (0), (2.67) using the divergence theorem in the second step. Since for arbitrary t: > 0, l2uVu l � c i Vu l 2 + (1/t:) l ul 2 , u E H 1 (0), ( 2. 6 8 ) and h E C0 (�n)n , one arrives at ( 2. 6 5 ) . Next we describe a family of self-adjoint Laplace operators -L\e,o in L 2 (0 ; dn x) indexed by the boundary operator 8. We will refer to as the generalized h +

0

-L\e,o

Robin Laplacian.

THEOREM defined2.by6 . Assume Hypothesis 2.2. Then the generalized Robin Laplacian, ( 2. 69) { E H 1 (0) I L\u E L2 (0 dnx); (-;:;N + e,D )u 0 in H- 112 (80) } , is self-adjoint and bounded from below in L2(0; dnx). Moreover, dom ( l - L\e, o l 1 /2 ) H 1 (0). (2 . 70) PROOF. We introduce the sesquilinear form ( , ) with domain H1 (D) H1 (0) by ( ) ( ) + ( !vu , e{v v ) , u , v E H 1 (0), ( 2.71 ) 1 12 where ( , ) on H 1 (D) H 1 (D) denotes the Neumann Laplacian form ( ) k dn ( Vu) ( ) ( V v)( ) , u, v E H 1 (0). ( 2.72 ) -L\e, o, - L\e , o -L\, dom( -L\e,o) = u =

;

=

=

a-�e. n u, v

a-�o . n

·

=

a _�9 n

a-�o. n u, v

·

a-�o.n u, v

x

=

x

x

·

x

·

·

x

1 16

F. GESZTESY AND M. MITREA

One verifies that

a-� e r .

J · , · ) is well-defined on H1 (D)

B(H1(D), H1 1 2 (8D))

x

H1 (D) since

1o E , e E B(H1 12 (8D) , H -112 (8D ) ) , (8 + (1 - ce)Ian) 1 12 E B(H1 12 (8D ) , £2(8!1; d"-1w) )

(cf. (B.43)). This also implies that

(8 + ( 1 - ee)Ian)1 121o

E

(2.73) (2.74) (2.75)

B(H 1 (!1) , £2 (8!1 ; dn-1w) ) .

Employing (2.1) and {2.2), a-�e.n is symmetric and bounded from below by ce . Next, we intend to show that a-� e . n is a closed form in L2 ( D; d"x) x £2 (!1; d"x) . For this purpose we rewrite (lo U , elo v)l/ as

( lo U, elvv\/2 =

(2 . 76)

2

(( 8 + ( 1 - ce)Ian)1121ou, (8 + ( 1 - ce)Ian ) 1121ov) £2( an ;d"-1w)

- (1 - ce) (lo u , lvV)£2 (80 ; d" - 'wJ •

u, v E H1 (!1) ,

(2.77)

(cf. (B.31), (B .32) ), and notice that the last form on the right-hand side of (2.77) is nonclosable in L2( D; dnx) since ID is nonclosable as an operator defined on a rlem;e subspace from £2(!1; dnx) into £ 2( 8!1; dn - lw) (cf. the discussion in connection with (B.44) ) . To deal with this noncloseability issue, we now split off the last form on the right-hand side of (2.77) and hence introduce

b_ �e.n (u, v) = (V'u, V'v) £2( 0-; d"x)" + ((8 + (1 - ce)Ian)1121ou, (8 + (1 - ce)Ian) 1121vv) £2(CJO;d"-'w) + db( u , v)L2(0;d"x) • u, v E H1 (D) , (2.78)

for db > 0. Then due to the nonnegativity of the second form on the right-hand side in (2.78) , b_�e.n is H1(!1)-coercive, that is, for some c > 0, where

llull � � (n}

=

1 b_�H u ( u, u) � cl ll u ll�l(O) • IIY'ulll2(!1;d"x}" + llulll2 ( fl; d" x) ' Next, we note that by

I ( (8 + ( 1 - ce)Ian) 1121ou, (8 + ( 1 - ce )Ian ) 112 !'oV) �

£2 (aO;d"-'w)

(2.79) (2.76),

I

li Ce + (1 - ce)Ian) 1 12 1o II �(Hl (n),P (an;d"-' w)) llu iiHl( n) llv i iHl en > , (2.80) u, v E H 1 ( !1). Since trivially, I I V' ull i2(0; d"x) + db ll u ii1,2(0;d"x) � cllu ll��(o) for some c > 0, one infers that b_�e.n i::; also H1 (D ) -bounded, that is, for some c2 > 0, (2.81) b_ �e .o ( u, u) � c2 llull�1 (!1)· Thus, the ::;ymmetric form b_ �e.n is H1 (D ) -bounded and H1 (!1)-coercive and hence densely defined and closed in L 2 ( !1 ; dnx) x L2(D; dnx) by the discussion following

(B.46). To deal with the nonclosable form bvu , /'o v)L2(80 ;d"- l w) > u , v E H1 (D), it suffices to note that by Lemma 2.5 this form is infinitesimally bounded with respect to the Neumann Laplacian form a-�o . o on H1 ( D) x H1 (!1), and since the form

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS

11 7

( ( 8 + 1 - ce) 1 1 2 rDu, ( 8 + 1 - ce ) 1 12 rDV) L 2 ( an;dn - l w) ' u, v E H1 (0) , is nonnegative, the form (/D u, rn v ) u (an dn - l w ) is also infinitesimally bounded with respect to the ;

form L t:. e , n . By the discussion in connection with (B.48) , (B.49) , the form a-t:. e, n (possibly shifted by an irrelevant real constant) defined on H1 (0) x H 1 (0) , is thus densely defined in L2(0; dnx) x L2 (0; dnx), bounded from below, and closed. According to (B.34) we thus introduce the operator - .6. e , n in L2 (0; dnx) by dom( - .6.e ,n)

=

{ v E H 1 (0) I there exists an Wv E L2 (0; dnx) such that

L dn x \i'w \i'v + ( /DW, e/DV )

1 /2

- .6.e , nu = Wu , u E dom( - .6.e,n).

=

}

L � X WWv for all w E H 1 (0) , (2.82)

By the formalism displayed in (B.20)-(B.43) (cf. , in particular (B.27)), self-adjoint in L 2 (0; dnx) and (2. 70) holds. We recall that

- .6. e,n

{u E H1 (0) l rDU = 0 on 80}.

(2.83)

L dnx v .6.u for all v E cgo (O), and hence Wu = - .6.u in

V'(O),

HJ (O)

=

Taking v E C0 (0) '---' H{j (O) '---' H1 (0) , one concludes

L dn X VWu

=

-

is

(2.84) with = C0 (0)' the space of distributions in 0. Next, we suppose that u E dom( -.6. e,n) and v E H1 (0) . We recall that : /D H1 (0) -> H112 (80) and compute

V'(O)

L dnx \i'v \i'u =

= =

-

L dnx v .6.u + (/D v , 1N u h; 2

L dn X VWu + (!DV, (1N + e,D )u \ / 2 - (rDV, e,D u \ / 2 L �x \7v \7u + (rDV, (1 N + EhD ) u) 11 , 2

(2.85)

where we used the second line in (2.82). Hence,

(2.86) Since v E H1 (0) is arbitrary, and the map /D : H1 (0) onto, one concludes that

->

H112 (80) is actually (2.87)

Thus,

dom( - .6.e,n)

�

{v E H1 (0) j .6.v E L2 (0; dnx) ; (1N + SrD ) v = 0 in H - 1 12 (80) } .

Finally, assume that u E {v E H1 (0) j .6.v E L2(0; dnx ) ; (1N + 8-·tn)v w E H1 (0) , and let Wu = �.6.u E L2 (0; dn x ) . Then,

L dnx wwu

= =

-

L dn x w div(\i'u)

L dnx \i'w \i'u - (ro w , 1N uh;z

(2.88) 0} ,

=

F. GESZTBSY AND M. MITREA

118

(2.89)

Thus, applying (2.82), one concludes that u E dom( -Lle , n) and hence dom( -Lle ,n) d { v E H1 (fl) I Llv E L2(fl; cf'x ) ; (::YN + e'Yn ) v = 0 in H-112 (80) } , (2. 90) finishing the proof of Theorem 2.6. 0 COROLLARY

2.7. Assume Hypothesis 2.2. Then the genemlized Robin Lapla

-� e , n , has purely discrete spectrum bounded from below, in particular, aess ( -b..e ,n) 0 . (2.91) PROOF. Since dom (l .6. ,n l 1 12 ) = H1 (n) , by (2.70), and H1 (fl) embeds com

cian,

=

e

pactly into L2 (fl; dnx) (cf. , e.g. , [37, Theorem V.4.17]), one infers that (-.6.e,n + In)-112 E B00 (L2(fl; dnx) ) . Consequently, one obtains ( -Lle,n + In) -1 E B00 (L2 (fl; dnx)), (2.92) 0 which is equivalent to (2 .91). -

The important special case where 8 corm'ipouds to the operator of multipli cation by a real-valued, essentially bounded function () leads to Robin boundary conditions we discuss next: COROLLARY 2.8. In addition to Hypothesis 2.1, assume that e is the oper

l

ator· of multiplication in L 2 (8D.; dn w) by the real-valued f1Lnction () satisfying () E L=(aD.; dn -1w). Then 8 satisfies the conditions in Hypothesis 2.2 resulting in the self-adjoint and bounded from below Laplacian - b.. fl ,n in L2 (0; d" x) with Rubin boundary conditions on 80 in (2.69) given by -

PROOF.

(::YN + B'Yn ) u = 0 in H-112 (8!1).

( 2.93)

H1 (fl),

(2.94)

By Lemma 2.5, the sesquilinear form ('Yo u, B'Yvvh;2 ,

u, v E

is infinitesimally form bounded with respect to the Neumann Laplacian form a -c.o n · By (B.48) and (B .49) this in turn proves that the form a-c.e . n in (2.71 ) is clos�d and one can now follow the proof of Theorem 2.6 from (2.82) on, step by step. 0 REMARK 2.9. ( i) In Lhe case of a smooth boundary afl, the boundary conditions in (2.93) are also called "classical" boundary conditions (d. , e.g., [91] ); in the more general case of bounded Lipschitz domains we also refer to [6] and [102, Ch. 4] in this context. Next, we point out that, in [62] , the authors have dealt with the case of Laplace operators in bounded Lipschitz domains, equipped with local boundary conditions of Robin-type, with boundary daj;a-in LP(8fl; dn 1 w ) and produced nontangential maximal function estimates. r'For the case p = 2, when onr setting agrees with that of [62], some of our rest'l.Jts in this section and the following are a refinement of those in [62] . Maximal ))•.:: regularity and analytic contraction semigroups of Dirichlet and Neumann Laplacians on bounded Lipschitz domains were studied in [106]. Holomorphic C0-semigroups of the Laplacian with Robin boundary conditions on hounded Lipschitz domains have been discussed in [1 03] . Moreover, Robin boundary conditions for elliptic boundary value problems on arbitrary open domains were first studied by Maz'ya [67] , [68 , Sect. 4.11.6], and subsequently in [29 ] (see also [30] which treats the case of the Laplacian). In -

,

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS

1 19

addition, Robin-type boundary conditions involving measures on the boundary for very general domains n were intensively discussed in terms of quadratic forms and capacity methods in the literature, and we refer, for instance, to [6] , [7] , [17] , [102] , and the references therein. In the special case () 0 (resp., § = 0), that is, in the case of the Neumann Laplacian, we will also use the notation (2.95) The case of the Dirichlet Laplacian -D.v,n associated with n formally corre sponds to e = 00 and so we isolate it in the next result: 2.10. 2.1. -D.v,o ,

(ii)

=

THEOREM defined by

Assume Hypothesis

- D.v,o = - D. , dom( - D.v,o) =

Then the Dirichlet Laplacian,

0 in H112 (an) } { u E H1 (D) I D.u E L2 (D; dn x) (2.96) { u E HJ (D) I D.u E L2 (D; �x) } , is self-adjoint and strictly positive in L2(D;dnx). Moreover, ; "fD U

=

=

(2.97)

PROOF. We introduce the sesquilinear form av,o( · , · ) on the domain HJ (D) a v ,o (u , v) l �x ('Vu)(x) ('Vv)(x), u, v E H6 (D). (2.98) x

HJ(D) by

=

Clearly, av ,o is symmetric, nonnegative, and well-defined on HJ (D) x HJ (D). Since n is bounded, that is, /D/ < oo, HJ (D)-coercivity of av,o then immediately follows from Poincare's inequality for HJ (D)-functions (cf., e.g., [105, Theorem !.7.6] ) . Next we introduce the operator -D.v,o in L2(D; by

dnx) E L2 (D;dnx) such that for all w E HJ (D) }•

{ v E HJ (D) I there exists an l dnx'Vw'Vv l (2.99) u E dom(-D.v,o). - D.v , ou = By the formalism displayed in (B.1)-(B.19), - D.v,o is self-adjoint in L2(D; dnx) and (2.97) holds. Taking v E C0 (D) HJ (D), one concludes l In x v D.u in D'(D) and hence -D.u in D'(D) . (2.100) Since v E H{j (D) if and only if v E H1 (D) and -y0v = 0 in H1 12 (8D) (cf., e.g., (48, Corollary 1.5.1.6 ]), and v E dom( - D.v,o) implies D.v E L2(D; dnx), one computes for u E dom( -D.v,n) and v E HJ (D) that (2.101) l dnx'Vv'Vu = - In dnx vD.u = l �xvwu. Thus, - D.u E L2 (D; dnx) and hence dom( -D.v,o) { v E HJ (n) I D.v E L2(D; � x ) } . (2.102) Wv

dom ( - D.v , n ) =

=

Wu ,

� X WWv

<--+

�X VWu

Wu

=

= -

�

Wu

�

=

1 20 Wu

F. GESZTESY AND

M. MITREA

E {v E HJ (O) j �v E L2 (0; (ftx) }, w E HJ (O) , -�u E L2(0; dnx) . Then,

Finally, assume that u =

k (ftx wwu -ln dn =

x

w div(

V ) u

=

and let

k dnx Vw Vu,

(2.103)

since /DW = 0 in £2 (80; dn -lw). Thus, applying (2.99), one concludes that dom( - �D.o) and hence dom( -Ll v,n ) 2

{ v E HJ (O) J Llv E £2 (0; �x)},

finishing the proof of Theorem 2 . 10.

uE

(2 . 104)

0

Since 0 is open and bounded, it is well-known that -Lln,o has purely discrete spectrum contained in (O, oo), in particular,

17ess ( - Ll n ,n )

=

1/J

(2. 105)

(this follows from (2.97) since HJ (O) embeds compactly into L2(0; dnx); the latter fact holds for arbitrary open, bounded sets 0 c !Rn , cf., e.g., [37, Theorem V.4.18]). While the principal objective of this paper was to prove the results in this section and the subsequent for minimally smooth domains 0, it is of interest to study similar problems when Hypothesis 2.1 is further strengthen to:

HYPOTHESIS 2.1 1 . Let n E N, n ?:: 2, and assume that 0 c IR" is a bounded domain of class C1•r for some 1/2 < r < 1 .

We refer to Appendix A for some details on G1•7·-domains. Correspondingly, the natural strengthening of Hypothesis 2.2 reads:

HYPOTHESIS 2.12. In addition to Hypothesis 2.2 and 2.11 assume that

e E Boo (H312 (80), H 1 12 (80)) .

(2. 106)

We note that a sufficient condition for (2.106) to hold is

e E B(H312-e (80) , H112 (80)) for some E > 0.

(2.107)

Notational comment. To avoid introducing an additional sub- or superscript into our notation of -Lle,o and -Lln,o , we will use the same symbol for these operators irrespective of whether the pair of Hypothesis 2.1 and 2.2 or the pair of Hypothesis 2.11 and 2.12 is involved. Our results will be carefully stated so that it is always evident which set of hypotheses is used. Next, we discuss certain regularity resu!ts-for fractional powers of the resolvents of the Dirichlet and Robin Laplacians, fir�f in Lipschitz then in smoother domains. LEMMA 2.13. Assume Hypothesis 2.1 \r;,.._s;!}nnection with - Lln,o and Hypothesis 2.2 in connection with -Lle,!l · Then the following boundedness properties hold for all q E [0, 1] and z E C\[0, oo ),

(2. 108) The fractional powers in (2. 108) (and in subsequent analogous cases) are defined via the functional calculus implied by the spectral theorem for self-adjoint operators. As discussed in [43, Lemma A.2] in the closely related situation of Lemma 2.14, the key ingredients in proving Lemma 2.13 are the inclusions

(2. 109)

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS

121

and real interpolation methods. The above results should be compared with its analogue for smoother domains. Specifically, we have: 2.14. 2.11 -D.v,n 2.12 -D.e,n . q [0, 1] z C\[0, oo), ( - D.v, n - zln)-q, ( -D.e,n - zln)-q E B( L2 (0; dnx ) , 2 q ( )) . (2.1 10) As explained in [43, Lemma A.2] , the key ingredients in proving Lemma 2.14 are the inclusions (2. 1 1 1) and real interpolation methods. Moving on, we now consider mapping properties of powers of the resolvents of generalized Robin Laplacians multiplied (to the left) by the Dirichlet boundary trace operator: 2.15. 2.1 E > 0, z E C\[O, oo) . In ( - D. e,n - zln)- ( l +e: )/4 E B(L 2 (0; �x), L2 (80; dn - 1 w)). (2.112) As in [43, Lemma 6.9] , Lemma 2.15 follows from Lemma 2.13 and from (2.7) and (2.38). Once again, we wish to contrast this with the corresponding result for smoother domains, recorded below. 2.16. 2.11 -D.v,n 2.12 E > 0, z E C\[O, oo) . -D.e,n, IN ( -D.n,n - zln) - (3H)/4 E B (L 2 (0; dn x), L2 (80; �- 1 w )), (2.113) )

LEMMA Assume Hypothesis in connection with andproperties Hypoth esis in connection with Then the following boundedness hold for all E and E H n

LEMMA

Assume Hypothesis and let

Then,

AssumewithHypothesisand letin connection with Then,and Hypoth esis LEMMAin connection rD ( - D.e,n - zln)- ( l + e: /4 E B ( L 2 (0; dnx) , L2 (80; dn - 1 w)).

As in [43, Lemma 6.9], Lemma 2.16 follows from Lemma 2.14 and from (2.7) and (2.38). In contrast to Lemma 2.16 under the stronger Hypothesis 2.12, we cannot obtain an analog of (2.1 13) for -D.v,n under the weaker Hypothesis 2.1. The analog of Theorem 2.6 for smoother domains reads as follows: 2.17. 2.12.

Assume Hypothesis Then the generalized Robin Lapla cian,THEOREM -D.e,n, defined by -D.e,n = - D. , dom (-D.e,n) = {u E H2 (0 ) I (rN + e,v)u 0 in H 112 (80) } , (2.114) is self-adjoint and bounded from below in L2 (0; dnx) . Moreover, (2.1 15) dom ( l - D.e,n l 1/2 ) = H 1 (0) . PROOF. We adapt the proof of [43, Lemma A.1], dealing with the special case =

of Neumann boundary conditions (i.e., in the case e = 0), to the present situation. For convenience of the reader we produce a complete proof below. By Theorem 2.6, the operator Te,n in L 2 (0; dnx), defined by (2.116) Te,n = - D. , 1/2 0 in H - (80.) } , dom(Te ,n) = H 1 ( 0) I E L2 (0; dnx); (;;;N + e,D ) is self-adjoint and bounded from below, and (2.1 17) dom ( e,n l 1/2 ) H 1 (0 )

{u E

u

D.u

IT

=

=

1 22

F.

GESZTESY AND M. MITREA

holds. Thus, we need to prove that dom(Te,n) <;;; H 2( f2) . Consider u E dom(Te,n) and set f = -D.u + u E L 2 ( n; d»x). Viewing f as an element in (H1 (n) ) *, the classical Lax-Milgram Lemma implies that u is the unique solution of the boundary-value problem

{

(-D. + In )u = f E L2 (n) '-+ (H1 (n) ) * , u E H1 (D.), (::YN + G")'D ) u = 0.

(2.1 18)

One convenient way to actually show that

u E H2(n),

(2.1 19)

is to use l ayer potentials. Specifically, let En(z; :c ) be the fundamental solution of the Helmholtz differential expression (-.6. z ) in JRn, n E N, n 2: 2, that is,

,

En(z,. x)

{

-

_ �(4) (27rlxl/z112 ) (2-n) /2 H((�� 2)12 (z112lxl) , n = 2 , z � C\{0},

-

n - 2, z - 0, n � 3 , z = 0,

2,. ln( lxl ) , (n - 2lwn-1 lxl2 -n ,

Im (z 1 12 ) 2: 0, x E IRn\{0}.

Here H51l ( · ) denotes the Hankel function of the first kind with index Sect. 9.1]). We also define the associated single layer potential

(Szg )(x) =

r

lan

v�

dn -lw(y) En (z ; X - y) g ( y) , X E n , z E C,

(2.120)

0 (cf. [1 ,

( 2.121)

where g is an arbitrary measurable function on an. As is well-known (the interested reader may consult, e.g., [73] , [101] for jump relations in the context of Lipschitz domains), if

(Kff'g )(x) = p.v.

{

lan

dn - 1w(y) 8vx En(z ; ;r, - y ) g(y) ,

x E an, z E
(2 . 122)

stands for the so-called adjoint double layer on an, the following jump formula holds (2. 123) 1?[;vSzg = ( -Van + Kff )g . It should be noted that

·��

K'f' E B (L2 (an; dn - 1 w )),

z E
(2.124)

whenever n is a bounded Lipschitz domain. See Lemma D.3. Now, if we denote by w the convolution of f E L2( D.; dnx) with E ( - 1 ; · ) in !1, then w E H2 (f2) and the solution u of (2.1 18) is given by

n

(2.125) u = w + $_1 g for a suitably chosen function g on an. Concretely, we shall then require that ('YN + e,D)S- lg = - (rN + EhD) w , (2.126)

or equivalently,

(- Fan + K�t) 9 + BrDS- 1 9 = - (/N + BiD ) w E H112 (an).

(2.127)

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS

By hypothesis, e E Boo (H3 12 (f2), H 1 12 (8r2)) and hence

1 23

(2.128) e/'vS- 1 E Boo (H 1 12 (f2), H 1 12 (8r2)) as soon as one proves that S_ 1 satisfies S- 1 E B(H 1 12 (8r2 ), H2 ( r2)) . (2.129) To prove this, as a preliminary step we note (cf. [74]) that (2.130) S- 1 : H- s (on) H - s+3 12 (n) is well-defined and bounded for each s E [0, 1], even when n is only a bounded ---r

},

Lipschitz domain. For a fixed, arbitrary j E {1, . . . , n consider next the operator Oxi S- 1 whose integral kernel is Oxi En ( -1; x - y ) = -Oyi En ( -1; x - y ) . We write

n 0 L vk (y ) 07kJ ( ) + vj ( y) v (y ) · Y'y k=1 where the tangential derivative operators 8j8Tk ,j = Vk Oj - v18k , j, Oyi

=

n L vk (y ) vk ( y )8Yi k=1

=

·

y

(2.131)

k

= 1, . . . , n, satisfy (A.17). Using the boundary integration by parts formula (A.24) it follows that

(2.132) where, for z E C,

Vzh (x ) = {an dn-1 w (y ) v ( y ) · V'y [En (z ; x - y) ]h (y ) , x E f2, l

(2.133)

is the so-called (acoustic) double layer potential operator. Its mappings properties on the scale of Sobolev spaces have been analyzed in [74] and we note here that (2.134) requires only that n is Lipschitz. Assuming that multiplication by (the components of) v preserves the space H 1 12 (8f2) (which is the case if, e.g., n is of class C 1 •r for some (1/2) < < 1; cf. Lemma A.5), the desired conclusion about the operator (2.129) follows from (2.130), (2.132) and (2.134). Going further, from Theorem D.8 we know that (2.135) r

so -�Ian + K'f!.1 + e/'vS_ 1 is a Fredholm operator in H 1 12(8f2) with index zero. This finishes the proof of (2.119). Hence, the fact that dom(Te,n) � H2(f2) has 0 been established.

Again we isolate the Neumann Laplacian - b. N , n , that is, the special case e = 0 in (2.114), under Hypothesis 2.11, - D.N,n = - D. , dom( - b. N,n) u E H2 (f2) I ::YN u = 0 in H 1 12 (8n) }. (2.136) Similarly, one can now treat the case of the Dirichlet Laplacian. This has originally been done under more general conditions on n (assuming the boundary of n to be compact rather than n bounded) in [43, Lemmas A.1]. For completeness we repeat the short argument below: =

{

124

F. GESZTESY AND M. MITREA

THEOREM 2.18. Assume Hypothesis 2.11. Then the Dirichlet Laplacian, -Lln, n , defined hy -Lln,n = -D., dom( -Lln,n) = { u E H 2 (rl) I 'Ynu = 0 in H312(orl) } , (2. 1 37) is self-adjoint and strictly positive in L2 (rl; dn.x) . Moreover, dom ( ( - Llo ,n ) 1 1 2 ) = HJ (n) . (2. 138)

PROOF. For convenience of the reader we reproduce the short proof of [43, Lemma A.1] in the special case of Dirichlet boundary conditions, given the proof of Theorem 2.17. By Theorem 2. 10, the operator To,n in L2(rl; dnx) , defined by

To , n = - D. ,

dom(Tn,n)

=

(2 . 139)

{ u E HJ ( n) I Llu E L2(rl; d''x); "/DU = 0 in L2 (orl; an- 1 w) },

is self-adjoint and strictly positive, and

(2. 140) HJ(n) holds. Thus, we need to prove that dom(Tn� n ) � H 2 (n) . To achieve this, we

dom((To,n)112)

=

follow the proof of Theorem 2. 17, starting with the same representation (2. 125 ) . This time, the requirement on g is that /nS_ 1 g = h = /DW E H312(o0.) . Thus, it suffices to know that (2.141) is an isomorphism. When on is of class coo, it has been proved in [97, Proposition 7.9] that ,0s_ 1 : H8 (o0.) -+ H8+ 1 (o0.) is an isomorphism for each s E � and, if n is of class C1·r with ( 1/2) < r < 1, the validity range of this result is limited to - 1 - r < s < r, which covers (2.141). The latter fact follows from an inspection of Taylor ':,; original proof of [97, Proposition 7.9] . Here we just note that the only significant difference is that if an is of class C1·r (instead of class C00), then s is a pseudodifferential operator whose symbol exhibits a limited amount of regularity in the space-variable. Such classes of operators have been studied in, e.g., [73] , [96, Chs. 1, 2] . 0 REMARK 2.19. We emphasize that all result8 in this section extend to closed Schrodinger operators (

(2. 142) - Lle,p + V, dom (He,o ) = dom( - Lle,n) for (not necessarily real-valu�d ) pot,entials V satisfying V E L00(rl; dnx), consis tently replacing - Ll by -D. + V, etc. More generally, all results extend to Kato He,n

=

- ___

Rellich bounded potentials V relative to -Lle, n with bound less than one. Exten sions to potentials permitting stronger local singularities, and an extensions to (not necessarily bounded) Lipschitz domains with compact boundary, will be pursued elsewhere. 3. Generalized Robin and Dirichlet Boundary Value Problems and Robin-to-Dirichlet and Dirichlet-to-Robin Maps

This section is devoted to generalized Robin and Dirichlet boundary value prob lems associated with the Helmholtz differential expression -Ll-z in connection with the open set 0.. In addition, we provide a detailed discussion of Robin-to-Dirichlet Af(O)D !lrl -1 maps, 1> S, ,f!> In L2 ( UH j dn W) . •

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 125 In this section we strengthen Hypothesis 2.2 by adding assumption (3.1) below: HYPOTHESIS 3.1. 2.2 (3.1) We note that (3.1 ) is satisfied whenever there exists some > 0 such that e (3.2) We recall the definition of the weak Neumann trace operator ::YN (2.40), (2.41) and start with the Helmholtz Robin boundary value problems: THEOREM 3.2. 3.1 -D.e )

In addition to Hypothesis suppose that E B(H1-c(an), L2 (an; rr-1 w)).

c

m

Assumedn-Hypothesis and generalized suppose thatRobin z E boundary C\tr ( value n 1w), the following Then for every g E L2(an; problem, { (-D. - _: )u = 0 in n , u E H312(n), (3.3) (::YN + 8rD)u = g on an, has a unique solution u ue. This solution ue satisfies /DUe E H 1 (an), ::YN ue E L2(an;dn- 1 w), (3.4) llrD ue iiH1 (80) + II ::YN ue ll£2(8!1;d"- 1w) • Cllgll£2(8!1;d" - 1w) and (3.5) ll ue ii H3/2(!1) Cllgll£2(8!1;d"- lw) • for some constant constant C = C(8, n, z) > 0. Finally, (3.6) (rD ( -D.e,n - zln) - 1 ) * E B(L 2 (an; rr - 1 w), H 312 (n)), and the solution ue is given by the formula (3.7) ue = (rD (-D.e,n - Z/n)- 1 ) * g. PROOF. It is clear from Lemma 2.3 and Lemma 2.4 that the boundary value problem (3.3) has a meaningful formulation and that any solution satisfies the first line in (3.4). Uniqueness for (3.3) is an immediate consequence of the fact that zcandidate E C\cr( -D.e,n). As for existence, as in the proof of Theorem 2.17, we look for a expressed as (3.8) u(x) = (Szh)(x), x E for some h E L2(an; dn-1w). This ensures that u E H312(n) and ( - D. - z)u 0 in n. Above, the single layer potential Sz has been defined in (2.121). The boundary condition (::YN + SrD )u = g on an is then equivalent to ,

.

=

:=:::

:=:::

f2

=

(3.9)

respectively, to

(3.10) (-� Ian + Kff) h + SrDSzh = g . Here Kff has been defined in (2.122). To obtain unique solvability of equation (3.10) for h E L2(an;dn- 1 w), given n- 1 w), at least when z E C\D, where D C is a discrete set, we gproceed E L2(an; d in a series of steps. The first step is to observe that the operator in question is Fredholm with index zero for every z E C. To see this, we decompose C

(3.11)

126

F.

GESZTESY AND M. MITREA

E 800(L2(8r!;dn-lw)) L2(8fl; dn-lw)

and recall that (Kff - Kt') (cf. Lemma D.3) and that with Fredholm index equal -� Ian + K! is a Fredholm operator in to zero as proven by Verchota [101]. In addition, we note that (3 .12) which follows from Hypothesis 3.1 and the fact that the following operators are bounded L 2 r! ; cf!'-1w) ---+ { u �u E (3.13)

Sz : (8 E H312(r!) I L2(r!; dnx)}, '"'/D : {u H312(n) 1 �u E L2(r!; dnx)} ---+ H1(8n), H312(Q) I �u E L2(r!; dnx)} i n (- �Ian Kff) EhvSz L2(8r!; d"-1w) (-�Ian Kff) + �hvSz L2(8Q; dn-1w). E

(where the space {u E is equipped with the natural graph norm u �---+ llu i H3/2 ( ) + l l �ull£2 (f!;d"x) ) · See Lemma 2.3 and Theorem D.7. + Thus, + is a Fredholm operator in with Fredholm index equal to zero, for every z E C. In particular, it is invertible if and only if it is injective. In the second step, we study the injectivity of on + For this purpose we now suppose that

Introducing w =

one then infers that w satisfies

Szk n {((;:.;-�N -_:)w in

6'"'!D )w

+

=

0 in 0 On of!.

r!, w E H3f2(f!),

(:3.15)

=

Thus one obtains, 0S

1 dnx

l'iwl 2 =

!!

= =

t J dnx Oj WOj'W - r dn x Aww t r cr - 1w ('"'!v 8pD) ln j= 1 lan � in dnx lwl 2 + bvw, '1N w)L2(8D;dn-lw) r d"x lwl2 ('YD'W, ;:.;N wh/2 zjn d"x lwl2 - ('"'!ow, Ehvw)112 . j= l n /-

1/j'"'/DW

+

=Z =

+

(3.16)

At this point we will first conHider the case when z E C\:IR (so that, in particular, Im(z) f. 0). In this scenario, recalling (2 .1) and taking the imaginary parts of the two most extreme sides of (3.16) imply that. fn d"x lwl2 = 0 and, hence, = 0 in

w

n.

To continue, let ;:.;'f;t and '"'lrr denote, respectively, the Neumann and Dirichlet traces for the exterior domain Also, parallel to (2 .121), set

Rn\n.

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS

127

where g is an arbitrary measurable function on 80. Then, due to the weak singu larity in the integral kernel of Sz , ext s x , z9 /D Sz 9 = rD e t

(3.18)

whereas the counterpart of (2.123) becomes �ext s z fN ext , 9 - ( 2l J80 + _

K#z ) 9

(3.19) Compared with (2.123), the change in sign is due to the fact that the outward unit normal to !Rn\f! is Moving on, if we set wext (x ) (Sext , z k ) (x ) for x E !Rn\f!, then from what we have proved so far. -v

.

=

(3.20)

Fix now a sufficiently large R > 0 such that !1 c B (O; R) and write the analogue of (3.16) for the restriction of wext to B (O; R) \D: dnx l wext l 2 (r'ir wext , ;y'fytwext h dnx I Y'wext l 2 = z f f 12 jB(O;R)\0_ jB (O;R)\0_ . V'wext (O. (3.21) f dn -l w(�) wext (�) i_ �� jl f. I == R _

_

In view of (3.20), the above identity reduces to dnx IY'wext l 2 f jB (O;R)\0_ dnx l wext l 2 z r jB(O;R)\0 =

-1

dn - l w(�) wext (�) i_ V'wext (�) . �� l f. I = R 0

(3.22)

Recall that we are assuming E C\R Given that, by ( C.17) ( and the comment following right after it ) , the integral kernel of Sext ,z k has exponential decay at infinity, it follows that wex t decays exponentially at infinity. Thus, after passing to limit R oo, we arrive at z

_,

f dnx IY'wext 1 2 = z f dnx l wext 1 2 . }H{n\f! }H{n \f!

(3.23)

-

-

Consequently, taking the imaginary parts of both sides we arrive at the conclusion that wext 0 in !Rn\f!. With this in hand, we may then invoke (2.123), (3.19) to deduce that (3.24) w we t given that , x vanish in 0 and !Rn\f!, respectively. Hence, one concludes that 0 in £2 (80; �-1w). This proves that the operator ( 1Iao + + erDSz is injective, hence, invertible on £2 (80; dn - 1w) whenever E C\R In the third step, the goal is to extend this result to other values of the param eter To this end, fix some z0 E C\IR, and for E C, consider (3.25) =

k

-

=

z

z.

Kff')

z

Observe that the operator-valued mapping Az E B (£2 (80; dn - 1w)) is analytic and, thanks to Lemma D.3, Az E Boo (L2 (80; dn- l w)) . The analytic Fredholm z �----+

128

F. GESZTESY AND M. MITREA

er

theorem then yields inv tibility of Thus,

I + Az

cep for

z

a

e t in discrete x

set D

C

C.

(3.26)

for z E where D is a discrete which, the inv rt bil ty esult roved in the previous paragraph, contained n R The above argument proves unique solvability of (3.3 ) for z E C\D, where D is a. di rete subset of JR. The (3.8) the fact that

isp invertible C\D is seti by e i i r sc representation and "YDSz : L2(80; dn- l w) --> H1 (80 ) then yields rDue H 1 (80) . Moreover, fN 'U.e = fNSzh = (-�Ian + KfF ) h L2 (80; dn-Iw) B(L2 (80; d"'- 1 w)). This ov z C\D , n t al estimate representati o m a om le b C\(D U a(-�e,n)) along two functions,on L2 (80; dn-lw ) . ue l v L2 ( D; dnx) and boundedly,

(3.27)

by (2.123) and (3.8) ,

E

E

(3.28)

sinee by (2. 124) , K'ff E pr es (3.4) when E C\D. For z E the a ur (3.5) is a consequence of the integral f r ula (3.8) and (D.28) . Next, ftx c p x num er z E with Also, let E so ve (3.3) . One computes g E

(ue , v ) D2 (D;dnx) = (ue , (- � - z) ( - �e .o - zln)-Iv) £2(!1;d" x) = (( -� - z)ue , ( - �e. o - Zlo) - 1 v) £2(1l;d"x)

+ (::YNue, /D(-�e,n - Zfn)-1v) £2(a!1 ;dn-Iw)

- (!Due, ::YN ( - �e . n - Zior 1 v) 1 12 (::YN ue, ID ( -�e,n - Zln) -1v ) L2(&!1;dn-Iw) + (rDue, erD ( -�e,!1 - zlo )- 1 v) l/2 = (::YNue, ID ( -�e,n - Zln)-1v) L2(&!1;dn -lw) =

-_t.kfD ( -�e ,!1 - zJn) -l v , G/D Ue )112

/

(= (1Nue, !D ( -�8,!1 - zfo)-1v) L2(8!1;dn- lw) \

"'-+-(EhDue , rD ( -�e.n - zln)- 1 v)112

( (::YN + GrD)ue, /D( -�e.n - zln)-1'v) 1 12

= (('1N + EhD )ne, ID ( -�e.o - Zin)-1v ) L2 (o!1:dn-'w) = (9, !D( -�e.n - Zln) -1v)L2(o!1 ;dn-lw) (3.29) = ((rv (-�e.n - zln) -1rg , v)L2(n;dnx) ' =

Since v

E

L2(0; �x) wa

s

arbitrary,

ue = (rD ( -�e. o - Zlo)-1 r g

this yields in L2(0; �x),

for

z E C\(D U a( -�e.n)),

(3.30)

which proves (3.7) for z E C\(D ua( - � e .n)). From this and (3. 5), the membership (3.6) also follows when z E C\(D U a( -�e.n)). The extension the more general case when z E C\a( -�e . n) then done by resorting to analytic continuation with respect to z. specifically, fix Zo E

to

is

More

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 129 C\a(Ae,o) . Then there exists r > 0 such that B ( zo, r) n a( -Ae,o) (3.31) ( B(zo , r) \ { zo } ) D n

=

0,

=

0,

since D is discrete and a( -Ae,o) is closed. We may then write

hv ( - Ae,o - zo lo)- 1 ]* = � J 2 rrt Jc(zo;r)

dz (z - zo)- 1 hv ( -Ae,o - Zlo)- 1 ] * (3.32) as operators in B(H - 1 12(80), L2(D; dnx)), where C(z0; r) C denotes the coun terclockwise oriented circle with center zo and radius r. (This follows from du alizing the fact 'YD( - Ae , o - zo lo ) - 1 E B(L2(D; dnx), H 1 12(8D)), which in turn follows from the mapping properties (-Ae,o - z0 Io) - 1 E B(L2(D;dnx), H 1 (8D)) and 'YD E B(H 1 (8D),H 1 12(8D)). ) However, granted (3. 3 1), what we have shown so far yields that hv ( - Ae,o - zlo)- 1 ] * E B(L2(8D; � - 1 w), H312(D)) whenever lz - zo l r, with a bound I hv ( -Ae,o - zlo)- 1 ]* II B(L 2 (80;dn - l w),H3/2(f!)) c C(D, Zo, r) (3. 3 3) independent of the complex parameter z E 8B (z0 , r). This estimate and Cauchy's representation formula (3. 3 2) then imply that hv(-Ae , o - zolo)- 1 ]* E B(L2 (8D;dn - 1 w),H312 (D)). (3. 34) This further entails that u hv ( -Ae,o - zo lo ) - 1 ]* g solves (3. 3 ), written with zo in place of z, and satisfies (3. 5 ). Finally, the memberships in (3. 4 ) ( along with natu rally accompanying estimates) follow from Lemma 2. 3 and Lemma 2. 4 . This shows that (3. 6 ), along with the well-posedness of (3. 3 ) and all the desired properties of the solution, hold whenever z E C\a(Ae,o). The special case 8 0 of Theorem 3.2, corresponding to the Neumann Lapla cian, deserves to be mentioned separately. OROLLARY 3 . Assume 2.1 andNeumann supposeboundary that z E value C\a( -AN,n). ThenCfor every g E3.L2(8D; dn- 1Hypothesis problem, w), the following { ;;;(-Au =- gz)uon 80,0 in D, u E H312(D), (3.35) N has a unique solution u UN . This solution UN satisfies 'YDUN E H 1 (8D) and lbv uN I I H'(80) + ll;yN uN ii £2(80;dn-lw) C II 9 II L2(80;dn - 1 w) (3.36) as well as l i uN II H3f2(f!) Cllg l i £2(80;dn-1w) > (3.37) for some constant constant C C(8, n, z) > 0. Finally, ['Yv ( -AN,o - zlo)- 1 ] * E B(L 2 (8D; dn - 1 w), H 312 (D) ) , (3.38) and the solution UN is given by the formula (3. 39) ue ('Yv ( -AN,O - zlo)- 1 ) * g. Next, we turn to the Dirichlet case originally treated in [46, Theorem 3.1] but under stronger regularity conditions on D. In order to facilitate the subsequent considerations, we isolate a useful technical result in the lemma below. c

=

:::;

e,

=

=

0

=

=

=

::=;

::=;

=

=

1 30

F.

LEMMA

GESZTESY AND M. MITREA

3.4. Assume Hypothesis 2.1 and suppose that z

( -.6-D, n - zln)-1 : L2(D ; dnx)

-.

E

C\a( -.6-D,n). Then

{ u E H312 (D ) 1 .6-u E L2 (D ; �x) }

(3.40)

is a well-defined bounded operator, where the space {u E H312 (D) 1 .6-u E L2(D; dnx)} is equipped with the natural graph norm u r--> ll u iiH i 2 (fl) + ll.6.u ll£2 (f!;d"x) · PROOF.

Consider

z

zln ) -1 f. It follows that

E ·u

3

C\a ( -.6-D,n ), f E L2 (D; dnx) and set w = ( -.6-D,n is the unique solution of the problem

(-.6. - z)w = f in D,

w E HJ (D).

(3.41)

The strategy is to devise an alternative representation for w from which it is clear that w has the claimed regularity in D. To this end, let j denote the extension of f by zero to IRn and denote by E the operator of convolution by En (z; ) . Since the latter is smoothing of order 2, it follows that v = (Ei) I n E H2 (D) and ( -.6. - z ) v = f in !'2. In particular, g = -"(DV E H1 (aD). We now claim that the problem ·

(-.6. - z )u

=

0 in D,

u

E

H312 (D) ,

rDU = g on a!t,

(3.42)

has a ::;olution (satisfying natural estimates). To see this, we look for a solution in the form (3.8) for some h E L2 (aD; dn- lw) . This guarantees that u E H312 (D) by Theorem D.7, and (-.6. - z)u = 0 in D. Ensuring that the boundary condition holds comC'.s down to solving rDSzh = g. In this regard, we recall that (3.43)

(cf. [101]) . With this in hand, by relying on Theorem D.7 and arguing as in the proof of Theorem 3. e can show that there exists a discrete set D C C such that

21£u \

rDSz : L2 (a!t;'cr+-'1 w) -. H1 (aD) is invertible for z E C\D.

(3.44)

Thus, a solution of (3.42) is given by (3.45) Moreover, by Theorem D. 7, this satisfies ll u iiH3/2 (f!J � C (D , z ) ll g iiHl(an ) � C (n , z ) II ! IIP(n; d"xl •

z

E

C\D.

(3.46)

Consequently, if z E C\(D U a( - .6- D,!J ) ) , then u + v solves (3.41 ). Hence, by uniqueness, w = u + v in this case. This shows that w = ( -.6-D ,n - zln)- 1 f belongs to H312 (n) and satisfies .6.w E £2 (.12; d" x) with

ll w ii H3/2 (0) + ll.6.w ll £2co;d...

x

)

� C(D, z ) II ! II L2(fl;dnx) >

z E C\(D U

a

( - .6- D ,n ) ) .

(3.4 7) In summary, the above argument shows that the operator (3.40) is well-defined and bounded whenever z E C.\(D U a(-.6-D.n)). The extension to z E C.\a( - .6.D , n ) is then achieved via analytic continuation (as in the last part of the proof of Theorem 0 3.2). Having established Lemma 3.4, we can now readily prove the following result.

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 131 LEMMA 3.5. Assume Hypothesis 2.1 and suppose that z E C\a( -�v.n) . Then (3.48) :YN ( -�v,n - zln) - 1 E B(L 2 (n; dn x), L 2 (an; dn- 1 w) ) , and [:YN ( -�v,n - zin)- 1 ] * E B(L 2 (an;dn - 1 w),L 2 (n;dn x) ) . (3.49) PROOF. Obviously, it suffices to only prove (3.48) . However, this is an imme0

diate consequence of Lemma 3.4 and Lemma 2.4.

We note that Lemma 3.5 corrects an inaccuracy in the proof of [46, Theorem 3.1] in the following sense: The proof of (3.20) and (3.21) in [46] relies on [46, Lemma 2.4] , which in turn requires the stronger assumptions [46, Hypothesis 2.1] on than merely the Lipschitz assumption on However, the current Lemmas 2.15 and 3.5 (and the subsequent Theorem 3.6) show that (3.20) and (3.21) in [46], as well as the results stated in [46, Theorem 3.1] , are actually all correct. After this preamble, we are ready to state the result about the well-posedness of the Dirichlet problem, alluded to above. 3.6. 2.1 z C\a(-�v,n).

n

n.

and suppose 1 (an), theHypothesis ThenTHEOREM for every f E HAssume following Dirichlet boundarythatvalueE problem, { (-� =- fz)uon an,0 in n, u E H3f2(n), (3.50) /DU has a unique solution u = uv . This solution uv satisfies :YN uD E L2 (an; � - 1w) and I !:YN uv ll u( an;d"-lw) Cv ll f i!Hl(Bf!) , (3.51) for some constant Cv Cv (n , z) > 0. Moreover, (3.52) llu v ii Ha/2 ( !1) :::; Cv llf ii H1 ( &n )· Finally, (3.53) [:YN ( -�v.n - zin)- 1 ] * E B(H 1 (an), H312 (n) ) , and the solution uv is given by the formula (3.54) uv [:YN ( -�v,n - zin)- 1 ] * f. PROOF. Uniqueness for (3.50) is a direct consequence of the fact that z E Existence, at least when E C\D for a discrete set D C, is implicit =

:::=:

=

=

-

C\a( -� v, n).

z

c

in the proof of Lemma 3.4 ( cf. (3.42)) . Note that a solution thus constructed obeys (3.52) and satisfies (3.51) (cf. Lemmas 2.3 and 2.4). Next, we turn to the proof of (3.54). Assume that z E C\(D U a( -� v , n)) and denote by uv the unique solution of (3.50). Also, recall (3.48)-(3.49). Based on these and Green's formula, one computes

(uv , V) £2(!1;d"x)

= =

= =

(uv, ( -� - z) ( -�D,f! - zfn)- 1 v) L2 (f!;dnx ) (( - � - z)uv, ( -�v,n - zin)- 1 v) L2(f!;dnx) (:YN uD, /D ( -� D,f! - zfn )- 1 v)U( 8!1;dn-1w) - ( !v uv, :YN (-�v,n - Zin)- 1 vh; 2 :YN ( -�D,O - ( (:YN ( -�D ,f! - Zfn) - 1 r V) L2(f!; dnx)

+ -(!,

zfn)- 1 v) l/2 f,

(3.55)

1 32

F. GESZTESY AND M. MITREA

for any v E L2 (n; d"x) . This proves (3.54) with the operators involved understood in the sense of (3.49) . Given (3.52), one obtains (3.53) granted that z E C\(D U

a ( -�o . n )) .

Finally, the extension of the above results to the more general case in which C\a( -�o .o ) is done using analytic continuation, as in the last part of the proof of Theorem 3.2. 0 z

E

Assuming Hypothesis 3.1, we introduce the Dirichlet-to-Robin map associated with ( -� - z) on n, as follows,

coJ n ( z) .. MD,e,

{

H 1 (an) __, L2 (an; d" -1w) , f ...... - (:YN e'YD ) uD,

where uo is the unique ::;olut.ion of

( -� - z ) u

=

0 in 0,

+

M�?e , o (z)

z E C\a ( -� o.o ) ,

u E H312 (n), /D 'U

=

f on

an.

(3.56)

(3.57)

Continuing to assume Hypothesis 3.1, we next introduce the Robin-to-Dirichlet map M�� b . o (z) associated with ( - � - z) on n, as follows, Af(o) "�

e , D,n (z)

where

ue

· .

{

£2 (an ; dn-lw) --> Hl (an) , 9 ...... , o u e ,

z E \1...""'\a ( -.u.e ,n ) ,

(3.5 8)

n, u E H312 (n), (:YN + 8--yo ) u = g on an.

(3. 59)

�

"

is the unique solution of

( -� - z)u

0 in

We note that Robin-to-Dirichlet maps h_51;ve abo been studied in [10] . We conclude with the following th rem, one of the main results of this paper: =

,O

THEOREM 3.7. Assume Hypothesis\i.J.:_/Then

Mg�,0(z) E B(H1 (an) , L2 (an; dn-lw)) , z E C\a(- �D .n) ,

(3.60)

and M�.� .o ( z ) = (:YN + e'i'D ) [ (:YN + EhD ). ( -�D.n - Zlo )-1] * ' z E C\a( - � , o ) . o

(3.61)

Moreover,

M��b, n Cz) E B(L2(an; dn - 1w) , H1 (an)) , z E C\a ( - �e .o ) ,

(3.62)

and, in fact, In addition,

M�b .n Cz) = rD ['i'D (-�e.o - Zlo ) - 1 r, z E C\a( -�e.n). Finally, let z E C\(a( -� D ,n) U a( -�e,n)). Then MS(O)D O (Z ) = - MD(O)S :O (Z ) - 1

(3.64) (3.65)

PROOF. The membership in (3.60) is a consequence of Theorem 3.6. In this context we note that by the first line in (2.96) , ID ( � o n - zln ) - l = 0, and hence ,

J

'

·

- .

'UD = - [:YN ( -�D,O - zJn)-l r f = - [ (:YN + e')'D ) ( - �D,f! - zfo ) - 1 ] * f

(3.66)

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 133 by (3.54). Moreover, applying - (;yN + e"Yv ) to uv in (3.54) implies formula (3.61) . Likewise, (3.62) follows from Theorem 3.2. In addition, since H 1 (80) embeds compactly into L2(80;dn- 1 w) (cf. (A.10) and [72, Proposition 2.4] ) , M��b,n (z), z E C\a( - �e.n), are compact operators in £2(80; dn-lw), justifying (3.63). Ap plying "YD to ue in (3.7) implies formula (3.64). There remains to justify (3.65). To this end, let g E £2(80; dn- 1 w) be arbitrary. Then

M�_le , n (z) M��b .n (z) g = M�_le,n (z)"Yvue = - (;yN + B"Yv ) uv, (3.67) f = "Yvue E Here ue is the unique solution of ( -� - z)u = 0 with u E and (;yN + Ehv)u = g, and uv is the unique solution of (-� - z)u = 0 with u E and "YDU = f E Since (uv - ue) E and "YDUD = f = "Yvue , one

H1 (80). H312(0)

H312(0)

H1 (80).

concludes

H312(0)

"Yv (uv - ue) = 0 and ( - � - z)(uv - ue) = 0.

(3.68)

Uniqueness of the Dirichlet problem proved in Theorem 3.6 then yields uv = ue which further entails that - (;yN + e"YD ) uv = - (;yN + e"YD )ue = - g . Thus, (3.69) M�.)e,n (z) M��b .n (z) g = - (;yN Ehv) uv = - g , implying M�_le , n (z)M��b.n (z) = -Ian · Conversely, let f E Then (3.70) M��b .n (z)M�_le ,n (z)f = M��b .n (z) ( - (;yN + Ehv)uv) "Yvue,

+

H1 (80). =

and we set

(3.71 ) Here uv, ue E are such that -� - z)ue = ( -� - z)uv = 0 in and "YDUD = (;yN e"Yv)ue = g . Thus (;yN Ehv) (ue + uv) = 0 , ( - � - z) (ue + Uniqueness of the generalized Robin problem uv) = 0 and (uv + ue) E proved in Theorem 3.2 then yields ue = -uv and hence "Yvue = -"Yvuv = -f . Thus, (3.72) Me( 0,)v , n z ) MD(0),e ,n z ) f - "Yvue -

( H312(0) J, + + H312(0). (

0

- J,

(

implying M�0b , n (z) M�� n (z) � -Ian · The desired conclusion now follows. '

'

'

0

3.8. In the above considerations, the special case e = 0 repre sents the frequently studied Neumann-to-Dirichlet and Dirichlet-to-Neumann maps . (0)N, n (z), respectively. That . MN(0,)D , n (z) = Mo(0,D) , n z) and MN( 0,)D ,n z) and Mv,

REMARK (

lS,

M�)N n (z) = M�� n (z). Thus, as a corollary of Theorem 3.7 we have (O ) n ( ) = - MD(O),N,n MN,D, whenever Hypothesis 2.1 holds and z E C\(a( -�v.n) U a ( -�N.n)). '

'

'

'

( Z) - 1 '

Z

(

(3. 73)

REMARK 3.9. We emphasize again that all results in this section extend to Schrodinger operators e , n = -�e.n + dom ( e ,n) = dom ( - �e.n) in for (not necessarily real-valued) potentials satisfying E or more generally, for potentials which are Kato-Rellich bounded with respect to -�e . n with bound less than one. Denoting the corresponding M-operators by

£2(0; dnx)

H

V,

V

H V

V L00(0; dn x),

F. GESZTESY AND M. MITREA

1 34

Mv,N,n (z) and Me , n , n (z), respectively, we note, in particular, that (3.56) -(3.65) extend replacing - � by - � + V and restricting z E C appropriately.

discussion of Weyl-Titchmarsh operators follows the earlier papers [43] and [46] . For related literature on Weyl-Titchmarsh operators, relevant in the context of boundary value spaces (boundary triples, etc. ) , we refer, for instance, to [3], [5] , [1 2] , [13] , [18]-[22] , [32]- [35] , [42] , [44] , [47 , Ch. 3], [49, Ch . 1 3] , [65] , [66] , [11 ] , [80] , [81] , [84], [85] , (88] , [89] , [100] .

Our

4. Some Variants of Krein's Resolvent Formula In this section we present our principal new results, variants of Krein's for mula for the difference of resolvents of generalized Robin Laplacians and Dirichlet Laplacians on bounded Lipschitz domains. We start by weakening Hypothesis 3.1 by using assumption (4. 1) below:

HYPOTHESIS 4. 1 . Jn addition to Hypothesis 2.2 SUPJ!PSe that

e E E= (H 112 (&n ) , H- 112 ( an r) .

We note that condition (4. 1 ) is satisfied if there exi�s. some e E E (H112-" ( 80) , H - 1 12 ( 80) ) .

E

(4. 1 ) > 0 such that

(4.2)

Defore proceeding with the main topic of this section, we will comment to the effect that Hypothcsil:i 3.1 is indeed stronger than Hypothesis 4.1, as the latter follows from the former via duality and interpolation, implying

e E E= (H8 ( 80), H 8 - 1 (80)) , To see this, one first employs the

fact

0 -::; s -::; 1 .

(4.3)

that

for s = ( 1 - B)so + Os 1 , 0 < (} < 1, 0 -::; so, s1 -::; 1, and s0 =/= s 1 1 where ( · )o,q denotes the real interpolation method. Second, one uses the fact that if T : Xj � Yj , j = 0, 1, is a linear and bounded operator between two pairs of compatible Banach spaces, which is = 0, then T E E= ( (Xo, Xl)o,p, ( Yo , Y1 )o,p) every (} E (0, 1). This is a result due to Cwikel [2 7] :

( 4.4)

·,

compact for

for

j

THEOREM 4.2 ( [2 7] ). Let Xj , Yj , j = 0, 1, be two compatible Banach space couples and suppose that the linear operator T : Xi � }j is bounded for j = 0 and compact for j = 1. Then T : (Xo, Xl )o,p � (Yo , Y1 )e,p is compact for all (} E (0, 1) and p E [1, oo] .

(Interestingly, the corresponding result for the complex method of interpolation remains open. ) In our next two results below (Theorems 4.3-4.5) we discuss the solvability of the Di ichlet and Robin boundary value problems with solution in the energy space H1 (0) .

r

THEOREM 4.3. A ssume Hypothesis 4.1 and suppose that z E C\a ( - �e .n) .

Then for every g problem,

E H- 1 12 (8fl), the following generalized Robin boundary value

{

( -� - � )u = 0 in 0, u E (;yN + 8·YD ) u = g on 80,

H 1 (U),

(4.5)

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 1 35 has a that unique solutionu ue. Moreover, there exists a constantC C ( , n , z) > 0 such (4.6) In particular, (4.7) ['yv( -�e.n - zin)-1 ] * E B(H - 1 12 (8!1), H 1 (D)), and the solution ue of (4.5) is once again given by formula (3. 7) . PROOF. The argument follows a pattern similar to that in the proof of Theorem 2.17. In a first stage, we look for a solution for (4.5) in the form (4.8) u(x) (Sz h ) (x) , E n, for some h E H - 1 12(8!1). Here the single layer potential Sz has been defined in (2.121), and the fundamental Helmholtz solution En is given by (2.120) (cf. also ( C .1)). Any such choice of h guarantees that u belongs to H 1(D) and satisfies (-� - z)u 0 in n . See (D.29). To ensure that the boundary condition in (4.5) is verified, one then takes =

e

=

=

X

=

(4.9 ) That the operator

(- � Ian + K'ff ) + G"(v Sz

:

H - 1 12 (8!1) --> H- 112 (8!1)

(4.10)

z,

is invertible for all but a discreet set of real values of the parameter can be established based on Hypothesis 4.1 by reasoning as before. The key result in this context is that the operator - �Ian + Kff E is Fredholm, with 0 Fredholm index zero for every E C.

z

B(H- 1 12(8!1))

The special case 8 = 0, corresponding to the Neumann Laplacian, is singled out below.

COROLLARY 4.4. Assume Hypothesis 2.1 and suppose that z E C\a(-�N.n) . 1 Then for every g E H- 12(8!1), the following Neumann boundary value problem, { '1(-�u - gz)uon=ail,O in n , (4.11) N has unique solution u = UN . Moreover, there exists a constant C C(D, z) > 0 suchathat (4.12) In particular, (4.13) ['yv ( -� N,n - :Zin)-1) * E B(H - 1 12 (8!1), H 1 (D)), and the solution u9 of (4.5) is given by the formula (4.14) uN ('Yv ( -�N,n - zin)- 1 ) * g . Finally, as a byproduct of the well-posedness of (4.11), the weak Neumann trace '1N in (2.40), (2.41) is onto. =

=

=

136

F. GESZTESY AND M. MITREA

In the following we denote the inclusion (embedding ) map of into a slight abUBe notation, denote the con tinuous inclusion map of into (HJ (n))* by the same symbol fn. We recall the ultra weak Neumann trace operator ;yN in Finally, assuming we

by fn contiof nuous we also H1(0) (H1(0)) *. HJ(O) By (2.60), (2.61). Hypothesis 4.1, denote by (4.15) -Lie,o B (H 1 (fl ) , (H1(0))*) the e o - t..e,n in (B.26). \i'v(x) ('YDU, EhDv)112 , H'(n) {u, - Li8,nv) (H'(fl))• = k u, E H1(0), (4.16) and -t..e,n is the restriction o -Li e,n L2(0: dnx) (B.27)). 4.5. 4.1 z E -t..e,n) . E (H1 (fl.)) * , ln inOV'(fl) , u E H1(fl), w { ;y(-t.-z)u =:_ (4.17) u, w) + erDu = N( = 'Ue,w· = z) > 0 (4.18) (L2(fl;-t..e,n -zlo)-1, z E -t..e,n), (-.6e,n - zln)-1 E B (L2(fl; (4.19) B ( (H1 ( )*, H1 (fl.) ), (4.20) ( - Lie ,n - zfn r1 E B( (H 1 (fl. ) )*, H 1 ( fl ) ) . (2.57). Hence, w E (H1(0.) )*, w (C.1) H1(fl) (-t.. i n V'(fl). sol u ti o n (4.17) s z)u0 w 0 u , u = 1 {(-t.. -B1v)u z)ul =1 O=in- ;y_,v (uo, Ew)H1(0), (4.21) e/ouo) E ( u o, w ) + ;yN(u, w ) = ;yN(( Uo, w ) (ul, 0) ) = N (u o , w ) U1 , 0) ho u l -B'Yo uo = -Ehou, (4.22) (2.64). solvable (4.18). n s for 4.3. (4.17) follows from 4.3. s operator (4.20) ( H 1 ( fl. ) ) * , (4.23) -.6 e , n), -t.. e ,o -zln H E

ext

ns i n of

accordance with

In particular,

cfl x \i'u(x) ·

f

THEOREM

Then for every w lem,

+

(cf.

to

v

Assume Hypothesis and supplse that C\a( the following generalized_ inhomogeneous Robin prob\,,.,

on 80. ,

has a un·ique solution ·u such that

Moreover, there exists a constant c

C(e, n,

llue,w iiHl (fl) .::::; Cll w llcH1(8fl))• ·

In particular, the operato1· as a bounded operator on

C\a(

dnx),

originally defined

dnx)) ,

n)

can be extended to a mapping in

which in fact coincides with

PROOF. We recall if taking the convolution of with En ( z; · ) in and then restricting back to fl. yields a function u0 E = for which in A of i then given by u where u1 satisfies +

fl.,

(;yN +

u1

+

Indeed, we have =

-€

+

;y

H-112 (80.) on {)fl.

+ ;yN(

=

;yN (

;yN u l

That the latter boundary problem is is guaranteed by Theorem We note that the solution thus constructed satisfies U nique es the corresponding uniqueness statement in Theorem Next, we observe that the inver e in is well-defined. To prove that 1 z E C\u( : (Q) �

by

ROBIN-TO-DIRJCHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS

137

is onto, assume that w E (H1 (0)) * is arbitrary and that u solves (4. 17) . Then, for every v E H1 (0) we have

H' (!1 ) (v , ( -Lie,n - zfn)u) (H'(!1))•

l dnx V'v(x) = l dnx V'v(x) =

=

·

·

l dnx v(x)u(x) + (!DV, e/Du) 112 V'u(x) - z l dnx v(x)u(x) - (/D v , -;:;N (u , w) ) I ;2

V'u(x) -

Z

H' (!1) (v, w) (H'(!1))• ,

(4.24)

on account of (2.61) , (4. 16), and (4. 17) . Since the element v E H1 (0) was arbitrary, this proves that ( -Lie ,n - zfn )u w , hence the operator (4.23) is onto. In fact, this operator is also one-to-one. Indeed, assume that u E H1 (0) is such that (-Lie ,n - zln)u = 0. Then, for every v E H1 (0) , formula (4. 16) yields =

0= =

H'(n) ( v, ( - Lie , n - zfn)u ) (H'(!1) )•

l dnx 'Vv(x)

·

V'u(x) - z

l dnx v(x)u(x) + (!DV, e/Du) 112 .

(4.25)

Specializing (4.25) to the case when E C0(0) shows that (-� - z)u = 0 in the sense of distributions in 0. Returning with this into (4.25) we then obtain (!DV, (-;:;N + e,D )u ) / 2 = 0 for every v E H1 (0). Given that the Dirichlet trace 1 /D maps H1 (0) onto H1 1 2 (80) , this proves that (-;:;N + e,D )u = 0 in H-1 1 2 (80) so that ultimately u = 0, since z E C\a( -�e ,n) . In summary, the operator (4.23) is an isomorphism. Finally, there remains to show that the operators (4. 19) , (4.20) act in a compat ible fashion. To see this, fix z E C\a( -�e ,n) and assume that w E L2 (0 ; dnx) (H1 (0)) * . If we then set u = (-Lie,n - zln)- 1 w E H1 (0) , it follows from (4. 16) that v

'---->

H' (!1) (v, w) (H'(!1))• =

l dnx 'Vv(x)

= ·

H'(n) ( v, ( - Lie ,n - zln)u \ H ' (!1)) •

V'u(x) - z

l dnx v(x)u(x) + (!DV, e/D u\12 ,

(4.26)

for every v E H1 (0). Specializing this identity to the case when v E C0 (0) yields ( -� - z)u = w E L 2 (0; dnx). When used back in (4.26) , this observation and (2.41) permit us to conclude that (/D v , (-;yN

+ Ehv)uh;2 = l �x V'v(x)

l dnx v(x )u(x) - H' (!1) (v, ( -�u - zu) ) (H' (!1))• + (!DV, E:),D u ) 1 12 = l dnx 'Vv(x) V'u(x) - z l dnx v(x)u(x) - H' (n ) (v, w )(H'( n ))• + (!D v , e,v u ) 1 1 2 ·

V'u(x) - z

�

·

= 0,

( 4.27)

138

F. GESZTESY AND M. MITREA

for every v E H1 (fl). Upon recalling that the Dirichlet trace /D maps H 1 (fl) onto H112 (8f!), this shows that C.:YN + EhD )u = 0 in H- 112 (80). Thus, u = ( -� e. n - z ln) -1 w , as desired. 0

REMARK 4.6. Similar (yet simpler ) considerations also show that the operator ( -�D.n - z ln ) - 1 , z E
(4.29)

Here - Li D ,n E B(H{j (fl), H-1 (0)) is the extension of -?� D.n in accordance with (B.26) . Indeed, the Lax-Milgram lemma applies and y�lds that (4.30) ( - LiD ,n - z ln ) : HJ ( o ) � (HJ ( o ) /\..:, H-:.� (0) is, in fact, an isomorphism whenever z E C\a( -�D , n) -

4.7. A.s.sume Hypothesi.s 4.1 and .suppo.se that z E C\a( -�e.n) . Then the operator M��)o , n (z) E B (L2 (80; dn-1w)) in (3.58), (3.59) extends ( in a compatible manner) to COROLLARY

M��b .n (z) E B(H- 112 (8fl) , H112 (8rt ) ) ,

z E C\a(-�e,n ) .

In addition, M��b, n (z) permits the representation -c o) (z) = D ( - � M e.n - zln) - 1ID , z E C\a( -�s .n)! e,D,n •

(4.31)

(4.32)

The same applies to the adjoint M��b,n (z)* E B(L2 (8fl; dn- l w)) of M��b .n (z) , resulting in the bo·unded extens·ion (M��b , 11 (z))* E 6(H-112 (Dfl), H112 (8fl) ) , z E
P ROOF. The claim (4.3 1) is a direct consequence of Theorem 4.3, while the claim (4.32) follows from the fact that

1L> : (H112 (8rt))*

(4.33) s -112(ao) � (H1 (rt))* in a bounded fashion (cf. (A.32), (4.20) and (3.64)) . The rest follows from dualizing these claims. 0 =

The following regularity result for the Robin resolvent will also play an impor tant role shortly.

LEMMA 4.8. A ssume Hypothesis 3.1 and suppose that z E
(4.34) is a well-defined bounded operator, where the space { u E H312 (f!) I �u E L2 (fl; dn x)} is equipped with the natural graph norm u r--> l l u i i H3/�(!1) + ll �u ii£2 (0;d"x) · PROOF. Consider f E L2(f!; d"'x) and set u that u is the unique solution of the problem

( -� - z)u = f in f2,

u E H1(fl) ,

=

( -�e.n - zln) - 1 f. It follows

(;yN + E},D ) u = O on 80.

(4.35)

The strategy is to devise an alternative representation for u from which it is clear that u has the claimed regularity in fl. To this end, let J denote the extension

139

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS

of by zero to JR.n and denote by E the operator of convolution by En (z ; · ) . Since the latter is smoothing of order 2, it follows that w = (EJ) i n E H2 (0) and ( -L:l - z)w = J in 0 . Also, let v be the unique solution of the problem

f

(-Ll - z)v = O in 0 , v E H312 (0) , (1N + e·-w)v = - (/'N + erD) w on 80. (4.36) That (4.36) is solvable is a consequence of Theorem 4.3. Then v+w also solves (4.35) so that, by uniqueness, = v + w. This shows that has the desired regularity properties and, hence, the operator (4.34) is well-defined and bounded. D

u

u

Under the Hypothesis 4 . 1 , (4.20) and (2.7) yield

/D ( - Lie,n - zin) - 1 E B((H 1 (0)) * , H 1 12 (&0)) .

(4.37)

Hence, by duality,

[rD ( - Lie,n - zin) - 1 ] * E B(H- 1 12 (&0) , H 1 (0)).

(4.38)

We wish to complement this with the following result.

COROLLARY 4.9. Assume Hypothesis 3 . 1 and suppose that z E C\a( -Lle,n) . Then (4.39) In particular, [rD (-Lle,n - zin)- 1 ] * E B (H- 1 (&0) , L 2 (0; dnx )) B ( L2 (&0; dn - 1 0) , L2 (0; dnx )). Co....+

(4.40)

In addition, the operator (4.40) is compatible with (4.38) in the sense that [rD ( -L:le,n - zin)- 1 ]* f = [rD ( - Lie,n - zin) - 1 ] * f in L2 ( 0; dnx ) , f E H- 1/2 (&0). (4.41) As a consequence, [rD ( - Lle , n - zin)- 1 ] * f = [rD ( - Lie,n - zin) - 1 ] * f in L2 ( 0; dnx ) , (4.42) f E L2 (&0; dn- 1 w) . PROOF. The first part of the statement is an immediate consequence of Lemma 4.8 and Lemma 2.3. As for (4.41 ) , pick f E H - 1 12 (&0) H- 1 (&0) and u E L2 (0; dnx) (H 1 (0)) * arbitrary. We may then write ([rD ( -Lle,n - zin) - 1 ] * f , u ) £2 (!l;dnx ) = ( f , !D ( -L:le,n - zin)- 1 u)t = ( f , /D ( -Lle,n - Zin)- 1 uh; 2 = ( f , rD ( - Lle,n - zin) - 1 u) 1 12 = Hl (!l) ( [rD ( - Lie, !! - zin) - 1 ] * f , u ) (Hl (!l))• = ( [rD ( - Li e,n - zin) - 1 ] * f , u ) £2 (!l;dn x) , (4.43) Co....+

Co....+

since (4.19) and (4.20) are compatible. This gives (4.41). Since L2 (&0; dn - 1 w) H - 1 12 (&0) , (4.42) also follows.

Co....+

D

1 40

F. GESZTESY AND M. MITREA

We will need a similar compatibility result for the composition between the Neumann trace and resolvents of the Dirichlet Laplacian. To state it, recall the restriction operator Rn from (2.58) . Also, denote by JIR" the identity operator (for ::;paces of functions defined in !Ftn ) . Finally, recall the space (2.59) and the ultra weak Neumann trace operator in (2.60), (2 .61). LEMMA

4.10. Assume Hypothesis

2.1. Then

( ( - Liv,n - zfn) -l o Rn, fiR, ) : (H 1 (!1 )) * --+ Wz (fl), I z E C \cr( -.6.v,n), (4.44) is a well-defined, linear and bounded operator. Consequ�ntly, , - H - 2 (8!1 )) , 7.N( ( - .6.v,n - zln) 1 o Rn, IIR") E 13( (H 1 (D ))*,\.,=r/ (4.45) z E C\o-( -.6.v,n ) , and, hence, �

[7N ( ( - Liv.n - zfnf 1 Rn , IR" ) ) * E !3 (H 1 12 (8n) , H 1 (0) ) , (4.46) z E C\cr( -.6.v,n)Furthermore, the operators (4.45), (4.46) are compatible with (3.48) and (3.49), respectively, in the sense that for each z E C\cr( -.6.v,n) , o

7N ( -.6.v,n - zlo)-1 f = 7N ( ( - Liv,n - zfn) - l o Ro , IR" ) f in H - 112 (8!1 ) , f E L 2(0 ; dn x), (4.4 7) and

- - 1 o Rn , JR, ) ] * f in L2 (0; d:"x), [7N ( -.6.v,n - zlo)- 1 ] • f = [7N ( ( - .6.v,n - zln) for every element f E H11 2 (80) . (4.48)

Let z E C\cr ( -.6.v,n ) . If f E (H 1 (0) ) + and u = ( -Liv,n-zlo) -1 (flo ) , then u E HJ (O) satisfies ( -.6. - z ) u = f i n in D'(n) . Hence, (u, f ) E Wz (O) which shows that the operator (4.44) is well-defined and bounded. Then (4.45) is a con sequence of this and (2.60), whereas (4.46) follows from (4.45) and duality. Going further, (4.47) is implied by Lemma 3.4, the compatibility statement in Lemma 2.4, and (2.62) - (2.64). There remains to justify (4.48). To this end, if f E H 112 (80) '-+ L2 (80; dn- 1 w) and u E L 2 (0 ; dnx) '-+ (H 1 (!1))* are arbitrary, we may write PROOF.

([7N ( -.6.v,n - zln) - l ] * f , u) L2 (!1;d"x) = (f , 7N ( -.6.v,n - zln ) - 1 u)0 = ( f , 7N ( - .6.v,o - z/n) - 1 u ) 1 2 u )I/ 2 1 = (! , 7.N ( ( - Liv,n - :zfnf1 o Rn, IR" )uh12 = Hl (fl) ( [7.N ( ( - Liv,n - zfn f 1 0 Rn , JIR,. ) r f , u) (HI (!1))• = ( [7.N ( ( - LS.v,n - zfnf 1 0 Rn , IIR") r f ' u) U (fl;d"x) '

(4.49)

where the third equality is based on (4.47) . This justifies (4.48) and finishes the 0 proof of the lemma.

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 141 LEMMA 4. 1 1 . Assume Hypothesis 4. 1 and suppose that z1 E C\(a( -Ae,n) U * H , Then the following resolvent relation holds on (0)) a(-An,n)) . ( ( - Ae,n - zln) -

-

1

=

( - An,n - zln) - o Rn + ( - Ae,n - zln) - 1 'Y1J:YN ( ( - An,n - zln) - 1 o Rn, IJRn ) . -

-

1

(4.50)

PROOF. To set the stage, we recall (2.40)-(2.41) and (4.33) . Together with and (4.29), these ensure that the composition of operators appearing on the

(4.20)

right-hand side of (4.50) is well-defined. Next, let ¢1 , 1jJ 1 E L2 (0; dnx) be arbitrary and define ¢ ( -Ae , n - zln)- 1 ¢1 E dom (Ae,n) c ( H 1 (0) n dom(:YN) ) , (4.51) 1/J ( -An , n - zln)- 1 1/J1 E dom( Av,n) c ( HJ (O) n dom(:YN)) . =

=

As a consequence of our earlier results, both sides of (4.50) are bounded operators from ( H 1 (0))* into H 1 ( 0 ). Since £2 (0; ( H1 ( 0 ))* densely, it therefore suffices to show that the following identity holds:

dnx)

<--t

( ¢1 , ( - Ae ,n - zln) - 1 1/J 1 ) L2(f!;dnx) - ( ¢1 , ( -An,n - zln) - 1 1/J1 ) £2(f!;dnx) (¢1 , ( -Ae,n - zln) - 1 'Yi:/YN( -A v,n - zln) - 1 1/JI ) £2(f!;dnx)· =

(4.52)

We note that according to (4.51) one has,

(¢ 1 , ( -A v , n - zln) - 1 1/JI ) p(f!;dnx ) ( ¢1 , ( -Ae, n - zln) - 1 1/Jl ) p( n;dnx)

= = = =

(4.53) (( -Ae,n - Zin) ¢, 1/J ) £2(f!;dnx ) > ( (( - Ae ,n - zln)- 1 ) * (4.54)

and, further,

(¢1, ( -Ae ,n - zln) - 1 'Y1J:YN ( -An,n - zln) - 1 1/J1 ) L2(f!;dnx) Hl(f!) ( ( -Ae,n - Zin)- 1 ¢1 , 'Y1J:YN( -An,n - zln) - 1 1/J 1 )(H l (f!))• = hn ( -A e ,n - Zin)- 1 ¢ , :YN( -A v,n - zln) - 1 1/J h; 1 1 2 hn

=

(4. 55)

Thus, matters have been reduced to proving that

(( -Ae,n - zln)¢, 1/J ) £2(f!;dnx) - (¢, ( -A v,n - zln) 'l/J ) £2(f!;dnx) = ('Yn¢, :YN'l/J h; 2 ·

(4.56)

Using (A.31) for the left-hand side of (4.56) one obtains

( ( -A e,n - Zln) ¢, 1/J ) L2(f!;dn x) - (¢ , ( -An,n - zln) 'l/J ) L2(f!;dnx) (4.57) -(L\
+

=

=

=

The stage is now set for proving the £2-version of Lemma 4.1 1 .

142 a(

F. GESZTESY AND M. MITREA LEMMA 4.12. Assume Hypothesis 3.1 and suppose that z E
-D.D,n )). (-D.e ,n -z n = (-b.D,O -zfn) -1

- D.e ,n)

-b.e,n -:zln)-1r [::YN(-ll.D,r! - D. , -zl_p ) -1]*[--rv (-D.e,n - zin) -1]. z fn ) - 1 ]

+ [rD( 1 = (-D.v,n - zin ) - + [::YN (

vn

U

(4 . 5 8 ) (4.59)

i{c�

PROOF. Consider the first equality .58) . To begin with, we note that the following operators are well-defined, linea�n� bounded: E B(L2 (D; dnx)) , (-D.� ,n - z!n)-1 E B( L2 (r2; �x)) , (4.60)

(-D.v,n - zln) -1 ::YN ( -D.v, n -zln)-1 -b.e,n - ] *

E B(£2 (0; �x), £2(00 ; dn-1w)), 1 E B(L 2 ( 8r2; dn -1w ) , L 2 (r2; �x))) . zin ) -

[in (

(4.61) (4.62)

Indeed, (4.60) follows from the fact that E
- b.v,n) -D.e,n )),

z

We note that the special case 8 = 0 in Lemma 4.12 was discussed by Nakamura [77] (in connection with cubic boxes 0) and subsequently in [43 , Lemma A.3] ( in the case of a Lipschitz domain with a compact boundary) . �( o)

-c o)

-

z

LEMMA 4.13. Assume Hypothesis 4. 1 and suppose that E
-D.e,n ).

[Me,v,n Cz)] Me,v,n (z) [Me.n,n z)] 1'vfe, v , n (z) . *

Then (4.63)

as operators in B( H - 112 (80) ; H112 (8r2) ) . In particular, assuming Hypothesis 3.1, then ( 0) (0) (4.64) = ( •

PROOF.

infers

where

u,

v

Let f, g E H-112 (80). Then using the definition of

u) \-fe,D ( 1-( ,n ( z ) f, g ) 112 = (rDU, ("'J'N + 8rD ) v)112 solve the Robin boundary value problems

Ai��b,n (z)

(4.65)

n, u E H1 ( r2) , (;;;N + 8rn)u = f on

{(-6 - :) u = Oin { ((::Y-6- _:)v N

and

one

(4.66)

Of! ,

0 in fl, v E H1 (fl), (4.67) + 6rn ) v = g O n 8[2, respectively. That this is possible is ensured by Theorem 4.3. Using (A.31) we may then 'vritc =

(::YN + Ehv) v) ; 1 2 + (rnu, Ehnv ) 11

(YDu, (rnv,, ::YN v h;2 ('Vu, 'Vv)L2(!1;d"x)" Hl(n)(u, b.v)(Hl(!!) • =

=

+

2

+ (rDU, Grn v\1

2

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS

1 43

= (\7u , \7v) L2 (!1;dnx)n - :Z (u , v) £2 ( !1 ; dnx)n + ("fo U, Eho v ) 1 12 = (\7v, \7u) £ 2 (!1;dnx)n - z (v, u) £2 ( !1;dnx)n + ('Yo U, Eho v ) 1 12 = (\7v, \7u) L2(!1;dnx)n + H l (!1) (v, u) ( H l (!1)) • + ( "fo u Jho v ) 1 12 ( "foV , "fNUhf 2 + ("fo u, fY"(o v ) 1 12 = ("fov, "(NUhf2 + w�"fDU, "(D V) 1 ;2 = bo v, 'iN uh; 2 + ("foV , 8 "(o u) 1 1 2

�

=

= ("foV , ('iN + G"fo )u ) 112 = (M��b.n (:Z) g, f ) l /2 '

(4. 6 8) - (!, Me,o,n (z)g) l/2 ' Now (4. 6 3) follows from (4. 6 5) and (4. 6 8). Finally, (4. 64) follows from (4. 63) by restriction of the latter to L2 (80; dn 1 -

-( o)

-

0

- w).

Next we briefly turn to the Herglotz property of the Robin-to-Dirichlet map. We recall that an operator-valued function M(z) E B(H), z E (where C+ = {z E j lm( z) > 0) , for some separable complex Hilbert space H., is called an if M( ) is analytic on and

C+ C operator-valued Herglotz function C+ (4.69) Im (M(z)) � 0, z E C+. Here, as usual, Im (M) = (M - M * )/(2i). LEMMA 4. 1 4. Assume Hypothesis 4.1 and suppose that z E C + . Then for every g E H- 1 12 (80), g -/= 0, - ,o (z) ] g) = Im(z) il ue ii£22 (!1;dnx) > 0, (4.70) 1 ( g, [Me, z) Me ( o 1 12 2i where ue satisfies {(-('iN�+-G"t_:)u =)u0=in 0on, 80.u E H1 (0) , (4. 71) g o In particular, assuming Hypothesis 3.1, then Im (M��b. n (z)) � 0, z E C+, (4.72) and hence M��b,n ( · ) is an operator-valued Herglotz function on L2 (80; dn-1 w) . PROOF. Let ue be given by the solution of (4. 7 1). Then M�0'b'n9 = "fD Ue by (3. 58) , and using self-adjointness of e (in the sense of (B.7)) and the Green's formula (A. 3 1), one computes, - ,o (z) - M-e , o (z) * ] g ) 1 ( g , [Me 1 12 2i 1 = 2i [( , Me ,o (z) g ) 1 2 - ( Me , o (z) g , g ) 1 1 ] 1 = 2i [( ('iN + 8"fo )u e, "fo ue) 1 /2 - ("fo Ue, ('iN + 8"to ) ue) 1 /2 ] ·

*

g

1

2

F. GES ZTESY AND M. MITREA

H4

=

;i [{::YNue, /nueh;2 - bn ue, ::YN ue)

+ =

:oince

g

1

2i

t

; 2]

[ (9'")'DUe , {DU8 ) 1 ;,2 - (inU e , 9{n ue ) 1 1J -

I

Im ((::YNue , /nue h;2)/ I = Im[('Vue, 'Vue)L2(n;Jzx) t Hl (n) (tiue , ne)(Hl(fl)) •l = Im(-z H' (n) (uA, uA) ( ;l-(n w ) = Im(z) H l (n) (ue , ue)(Hl (fl))• = Im( z) llue lli2 (f!;d" x ) > 0 =j:. 0 implies ue

=j:.

(4.73)

0. This proves (4.70). Restriction of (4.70) to

L2 (ofl; dn-1w) then yields (4.72) .

g E

0

Returning to the principal goal of this section, we now prove the following variant of a Krein-type resolvent formula relating Lie , n and Lin ,n: THEOREM 4.15. Assume Hypothesis 4.1 and suppose that z E
- 1 - -1 ( - tie ,n - zln. ) = ( - An ,n. - zl0 ) - o Rn 1 + [::YN ( ( - Lin,n. - zfn.) - o Rn. , JR., )] * M��b,n. (z) [::YN -

-

1

x ( ( - tin,n - zlnr o Rn, IIRn)] .

U

(4.74)

PROOF. Applying ID from the left to both sides of (4.50) yields - 1 .. - -1 ID ( - Ae,n - zlo) - 1 = {D ( - tie,n - zln) - InIN ( ( - Ao,n - zln) o Rn , IJR.n) (4.75) since In ( - Lin ,n - zYnr1 = 0. Thus, by (4.32), -1 �(0) - -1 ln ( - tie ,n - zlo ) = Me ,n ,0(z)::YN ( ( - tin,n - zln) o Rn , JRn ) , (4.76) as operators in !3 ( (H 1 (D)) * , H11 2 (8f2)) . Taking adjoints in (4.76) (written with z in place of z) then leads to 1 ( - Lie,n - zlo)-1io = [In ( - Lie,n - zlnr ] * - - 1 -(o) = ['YN ( - tio,n - Zin) ] [Me,n,o (z) ] • * -(o) _- - 1 (4.77) = [iN( ( - A n,n - z lo) o Rn , IR" ) ] Me,n,n (z) , by Lemma 4.13. Replacing this back into (4.50) then readily yields (4.74) . 0 *

_

The £2 (D; dn x)-analog of Theorem 4.15 then reads as follows:

THEOREM 4.16. Assume Hypothesis 3.1 and s·uppose that z E
PROOF. This follows from Theorem 4.15 and the compatibility results estab0 lished in Lemma 4.10.

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS

145

An attractive feature of the Krein-type formula (4.78) lies in the fact that M��b , n (z) encodes spectral information about �e.n . This will be pursued in future work. Assuming Hypothesis 2 . 1 , the special case 8 = 0 then connects the Neumann and Dirichlet resolvents,

- ) -1 R ( - �N,n - zln ) - 1 = ( - �D,n - zln n o -( 1 Rn , IJRn )] * MN,)D , n (z) (4.79 ) + bN ( ( - �D , n - zln ) x [;yN ( ( - b.v,n - z fn ) -1 Rn, IJRn) ] , z E C\( a( -�N.n) U a( -�v.n ) ) , on ( H 1 (0 )) * , and similarly, (4.80) ( -�N.n - zln )-1 ( -�v.n - zln ) - 1 + [;yN ( -�v . n - zln )- 1 ] * M�,b ,o (z) [;yN (-�v.n - z ln )- 1 ] , z E C \(a( -�N,n ) U a( -�v,n )), o

0

o

=

o n L2 (0; dnx). Here M
R that

'

EMARK 4.17. In the case when Hypothesis 2 . 1 1 is enforced, it can be shown

M;J,b ,n (z) E B( H 1 1 2 (80), H31 2 (80)), z E C\(a(-�N.n ) U a(-�v.n)),

(4.81)

and

Note that, by duality, the latter membership also entails

(-�v.n - zln )- 1 E B(( H2 ( 0 ))* , L2 (0; dnx) ) , z E C \a(-�v.n ),

(4.83)

and, given (2.38),

rN ( -�v.n - zln)- 1 E B (L 2 ( 0; dn x), H 1 1 2 (80)), z E C\a( -�v.n) .

(4.84)

Since, in the current scenario, we also have

TN E

B ( H - 1 12 (80) , (H2 (0))*) ,

( 4.85)

it follows that (4.80) takes the form

( -�N,n - zln)- 1 = ( -�v.n - zln )- 1 ( -�v.n - zln) - 1 riVM�,b ,n (z)rN ( -�v.n - zln)- 1 , (4.86) z E C\(a( -�N,n) U a( -�v , n )), o n L2 (0; dnx), where the composition of the various operators involved is well

+

defined by the above discussion. Formula (4.86) should be viewed as a variant of (4.78) in which the Neumann trace operator can be decoupled from the two resolvents of -�v.n in the second term on the right-hand side of (4.78).

146

F. GESZTESY AND M. MITREA

Due to the fundamental importance of Krein-type resolvent formulas (and more generally, Robin-to-Dirichlet maps) in connection with the spectral and inverse spectral theory of ordinary and parti9-Ydifferential operators, abstract versions, connected to boundary value spaces ( oundary triples) and self-adjoint extensions of closed symmetric operators with equ'a.L(possibly infinite) deficiency spaces, have received enormous attention in the literature. In particular, we note that Robin to-Dirichlet maps in the context of ordinary differential operators reduce to the celebrated (possibly, matrix-valued) Weyl-Titchmarsh function, the basic object of spectral analysis in this context. Since it is impossible to cover the literature in this paper, we refer, for instance, to [2, Sect. 84] , [4] , [8 ] , [9] , [12] , [14] , [15] , [20] , [22] , [23] , [41] , [44] , [49, C h. 1 3 ] , [52] , [54]-[61] , [64] , [65] , [71 ] , [78]-[85] , [90], [93] [95] , and the references cited therein. Vle add, however, that the case of infinite deficiency indices in the context of partial differential operators (in our concrete case, related to the deficiency indices of the operator closure of - � rC.J"' (!I) in L 2 (0; dnx ) ) , is much less studied and the results obtained in this section, especially, under the assumption of Lipsch it:;r, (i.e., minimally smooth) domains, to the best of our knowledge, are new. Finally, we emphasize once more that Remark 3.9 also applies to the content of this section (a.<;suming that V is real-valued in connection with Lemmas 4.13 and

�

4.14).

Appendix A. Properties of Sobolev Spaces and Boundary Traces for C 1 •r and Lipschitz Domains

The purpose of this appendix is to recall t:>ome basic facts in connection with Sobolev spaces corresponding to Lipschitz domains n C IRn , n E N, n � 2, and on domains satisfying Hypothesis 2. 11 . In thit:> manuscript we use the fol1owing notation for the standard Sobolev Hilbert spaces (s E IR) , H S (JRn )

=

{

U E S (!R n )'

H s (n ) = {u

E D'(O)

I

I I U I �•(JR" )

u =

=

i_, dn� ttJ (�) I 2 (1 + I�J2s)

Uln for some U

H0 (0 ) = { u E H8 (IRn ) supp (u ) s::; 0}.

I

E

H8 (!Rn) } ,

< 00

},

(A. I)

(A.2) (A.3)

Here D' (D) denotes the usual set of distributions on n <;;; IRn , n open and nonempty, S(IR.n)' is the space of tempered distributions on !Rn, and fi denotes the Fourier transform of U E S(IRn )'. It is then immediate that H81 (D) <--+ H80 (0) for - oo <

so

::; s1 < +oo,

(A.4)

continuously and densely. Next, we recall the definition of a C1·r -domain 0 s::; IRn , 0 open and nonempty, for convenience of the reader: Let N be a space of real-valued functions in !Rn- 1 . One calls a bounded domain n C IR" of class N if there exists a finite open covering {OJ hSjSN of the boundary 80 of 0 with the property that, for every j E { 1 , . . . , N } , Oj nn coincides with the portion of oj lying in the over-graph of a function :Pj E N (considered in a new system of coordinates obtained from the original one via a rigid motion) . Two special cases are going to play a particularly important role in the sequel. First, if N is Lip (!Rn-l ), the space of real-valued functions satisfying a (global) Lipt:>chitz condition in JRn -l, we shall refer to D as being a Lipschitz

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS

1 47

domain; cf. [92, p. 1 89] , where such domains are called "minimally smooth" . Sec ond, corresponding to the case when N is the subspace of Lip (JR.n-1 ) consisting of functions whose first-order derivatives satisfy a (global) Holder condition of order T E (0, 1), we shall say that 0 is of class C1•r . The classical theorem of Rademacher of almost everywhere differentiability of Lipschitz functions ensures that, for any Lipschitz domain 0, the surface measure dn-1w is well-defined on 80 and that there exists an outward pointing normal vector v at almost every point of 80. Call a bounded, open set 0 c JR.n a star-like Lipschitz domain with respect to a point x* ( called center of star-likeness) if 0 is Lipschitz domain and

x * + t(x - x * ) E 0

for every

x E 0 and t E [0.1] .

(A.5)

The above geometrical characterization of Lipschitz domains can be used to show that, given a bounded Lipschitz domain 0 c JR.n then there exists a finite family of open sets 0j , 1 s:; j s:; N, such that N (A.6) 0 Oj , Oj star-like Lipschitz domain, 1 s:; j s:; N. =

U

j=1

0 c JR_n it is known that (H8 (0) ) * H - 8 (0), -1/2 < s < 1/2.

For a Lipschitz domain

(A.7)

=

See [99] for this and other related properties. We also refer to our convention of using the (rather than the dual) space X * of a Banach space X as described near the end of the introduction. Next, assume that 0 c JR.n is the domain lying above the graph of a function r.p : JR.n- 1 -+ JR. of class C1·r. Then for 0 s:; s < 1 + r, the Sobolev space H8 (80) consists of functions f E L2 (80; dn-1w) such that f(x' , r.p(x')) , as a function of x' E JR_n-1 , belongs to H8 (JR.n- 1 ) . This definition is easily adapted to the case when 0 is a domain of class C1·r whose boundary is compact, by using a smooth partition of unity. Finally, for -1 - r < s < 0, we set H8(80) = (H-8(80)) *. The same construction concerning H8 (80) applies in the case when 0 C JR.n is a Lipschitz domain (i.e., r.p : JR_n-1 -+ JR. is only Lipschitz) provided 0 ::::; s ::::; 1. In this scenario we set H8 (80) = (H -8 (80)) * , - 1 ::::; s ::::; 0. (A.8)

adjoint

It is useful to observe that this entails

II J IIH- • (8!1) � 11\/1 j\i' r.p( ) 1 2 J ( · ) r.p ( · )) IIH-•(JRn -1) , 0 S:: S:: 1 . To define H8 (80), 0 s:; s ::::; 1, when 0 is a Lipschitz domain with

+

·

S

(A.9)

compact boundary, we use a smooth partition of unity to reduce matters to the graph case. More precisely, if 0 s:; s s:; 1 then f E H8(80) if and only if the assignment JR.n-1 3 x' t---> ('!j;f) (x', r.p(x' ) ) is in H8 (JR.n-1 ) whenever 'lj; E COO (JR.n) and r.p : JR_n-1 -+ JR. is a Lipschitz function with the property that if I: is an appropriate rotation and translation of { (x', r.p( x' )) E JR.n I x' E JR_n-1 }, then ( supp ( 'lj;) n 80) C I: (this appears to be folklore, but a proof will appear in [72, Proposition 2.4] ) . Then Sobolev spaces with a negative amount of smoothness are defined as in (A.8) above. From the above characterization of H8 (80) it follows that any property of Sobolev spaces (of order s E [ - 1 , 1]) defined in Euclidean domains, which are invariant under multiplication by smooth, compactly supported functions as well as composition by hi-Lipschitz diffeomorphisms, readily extends to the setting of

F. GESZTESY AND M. MJTREA

148

H8 (80.) domain

(via localization and pull-back) . As a concrete example, for each Lipschitz

0. with compact boundary, one has 1 H8 (80.) L2 (80.; dn- lw) coAwactly if 0 < /� --

<-4

� ..

8

S

(A. 10)

1.

For additional background information in this context we refer, for instance, to [10] , [11], [37, Chs. V, VI] , [48, Ch. 1] , [69, Ch. 3] , [105, Sect. 1.4.2]. For a Lipschitz domain 0 c JRn with compact boundary, an equivalent defini tion of the Sobolev space H 1 ( an) is the collection of funct i ons in L2 ( an; dn-1w) with the property that the (pointwise, Euclidean) norm of their tangential gradient belongs to L 2 (an; cfl-1 w ). To make this precise, consider the first-order tangential derivative operators a;arj,k, 1 s j, k s n, acting on a fwtction '1/J of class C1 in a neighborhood of an by 81/J j arj, k

vi (8k1/J) I an - vk (aJ'r/J) Ian · For every J E L1 (an) define the functional aJIaTj , k by setting 3

{

'1/J ......,

r

(A.12) dn-lw f (a'lj;jOTk,j ) la n has of j{)Tj, k E £1 (80.; d"-1w), the following integration

f)j / fhj,k : C1 (1Rn) When j E L1 (an; dn- 1 w) by parts formula holds:

(A.ll)

=

lan �- 1 w f (8'1j;jark,j )

=

{

lan

d"- 1 w (afjaTj,k ) 'r/J,

't/J E C1 (1R" ) .

We then have the Sobolev-type description of H 1 (an): H 1 (a0. )

=

{!

E

£2(80. ; d" -1w) I af/aTj, k E L2 (an ; �-1w ),

j, k

=

(A.13)

1, . . . ' n } (A. 14)

with n

IIJIIH1 (8fl) � IIJIIP(ofl;rfn-1w) +

L IIDJ/Orj, k iiP(&f!;d"-lw) •

j,k =l

(A.15)

( � denoting equivalent norms), or equivalently,

H 1 (an) =

{

I

f E L2 ( 80.; �- 1 w ) there exists a constant c > 0 such that for every

v E C0 (1Rn ) ,

I !an �- 1 wf 8 /8 j,k i v

T

Let us also point out here that if 0. for any j, k E {1, . . . , n} the operator

S c llvll£2(&n ;d"-'w) , j , k C

=

1, . . . , n

}·

(A. 16)

!Rn is a bounded Lipschitz domain then

8/oTj,k : H5 (80.) -> H8-1 (80.), 0 S s S 1 ,

(A.l7) is well-defined, linear and bounded. This is proved by interpolating the case 8 = 1 and its dual version. In fact , the following more general result (extending (A.14)) is true.

A . l . Assume that n c !Rn is a bounded Lipschitz domain. Then for [0, 1], f j,k E H8 - 1 (80.), 1 S j, k S n} (A. J 8) H8 (80.) = {f E £2 (80; dn - 1w) J 8f/ h LEMMA

every 8

E

and

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 149 II

I J H • ( an )

�

ll f ll £2 (8!1;dn-lw)

PROOF.

n a + jL ,k= 1 ll JjaTj,k ii H•-1(8!1) ·

(A.19)

The left-to-right inclusion in (A. 18) along with the right-pointing in equality in (A.19) are consequences of the boundedness of (A. l 7) . As for the opposite directions, we note that using a smooth partition of unity and making a rigid transformation of the space, matters can be localized near a boundary point where coincides with the graph of a Lipschitz function JRn- 1 --> R Then for each sufficiently nice function --> IR the Chain Rule yields 1 1 :S j :S n - l . (A.20 )

an cp : f : an ( 8Taf ) (x, cp(x)) = Y' 8xa [f(x, cp (x)) ] , j,n Jl + I cp(x)l 2 j

On account of this and (A.9) , we then deduce (upon noticing that that

-1 j=1

n

n

L ll aJ / 8Tj, n ii H •-1 (8!1) � L ll aj [J (

j=1

Furthermore, we also have

·

' cp( ' ))J IIH•-l (JRn-1 ) ·

aj8Tn,n

=

0)

(A.21)

(A.22) Next, we recall the general Euclidean lifting result I {! E E Hs -1 (1Rn - 1 ) , 1 :::; j :::;

n - 1 } , s E IR, Hs - 1 (1Rn- 1 ) aj j (A.23) which can be found in [86 , Section 2 . 1 .4] . Now, the right-to-left inclusion in (A.18), as well as the left-pointing inequality in (A. 19), follow based on (A.21 ) , (A.22) and HS (JRn- 1 )

=

the estimate which naturally accompanies (A.23) .

0

LEMMA A.2. Assume Hypothesis 2 . 1 . Then for every s E [0, 1] and j, k E { 1 , . . . , n} (A.24) (ajjaTj,k , g/ 1 -s ( f , 8gjaTk,j h -s 1 s for every f E H8(an) and g E H - (an). PROOF. Since for every s E [0, 1] C00 (!Rn ) l an H8 (an) densely, (A.25) it suffices to prove (A.24) in the case when f u l an and g = vl an for u, v E coo (JRn). In this scenario, we need to establish that { dn- 1 w (8ujaTj,k )v J{an dn - 1 w u(avj8 ,j ) , 1 :S j, k :S n. (A.26) lan =

<--+

=

Tk

=

To this end, we rely on Green's formula (valid for Lipschitz domains) to write

r dn

lan

- 1 (8ujaTj,k )V w

8k - llk aju)v = In [aj(v8ku) - ak (v81 u)] In dn x [ (8j v)(8k u) - (ak v)(aju)]. =

=

r dn- 1 W ( j lan dn x v

U

(A.27)

F. GESZTESY

1 50

-')!"(��MITREA \

One observes that the right-most integrand-above is an antisymrnetric expression in the indices j, k. ConsequenLly, so is the left-most integral in (A.27). This, however, 0 is equivalent to (A.26) . Moving on, we next

{{

(w , f)

E

L2 (!.l; dnx)n

consider x

the following bounded linear map

(H1 (0)) * I div(w) = fi n } __. H- 1 12 (00) = (H1 i 2 (80)f w � v · (w, f)

(A.28)

by setting

H l /2 (8fl) (¢ , v · ( w ,f) ) (Hli2 (8fl) • =

L dnx V'll> (x) w(x) + Hl (fl) (, /) (Hl( fl) )• (A.29) ·

H112(80) and E H1 (!.l) is such that 'Yvll> = ¢. Here the pairing H l (O) ( , f) (H l ( fl) ) • in (A.29) is the natural one between functionals in (H1 (0) )* and elements in H1 (0.) (which, in turn, is compatible with the (bilinear) distribu

whenever ¢

E

tional pairing). It should be remarked that the above definition is independent of the particular extension E H1 (n) of ¢. Going further, one can introduce the ultra weak Neumann trace operator 'iv as follows:

IN '

_

·

{ { (n, f)

E H1 (0)

x

(H1 (0)) * l 6u = flo} -> H-· l /2 (80.) u f-4 ;;;N( u, f) = v (\l u , f), ·

(A.30) with the dot product understood in the sense of (A.28). We emphasize that the ultra weak Neumann trace operator ;yN in (A.30) is a re-normalization of the operator rN introduced in (2.38) relative to the extension of 6u E H-1 (0.) to an element f of the space (H1 (f2)) * = {g E H-1(1Rn ) I supp (g ) � 0} . For the relationship between the weak and ultra weak Neumann trace operators, see (2.62)-(2.64). In addition, one can show that the ultra weak Neumann trace operator (A.30) is onto (indeed, this is a corollary of Theorem 4.5). We note that (A.29) and (A.30) yield the following Green's formula

bv, ;yN (u , f ) h; 2

=

(V' , 'V u )P (fl;d"x)n

+ Hl (fl) (, f) (JJl(rJ) )• ,

(A . 31)

valid for any u E H1 (0.), f E (H1 (f2)) * with .6.u = f i n , and any E H1(0). The pairing on the left-hand side of (A.31) is between functionals in (H112 (&n)f and elements in H112(80), whereas the last pairing on the right-hand side is between functionals in (H1 (!.l))* and elements in H1 (f2). For further use, we also note that the adjoint of (2.7) maps boundedly as follows (A.32)

REMARK A.3. While it is tempting to view rD as an unbounded but densely defined operator on £ 2 ( 0; dnx) whose domain contains the space C0 (f2) , one should note that in this case its adjoint r'D is not densely defined: Indeed (cf. [43, Remark A.4]), dom(rj)) = { 0} and hence 'YD is not a closable linear operator in L2(0; dn x).

Next we recall the following result from [46] (and reproduce its proof for sub sequent use in the proofs of Lemmas A.5 and D.3).

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 151 LEMMA A.4 ( cf. [46] , Lemma A.6) . Suppose JR.n, n 2, is an open Lipschitz domain with a compact, nonempty boundary 80 . Then the Dirichlet trace operator "YD (originally considered as in (2. 7)) satisfies (2.9) . PROOF. Let u E H(3/2)+c (f2), v E COO (JR.n), and Up (E3/2coo (n) H(3/2) +c(f2), £ E N, be a sequence of functions approximating u in H l + " ( f2 ) . It follows from (2.7) and (A.4) that "YDU, "YD ( V' u) L 2 (8D; dn - l w ) . Utilizing (A.13), one computes for all j, k 1, . . . f2

;::::

c

'--'

E

=

, n,

av aue I I e . oo 1an dn-1w v -11 an dn- OTjav,k I I f-+oo 1an dn- w ue -OTj,k OTj,k I I f�� ian dn-1 W V "YD (V'ue) I ( u l b (an;dn-lw) ll vll £2 (an;dn-lw) 1 W "YDU --

1

lim

=

:::; C

=

hm .....

:S: C lbD V' )

Thus, it follows from (A.16) and (A.33) that "YDU E

·

(A.33)

D

H 1 (80 ) .

Next, we prove the following fact:

;::::

!Rn' nis a2,module is a bounded Lipschitz domain. Then for eachLEMMA r E (1/2,A.5.1), Suppose the spacencr (aD over H 1 12 (8D) . More precisely, if ) Mt denotes the operator of multiplication by f, then there exists C C ( D , r) > 0 such that Mt E B(H 112 (8D)) and IIMt ii5 (H 11 2( anJ ) Cll f llcr( an) for every f E Cr (80 ) . (A.34) Asthe aNeumann consequence, i f is actuall y a bounded C1 • r -domain with r E (1/2, 1), then and Dirichlet trace operators "Y , "YD satisfy c

=

:S:

f2

N

"YN

and

E B(H2 (D ), H 1 12 (8D ))

(A.35)

(A . 36)

PROOF.

The first part of the lemma is a direct consequence of general results about pointwise multiplication of functions in Triebel-Lizorkin spaces ( a scale which contains both Holder and Sobolev spaces ) ; see [86, Theorem 2 on p. 177] . Then (A.35) follows from this, (2.7), the fact that "YN = II . 'YD . and II E cr (an) . Next, one observes that for each u E H2 (0) one has "YDU E H1 (8D ) by Lemma A.4. In addition, .<:l 0 UTj,k

("YD u)

=

(IIJ'YD (aku) - llk"YD (aj u)) E H 1 12 (8D) ,

(A.37)

with a naturally accompanying estimate, by (2.7) and the fact that, as observed in the first part of the current proof, multiplication by llj ( for 1 :::; j :::; preserves H 1 12 (8D). Consequently, (A.36) follows from this and (A.38), (A.39) below. D

n)

Our next result should be compared with (A. 14) and Lemma A.l. LEMMA A.6.

and

H312 (8D)

=

f2

E (1/2, 1) JR.n C1 •r { ! E H 1 (80) I o f jaTj, k E H 1 12 ( 80 ) 1 :S: j, k :S:

If

C

is a bounded -domain with r

n}

n II J II H3/2(an) � II ! II H1(an) + L ll af/8Tj, k i1 Hll2(an) · j,k=l

then

(A.38) (A.39)

152

F. GESZTESY AND M. MITREA

PROOF. To justify (A.38) and (A.39) we use a smooth cut-off function to lo calize the problem near a boundary point where an coincides with the graph of a C1•r function rp : �n-l � R In this setting, the desired conclusions follow from (A.20), (A.34) , and (A.23) used with s = 3/2. D

Appendix B. Sesquilinear Forms and Associated Operators In this appendix we describe a few basic facts on sesquilinear forms and linear operators associated with them. Let H. be a complex separable Hilbert space with scalar product ( · , · ht (an tilinear in the first and linear in the second argument), V a reflexive Banach space continuously and densely embedded into H. Then also H. embeds continuously and densely into V*. (B. I) V '----) H. '----) V * . Here the continuous embedding 'H. H.

'----)

V* is accomplished via the identification

3 U !----}

( , u)H E V * ,

(B.2)

•

and we recall the convention in this manuscript (cf. the discussion at the end of the introduction) that if X denotes a Banach space, X* denotes the adjoint space of continuous conjugate linear functionalH on X, also known as the conjugate dual of X. In particular, if the sesquilinear form

v { , · ) v· : V x V* � C denotes the duality pairing between V and V*, then v { u, v)v· = ( u , v ) H , u E V, v E H. '----) V* ,

( B.3)

·

(B.4)

that is, the V, V* pairing v { · )v• is compatible with the scalar product ( · , · ) rt in H. Let T E B(V, V*). Since V is reflexive, (V* )* = V, one has · ,

T : V � V*,

T* :

V

___...

V*

(B.5)

and

v (u, Tv ) v· = v· (T* u, v)( v• )· = v· (T*u, v)v = v (v , T*u)v· . Self-adjointness of T is then defined by T T* , that is, v ( u , Tv)v · = v· (Tu , v ) v

nonnegativity of T is defined by

(B.6)

=

=

v ( v, Tu) v• , u, v E V,

(B.7)

v(u, Tu)v• 2: 0, u E V, and boundedness from below of T by cr E � is defined by

(B .8)

(B.9) v { u , Tu)v· 2 crllu l l�, u E V . (By (B.4) , this is equivalent, to v (u, Tu) v· 2 cr v (u, u) v • , u E V.) Next, let the sesquilinear form a ( , ) : V x V ----> C (antilinear in the first and

linear in the second argument) be V- bounded, that is, there exists a that l a(u, v ) l � ca ll u llv ll v llv , u, v E V. ·

·

ca

> 0 such (B. lO)

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 153 Then A defined by -> V-* ' (B.ll) A:- {v--V f---t AV = a(· ,v), satisfies A E B(V, V * ) and v ( u,Av ) v • = a(u,v), u,v E V. (B.12) Assuming further that a( , · ) is symmetric, that is, (B.13) a(u,v) = a(v,u), u,v E V, and that a is V-coercive, that is, there exists a constant C0 > 0 such that (B.14) a(u, u) � Co ll u ll �, u E V, respectively, then, A: V ----> V * is bounded, self-adjoint, and boundedly invertible. (B.15) Moreover, denoting by A the part of A in H defined by dom( A) = { u E V I Au E H } H, A = A idom(A) : dom(A) ----> H, (B.16) then A is a (possibly unbounded) self-adjoint operator in H satisfying A � Colrt , (B.17) dom (A 1 1 2 ) = V. (B.18) In particular, A - 1 E B(H). (B.19) The facts (B.1)- (B.19) are a consequence of the Lax-Milgram theorem and the second representation theorem for symmetric sesquilinear forms. Details can be found, for instance, in [31, §VI.3, §VII.l] , [37, Ch. IV] , and [63] . Next, consider a symmetric form b( · , ) V V C and assume that b is bounded from below by cb E JR, that is, (B.20) b(u, u) � cbl l u l � , u E V. Introducing the scalar product ( · , · )v ( b) : V x V ----> C (with associated norm denoted by I l v < bJ ), (u,v)v(b) = b(u,v) + (1 - cb)(u,v) H , u,v E V , (B. 2 1) turns V into a pre-Hilbert space (V; ( · , ) v ( b) ), which we denote by V(b) . The form bb isis called closed if V(b) is actually complete, and hence a Hilbert space. The form called closable if it has a closed extension. If b is closed, then l b(u, v) + (1 - cb )(u, v)rt l l u l v(b) l v l v(b) , u, v E V, (B.22) and (B .23) l b(u,u) + (1 - cb) l u l � l = l u l �(b) ' u E V, show that the form b( · , · ) + (1 - cb )( · , · ) H is a symmetric, V-bounded, and V coercive sesquilinear form. Hence, by (B.ll) and (B.12), there exists a linear map { V(b) f---t V(b)*, Bcb .· (B.24) V Bcb v = b( · ,v) + (1 - cb )( · ,v) H , ·

<:;;;;

·

·

·

�

---t

:

x

---->

F. GBSZTESY AND M. MITR.EA

154 with

Bch E B(V(b) , V(b) *) and V (u) (u , Bcb v ) V (b) + = b(u, v) + ( 1 - cb)(u, v)1-t , u, v E V. (B.25)

Introducing the linear map

B = Bcb + (cb - 1 )1: V(b) ---+ V(b)*,

(B.26)

where Y: V(b) <-+ V(b) * denotes the continuous inclusion (embedding) map of V (b) into V(b)*, one obtains a self-adjoint operator B in 1-{ by restricting B to 1-i, dom{B) = { u E

V I Bu E 7-i}

k 'H.,

B=

satisfying the following properties:

B � Cbl')-t,

dom ( I B I 1/2 )

Bjdom( B ) : dom(B)

-+

1-i,

{B.27) (B.28)

=

dom ((B - cbl'H ) 112)

b(u, v) = ( I B I 1 1 2 u , UB IBI 112v) 1t

=

V,

(B.29)

= ( (B - cb l1-t ) 1 12 7J., (B - cblrt)11 2 v) 'H + Cb ( u , v)rt = V (b) (u, Bv )V(b)+ ' u, v E V,

b(u , v) = (u, Bv)rt, u E V , v E dom(B), dom(B) = { v E V I there exists an fv E 'H. such that b(w , v) = (w , fv )'H for all w E V}, Bu = fu : u E dom{B) ,

dom(B) is dense in 'H.

and in

V(b).

(B.30) (B.3 1 ) (B.32) (B.33) (B.34) (B.35)

Properties (B.34) and (B.35) uniquely determine B. Here UR in (B.3 1 ) is the partial isometry in the polar decomposition of B, that is,

B

=

UB IBI,

IBI = (B* B) 1 12 .

The operator B is called the operator associated with the form The norm in the Hilbert space V(b)* is given by

ll f ll vcb) ·

=

sur { lv ( b) ( u , l}vcw l l ll u llvc bl � 1 },

f

E

(B.36)

b.

V(b)*,

(B.37)

E

(B.38)

with �sociated scalar product,

(f1 , f2 ) vc w Since

=

v cbl ( (B + ( 1 - cb) J) - 1 -t1 , f2 )v cw • e1 , e2 II ( B + ( 1 - q, ) l)v ll v(b)• = ll v l l vcb) •

the Riesz representation theorem yields

V(b) * .

v E V,

(B + ( 1 - cb) J) E B(V(b), V(b) * ) and (B + (1 - cb ) J) : V(b) ---+ V(b)*

In addition,

(B.39) is unitary.

(B.40)

1/2 1/2 V(b) ( u , (B + ( 1 - cb)I)v)V(b )• = ( ( B + ( 1 - cb ) Irt) u, (B + ( 1 - cu ) I'H) v) rt = (u , v ) v ( b) • u, v E V(b). (B . 4 1 )

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 155 In particular, (B.42) and hence (B + ( 1 -

cb)Irt) 1 12 E B(V(b) , 'H.) and (B

(1 -

cb)Irt) 1 12 : V(b)

1i is unitary. (B.43) The facts (B.20)-(B.43) comprise the second representation theorem of sesquilinear forms (cf. [37, Sect. IV.2] , [40, Sects. 1 . 2-1 .5] , and [53, Sect. VI.2.6] ) . A special but important case of nonnegative closed forms is obtained as fol lows: Let 'Hj , j = 1 , 2, be complex separable Hilbert spaces, and T : dom(T) -> 1i , dom(T) � 1i 1 , a densely defined operator. Consider the nonnegative form ar : dom(T) x dom(T) -> C defined by +

->

2

(B.44) ar(u,v) (Tu,Tv)rt2, u,v E dom(T). Then the form ar is closed (resp. , closable) if and only if T is. If T is closed, the unique nonnegative self-adjoint operator associated with ar in 1i , whose existence =

1 is guaranteed by the second representation theorem for forms, then equals T*T. In particular, one obtains

ar(u,v) ( !T ! u, !T!v)HP u,v E dom(T) dom(I TI ) . (B.45) In addition, since b(u, ) + ( 1 - cb)(u, v) H ( (B + ( 1 - cb)Irt) 1 12 u, (B + (1 - cb)Irt) 1 12 v) H , u, v E dom(b) dom ( I B I 1/2 ) V, (B.46) and (B + (1 cb )Irt) 1 12 is self-adjoint (and hence closed) in 1i, a symmetric, V =

=

v

=

=

=

-

bounded, and V-coercive form is densely defined in 'H. x 1i and closed (a fact we used in the proof of Theorem 2.6). We refer to [53, Sect. VI.2.4] and [104, Sect. 5.5] for details. Next we recall that if aj are sesquilinear forms defined on dom(aj ) x dom(aj ) , j = 1, 2, bounded from below and closed, then also

(a1 ) { (u, v) + a2 :

(dom(al ) n dom(a2 )) X (dom(al ) n dom(a2 )) v) = a + a2 v ) >----+ ( a 1 + a2 )

(u,

1 (u, v) (u,

-t

C,

(B.47)

is bounded from below and closed (cf. , e.g., [53, Sect. VI. 1 .6] ) . Finally, we also recall the following perturbation theoretic fact: Suppose a is a sesquilinear form defined on V x V, bounded from below and closed, and let b be a symmetric sesquilinear form bounded with respect to a with bound less than one, that is, dom(b) :;2 V x V, and that there exist 0 � a < 1 and f3 ;;::: 0 such that Then

l b(u, u) l � a ! a(u, u) l + fJI I ul � , u E V. (a + b) :

{

V x V -> C, (u, >----+ (a + b) (u,

v)

v) a(u, v) + b(u, v)

(B.48)

(B.49)

=

defines a sesquilinear form that is bounded from below and closed (cf., e.g., [53, Sect. VI. 1 .6] ) . In the special case where a can be chosen arbitrarily small, the form b is called infinitesimally form bounded with respect to a.

1 56

F. GESZTESY

AND M. MITREA

Appendix C. Estimates for the Fundamental Solution of the Helmholtz Equation

The principal aim of this appendix is to recall and prove some estimates for the fundamental solution (i.e., the Green's function) of the Helmholtz equation and its x-derivatives up to the second order. Let En ( z; x) be the fundamental solution of the Helmholtz equation ( - L\. z)'!j!(z; ) = 0 in �n, n E N , n � 2, already introduced in (2. 1 20) , and reproduced for convenience below: (2- n)/2 (1 ) i 21T/xl H(n- 2 );2 (z 1/2 lxl) , n � 2, z E C\{0}, � zi/2 _ . En (z , x) - __!ln(lxl ( C.1) n = 2' z = O' ), 21r

{(

·

)

(n 2�Wn-l l x l2-n ,

n ;?: 3, = 0, Im ( z112) 2: 0, X E �n\{0} , Z

where H�1 ) ( ) denotes the Hankel function of the first kind with index v 2: 0 (cf. [1 , Sect. 9.1]) and Wn - l = 2rrn1 2 jr(n/2) (r( · ) the Gamma function, cf. [1, Sect. (l.l]) represents the area of the unit sphere sn- 1 in �n. As z ----; 0, En (z , x ) , x E lRn\{0} is continuous for n 2: 3, x 2 -n En (z, x) z-->0 = En(O, x) = (n 2)1 (C . 2 ) - Wn-l l l , X E R." \ { 0} , n � 3 , ·

but discontinuous for n = 2

as

-1 1 2 ln(z 112 lxl/ 2) [1 + O (z lxl 2 ) ] + E2 (z, x) z-+0 = 27r 'I/J(1) + O( zlxl ), 27r (C.3) X E R.2 \{0}, n = 2. Here '1/J (w) = r'(w)jr(w) denotes the digamma function (cf. [1, Sect. 6.3]). Thus, we simply define E2 (0; x) = ;-;ln( lxl), x E R.2\{0} as in (C.1). To estimate En we recall that (cf. (1 , Sect. 9.1)) (C.4) Hc(!� 2)/2 ( · ) = Jcn-2)/2( · ) + iY(n - 2)/2 ( · )

with J�.� and Yv the regular and irregular Bessel funcLions, respectively. We start considering small values of lxl and for this purpose recall the following absolutely convergent expansions (cf. (1, Sect. 9 . 1]) : ,

J�.� (( ) =

( ( ) t; 2

" 00

( - 1 ) k (2k + k + 1) ' k!r(v k 4

( E C\( - oo, 0] ,

Lm (() = ( - 1) m Jm ((), ( E C, m E No , Y.l/ (1') __lv ( () cos(znr) - L., (()

v E R.\( -N) ,

]

)\ . , ( E C \( -oo, 0 , v E ( 0, oo N, sm(vtr) m (- m - 1 (m - k - 1) ! (2k 2 4k + ; Jm (( ) ln((/2) k! Ym (( ) = - 2m 7r L k=O (- 1 ) k (2k (m oo , -,

- 2m L['lp ( k + 1) + '!j!(m + k 1)] 4k k!(m + k )! ' tr k=O ( E C\( OJ , m E No. =

- oo ,

(C.5) (C.6) (C.7)

(C.8)

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 157 We note that all functions in (C. 5 ), (C. 7), and (C. 8 ) are analytic in C\ ( -oo, 0] and that Jm ( ) is entire for m E In addition, all functions in (C. 5 )- (C. 8 ) have continuous nontangential limits as ( < 0, with generally different values on either side of the cut ( -oo, 0] due to the presence of the functions (v and ln( ( ). (We chose E lR and subsequently usually � 0 for simplicity only; complex values of are discussed in [1, Ch. 9]. ) Due to the presence of the logarithmic term for even dimensions we next dis tinguish even and odd space dimensions n: (i) n = 2m + 2, m E No, and z E C\{0} fixed: m E2m+2 (z,. x) - !:_4 ( 2z7T1/ix2 l ) - Hm(1 ) (z 1/2 i x i ) = � ( 2z�� ) -m [Jm (z 1 !2 l x l ) + iYm (z 112 l x l ) ] = � ( 2z��l ) -m { O( lx l m) � ln ( z 1/;lx l ) o ( lx l m) (C.9) m 1 1 12 i z 1 x ) ( - ;: -2- (1 - <5m,o ) [( 1)! + (1 - <5m, 1 )(m - 2)! ( -z l4x-l 2 ) + O(lx l 4) ] } (ii) n = 2m + 1, m E N, and z E C\{0} fixed: . _- 4i ( 2z7T1/i x2 l ) ( 1 /2) - mHm- (1/2) (z1 /2 l x l ) E2m+ l (z,x) = � zm/2 7T-m2 1 -m l xl 1 -m h�� 1 (z1/2 1x l ) · (m + k - 1)! z_ . =_ 2z1/2 (2 I x 1 ) -meizl/2 1xl � k!(m - k - 1)! ( - 2z'z 1/21x 1 ) = 1 , (C.lO) { (47Tix l ) - 1 [1 + iz1112 i1x i +m 0 ( l xl 2 )], lxi"=+o [(2m - 1)w2m ] - l x l - 2 [ 1 + O(l x l 2 ) ] , m � 2, with h�1 ) ( · ) defined in [1, Sect. 10.1]. Given these expansions we can now summarize the behavior of En (z; x) and its derivatives up to the second order as l x l 0: LEMMAequation C.l. Fix( z E C\{0}. Then the fundamental solution En (z; ) of the Helmholtz -� - z)'!f; ( z; ) = 0 and its derivatives up to the second order satisfy the following estimates for 0 < l x l < R, with R > 0 fixed: n = 2,3, (C.ll) I En (z; x) - En (O ; x) i � C (l ln(l x l ) l + 1], n = 4, -n C( lx l 4 + 1] , n � 5 , n = 2,3, (C .1 2) n � 4, Z.

·

--+ TJ

v

v

v

_

+

m -

·

(1)

-k

L..t

m

k=O

m

_

--+

·

·

{c,

F.

158

GESZTESY AND M. MITREA

< I DJ· ak En (z · x ) - aJ. ak En (0 :, x) [ '

aj

=

=

+

1], n = 2, n � 3.

(0.13)

+ 1] ,

C(R, n, z ) represent various different constants in (C.ll)-(C. l3) and

a;axj , 1 � j �

Here C

{ CC[Il[lxln(lx2-ni ) l

n.

PROOF. The estimates in ( C . l l ) follow from combining (C. 1), (0.9) , and (0.10) . The estimates in (C. 12) follow from the fact that z E C\ {0} , x E lRn\{0} , 1 � j � n, n ;:?: Oj E, ( z; x ) = - 27rxjEn+ 2 ( z ; x ) , (0.14) which permits one to reduce them essentially to (C. l l ) with n replaced by n + 2. The recursion relation (0.14) is a consequence of the well-known identity (cf. [1 , Sect. 9. 1])

2,

(0. 15)

where C"( · ) denotes any linear combination of Bessel functions of order and v independent coefficients. Iterating (0.14) yields aj ak En (z; x ) = 41f 2 Xj X kEnH(z; x) - 21fOj,kEn+2 (z; x ) , Z E C\{ 0 } , x E lRn \ { 0 } , 1 � j , k :{,: n, n ;:?: 2 . Combining (0. 11) and (0. 16) then yields (0.13).

v

with ( {0.16)

0

Finally, we mention for completeness that for large values of l x l , (C.l) implies the following simple asymptotic behavior (cf. [1, Sect. 9.1]): En (z; x )

=

. (2�I ' ) (1-n)/2 1 /2 1/2

_ z _

jxj ---; oo z

z

ei

(zl/2 1xl- r.((n-1) /4) ) [1

+ O( l x l - 1 ) ] '

(0.17) 1 0. � 12 I z E C\{0} , m (z ) IC \ [O, oo) (and hence Im(z1 12 ) > 0), En (z; :r ) decays

In particular, as long as z E exponentially with respect to :z: ru:; l:cl ---. oo .

Appendix D. Calder6n-Zygmund Theory on Lipschitz Surfaces

This appendix records various useful consequences of the Calder6n-Zygrrmnd theory on Lipschitz surfaces. Our first re�mlt, Lemma D.l below, is modeled upon a more general result in [50]. For the sake of completeness we include the full argument.

LEMMA D.l. Let n c JR n be a Lipschitz domain with compact boundary and let k( · , ) be a real-valued, measurable junction on an X {)[! satisfying 1J) (Ix - y i ) l k( X , Y ) I < (D.l) - - ln-1 ' X, Y E an ' Ix y where '1/J is monotone increasing and satisfies ·

Consider

dt 'lj; ( t ) {1 t }0

(K f ) (x) =

{

len

<

(D . 2 )

00.

dn - 1 w (y ) k( :z: , y) f(y) ,

x

E an .

(D.3)

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS Then for each E ( 1 , oo ) . PROOF. For a fixed, arbitrary E > 0, decompose k(x, y ) where { 0,k(x, y) , lxx - Yl >� E., kc: (x, y ) l - Yl E

1 59

(D.4)

p

=

kc: (x, y ) + kb(x, y) ,

(D.5)

_

Then K = Kc: +Kb, where Kc:, Kb are integral operators on &0. with integral kernels kc: (x, y) and kb(x, y) , respectively. Setting Sj (x) = { y E &0. I Ti - 1 � l x - Y l < Ti } , x E &0., j E N, for each x E &0. we may then compute (with the logarithm taken in base 2)

(D.6)

�(I �Yl-n�� c: 'lj; t) (D.7) � C 2: 'lf;( e-i ) � C 1 dt � . 0 j?_log 1/ c: Of course, there is a similar estimate for fao dn - 1w( x ) l kc: (x, y) l , uniformly for y E &0.. Schur's lemma then yields (D.8) -tt-) --+ 0 as E --+ 0. o dt 'lf;( II Kc: IIB( LP(8r!;dn-lw)) � C r J

1ao

dn - 1 w(y)

l kc:(x, y) l �

c

2:

r

j?_iog 1/c: Jsj (x )

dn - 1 w(y)

Thus, it suffices to show that Kb is compact on each LP (&O.; dn - 1w) space, for E ( 1 , oo), under the hypothesis that kb ( x, y) is bounded. First note that Kb is compact on L2 ( &0., dn - 1w), since it is Hilbert-Schmidt, due to the fact that w(&O.) < oo. The compactness of Kb on LP(&O., dn - lw) for each p E ( 1 , oo) then follows from an interpolation theorem of Krasnoselski (see, e.g., [16, Theorem 2.9, 0 p. 203] ) .

p

We now record a basic result from the theory of singular integral operators of Calder6n-Zygmund-type on Lipschitz domains. To state it, we recall that j denotes the Fourier transform of appropriate functions !Rn --+
0,_ f: =

>0

( M ) (x ) = sup { l u ( y) l l y E r � (x ) } u

(with the choice of sign depending on whether

u defined in n± , we set bn.t.u) (x) =

lim

y�x

yEr;!' (x)

u

0,_)

D

( .9)

_

is defined in 0.+ , or 0. ) and, for

u(y) for a.e.

x E &0..

(D.10)

For future reference, let us record here a useful estimate proved in [36] , valid for any Lipschitz domain 0. c IRn which is either bounded or has an unbounded boundary. In this setting, for any p E oo) and any function u defined in 0.,

(0,

(D.ll)

F. GESZTESY AND M. MITREA

160

THEOREM D.2. There exists a positive integer N = N(n ) with the following significance. Let n c IR71 be a Lipschitz domain with compact boundary, and assume that k E C N (!Rn\{0}) with k(-x) = -k ( x)

and k(.Ax) = .A -(n-l ) k (x) , A > 0, X E !Rn \{ 0 }. Define the singular integral operator

( Tf)(x)

=

{

Jan

d n-1w (y) k(.r- - y) f( y ) ,

x E IR"'\an.

(D.12)

(D.13)

Then for each p E { 1 , oo) there exists a finite constant C = C(p, n, on) > 0 snch that (D.14) IIM(Tf)II LP(8!1;d"-lw) S Cllklsn-t l l cN IIJIILP(8!1;dn-lw) · Furthermore, for· each p E (1, oo ) , f E LP (an; dn - lw), the limit

(Tf)(x)

= p .v.

{

Jan

dn -l w (y) k(x - y)j(y) = lim

exists faT a. e. X E an' and the jump-formula

"�o+

1

lx-yl >" yEan

dn- lw(y) k (x - y)f(y)

"fn.t. (Tf)(x) = !� (Tf) (z ) = ±-1;k (v(x))f(x) + ( Tf ) (x)

z Er;!' (x)

(D . 15) (D.l6)

is valid at a. e. X E on, where v denotes the unit normal pointing outwardly relative to n (recall that 'hat ' denotes the Fourier transform in !Rn). Finally,

liT!II H•I2(n ) See the discussion in [24] , [25] , [73]. LEMMA

and

S

Cllf iiP can;dn-lwJ·

(D. l7)

D.3. Whenever n is a Lipschitz domain with compact boundar·y in IR.71 , (D. l8) Kf' E B (L2 (8n; d71-1w)) , z E C,

(K"ff: - K�)

"YDSz

E

E

B00 (L2(an; dn- l w)) ,

B (L2 (8n; dn- 1 w ) , H 1 (8D) ) ,

z1 , z2 E C,

z E C.

(D.l9) (D.20)

PROOF. We recall the fundamental solution En (z; · ) for the Helmholtz equa = 0 in IR71 introduced in (2. 120). Then the integral kernel of the operator Kf - K/! is given by (D.21) k(x, y) = v(x) . (� En ( z ; X - y) - �En (O ; X - y)) , x, 'Y E an. By (C .l2) we therefore have l k(x, y) l S C lx - y l 2 - n , henec (D.l) holds with '1/J (t ) = t. �otc that (D.2) is satisfied for this choice of 1/J, so (D. 19) is a consequence of Lemma D.l. In addition, (D.18) follows from (D. 19) and Theorem D.2, according to which K! E B (L2 (80.; d"-1w)) . Finally, the reasoning for (D.20) is similar (here (A. l5) is useful) , 0 tion ( -ll - z)�(z; · )

LEMMA

then

D.4, If n is a Cl,r,

r

> 1/2, domain in R" with compact boundary,

(D.22)

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 16 1 PROOF. The integral kernel of the operator Kf - Kt/ is given by (D.21 ) . By Lemma A.5, the operator of multiplication by components of E [Cr (8!1)]n belongs to B(H 1 12(8!1)). Hence, it suffices to show that the boundary integral operators whose integral kernels are of the form (D.23) 8jEn (z ; x - y) - 8jEn ( O; x - y)) , x,y E 8!1, j E {1, . . . , n}, belong to B(L2(8!1;dn- 1 w),H 1 (8!1)). This, however, is a consequence of ( A.18 ) , (A. 19) (with s = 1 ) , (C.13), and Lemma D . 1 (with '!f; (t) = t). LEMMA D.5. Let 0 < a < (n - 1) and 1 < p < q < be related by ! = ! - (a + !) .!. . (D.24) pn q p Then the the operator Ja defined by 1 Ja f( X ) = { dn - 1 Y !Rn , E LP(JRn- 1 .,dn-1 x) , ( ) lx - Yl n - 1 - a J Y ' X E + J }JRn- 1 is0, bounded from LP(!Rn- 1 ; dn- 1 x) to Lq (!Rf-; dnx), that is, for some constant C(D.25) a > q v

0

oo

,

p,

(D.26)

PROOF. A direct proof appears in [75] . An alternative argument is to ob serve that M(Jaf) (x) ::; CJa ( I JI ) (x), uniformly for X 81Rf- , and then to invoke the general estimate (D. l l ) in concert with the classical Hardy-Littlewood-Sobolev 0 fractional integration theorem (cf., e.g. , [92] , Theorem 1 on p. 119).

E

Next, we record a lifting result for Sobolev spaces in Lipschitz domains in [51] .

Let n !Rn be a Lipschitz domain with compact boundary. Then,THEOREM for everyD.6. a > 0, the following equivalence of norms holds: (D.27) Lipschitz domain. Then for every z E C,THEOREM D.7. Let n !Rn 2be a bounded (D.28) Sz E B(L (8!1;dn- 1 w),H312 (!1)), and (D.29) Sz E B(H - 1 (8!1), H 1 12 (!1)). In particular, Sz E B(H8 -1 (8!1), Hs + ( l / 2l(!1)), 0 s 1 . (D.30) PROOF. Given f E L2(8!1;dn - 1 w), write Szf = Sof + (Sz - So)f. From (D.17) and Lemma D.6 we know that II Sof i i H3/2 (fl) :S C II J I I £2 (80;dn- lw) > for some constant C > 0 independent of f. Using (C. 13) and Lemma D.5 (with a = 1) one concludes that (D.31) V'2 (Sz - So) E B(£2 (8!1 ; dn - l w), £2 (!1; � x) ) , and (D.28) follows from this. The proof of (D.29) , is analogous and has as starting point the fact that S f E B(H - 1 (8!1), H112(!1)), itself a consequence of (D. 1 7) and the following description of H - 1 (8!1): H- 1 (8!1) = {g + :s;L (81J,k /8rj,k ) I g, hk E L 2 (8!1; dn - 1 w) }· (D.32) j ,k$ c

c

�

0

1

n

�

1 62

F.

GESZTESY AND M. MITR.EA

Then (D.31 ) ensures that

(Sz - So) E B(H-1 (80), H1 (D)) ,

(D.33)

0 and (D.29) follows. We recall the adjoint double layer on an introduced in (2. 122) and denote by

( Kzg)(x)

=

p.v. r dn- lw(y) Ovy En ( z; y - x )g(y) , X E an ,

lao

(D.34)

its adjoint. It is well-known (cf., e.g., (101]) that (D.35) Hypothesis 2.1 � K E B(L2 (80.; dn - 1w) ) n B (H1 (8n)) and (cf. [38] and (D. 1 9)) that (D. 36) 0. a bounded C1-domain � Kz E Boo (L2(80.; �-1w)), z E C. It follows from (D.35), (D.36), (4.4), and Theorem 4.2 that (D.37) f2 a bounded C1-domain � Kz E Bx (H8(of2)), s E (0, 1) , z E C . We wish to complement this with the following compactness result. THEOREM D.8. If 0. C lRn is a bounded C1·r -domain with r E (1/2, 1) then (D.38) K'ff E Boo (H 1 12 (8n)) , z E 0, f a Lipschitz function with compact K(x, y) lx_!yi" F (

A(fl:::�(y)),

support in lRn , define the truncated operator =

(T.J) (x ) =

J

ix-yj > E

dny K (x, y)f(y),

x E lRn .

(D.39)

y ( Rn

Then, for each 1 0} is bounded on LP(IRn; rflx ) . (ii ) If 1 0

(D.40)

bounded on LP(lRn; dnx) .

A proof of this result can be found in [70] .

THEOREM D.lO. Let A : lRn ----> lRm , B = (B1 , , Be ) : lRn --7 IRe be two Lipschitz functions and let F : lRm x lRe ---) IR be a CN {with N = N(n, m, f) a sufficiently large integer) odd function which satisfies the decay conditions (D.41) IF(a, b) l :::; C ( l + lbl) - n, • • •

IVI F(a, b)l ::; C,

(D.42)

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 163 (D.43) l V'u F(a,b)J ::::; C(1 + J b l )- 1 , uniformly for a in compact subsets of JR.n and arbitrary b E JRR. (Above, and \7 denote the gradients with respect to the first and second sets of variables.) For x, y E !Rn with x =f. y and t > 0 we set B1 (x) - B1 (y) + t . . . ' Be (x) - Be (y) + t ) . Kt ( x, y) = lx -1 y J n F ( A(x)lx -- A(y) Yl lx - Yl Jx - Yl (D.44) In addition, for each t > 0 we introduce \7 I

I1

'

'

(D.45)

and, for some fixed, positive K,

( D.46)

Then, for each 1 < < oo, the following assertions are valid: (1) The nontangential maximal operator T** is bounded on LP(JR.n; dnx). ( 2 ) For each f E LP(!Rn;dnx), the limit ( D.47) (Tf)(x) = lim 0 (Tt f)(z) ----+t exists at almost every x E !Rn and the operatorT is bounded on LP(JR.n; dnx). PROOF. Fix p E ( 1 , oo) . For x, y E !Rn with x =f. y consider the kernel K( x, y) = lx -1 yJ n F ( A(x)l x -- YlA(y) ' B(x)lx -- YlB(y) ) ' (D.48) and let T, T* be the operators canonically associated with this integral kernel as in Theorem D.9. The crux of the matter is establishing the a.e. pointwise estimate p

! x - z l < ,d z�x,

f LP(!Rn; dnx), M x, z !Rn, t M

( D.49 )

uniformly for E is the Hardy-Littlewood maximal operator where in Then the first claim in the statement of the theorem follows from Theorem D.9 and the well-known fact that is bounded on To this end, fix < and let a > 0 be a large E > 0 such that constant, to be specified later. Then

!Rn.

Kt, dnx). J x -z l LP(JR.n;

I }ff{n{ dny Kt(z, y)f(y) - 1 dny K(x, y)f(y) l ::::; 1 dny J Kt (z, y)J i f (y)J + 1 dny J Kt (z, y) - K(x, y)J i f (y)J

(D.50 )

l x - y i > at

lx-y i < a t

= I+ II.

Clearly, it suffices to show that IJI, il

l x - y i > at

(D.51 )

J l ::::; CMJ. To see this, first observe that J Kt (z, y)J ::::; Ccn uniformly for any z, y E !Rn , z =f. y (D.52) (in fact, this also justifies that Tt is well-defined) . Indeed, using the fact that for each j E {1 , ... , £} one has Ct ::::; l Bj (z) - Bj (y) +t l + l z -y l (easily seen by analyzing

164

F. GESZTESY

AND M. MITREA

the c�et> lz - Y l 2: 2uvij llr, oo and lz - y J � 211vii iiL ), we may infer that '"' n J Bj ( z ) - Bj (y) + tl ) - � C ( -t- ) -n + (D.53) /z - y / lz - yl j= l

(1

t

With this at hand, the estimate (D.52) is a direct consequence of (D.41). Returning to I, from (D.52), we deduce that III � CMJ(x) . Thus, we are left with analyzing II in (D.51 ) . To begin with, we shall prove that (D.5 4) IKt(z, y) - K(x, y) l � Ctlx - Y l-n - l for lx - Yl > at. Let Gy(x , t ) = Kt(x , y). Then (D.55) JKt(z, y) - K(x, y) l = IGy(z, t ) - Gy (x, O) I can be estimated using the Mean Value Theorem by (D. 56) Ct( I'VIGy(w, s) l + /'VnGy(w, s)l) , where w = ( 1 - O) z + Ox , s = (1 - B )t for some 0 < f) < 1 . Next, I'V 1Gy (w , s)l C < ( A(w) - A (y) , B 1 (w) - B1 (y) + s , . . . , Bt(w) - Be(Y) + - lw - yJ n+ l /w - Yl Jw - y j / w - yj C 'V 1 F ( A(w) - A(y) ' B1(w) - B1 (y) + s ' ... ' Bt(w) - Be(y) + s ) I + I l + w l w - Yl l w - Yl Yln l w - Yl l A(w) - A(y) B1 (w) - B1 (y) + s . . , Bt (w) - B�.(y) + s ) C II F ( l lw - yj lw - Yi n l w - Y/ ' '. lw - Y l

s)

IF

+

x

'V I C(Jw - Yl

+

t /Bi(w)lw--BYli (2Y) + s J )

I

(D.S?)

i= l Keeping in mind the restriction.ti on the size of the derivatives of the function F stated in (D.41 )-(D.43) , conclude that the above expression is bounded by Clw - Y l - (n+ l) . Similarly, it can be shown that .

we

IV' uG y(w, s)l �

c

. (D.58) lw y J l To continue, one observes that if we choose a > r;, then, in the current context,

lw - xl � lz - xl � r;,t = and l w - xl + lw - yj 2: lx - y/ . Hence,

_

n+

(;) o:t < (;) lx - yj,

( ;) lx - yl,

lw - Yl 2: 1 -

and, therefore,

(D.59)

(D.60)

Next, we split the domain of integration of JJ (appearing in (D.51)) into dyadic annuli of the form o:t � l x - Y l � 2J+ l at, j = 0, 1, 2, .. . . Then (D.61)

2J

1lx-yl>od dny IKt(z, y) - K(x, y)l /f(y) l

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 165 (D.62) :::; C L rJ (MJ)(x) = C(Mf)(x). This yields the desired inequality, that is, I III :::; CMJ(x). The proof of last second claim in the statement of the theorem utilizes a well 00

j =O

known principle (cf. , e.g. , [38]) to the effect that pointwise convergence for a dense class along with the boundedness of the maximal operator associated with the type of convergence in question always entails a.e. convergence for the entire space Thus, it suffices to identify a dense subspace V of such that for any E V the limit in question exists for almost every E Then the boundedness of the maximal operator associated with the type of convergence under discussion ensures that this limit exists for any E at almost every E In our situation, we may take V and observe that

LP(JRn; dnx).

LP(JRn;JRn.dnx) x f LP(JRn; dnx)

f x lRn.

=

z---+ x , t---+ 0

where

(T f)( ) = lim t

lim

l x - z l <�
CJ (JRn)

z

lim

c---+ 0 l x - z l < �
[I + II + III] ,

I = 1 dn y ( y)J(y), II = J dny Kt (z, y)[f(y) - f(x)], III = f(x) J dnyKt (z,y). t

(D.63)

z,

K

lx-v l > l

l > l x -y l > e:

(D.64)

l > l x - v l > e:

Consequently, lim

lim

lim

lim

e:-->0 l x - z l < �t z---+ x , t -+0 E:-->0 l x - z l < �t z-+x , t -+0

whereas

I = 1 �y K(x, y)f(y), II = ! dn yK(x,y)[f (y) - f(x)], lx-yl > l

(D.65) (D.66)

l>lx-yl

III = lim J

dn yK(x,y). (D.67) Now, this last limit is known to exists at a.e. x E JRn (see, e.g., [76]). Once the pointwise definition of the operator has been shown to be meaningful, the boundedness of this operator on LP(JRn; dnx), 1 < p < oo, is implied by that of T** · THEOREM D . l l . There exists a positive integer N = N(n) with the following signi that ficance: Let n ]Rn be a Lipschitz domain with compact boundary, and assume k E CN(JRn \{0}) with k( -x) = -k(x) (D.68) and k(>.x) = >.- (n- l) k(x), >. > 0, x E lRn \{0}. lim

lim

e:-->0 l x - z l < � t z---+ x , t---+ 0

e:-->0

l > l x - y l > e:

T

0

C

166

Fix ry E

Then

F. GESZTESY AND M. l'v1I'T'REA

eN (JR n ) and define the singular integral operator

(Tf)(x)

=

p.v.

l{an dn - 1w (y) (ry(x) - ry (y) )k(x - y)f(y) , T

x E oft.

E

!3(L2 (8fl; d"-1w) , H1 (80)). PROOF. Fix an arbitrary f E L2 (8fl; d'' - 1w) and consider u(x) =

{

lan

d''- 1 w(y ) (ry(x) - ry(y))k( .r, - y)f(y) , x E fl .

(D.69) (D.70) (D.71)

Since Tf = ul an , it suffices to show that (D.72) IJM( V'u) II P(afl;d"-lw) S CJJJIIL2(ofl:dn -lw) > for some finite constant C = C (O ) > 0 (where the nontangential maximal operator M is as in (D.9) ) . With this goal in mind, for a given j E {1, . .. , n}, we decompose (oj u) (x) = 'U t (x) + u2 (x), X E 0, (D. 73) where (D.74) dn - l w (y) k(x - y)j(y) Ut(X) = (Oj1'J)(x) an and (D. 75) dn- 1 w ( y) (ry(x) - ry(y) ) ( ojk)(x - y)f(y). u2 (x) = an Theorem D.2 immediately gives that (D.76) II M ut llu(afl;d"-lw) S CIIJIIL2(ofl;d"-lw)• so it remains to prove a similar estimate with u2 in place of u 1 . To this end, we note t.hat the problem localizes, so we may assume that 11 is compactly supported and 0 is the domain above the graph of a Lipschitz function ;,p : IR.n -t -+ R In this scenario, by pasHing to Euclidean coordinates and denoting g( y' ) = f (y' , ;,p (y' )) , y' E IR."- 1 , it suffices to show the following. l 0, set

lr

l{

v(x' , t) =

kn-l dn- 1 y' (ry(x' ,
and, with

;;;

> 0 fixed, consider v (x' ) ••

(Dj k ) (x ' - y', ;,p(x') - c.p( y' ) + t) g(y' )

= sup {Jv( z', t) l l lx' - z' l

<

;;;t} ,

x' E IR.n-t.

(D.77)

(D.78)

(D. 79) llv • • IIL2(JR:n-l;d"-lx'l s CllgiiP(JR"-l;d"- lx' l · n To establish (D.79) , fix a smooth, even function '1/J defined in IR. , with the property that 1/J = 0 near the origin and 'lf;(x) = 1 for J x l 2: �· We then further decompose (D.80) v(x' , t) = v1 (x', t) + v 2 (x' , t) where Then

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS 167 v 1 (x1, t) = l dn - l Y1 ( 1J (X1,

1

=

f

=

=

As a consequence, it is enough to prove that

llvL IIL2(Rn - 1 ;dn-1x') :::; CllgiiL2(JRn- 1 ;dn-1x' ) >

(D.86)

for j = 1, 3. We shall do so by relying on Theorem D.10 (considered with n replaced by n - 1 ) . When j = 1, we apply this theorem with

a = (a 1 , a2 ) E � x �n - 1 , b E �, (D.87) = A= When j = 3 , Theorem D . lO is used with (again n - 1 in place of n ) and m = n, £ = 2, b = b b E � x � a E � (D.88) = F (a, b) = (( b1 - b2 )b2 (1/J 2 = 0 , A = IJRn - 1 , where ( E C0 (�) is an even function with the property that ( 1 on [-M, M] (where M is the Lipschitz constant of In each case, the hypotheses on F made m = n, £ = 1, F (a, b) =

a 1 (¢8j k)(a2 , b), B
=

168

F. GESZTESY AND M. MITREA

in the statement of Theorem D.10 are verified, and part (1) in Theorem D.10 yields the corresponding version of (D.86) . This finishes the proof of Theorem D . 1 1 . 0 After these preparations, we are finally ready to present the following proof:

D.8 . We work under the assumption that n is a C l ,r_ domain, for some r > 1/2. In particular , v E [C''(8D) ] " . Thanks to Lemma D.4 it suffices to show that (D .38) holds for z = 0. To this end, we write PROOF OF THEOREM

Kt = Ko + (Kt - Ko)

and observe that the integral kernel of the operator R = Kt ( v (x) - v (y)) · V'E71(0 ; x - y) ,

x, y E o!l.

(D.89)

- K0 is given by (D.90)

Let flo. E [C00(lR")]", a E .N, be a sequence of vector-valued functions with the property that (D.91)

and denote by Ro. the integral operator with kernel (TJo. (x) - ry0 (y)) Y'En (O; x - y), ·

x, y E 8fl .

(D.92)

From (2. 129) we know that (D.93)

which implies that for each j E { 1 , ... , n }, the principal-value b oundary integral operator with kernel 81En (O; x - y) maps H112 (8fl) boundedly into itself. From this, (D.90), (D.91), and Lemma A.5 we may then conclude that (D.94)

Also, from Theorem D . l l we have

From (D.94) and (D.95) we may then conclude that (D.96)

hence, ultimately,

Kt by ( D.89) , (D.96) and (D .37) .

E Boo (H1 12 (8fl) ) ,

(D.97)

0

Acknowledgments. We wish to thank Gerd Grubb for questioning an inaccurate claim in an earlier version of the paper and Maxim Zinchenko for helpful discussions on this topic.

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS

1 69

References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. [2] N. I. Akhiezer and I . M. Glazman, Theory of Linear Operators in Hilbert Space, Dover, New York, 1993. [3] S. Albeverio, J. F. Brasche, M. M. Malamud, and H. Neidhardt, Inverse spectral theory for symmetric operators with several gaps: scalar-type Weyl functions, J . Funct. Anal. 228, 144-188 (2005) . [4] S. Albeverio, M. Dudkin, A . Konstantinov, and V. Koshmanenko, On the point spectrum of 'H- 2 -singular perturbations, Math. Nachr. 280, 20-27 (2007) . [5] W. 0. Amrein and D. B. Pearson, M operators: a generalization of Weyl-Titchmarsh theory, J. Comp. Appl. Math. 171, 1-26 (2004). [6] W. Arendt and M. Warma, Dirichlet and Neumann boundary conditions: What is in be tween?, J. Evolution Eq. 3, 1 1 9-135 (2003) . [7] W. Arendt and M. Warma, The Laplacian with Robin boundary conditions on arbitrary domains, Potential Anal. 19, 341-363 (2003). [8] Yu. M . Arlinskii and E. R. Tsekanovskii, Some remarks on singular perturbations of self adjoint operators, Meth. Funct. Anal. Top. 9, No. 4, 287-308 (2003) . [9] Yu. Arlinskii and E. Tsekanovskii, The von Neumann problem for nonnegative symmetric operators, Int. Eq. Operator Theory, 51, 319-356 (2005). [10] G. Auchmuty, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Num. Funct. Anal. Optimization 25, 321-348 (2004) . [11] G. Auchmuty, Spectral characterization of the trace spaces H 8 (80) , SIAM J. Math. Anal. 38, 894-905 (2006). [12] J. Behrndt and M. Langer, Boundary value problems for partial differential operators on bounded domains, J. Funct. Anal. 243, 536-565 (2007). [13] J. Behrndt, M. M. Malamud, and H. Neidhardt, Scattering matrices and Weyl functions, preprint, 2006. [14] S. Belyi, G. Menon, and E. Tsekanovskii, On Krein's formula in the case of non-densely defined symmetric operators, J. Math. Anal. Appl. 264, 598-616 (2001). [15] S. Belyi and E. Tsekanovskii, On Krein's formula in indefinite metric spaces, Lin. Algebra Appl. 389, 305-322 (2004) . [16] C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, Vol. 129, Academic Press, Inc., Boston, MA, 1988. [17] M. Biegert and M. Warma, Removable singularities for a Sobolev space, J. Math. Anal. Appl. 313, 49-63 (2006) . [18] J. F. Brasche, M. M. Malamud, and H. Neidhardt, Weyl functions and singular continuous spectra of self-adjoint extensions, in Stochastic Processes, Physics and Geometry: New Inter plays. II. A Volume in Honor of Sergio Albeverio, F. Gesztesy, H. Holden, J. Jost, S. Paycha, M. Rockner, and S. Scarlatti (eds.) , Canadian Mathematical Society Conference Proceedings, Vol. 29, Amer. Math. Soc., Providence, RI, 2000, pp. 75-84. [19] J. F. Brasche, M. M. Malamud, and H. Neidhardt, Weyl function and spectral properties of self-adjoint extensions, Integral Eqs. Operator Theory 43, 264-289 (2002). [20] B. M . Brown, G. Grubb, and I. G. Wood, M -functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems, preprint, 2008, arXiv:0803.3630. [21] B. M. Brown and M. Marietta, Spectral inclusion and spectral exactness for PDE's on exterior domains, IMA J. Numer. Anal. 24, 21-43 (2004) . [22] B. M. Brown, M. Marietta, S. Naboko, and I. Wood, Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices, J. London Math. Soc. , to appear. [23] J. Briining, V. Geyler, and K. Pankrashkin, Spectra of self-adjoint extensions and applications to solvable Schrodinger operators, preprint, 2007. [24] R.R. Coifman, G. David and Y. Meyer, La solution des conjecture de Calderon, Adv. in Math. 48, 144-148 (1983) . [25] R.R. Coifman, A. Mcintosh and Y. Meyer, L 'integrale d e Cauchy definit u n operateur borne sur L2 pour les courbes lipschitziennes, Ann. of Math. (2) 1 16, 361-387 (1982).

F. GESZTESY AND M. MITREA

170

(1998).

[26]

M. Costabel and M. Dauge, Un resull.ai de densite pour les equations de Maxwell regularisees

[27]

M . Cwikel, Real and complP.x interpolation and extrapolation of compact operators, Duke

dans un domaine lips chitzien, C. R. Acad. Sci. Paris Ser. I Math., 327, 849-854

Math. J . 65,

333-343 (1992).

[28]

B . E . J . Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65,

[29]

E. N . Dancer and D . Dancrs, Domain perturbation for elliptic equations subject to Ro b·in

(1977).

275-288

boundary conditions, J. Diff. Eq. 138, 86-132 (1997). [30] D. Daners, Robin boundary value problems on arbitrary domains, Trans. Amer. Math. Soc. 352, 4207-4236 (2000). [31] R. Dautr ay and J.-L. Lions, Mathematical Analysis and Nu·merical Methods for Science and Technology, Volume 2, FUnctional and Variational Methods, Springer, Berlin, 2000. [32] V. A. Derkach, S. Hassi, M. M. Malamud, and H. S. V. de Snoo, Generalized resolvents of symmetric operators and admissibility, M eth . Funct . Anal. Top. 6, No. 3, 24-55 (2000). [33] V. Derkach , S. Hassi, M. Malamud, and H. de Snoo, Boundary relations and their Weyl famil·ies, Trans. Amer. Math. Soc. 358, 5351-5400 (2006) . [34] V. A. Derkach and M. M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 , 1-95 ( 19 9 1 ) . [35) V. A. Derkach and .M . .M. Malamud, The extension theory of Hermitian operators and the moment problem, J. Math. Sci. 73, 141-242 (1995). [36) M . Dindos and M . :'vlitrea, Semilinear Poisson problems in Sobolev-Besov spaces on Lipschitz domains, Pub!. Mat. 46, 353-403 (2002). (37] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1989. [38] E. Fabes, M. Jodeit and N. Rivi ere , Potential techniques for boundary value pmblems on C1 domains, Acta M ath . , 141 (1978), 165-186. [39) E. Fa.bes, Layer potential methods for boundary value problems on Lipschitz domains, Po tential theory-stU"veys and problems (Prague, 1987), pp. 55-80, Lecture Notes in Math., Vol. 1344, Springer, Berlin, 1988. [40) \V. G. Faris, Self-Adjoint Operators, Lecture Notes in .Mathematics, Vol. 433, Springer, Berlin, 1975. [41] F. Gesztesy, K. A . Makarov, and E. Tsekanovskii, An addendum to Krein's formula, J. Math. Anal. Appl. 222, 594-606 (1998). [42] F. Gesztesy, :'11 . J. Kalton, K. A . Makarov, and E. Tsekanovskii, Some Applications of

[43] [44]

Operator- Valued Herglotz Functions, in Operator Theory, System Theory and Related Topics. The Moshe Livsic Anniversary Volume, D. Alpay and V. Vinnikov (eds. ) , Operator Theory:

Advances and Applications, Vol.

123,

Birkhiiuser, Basel,

200 1 ,

pp.

determinants, and .� ome applications, Russ. J . .Math, Phys. 12,

443-471 (2005).

extension theory point of vie w and Weyl-Titchmarsh functions, in prep aration . Gesztesy and M. Mitrea, Robin-to-Robin maps and Krein-type resolvent formulas for Schrodinger operators on bounded Lipschitz domains, in Modern Analysis and Applications. The Mark Krein Centenary Conference, V. Adamyan, Yu. Berezansky, I. Gohb erg , M. Gor

F. Gesztesy and M. M. Malamud , Boundary value problems for elliptic operators from the

[45] F.

bachuk, V. Gorbachuk, A. Kochubei,

H.

Langer, and G. Popov

Advances and Applications, Birkhauser, to appear.

[46]

271-321.

F . Gesztesy, Y. Latushkin, M. Mitrea, and M. Zinchenko, Nonselfadjoint operators, infinite

(eds. ) ,

Operator Theory:

F. Gesztesy, .M. Mitrea, and M. Zinchenko, Variations on a Theme of Jost and Pais, J.

Funct. Anal. 253,

399-448 (2007) .

B oun dary Value Problems for Operator Differential

[47]

V. I. Gorbachuk

[48] [49)

G . Grubb, Distributions and Operators, Graduate Texts in Mathematics, Springer, in prepa

and .M. L. Gorbachuk , Equations, Kluwer, Dordrecht , 1991.

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston,

1985.

ration.

[50]

S. Hofmann,

M. Mitrea and M. Taylor, Singular integrals and elliptic boundary problems on

regular Semmes-Kenig-Toro domains, preprint,

(51]

2007.

D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J.

Funct. Anal.

130,

161-219 ( 1995) .

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS

171

[52] W. Karwowski and S. Kondej, The Laplace operator, null set perturbations and boundary conditions, in Operator Methods in Ordinary and Partial Differential Equations, S. Albev erio, N. Elander, W. N. Everitt, and P. Kurasov (eds.), Operator Theory: Advances and Applications, Vol. 132, Birkhauser, Basel, 2002, pp. 233-244. [53] T. Kato, Perturbation Theory for Linear Operators, corr. printing of the 2nd ed. , Springer, Berlin, 1980. [54] V. Koshmanenko, Singular operators as a parameter of self-adjoint extensions, in Operator Theory and Related Topics, V. M. Adamyan, I. Gohberg, M. Gorbachuk, V. Gorbachuk, M. A. Kaashoek, H. Langer, and G. Popov (eds. ) , Operator Theory: Advances and Applications, Vol. 1 18, Birkhauser, Basel, 2000, pp. 205-223. [55] M. G. Krein, On Hermitian operators with deficiency indices one, Dokl. Akad. Nauk SSSR 43, 339-342 (1944). (Russian). [56] M. G. Krein, Resolvents of a hermitian operator with defect index (m, m ) , Dokl. Akad. Nauk SSSR 52, 657-660 (1946) . (Russian) . [57] M. G. Krein and I. E. Ovcharenko, Q -functions and sc-resolvents of nondensely defined hermitian contractions, Sib. Math. J. 18, 728-746 ( 1 977) . [58] M. G. Krein and I. E. Ovcarenko, Inverse problems for Q-functions and resolvent matrices of positive hermitian operators, Sov. Math. Dokl. 19, 1131-1134 (1978) . [59] M. G. Krein, S. N. Saakjan, Some new results in the theory of resolvents of hermitian oper ators, Sov. Math. Dokl. 7, 1086-1089 (1966) . [60] P. Kurasov and S. T. Kuroda, Krein's resolvent formula and perturbation theory, J. Operator Theory 51, 321-334 (2004). [61] H. Langer, B. Textorius, On generalized resolvents and Q -functions of symmetric linear relations (subspaces) in Hilbert space, Pacific J. Math. 72, 1 35-165 ( 1977). [62] L. Lanzani and Z. Shen, On the Robin boundary condition for Laplace 's equation in Lipschitz domain, Comm. Partial Diff. Eqs. 29, 91-109 (2004) . [63] J. L. Lions, Espaces d 'interpolation et domaines de puissances fractionnaires d 'operateurs, J. Math. Soc. Japan 14, 233-241 ( 1962) . [64] M. M.Malamud, On a formula of the generalized resolvents of a nondensely defined hermitian operator, Ukrain. Math. J. 44, 1522-1547 ( 1992). [65] M. M . Malamud and V. I. Mogilevskii, Krein type formula for canonical resolvents of dual pairs of linear relations, Methods Funct. Anal. Topology, 8, No. 4, 72-100 (2002) . [66] M. Marietta, Eigenvalue problems on exterior domains and Dirichlet to Neumann maps, J. Comp. Appl. Math. 171, 367-391 (2004) . [67] V. G. Maz'ja, Einbettungssiitze fiir Sobolewsche Riiume, Teubner Texte zur Mathematik, Teil 1 , 1979; Teil 2, 1980, Teubner, Leipzig. [68] V. G. Maz'ja, Sobolev Spaces, Springer, Berlin, 1985. [69] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge Univer sity Press, Cambridge, 2000. [70] Y. Meyer, Ondelettes et operateurs, Hermann, Paris, 1990. [71] A. B. Mikhailova, B. S. Pavlov, and L. V. Prokhorov, Intermediate Hamiltonian via Glaz man's splitting and analytic perturbation for meromorphic matrix-functions, Math. Nachr. 280, 1376-1416 (2007) . [72] I. Mitrea and M. Mitrea, Multiple Layer Potentials for Higher Order Elliptic Boundary Value Problems, preprint, 2008. [73] D. Mitrea, M. Mitrea, and M. Taylor, Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds, Mem. Amer. Math. Soc. 150, No. 713, (2001). [74] M. Mitrea and M. Taylor, Potential theory on Lipschitz domains i n Riemannian manifolds: Sobolev-Besov space results and the Poisson problem, J. Funct. Anal. 176, 1-79 (2000). [75] M. Mitrea and M . Taylor, Sobolev and Besov space estimates for solutions to second order PDE on Lipschitz domains in manifolds with Dini or Holder continuous metric tensors,

Comm. Part. Diff. Eq. 30, 1-37 (2005). [76] M. Mitrea and M . Wright, Layer potentials and boundary value problems for the Stokes system in arbitrary Lipschitz domains, preprint, 2008. [77] S. Nakamura, A remark on the Dirichlet-Neumann decoupling and the integrated density of states, J. Funct. Anal. 179, 136-152 (2001).

F. GESZTESY AND M. MITR.EA

172

[78] G. Nenciu, Applications

of the Krein resolvent formula to the theory of self-adjoint extensions

of pos�tive symmetric operators, J.

Operator Theory 10, 209-2 1 8 (1 983). [79] K. Pankrashkin, Resolvents of self- adjoint extensions with mixed boundary condit�ons, Rep. Math. Phys. 58, 207-221 (2006) . [80] B. Pavlov , The theory of extensions and explicitly-soluble models, Russ . Math. Surv. 42:6, 127-168 (1987). [81] B. Pavlov, S-matrix and Dirichlet-to-Neumann operatv r·s , Ch. 6.1.6 in Scattering: Scattering and Inverse Scattering in Pure and Applied Scien ce, Vol. 2, R. Pike and P. Sabatier (eds.) , Academic Press, San Diego, 2002, pp. 1678-1688. [82] A. Posilicano , A Kr-e in-like formula for singular perturbations of self-adjoint operators and applications, J. Funct. Anal. 183, 109-147 (2001). [83} A. Posili cano , Self-adjoint P.:r.tP. n8ion., by additive p e rturbati o ns, Ann. Scuola Norm. Sup. Pisa Cl. Sci (5) Vol. II, 1-20 (2003). [84] A. Posilicano, Boundary triples and We yl functions for singular perturbations of self-adjoint operators, Meth. Funct. Anal. Topology 10, No. 2, 57-63 (2004). [85] A. Pos ilicano, Self-adjoint extensions of restrictions, Operators and Matrices, to appear; arXiv:rnath-ph/0703078. [86] T. Runst and W. Sickel, Sobolev Spaces of F'mctional Ord er, Nemytskij Operators, and Non linear partial Differential Operators, de Gruyter, Berlin, New York, 1996. [87] V. S. Rychkov, On restrictions and extensions of the Besov and 'Ih ebe l-Lizorkin spac�.• with respect to Lipschitz domains, J. London Math . Soc. (2) 60, 237-257 ( 1 999). [RR] V. Ryzhov, A general boundary value p roblem and its Weyl function, Opuscula Math. 27, 305-331(2007). [89] V. Ryzhov, Weyi-T�tchmarsh function of an abstract boundary value problem, operator col ligations, and linear systems with boundary control, Complex Anal. Oper ator Theory, to appear. [90] Sh. N. Saakj an, On the theory of the resolvents of a symmetric operator with infinite defi ciency indices, Dokl. Akad. Nauk Arm. SSR 44, 193-198 ( 1965). (Russian.) [91] B. Simon, Classical boundary co nditi on.s as a to o l in quantum phystcs, Adv. Math. 30, 268281 (1978). [92] B. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math ematical Series, No. 30, Princeton Universi ty Press, Princeton, N J , 1970. [93] A. V. Straus, Generalized resolvents of .•ymmetri c operators, Dokl. Akad. Nauk SSSR 71, 241-244 (1950). (Russian.) [94] A. V. Straus, On the ge neralized resolvents of a symmetric operator, lzv. Akad. Nauk SSSR Ser. Math. 18, 51 86 (1954). (Russian .) [95] A. V. Straus, Extensions and generalized resolvents of a non-densely defined symmetric operator, Math. USSR lzv. 4, 179-208 (1970). [96] M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Progress in Mathemat ics Vol. 100, Birkhiiuser, B oston , MA, 1 99 1 . [97] M. E . Taylor, Partial Dij]e1-ential Equations, Vol . II, Springer , New Yo rk , 1996. [98] M. E. Taylor, Tools for Partial Differential Equations, American Mathematical Society, Prov idence, Rl, 2000. [99] H. Triebel, Function spaces in Lipschitz domains and on Lipschitz manifol ds. Characteristic functions as pointwise multipliers, Rev. Mat. Complut. 1 5 , 475-524 (2002). [100] E. R. Tsekanovskii and Yu. L. Sh mul ' yan , The theory of bi-exte nsions of operators on rigged Hilbert spaces. Unbounded operator colligations and chn.ractP.ri.stic functions, Russ. Math. Surv. 32:5, 73-131 (1977). [101] G . Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace 's equation in Lipschitz domains, .J. Funct. Anal. 59, 572 ·611 (1984). [102] M. Warma, The Laplacian with general Rob�n boundary conditions, Ph.D. Thesis , University of Ulm, 2002. [ 103] M. Warma, The Robin and Wentzell-Robin Laplacians on Lipschitz domains, Semigroup Forum 73, 10-30 (2006). [104] J . Weidmann, Linear Operators in Hil bert Spaces, Springer, New York, 1980. [105] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987. [106] I. Wood, M=imal U'-rcgvlarity for the Laplac i an on Lipschitz domains, Math. Z. 255, 855-875 (2007).

ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSOURI, COLUMBIA, MO

652 1 1 , USA

E-mail address: fri tz«<math . missouri . edu URL: http : //www . math . missour i . edu/personnel/faculty/gesztesyf . html DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSOURI, COLUMBIA, MO

E-mail address: marius«<math . missouri . edu URL: http : //www . math . missouri . edu/personnel/faculty/mitre am . html

6521 1 , USA

173

Proceedings of Symposia in Pure Mathematics Volume 79, 2008

A local Tb Theorem for square functions Steve Hofmann Dedicated to Prof. V. Maz'ya on the occasion of his 10th birthday. ABSTRACT. We prove a "local" Tb Theorem for square functions, in which we assume only Lq control of the pseudo-accretive system, with q > 1 . We then give an application to variable coefficient layer potentials for divergence form elliptic operators with bounded measurable non-symmetric coefficients.

1 . Introduction, statement of results, history The Tb Theorems of Mcintosh and Meyer [McM] , and of David, Journe and Semmes [DJS] , are boundedness criteria for singular integrals, by which the boundedness of a singular integral operator T may be deduced from sufficiently good behavior of T on some suitable non-degenerate test function A "local Tb theorem" is a variant of the standard Tb theorem, in which control of the action of the operator T on a single, globally defined accretive test function is replaced by local control, on each dyadic cube Q, of the action of T on a test function which satisfies some uniform, scale invariant LP bound along with the non-degeneracy condition

£2

b. b, b , Q

(1.1) for some uniform constant C0. A collection o f such local test functions, ranging over all dyadic cubes Q (or over all cubes or balls) is called a "pseudo-accretive system" . The first local Tb theorem, in which the local test functions are assumed to belong uniformly to is due to M. Christ [Ch] , and was motivated in part by applications to the theory of analytic capacity; an extension of Christ's result to the non-doubling setting is due to Nazarov, Treil and Volberg [NTV] . A more recent version, in which Christ's L00 control of the test functions is relaxed to Lq control, appears in [AHMTT] , and this sharpened version (see also [AY] , and the unpublished manuscript [H] ) has found application to the theory of layer potentials associated to divergence form, variable coefficient elliptic PDE (see [AAAHK]) . It is also of interest to consider local Tb theorems for square functions (as opposed to singular integrals) . These have found application to the solution of the

£00,

2000 Mathematics Subject Classification. Primary 42B25; Secondary 35J25 . 8. Hofmann was supported by the National Science Foundation. @2008 American Mathematical Society

175

STEVE HOFMANN

1 76

Kato problem [HMc] , [HLMc] , [AHLMcT] (tree also [AT] and [SJ for related results), and to variable coefficient layer potentials [AAAHK] . In this note, we consider the square function estimate ( 1.2)

where Btf(x) :=

r

./R "

'Wt( X , y)J(y)dy

and {?ft(x, y)}tE (O,oo) > satisfies, for some exponent a > 0, t<> I 'Wt ( X, y) J � C (t + J x - n+ a yJ) and ( a)

( b)

11/•t (X, Y

I 'W

+ h) - 1/Jt (X , y) J

� C

(1 .3) J h J<>

a J h J<> t (X + h , y) - 1/lt (X , y) J � C + J (t x y l)n + a (t + J x

_

y J )n +

(1 .4)

_

whenever J hl � t/2. Our main result. in this paper is the following:

THEOREM 1 . 1 . Let Ot f(x) : = J 'l/J1 (x, y)f(y)dy, where 'l/J 1(x , y) satisfies (1.3 ) and ( 1 . 4 ) . Suppose also that there exists a constant C0 < oo, an exponent q > I and a system {bq} of functions indexed by dyadic cubes Q � R", S'uch that for each dyadic cube Q (i) JIR" Jbq Jq :<:; Co JQJ (ii) J0 IQI � J� bq l 12 (iii) dx � Co J Q J . IBt bq (x W� Then we have the square function bound (1.2).

[ Q JQ (I�C )

r

Here, and in the sequel, we use the notation f(Q) to denote the side length of a cube Q. The case q = 2 of this theorem was already known, and requires only the first inequality in (1 .4) (smoothness in the y variable). See [H2] and [A] for explicit formulations in that case, although in fact the result and its proof were already implicit in [HMc], [HLMc] and [AHLMcT] . As mentioned above, analogous results for singular integrals (as opposed to square functions) were obtained for q = oo in [Ch] and [NTV] , for q > 1 in the "perfect dyadic" case treated in [AHMTT] , and, bal:ied on [AHMTT], for q = 2 (or q = 2 + E) in the case of st.andard singular integrals in [AY] and [H] . It remains an open problem to treat the case q < 2 for singular integrals that are not of perfect. dyadic type, but rather satisfy standard Calderon-Zygmund conditions. The paper is organized as follows: in the next section we prove Theorem 1 . 1 , and in Section 3 we present an application to the theory of layer potentials for vari able coefficient divergence form operators with bounded measurable non-symmetric coefficients.

Tb

A LOCAL THEOREM

177

2. Proof of Theorem 1 . 1 We begin by recalling the following well known fact, due explicitly t o Christ and Journe but also implicit in the work of Coifman and Meyer

[CJ], [CM]. 2 . 1 . [CJ] Let ()t f(x) fJRn 'l/Jt (x , y)f(y)dy, where 'l/Jt(x, y) satis fies PROPOSITION (1.3) and ( 1.4) (a). Suppose that we have the Carleson measure estimate 1IQT Jro£(Q)Jr IBt l(x) l 2 dxdtt C. (2.1) s p � Q Then we have the square function bound ( 1 .2) . Remark. The converse direction (i.e. that (1.2) implies (2.1)) is essentially due to Fefferman and Stein [ F S). Thus, to prove Theorem 1. 1 , it is enough to establish ( 2 . 1 ) . In fact, by covering an arbitrary cube by finitely many dyadic cubes of comparable side length, it is :=

-

"5:.

enough to establish a version of (2. 1 ) in which the supremum runs over dyadic cubes only. To this end, we shall use the following lemma of "John-Nirenberg" type.

LEMMA 2.2. Suppose that there exist E (0, 1) and 01 < oo, such that for every E .!Rn dyadic Q, withcube Q , there is a family { Qj} of non-overlapping dyadic sub-cubes of ( 2.2 ) and r}Q (1TQ£((x)Q) IBt l(x) l 2 dtt ) q/2 dx C1 1 Q I , ( 2.3 ) where TQ (x) := 2::: 1 Q1(x)€(Qj)· Then (2 . 1 ) holds. TJ

"5:.

1. 1 .

1

We shall defer momentarily the proof of Lemma 2.2, and proceed to the proof of Theorem We may suppose without loss of generality that < q < 2 , as the case > 2 may be reduced to the known case 2 by Holder's inequality. We claim that, in the spirit of and (but using also Lemma 2.2 ) , it is enough to prove that for each dyadic cube there is a family of non-overlapping dyadic sub-cubes of satisfying ( 2.2) for which

q

[S]

q

[AQ,T]

=

{ Qj } Q Q£( ) IBt l(x)l2 _!d ) q/2 dx C 1 (Q l(lt l(x) Atb (x) l 2 _!d ) q/2 dx, (2.4) Q t 1Q (1TQ (x) t Q (1R.0 ) where A t denotes the usual dyadic averaging operator, i.e., At f(x) I Q (x, t) l 1 }rQ x t f, ( ) x with side length at least and Q(x, t) denotes the minimal dyadic cube containing t. Indeed, given (2.4) , we may follow [CM] and write Bt lAt = (Bt l) (At - Pt ) (Bt lPt - Bt ) Bt := R?l R�2 Bt . where Pt is a nice approximate identity, of convolution type, with a smooth, com pactly supported kernel. By hypothesis (iii) of Theorem 1. 1 , the contribution of to the right hand side of ( 2.4) , is controlled by CI Q I , as desired. Moreover, "5:.

:=

+

-:'7 bQ .

-

,

+

+

l

+

178

STEVE HOFMANN

R�2) 1 = 0 , and its kernel satisfies (1.3) and ( 1 .4). Thus, by standard Littlewood Paley/vector-valued Calder6n-Zygmund theory, we have that

fa (1

l(Q)

I R (2) bq (x) j 2

) � t

q/ 2

dx

S Cq llbd � S CJ Q J ,

(2.5)

where in the last inequality we have used hypothesis (i) of Theorem 1 . 1 . Further

more, the same Lq bound holds for RF> (even though (1 .4) fails for this term), as may be o;een by following the interpolation arguments of [DRdeF] . We omit the details. Thus, the right hand side of (2.4) is bounded by C J Q J , so that the conclusion of Theorem 1 . 1 then follows by Lenm1a 2.2 and Proposition 2.1 . Therefore , it is enough to establish (2.4), for a family of dyadic sub-cubes of Q satisfying (2.2) . To this end, we follow the stopping time arguments in [HMc] , [HLMc] and [AHLMcT] (but see also [Ch] , where a similar idea had previously appeared ) . Our starting point is hypothesis (ii) of Theorem 1.1. Dividing by an appropriate complex constant , we may suppose that (2.6) We then sub-divide Q dyadically, to select a family of non-overlapping cubes { Q1 } which are maximal with respect to the property that �e

�

l il

kJ

(2. 7)

bq S 1/2.

By the maximality of the cubes in the family { Q1 } , it follow::> that 1 2 S �e Atbq (x), tf t > Tq (x), 0

so that (2.4) holds with C = 2q. It remains only to verify that there exists "' > 0 such that (2.8) l E I > ryJ Q I , where E = Q \(U Qj ) · By (2.6) we have that IQI

=

r bq = �e r

)q

s I E I 1/q'

)q

(h

bq

J bq jq

)

=

l/q

�e +

r bq + �e L r j

JFJ

)qj

bq

� L IQj j ,

when in the last step we have used (2.7). From hypothesis (i) of Theorem 1.1, we then obtain that ' IQI s CJEI 11q IQI 11q + I Q I ,

�

and (2.8) now follows rea(tily. This concludes the proof of Theorem 1 . 1 , modulo Lemma 2.2, whose proof we now give. PROOF OF LEMMA 2.2.

We

begin by stating

LEMMA 2.3. Suppose that there exist N < oo and {J E (0, 1) such that for every dyadic cube Q, (2 .9) i {x E Q : gq (x) > N } l S ( 1 - {J) I QI ,

Tb

1 79

A LOCAL THEOREM where

(1R(Q) IBt1(xW �t ) 1/2

gq(x) : =

Then (2.1) holds.

We take this lemma for granted momentarily, and prove Lemma 2.2. Fix a dyadic cube Q. For a large, but fixed to be chosen momentarily, let

flN

N

{x E Q : gq(x) > N}.

:=

Under the hypotheses of Lemma 2.2, with E := Q \ (uQ1), we have < N}l ) 1/ (1 - ry) I Q I + l {x E Q : 1 I Bt1(xW dtt 2 > N} l ( 1 - ry) I QI + q I QI , N

L

( R.(Q)

rq (x)

<

c1

where in the last step we have used Tchebychev's inequality and (2.3) . Choosing N so large that CI/Nq ::; ry/2, we obtain (2.9) with f3 = ry/2. Thus, Lemma 2.3 implies Lemma 2.2. In turn, to prove Lemma 2.3, we proceed as follows. We momentarily fix f E (0, and let f3 be as in the hypotheses of Lemma 2.3. For a dyadic cube Q , set

1),

N,

gq ,, (x)

:=

(jmin(£(Q),1/•) IBt1(x)l2 tdt ) 1/2 ' '

where we take this term to be 0 if £( Q) ::;

E.

Define

K(�:) : = s�p l�l � g� ·"

where the supremum runs over all dyadic cubes Q. By the truncation, for each fixed E, and our goal is to show that

0<•<1 K(�:) sup

K(�:) is finite

< oo.

Now fix a dyadic cube Q, and set

nN,E

{x E Q : gq ,, (x) > N}. By the truncation, and (1. 4 ) (b), gq , , is continuous, so we may make a Whitney decomposition Then, if FN, E

:=

:=

Q \ nN, f l we have

(2. 1 0)

1 80

STEVE HOFMANN

But where in the last step we have used (2.9} and the fact that 9Q ,e ::; so that nN, E � O N . We now claim that

(2.1 1 } gq

for every Q,

(2. 12} Assuming momentarily that the claim holds, we deduce from (2. 12} that

(2.10}, (2.11) and

� k 9b,e :S: C + K(E} ( 1 - /3) .

� �

Taking the supremum over all dyadic cubes Q, and using that K (E) is finite for each fixed E, we obtain that c K(E) ::; , /3 uniformly in E. We may then let E --. 0 to obtain the conclusion of Lemma 2.3. It therefore remains only to prove (2.12}. Sin ce Qk is a ¥/hitney cube, there is a point Xk E FN ,e such that

(2. 1 3} The

left hand side of (2.12) is then bounded by an absolute constant times

1 1Cnl(Qk) qk

t( Q k)

1 1e( Q ) dt dt 2 (x dx IOtl(x) - Bt l (xk) l2 -dx + l llM ) t t Qk c,e(Qk) dt 1 /min(e( Q },l/e) + IBt 1 (xkW -dx =: I + II + III, Qk

E

t

where Cn is a purely dimensional constant that will be chosen momentarily. By ( 1 . 3 ) , II Btl lloo :S: C, so that Since

x,. E FN,. , we have that III :S: N2 1Qk l ·

Finally, by

for all

(1.4) (b) and (2.13), for Cn chosen large enough we have that

x E Q,.. Hence,

This concludes the proof.

0

A LOCAL Tb THEOREM

181

3. Application to variable coefficient layer potentials We consider here the matter of L 2 boundedness of layer potentials associated to divergence form elliptic equations L u = 0 , in the domains JR.±+ l , where

L = - div A

\7

=

n+l a

- I: ax . "

(Ai,j a ) ax .

i ,j 1 JR.}, � 1 (we use the notational convention =

J

n(n+ (n +

is defined in JR.n+l = { (x , t) E JR.n x that Xn+ l t) , and where A = A(x) is an 1) x 1) matrix of real-valued L00 coefficients, which we allow to be non-symmetric, defined on JR.n (i.e., independent of the t variable) and satisfying the uniform ellipticity condition =

n+ 1 (3.1) I: Aij (x) �j �i , II A IILoo(JRn) :S A , i ,j = 1 for some ,\ > 0 , A < oo, and for all � E JR.n+I , x E JR.n. The divergence form equation is interpreted in the weak sense, i.e. , we say that L u 0 in a domain n if u E W1�·; (n) and A'Vu · 'VI¥ = 0 :S

.AI�I 2

(A(x) � , �)

=

=

J

for all W E CQ"(D) . Although the case of real symmetric "radially independent" (i.e. , in our context, t-independent) coefficients is now rather well understood (see [JK] , [KP] , [K] and also [AAAHK]) , in general it remains an open problem to establish solvability results, with data, for boundary value problems associated to equations in JR.±+ 1 . However, in the case = 1 , i.e. , in the domains JR.1, solvability of the Dirichlet problem ( ) has been established in [KKPT] for p sufficiently large (but finite) , while solvability of the Neumann ( and Regularity ( "Rp" ) problems with p near 1 (in fact, dual to the Dirichlet exponent) was obtained in [KR] . We refer to those papers for detailed statements of the boundary value problems (this is not our main emphasis here), but we note bound, p- 1 q- 1 = 1 , that solvability of is equivalent to a scale invariant for the Poisson kernel. To b e precise, fix a boundary cube Q C JR.n, and let A� denote the upper and lower "corkscrew points" associated to Q , i.e., if xq denotes the center of Q , then A � := (xq , ±C(Q)) E JR.±+ l ·

LP

non-symmetric "Np")

LP

"Dp"

Dp

X

n

LP

Lq

+

For a point E JR.±+ 1 (we shall adopt the notational convention that capital letters := ( x, t) , Y : = may be used to denote points in JR.n+ 1 ) , let denote the Poisson kernel for L with pole at in JR.±+l . It turns out that is solvable for L in the domain JR.±+ 1 if and only if there is a constant B such that the following scale invariant bound holds for every cube Q c JR.n:

X

(y, s)

kf,±

X

Ln (k�.� r

Dp

:::; B

I Q i l- q ,

(3.2)

with p - 1 + q- 1 = 1 (see, e.g. , [KKPT] or [K] ) . It is shown in [KKPT] that for every L as above in JR.1 , there is a q := q(L) > 1 such that (3.2) holds. The proof in [KR] of the solvability of and in JR.1 , with p near 1 , uses in a crucial way the L 2 boundedness (but not invertibility) of the layer potentials associated to L. In this paper, we present an alternative (and rather short) proof

NP Rp,

STF:VR HOFMANN

182

of this boundedness, based on the local Tb Theorem proved in Section 1 . The proof in [KR] also uses Th theory (to be precise, Lhe result of [DJS] ) , but is tied very closely to the 2-dimensionality of the domain. Our proof is in principle not dimension dependent, but rather relies only on the Poisson kernel estimate (3.2). Of course, at present, (3.2) is known to hold for non-symmetric operators only when n + 1 = 2. We conjecture that (3.2) remains true for non-symmetric t-independent operators in all dimensions. The idea to use estimates like (3.2) to prove layer potential bounds (in the set.ting of symmetric coefficients) has appeared previously in [AAAHK], but the argument there was limited to the case q = 2, as it depended on the local Tb theorem for singular integrals [AHMTT] , as extended to st.andard Caldcr6n-Zygmund operators in [H] or [AY] . In the present paper, we will use the square function/non-tangential maximal function estimates of [DJK] to reduce matters to square functions treatable via Theorem 1 . 1 . We now recall t.he method of layer potentials . For L as above, let L * : = - div A*\7 denote the transpose operator (which is also the adjoint, since we are dealing with real coefficients here) , and let r(X, Y) and f * (X , Y) : = r(Y, X) denote the corresponding fundamental solutions in JRn +I . Thus,

L x r(X, Y) = 8y , L y f* ( Y, X) := Lv f( X, Y)

<5x ,

where <5x denotes the Dirac mass at the point X. By the t-independence of our coefficients, we have that =

r(x , t , y , s ) = r(x, t - s , y , 0) .

( 3.3)

{ f (x, t , y, 0) f(y) dy, t E IR s; f (.'E) = { r • (x, t , y, 0) f (y) dy , t E IR, ./[Rn

(3 .4 )

sup I I V'x,t StfiiP(JRn) + sup II Y' x,t s; f i i £2 (JRn ) ::; C IIJIIL (JRn) 2 t

(3 . 5)

vVe define the single layer potential operators for L and L * by

St f (x ) :=

}[R n

(we apologize for this notation: s; denotes the single layer potential for L"' , and is not, in general, equal to the adjoint of St) and our goal is to show that (the latter estimate implies £2 bounds for the corresponding double layer potentials via duality) . To be precise, we have the following t

THEOREM 3 . 1 . Suppose that L

'iS

that there are exponents q(L ), q(L* ) A±

an operator of the type described above and

> 1 and a constant B such that kL ..� and A±

kL�± (the Poisson kernels for L and L*, respectively) satisfy (3.2) for every cube Q C JR". Then the layer potential bound (3.5) holds, with a constant depending only on dimension, A., A, B and min ( q ( L ) , q( L * ) ) . Remarks. In particular, since (3.2) always holds for such operators when n = 1 [KKPT] , we recover the boundedness result of [KR] . We also observe that our proof will require that (3.2) hold for both L and L •, even if we restrict our attention to the bound for St .

PROOF. \Ve begin with some preliminary reductions. We treat only St in the case t > 0, as the same argument carries over mutatis mutandi to the case t < 0

1 83

A LOCAL THEOREM Tb

and to s; . By Lemma 5.2 of [AAAHK] , it suffices to prove (3.6) and (3.7) Moreover, the same lemma shows that (3.7) follows from

!1

dx dt It (8t ) 2 St f( x) i 2 1 t 1 :::; C I I ! I I 2L2(JRn ) · IR±+I -

(3.8)

;�

In addition, from (3.2) for k , , along with the results of [DJK] applied to the solution u(x , t ) := 8t St f( x ) , we have that (3.7) implies (3.6) (we use here that the arguments of [DJK] carry over to the non-symmetric case - see the comments in the introduction to [KKPT]) . Thus, it is enough to prove (3.8). To this end, we first note that by [GW] (if n + 1 ? 3), or (if n + 1 = 2) as a consequence of the Gaussian bounds and local Holder continuity of the kernel of the heat semigroup e - rL (see, for example [AMT]) , we have that

'1/Jt (X, y ) : = t (8t ) 2 f( x , t , y , 0),

t

the kernel of Bt := (8t ) 2 St , satisfies (1 .3) and (1.4) . Thus, it is enough to construct a pseudo-accretive system { bQ } satisfying the hypotheses of Theorem 1 . 1 . We now set

bQ

=

I Q I kLA�- - ·

Observe that condition (i) of Theorem 1 . 1 follows immediately from (3.2). Moreover (ii) is an immediate consequence of the following well known estimate of Caffarelli, Fabes, Mortola and Salsa [CFMS] , extended to the case of non-symmetric coeffi cients (as may be done: see the comments in [KKPT] concerning the validity of the results of [CFMS] in the non-symmetric setting) :

A-

JrQ kL � - ( y) dy ? c · 1

It remains to establish condition (iii) of Theorem 1 . 1 . Let ( x , t) E R� = Q (0, f ( Q )) . Then, since for fixed (x, t ) E JR.�+l , we have that 8lf(x, t, ·, ·) is a solution of L*u = 0 in lR.'.:_+ l , we obtain x

IBtbQ ( x ) l

=

IQI tl

j (8t )2 r (x , t , y , o) k;�_ (y) dy i =

I Q I t l (8t ) 2 r(x, t , Aq ) l =

�

�

I Q I + ( 1 '1/Jt +t (Q) (x, x Q ) I :::; c e( ) " t Q)

where in the last two steps we have used (3.3) and then ( 1 .3). Hypothesis (iii) now follows readily. This concludes the proof of Theorem 3 . 1 . 0

184

STEVE HOFMANN

[AAAHK] M. Alfonseca, P. Auscher, A. Axelsson, S. Hofmann and S. Kim, A nalyti ci ty of layer potentials and L2 solvability of boundary value pmblems for divergence form elliptic equations

References

with complex £= coefficients, preprint .

[A] P. Auscher, Lectures on the Kato square root problem, Surveys in analysis and operator theory (Canberra, 200 1 ) , Proc. Centre Math. Appl. Austral. Kat. Univ. 40, Austral. Kat. Univ., Canberra, 2002, pp. 1-18. [AHL McT] P. A usch er, S. Hofmann , M. Lacey, A. Mci ntosh, and P. Tch amitchi an , The solution of the Kato Square Root Proble m for Second Order Elliptic op erat or.9 on lRn,

156

Annals of Math.,

[AHMTT] P. Auscher, S. H ofma nn , C. M uscalu , T. Tao, C. Thiele, Carleson measures, trees , extrapolation, and T(b) theorems, Pub!. M at . , 46 (2002), no. 2, 257-325. [AMT] P. Auscher, A. Mcintosh and P. Tchamitchian, Heat kernels of second o rder complex el liptic operators and applications, J. Fu ncti onal A nalysis , 152 ( 1 998), 22- 73 . [AT] P. Auscher and Ph. Tcharnitchian, Square root problem for divergence operators and relat ed topics, Asterisque Vol. 219 (1998), Societe MatMmat ique de France . [AY] P. A us cher and Q. X. Yang, On local T(b) Theorems, preprint . [CFMS] L. Caffarelli, E. Fabe s , S. Mortola and S. Salsa, Boundary behavior of nonnegative so lutions of elliptic operators in divergence form, Indiana Univ. Math. J . , 30 ( 198 1 ) , no. 4, (2002) , 633-654.

C hrist , A T{b) th eo rem with remarks on analytic capacity and the Cauchy integra� Collo quium Mathematicum, LX/LXI ( 1 990) 601-628. (CJ] M. Christ and J.-1. Journe , Poly nomi al growth estimates for multilinear s ingular int egral operator·s, A cta M ath . , 159 ( 1 987), no. 1-2, 5 1-80. [CM] R. Coifman and Y. Meyer, Non-linear harmoni c analysis and PDE, E. M. Stein, editor , Beijing Lectures in H armonic A nalys is, vol. 1 1 2, Annals of M at h. Studies, Princeton Univ. 621-640.

(C h] M.

[DJK] B. Dah lb erg, D. Jerison and C. Kenig, Area int egral estimates fur elliptic differential operators with no m moo th coeffir.i en t,s, Ark. Mat., 22 { 1 984) , no. 1, 97-108. [DJS] G. David, J.-L. Journe, and S. Sem mes , Operateurs de Calder6n -Zygmund, fonr.tions para-accretives et interpo lat ion, Rev. M at . 1beroamericana, 1 1-56, 1985. [DR.deF] J. Duoandikoetx.ea and J. L. Rubio de Francia, Maximal and singular integral operator·s via Fourier transform estimates, I nvent. Mat h . , 84 (1986), 541-561. [FS] C . Fefferman, and E. M. Stein, HP spa ces of several v ari ables, Ac ta M at h . , 129 ( 1 9 72) , no. Press, 1986.

3-4, 1 37-193 [GW] M. Griiter and K. 0. Widman , The Green functio n for uniformly elliptic equations, M anuscripta Math . , 37 (1982), 303-342. (H] S. Hofmann, A proof of the local Tb Theorem for stan dard Calder6n-Zygrnund operators, un p u blish ed manuscript, http :/ fwww.rnath.missouri.edu/ - hofmann/ (H2] S. Hofmann, Local Tb Theorems and applications in PDE, Proceedings of the ICM Madrid, Vol. II, pp. 1375-1392, European Math.

(H LMc]

S.

Hofmann,

form elliptic

Soc., 2006. M. Lacey and A. Mcintosh, The solution of the Kato problem for d·ivergence operators with Gaussian heat kernel bo un ds, Annals of Math . , 156 {2002), 623-

(HMc] S. Hofmann and A. M cint os h , The solution of the Kato problem in two dimensions, Pro ceedings of the Conference on Harmon i c Analysis and PDE held in El EscoriaL Spain in July 2000, PubL Mat., VoL extr a, 2002, pp. 143- 160. [ JK ] D. Jer iso n and C. K enig , The Dirichlet pr·oblern in rwnsmooth domains, Ann. of Math. (2) , 113 ( 1 981), no. 2, 367-382. [K] C. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, 83. Published for the Conference Board of the Mathematical Sciences, Washington, DC, A merican Mathematical Society, Providence, R.I, 1994 [KKPT) C. Kenig, H. Koch, H. J. Pipher and T. Toro, A new approach to absolute continuity of elliptic measure, with applica tions to non-:;ymmetric equations, Adv. Math., 153 (2000), no. 2, 231-298. 631.

1 85

A LOCAL Tb THEOREM

ficients, Invent . Mat h. , 113 (1993) , no. 3, 447-509.

[KP] C. Kenig and J. Pipher, The Neumann problem for elliptic equations with nonsmooth coef [KR] C. Kenig and

D.

Rule, The regularity and Neumann problems for non-symmetric elliptic

operators, preprint.

[McM] A. Mc intosh and

Y.

Meyer, Algebres

d'operateurs

C. R. Acad. Sci. Paris, 301 Serie 1 395-397, 1985.

[NTV]

F.

113 ( 2002 ) ,

par des

integmlf:'s singulier·es,

Nazarov, S . Treil and A. Volberg, Accretive system Tb-thcorcms on nonhomogeneous

spaces, Duke Math. J . ,

[S]

definis

no.

S. Semmes, Square function estimates

(1990) , no.

3, 721 726.

address:

and

the T(b) Theorem, Proc. Amer. Math. S oc ., 110

2, 259-312.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSOURI, COLUMBIA, MISSOURI 652 1 1 , E-mail

hofmannillm at h . missouri . edu

USA

Proceedings of Syntpu�ia in Pure :Ma.therna.tics Volume 79, 2008

Partial differential equations, trigonometric series, and the concept of function around 1 800: a brief story about Lagrange and Fourier Jean-Pierre Kahane

Dedicated to Vladimir Maz'ya and Tatyana Shaposhnikova.

ABSTRACT. Functions of real variables, PDE's and trigonometric series have strong relations. A brief history of these relations as they appeared around 1750 and developed around 1800 is given in the first part of the article. The controversy on vibrating strings, involving d'Alemhert , Euler, Daniel Bernoulli and Lagrange, is well know, and also the birth of Fourier series with the an al ytic theory of heat. Among the many references quoted in the article the main source is the thesis of Riemann on trigonometric series. Riemann showed how difficult is was to acce pt the idea that an arbitrary fWlction could be expressed by a trigonometric series, and he mentioned the strong opposition of Lagrange to the statements of Fourier. A sentence in Riemann's thesis is the source of the second part of the paper, where the author describes his search of two handwritten pages in the collection of manuscripts of Lagrange, supposed to express this opposition, and tries to explain what he found.

Vladimir Maz'ya and Tatyana Shaposhnikova made a significant contribution to the history of mathematics with their authoritative biography "Jacques Hadamard : A Universal Mathematician" [11] . Thus it seems appropriate to include the history of mathematics as one of the themes for the present volume. I will be concerned with the period 1750-1850, and will focus on how the notion of "function" was influenced by the study of partial differential equations (PDEs) and trigonometric series. There are two parts in this article. The first relies on previous studies, it is just a way to look at a well known story. The second relies on the first, it contains a new material, and it is a tentative answer to a series of puzzling questions about a historical document. The study of the relations among PDEs, trigonometric series, and the possible notions of function remains active in modern Limes. A classic example is the theory of distributions developed by Laurent Schwartz in the middle last century. This 2000

Ma.thematir.8 Subjer.t Classification.

Primary 01A50; Secondary 01A55. Lagrange, Fourier ,

Key words and phmses. d'Alembert, Euler, Daniel Bernoulli,

187

Riemann

.

188

JEAN-PIERRE KAHANE

theory extented the notions of functions, derivatives, and Fourier transform [13] and it was first applied to the study of PDEs. Todays, classes of functions and their generalizations play an essential role in research on PDEs, while trigonometric series and Fourier transforms enter as important tools. This story is far from over , and I believe that a look at some of the beginning contributes to our appreciation of current research. 1. Part I I begin with a well known episode, the controversy around vibrating strings af ter 1747. It deals with PDEs as well as trigonometric series [16] , and both subjects introduced important ideas and discussions about functions. The principal charac ters of the story are d' Alembert (1717-1783) , Euler ( 1 707-178:�), Daniel Bernoulli (1700-1782) , and Lagrange (1736-1813). The story was told in many ways, first by the actors themselves, then by many other authors. A volume of the collected works of Euler contains the deo;cription of the debate by C. Truesdell [14] , and this is the most current source of information. The earlier, thorough study by H. Burkhardt on series and PDE (1804 pages) was essentially related to the math ematico; in question, and a large part was devoted to the period we consider [2] . A short and illuminating article of A.P. Youschkevitch (10 pages) appeared in 1975, with the principal niferences on the subject of vibrating strings and the use of "dis continuous" functions [15] . A French thesis has been defended recently in Lyons by Guillaume Jouve ; it contains much new material, comments, translations, un published papers of d' Alembert, together with the relevant part of d'Alembert's Opuscules rnathematiques [8] . The languages used by these authors are English, Germm, Russian, and French. I shall just sketch the story, and I recommend Jouve as a source of further information. The second episode is related to the heat equation. It is a fascinating story, involving mainly Fourier and (again) Lagrange with the participation of many of their contemporaries. I will just sketch the story, but I wish to highlight the appear ance of trigonometric series as a tool, as a mathematical object, and as a source of ideas and problems. The main reference here, apart from Fourier's book "Theorie analytique de la chaleur" [5] , is the historical part of Riemann's dissertation on trigonometric series [1 2] . In fact, this historical part is the best exposition of whole subject, from vibrating strings to Fourier and Dirichlet, that I know. Riemmn's dissertation contains many ideas that have been significant for the development of mathematics in general. His appreciation of the people and their work is well in formed and accurate. I will devote a section to comments on this dissertation, and this will lead to a comment on Dirichlet's ideas. A statement by Riemann serves as motivations for Part II. 1 . 1 . As I already said, the first part of the story, the controversy about vibrat ing strings, is well known. It is kind of dramatic play. The first act begins with d' Alembert in 17 4 7 and Euler in 1748. Both considered a string fixed at two points, say 0 and e on the x--axis, and ordinates y(t, x) above x at time t. Both established the PDE (written here in modern notation) (1)

[J2 y [J2 - w2 y2 2 8t - 8x ·

PARTIAL DIFFERENTIAL EQUATIONS

189

Both found a solution (2)

F(wt + x) - F(wt - x) ,

(3)

F(:r) - F( -:z:) = p(x) , w(F'(x) - F'( -x) ) v(x) .

where F is a 2 e-periodic function. F is well defined by the initial conditions (po sition and velot:ity at time 0) :

=

For d 'Alembert the functions of the form (2) were a particular class of functions, and the functions defined in (3) were particular as well. Therefore (2) provided a solution for a special class of initial data. For other initial data, he said that. t.here could be other solutions, and when the data are not regular enough he said that it rnight become a question of physics, not of mathematics. For Euler, on the contrary, the solution was general for any kind of initial data. Euler's motivation is clear : for any initial data one can compute F, therefore it has to be the solution of the problem. (At first , Euler assumed that the initial velocity vanishes, and in this case (3) means that F is an odd function such that '2F(x) = p(x) when 0 < x < £) . No "continuity" is needed for the initial data ; in particular, broken lines could be allowed. Here is a quotation by d'Alembert, discussing Euler's point of view: On ne trouve la solution du pmbleme que po·u,- le.s cas ou les differentes figtLTes de la corde vibrante peuvent etre renfermee dans une seule et meme equation ([8] , II, p. 72). (One obtains the solution only in the cases when the different forms of the vibrating string can be expressed by one specific equation.) Already here there are two conceptions about the functions you can consider in mathematical analysis. For d'Alembcrt, they should have a well defined expression in terms of known functions. For Euler, they can be defined as well by any graphical representation. A new scene appeared with Daniel Bernoulli. Since a sound is a superposition of harmonics, the general solution of the problem of vibrating strings should be a series of the form (again, I use modern notations)

(4)

� "" k

(

ak

brwt

cos -e

+

)

. knwt . k1rx bk sm -- sm e e

-

[ 1] . Now neither d'Alembert nor Euler would agree. For d'Alembert, and for Euler as well, trigonometric series would represent only a very special class of functions. The last character in this first episode is Lagrange. He wru; able to treat the problem in a complete form when the string is replaced by equidistant weighted points distributed on a thread. Then, finer and finer discretisations of the initial data result in discrete solutions tending to the solution proposed by Euler. It appeared as a justification for Euler's point of view, although it was rejected by d'Alembert. And that is the end of this part of the story [9] . 1.2. The second episode involves Joseph Fourier (1 768-1830) as a principal character. Fourier sent a memoir to the French Academy of Sciences (then called the first class of the Institut de France) in 1807, on t.he propagation of heat in solid bodies. The memoir was read by a committee consisting of La.gmnge, Laplace, Lacroix and Monge. It was not published. The Academy then proposed the subject for a competition. Fourier extended his study and sent his contribution with the

19 0

JEA:"-PIERRE KAHANE

beautiful subtitle "et ignem regunt numeri" (heat also is governed by numbers) . Laplace. Lagrange, and Lacroix were again examiners. Fourier won the Prize, but there were severe reservations, and the work again was not published. Only after 1817, when Fourier became a member of the Academy, did a printed version appeared ; an extended version took the form of an important book, la Theorie analytique de la chaleur, the analytic theory of heat, published in 1822 [5] . The book consists of an introduction, Discours preliminaire, and nine chapters. The fhst chapter expounds the physical aspectH of heat propagation. The second gives the differential equations, first as examples, then in a general way : inside a homogeneous body the heat propagation is governed by the equation

(5 )

where K, C, D are physical constants depending on the body. If we forget the constant, it is what we now call the heat equation. Moreover there are boundary and initial conditions , a.<; for vibrating strings. The third chapter introduces a method, the use of trigonometric series, for solving a special equilibrium problem. Here is the problem. Take an infinite rectangular body, limited by a horizontal strip and two parallel vertical half-planeH, say, (6)

o :::; y < oo .

The strip part is at the temperature of boiling water, the temperature of melting ice, that is

u(x, O) = l u ( - � ' y) = u ( � ' y)

(7)

=

vertical

edges at the

( - � < x < �) , 0 (y > 0) '

where we write 1 for the temperature of boiling water. The temperature inside the body is given by the heat equation in a reduced form : (8) Nowadays we call this a Dirichlet problem. The treatment by Fourier consists of looking first for solutions of (8) in the form u(x, y)

=

f(x) g(y) ,

then, taking into account the boundary conditions on the vertical edges and the fact that temperatures are bounded in the body, considering as general candidates

u (x , y)

= a

exp(-y) cos x + b exp (-3y) cos :h; + c exp(-5y) cos 5x + · · · .

It remains to express that u(x, 0) = 1 when -1T /2 < x < 1T/2, that is 1 = a

cos x + b cos 3x + c cos 5x +

· · ·

(- 1T/2 < x < n/2) .

191

PARTIAL DIFFERENTIAL EQUATIONS

Fourier finds the values of the coefficients and he is pleased with the solution : (9)

(10)

x - �3 cos 3x + �5 cos 5x - �7 cos 7x + ( - �2 < x < �2 ) ' 1 1 1 -u(x. y) = e-y cos x - - e-3Y cos 3x + - e-5Y cos 5x - - e-7Y cos 7x + 3 5 7 4 ( - � < x < �,y > o) .

� u (x 0) = cos ' 4 7r

·

·

·

·

· ·

'

Fourier knows what convergence means and explains that these series converges, indicating the sum of the series (9) on different intervals (n° 177) with a full proof using the so-called Dirichlet kernel (n° 1 79) ; "the limit of the series is positive and negative alternatively. By the way, the convergence is not rapid enough in order to provide an easy approximation, but it suffices for the truth of the equation" (n° 1 79) . The Fourier's main interest is the second series because it is "extremely convergent" and gives a good estimate of the temperature inside the body by using only a few terms (n° 191). Then there is a long digression in the book. Before considering the propaga tion of heat in other domains (chapters 4 to 9) , he spends 50 pages playing with particular functions and their expansions into series of cosines or serieH of sines, thereby giving different extensions of functions defined on an interval. His conclu sion is that arbitrary functions can be represented by trigonometric series and that all series converge. (That was a mistake, but a very fruitful mistake.) He observes that his analysis applies to vibrating strings, therefore justifying the approach of Daniel Bernoulli. As far as the notion of function is concerned, it is clear after Fourier that a function is associated with a domain and that there is no canonical way to extend a function. Vve shall see in part II how trigonometric series played a role in this clarification. 1.3. The first historical study of these matters is due to lliemann (1826-1866). It is the first chapter of his dissertation on trigonometric series [12] . The second chapter introduces the Riemann integral, together with a characterization of the Riemann-integrable functionH. The third chapter is a firework of ideas, methods, examples , and general results. It contains a characterization of the functions ob tained as sums of everywhere convergent trigonometric series. Starting form the series and not from the function forbids the use of integration for computing the coefficients (or needs another definition of the integral, as Denjoy made much later [4] ) . ThiH iH now called the Riemann theory of trigonometric HerieH [16] . The Rie mann theory was completed by George Cantor ( 1845-1 918) ; he proved that if the sum of the series is zero everywhere, it is the null series. The he extended the theorem and proved that the conclusion still holds when "everywhere" is replaced by "everywhere except on some particular set" . This extension is the first paper by Cantor on real numbers and set theory [3] . Riemann's third chapter is a jewel mine, but my purpose here is to use and comment the first chapter. The first chapter is divided into three sections, whose subjects are vibrating strings, Fourier, and Dirichlet ( 1805-1859). The first section is a very clear ex position of the controversy about vibrating strings : d' Alembert rej ecting his own solution when arbitrary initial positions and velocities are given ; Euler claiming that no restriction is needed ; Bernoulli assuming that the motion of vibrating

192

JEAN-PIERRE KAHANE

strings is a superposition of harmonic motions ; and Lagrange's approach, from finite to infinite, supporting Euler's claim. D' Alembert did not agree with Euler and Lagrange, and the three of them rejected the claim of Bernoulli. Riemann says at the beginning of the second section that a new area began with Fourier, namely with the couple of formulas

{

J(x) (11)

=

{

sin x + a2 sin 2x + · · · +2 bo + b 1 cos x + b2 cos 2x + · · .

a1

1

an

= -

bn

=

1

j" f(x) sin nx dx ,

� �-.: j (x ) cosnx dx .

1r

- 7r

He then explains that Lagrange strongly opposed Fourier's method. Let me quote Riemann : Als Fourier in einer seiner ersten Arbeiten uber die Warme, welche er

der franzosische Akademie vorlegte (21 Dec. 1807) zuerst den Satz aussprach, dass eine willkurklich (graphisch) gegebene Function sich durch eine trigonometrische Rcihc ausdruckcn lasse, war diese Behauptung dem greisen Lagrange so unerwartet, dass er ihr auf das Entschiedenste entgegentmt. Es soll sich hieruber noch ein Schrijtstuck im A rchiv der Pariser Akademie befinden.

( When Fourier in one of his first works on heat, communicated to the French Academy on Dec 21 1807, stated that an arbitrary function (given in a graphic way) could be expressed by a trigonometric series, this statement was to the old Lagrange ::;o unexpected that he opposed it in the strongest way. There should still be a written document about this in the Archives of the Parisian Academy. ) Let me explain the phrase, "dem greisen Lagrange. " In December 1807, La grange was 71 ; the other members of the committee were much younger : Monge 61, Laplace 58, Lacroix 42, and Fourier was 39. Concerning the "Schriftstiick" , a footnote explains that the information carnes from Dirichlet, who had known Fourier in Paris. I have looked for this document, and the second part of this article describes what I found. Riemann then discusses matters of priority and concludes :

Durch Fourier was nun zwar die Natur der trigonometrischen Reihen vol lkurnmen ·richt·ig erkannt i sie wurden seitdem in der rnathematischen Physik zu Darstellung willkiirlicher Funktionen vielfach angewandt, und in jedem einzelne Falle iiberzeugte man sich leicht, dass die Fourier 'sche Reihe wirklich gegen den Werth der FUnction convergiere i aber es dauerle lange, ehe d·ieser w·ichtige Satz allgernein bewiesen wurde.

( Through Fourier indeed was the nature of trigonometric series fully understood ; since then they were applied many times in mathematical physics for representing arbitrary functions, and in each case one was easily convinced that the Fourier series really converges to the function ; but it lasted long before this important theorem was proved in full generality. ) As a last comment on this section, the term of "Fourier series" was not classical when Riemann wrote his thesis. He was the first to emphasize the importance of the notion. Nowadays a Fourier series is a trigonometric series whose coefficients are given by the Fourier formulas. Therefore it depends on the kind of integral

PARTIAL DIFFERENTIAL EQUATIONS

193

we consider, and there are indeed many to consider Fourier-Riemann, Fourier Stieljes, Fourier-Lebesgue (the most important now) , Fourier-Denjoy, Fourier Wiener, Fourier-Schwartz series, Haar-Fourier series on compact abelian groups, etc. Before going to Dirichlet in the third :->ection, Riemann mentions several com petitors and several mistakes. Cauchy's mistakes were surprisiug a,ud fruitful : one was about convergent series, another on analytic functions, the very domains where Cauchy made such brilliant contributions. Both were pointed out by Dirichlet and raised important observations by Riemann. The work of Dirichlet on <":onvergencc of the Fourier series was published in 1829. Dirichlet was stimulated by the Cauchy's errors, and his article is a model of rigor in mathematical analysis. lliemann explains the method, and here is his appreciation : Durch die Arbeit Dirichlet's war einer grossen Menge wichtiger analytischer Untersuchungen eine feste Grundlage gegeben. Es war him gelungen, indem er den Punkt, wo Euler irrte, in volles Licht bmchte, eine Frage z·u erledigen, die so viele ausgezeichnete Mathematker seit mehr als siebzig Jahren (seit dem Jahre 1 753) beschaftig hatte. In der That fur alle Falle der Natur, um welche es sich allein handelte, war sie vollkommen erledigt, denn so gross auch unsere Unwissenheit dariib er ist, wie s·ich die Kriifte and Zustiinde der Materie nach Ort und Zeit im Unendlichkleinen andern, so konnen wir doch sicher annehmen, da.�s die Functio nen, auf welche sich die Dirichlet'sche Untersuchung nicht erstreckt, in der Natur nicht vorkommen. (Dirichlet's work gave a solid ground to a large number of important analytical investiga-tions. It was given to him to fully clarify a point on which Euler was mistaken, and to settle a question investigated by so many eminent mathematicians for more than 60 years (since 1753 ) . Actually the question was settled completely for all cases we can encounter in Nature, since, even ignorant as we are of the evolution of forces and states of matter according to space and time, we can be sure that functions on which Dirichlet 's investigation does not apply do not exist in Nature. ) Euler's error was the common error of d 'Alembert , Euler and Lagrange , who all considered Bernoulli's approach as hopeless. There is an inaccuracy in the lliemann's statement that functions occuring in Nature are only of the type Htudied by Dirichlet (a finite number of maxima and minima on every interval). Brownian motion is a counterexample, which Rjemann couldn't gueHs. But Riemann also says that mathematicians should not restrict the range of functions to those occurring in mathematical physics. For example, func tions arising in number theory are also worthy of study. And that is the motivation for the last and most important. part of his dissertation, where he investigates the very strange functions that trigonometric series provide. Riemann did not mention an observation made by Dirichlet at t.he end of his article. Dirichlet observed that only integrable functions can be considered in his study. For him, an integrable function is necessarily continuous on some interval (it is the notion of an integral described by Cauchy). Then he gives the famous counterexample, a function taking one value on the rational:-> and another value on the irrationals. ·with Fourier, Dirichlet, and Riemann we are not at the end, and the relations between function theory, trigonometric series, and POE are renewed constantly.

194

JEAN--PIERRE KAHANE

But we have seen the enormous change made between 1750 and 1850 in the con ception of what a function is. For d'Alembert, it was an analytic expression, in order for the equation of vibrating strings to make sense ; for Euler, a graphic representation was sufficient, as suggested by the form of a string at initial time ; for Bernoulli, relying to a physical interpretation of sound by means of harmonics, infinite trigonometric series would occur ; for Lagrange, finite to infinite had to start with a discrete analogue of the vibrating strings, but "discontinuous" func tions (like broken lines) were not excluded. With Fourier it became clear that the domain has to be specified when a function is introduced, and that there are several natural extensions of a function given on an interval. With Dirichlet a first model is given of a complete proof in the domain of functions of a real variable, and a first example of a function that cannot be defined by a graphical representation. Finally Riemann was able to summarize all the main points and consequences of the di�ClL."iSions about vibrating strings and trigonometric series, and to pave the way to Cantor and others for the foundations of the theory of functions of a real variable. 2. Part II

There are amateurs in mathematics, in the sense that they like to read mathe matics and also to play with it ; they enjoy mathematics, and sometimes they make a small contribution. I am an amateur in the history of mathematics, and here is a small contribution. 2.1. Consulting the archives of Academie des sciences and the Library of In stitut de France, with the help of the archivists and librarians, I was able to found a document that may be the "Schriftsti.ick" of Lagrange mentioned by Riemann. It is just one page long, and I shall reproduce it below. There is no date on it. Obviously it is negative comment on a formula written by Fourier. It contains el ementary computations, leading to an apparent contradiction. Here it is ((a) , (b), (c) are in the handwritten paper ; (i), (ii), (iii) , (iv), (v) are signs I introduced in order to avoid repeating the formulas later) : L 'equation

1 . 1 . 1 , 1 . = sm x - - sm 2x + - sm 3x - - sm 4x + 4 2 2 3 n'a lieu qne depuis x = 0 jusqu 'a x = i = 90° . Supposons qu 'elle ait lieu au dela. Dans (a) faisons X = (a)

-x

on aura

· · ·

7r

-

y y etant

<

1f - y 1 . 1 . . y 1 . sm + - sm 2y + - sm 3y + - sm4y + · · · 4 2 2 3 Changeons y en x ce qui est permis on aura done aussi

(i )

-- =

1 . . 1 , 1 . sm x + - sm .') x + - sm 3x + - sm 4 x + 4 2 2 3 Or cette section ne peut s 'accorder avec la le:re (a) car la JeTe di.fferentiee donne 1 - = cos x - cos 2x + cos 3x - cos 4x + · · · ( ii) 2

(b)

1!' - X

--

=

· ·

·

i

l95

PARTIAL DIFFERENTIAL EQUATIONS

Fa'isons (•••) .. '

X = 7r

- y on aura y etant < 90" 1 2 = - cos y - cos 2y - cos 3y - cos 4y

-

Changeant y en x on aura 1 - 2 = cos x + cos 2x + cos 3x + cos 4x + ( iv)

... ·· ·

Mul(tiplions) par dx et integrons 1 1 1 1 canst - -x = sin x + - sin 2x + - sin3x + - sin 4x + · · · ( v) 2 2 3 4 Pour determiner la constante on fait x = 0, on a Canst = 0. Done 1 1 . 1 . 1 . . ( c) - 2 x = sm x + 2 sm 2x + 3 sm 3x + 4 sm 4x + · · · En comparant (b) et (c) on voit qu 'elles ne peuvent subsister.

Here is an abbreviated translation.

Equation (a) holds only from x = 0 to x = � . Supposing it extends outside, take x = 1r - y . Since y < � we shall have (i). Changing y into x we have also (b) . But this cannot agree with (a) for differentiating (a) gives (ii). Writing x = 1r - y we get (iii), then changing y into x we get (iv), then multiplying by dx and integrating we obtain (v) . In order to define the constant we take x = 0 , and that gives const = 0. Hence (c). Comparing (b) and (c) we see that they can't hold together. Lagrange was mistaken : (b) is not valid for x = 0. The root of the mistake lies in the word "au dela" , just after (a) ; by the way "au deh't" is not easy to read in Lagrange's handwriting. It means that Lagrange extends (a) from the interval (0, �) to what seems natural to him. Actually both sides of (a) have a natural extension, but they are not the same. 2.2. I was not satisfied with this page : it is not as important as I expected, nor does it express what Riemann said about Lagrange's strong opposition towards Fourier. I already knew from the catalogue in the Archives that there were two pages and not one. Would there exist another page of Lagrange more explicit about this opposition ?

Yes and no. The handwritten papers of Lagrange are kept in a series of vol umes collecting what Lagrange wrote on different subjects. Volume n° 906 contains mainly contributions to interpolation methods and recurrent series. Each contri bution contains a number of sheets of paper, this number is written at the back of the last page and four signatures follow : Lacroix, Legendre (written Le Gendre), Prony (written De Prony) and Poisson, a committee of academicians. Sometimes a title is given at the back of the last page as well. Actually there are two sheets of paper with the title "papiers relatifs au memoire de Fourier" and "deux feuilles" written at the back of the second. The first is what I just copied. The second ha..., a quite different handwriting, and it is the only piece in the volume that was not written by Lagrange. It is endorsed by the four academicians, and I thought first that they took the decision to react to Lagrange 's mistake. I shall reproduce the

196

JEAN-PIERRE KAHANE

text below, but I can't reproduce the aspect of this sheet, full of crossings out, easy to read at the beginning, and completely squeezed at the end. It is nothing but a rough draft. This excludes the assumption that it was produced by the academic committee. But why did they endon;e this draft ? Reading the draft I noticed that it was exactly in the spirit of Fourier. Actually there is no doubt that it is his handwriting. Before going further, let us look at what he wrote. On top of the page is the formula -w - x 1 . 1 . 1 . -- = Slll X + - Hlil 2x + - Hlil 3:J; + · · · = -

2

2

""

3

2

.

/ sin(m + �x) dx . 1 sin ( 2x)

Then : 1 . La valeur de la constante dans l 'equation 1 2

.

1 . 2

1 . 3

1 . 4

C - - x = sm x + - sm 2x + - sm 3x + - sm 4x + · · · n'est pas nulle. En effet le calcul fait voir que le reste de la serie consideree comme function de x et du nombre m de termes ne devient pas nul lorsqu 'on fait x = 0 et m infini. Mais si l'on donne a X une valeur quelconque plus grande que 0 et plus petite que 2-w, le meme calcul montre que la valeur du reste devient nulle lorsqu'on suppose m ·infini. Jl suit de la que pour determiner la constante il jaut donner a X une valeur quelconque comprise entre 0 et 2-w. 2. L 'equation 1 1 1 . 1 . . -x = sm x - - sm 2x + - sm 3x - - sm 4x + 4 2 2 3 .

· · ·

est vraie pour to·utes les valeurs positives moindres que -w et pour toutes les valeurs negatives plus grandes que --w. C'est-a-dire que si l 'on met dans le second membre une valeur de x moindre que -w et pl-us grande que -!V ce second membre aura une limite determinee dont on approchera sans fin a mesure que le nombre de termes augmentera et cette limite sera �x. Lorsqn 'au lieu de x on met w - y l 'equation a lieu entre les limites w - y = -w et w - y = w c 'est-a-dire pour toutes les valeurs de y moindres que 2w et plus grandes que 0. C'est pourquoi si dans /'equation (b)

1 . 1 . r;;;J - X . 1 . = sm x + - sm 2x + - sm 3x + - sm 4x + · · · 2 2 4 3

--

on met au lieu de x une valeur plus grande que 0 et moindre que 2w, le second membre aura toujours pour limite la quantite w2x . ( trois lignes raturees) . Si l 'on differentie l 'equation (b) on a

1 - '2 = cos x + cos 2x + cos 3x + cos 4x +

.

.

·

comme on trouve dans la note. Si on l 'integre il faut determiner la constante en donnant a X une valeur quelconque moindre que 2w et plus grande que 0. Ainsi dans l 'eqnation

('

1 1 . 1 1 . . , - - x = sm x + - sm 2x + - sin 3x + - sm 4x + · · 4 2 2 3

·

1 97

PARTIAL DIFFERENTIAL EQUATIONS

il faut donner a x une valeur > 0 et < 2ro. Si par exemple on fait x = w on aura C = !w ce qui est la veritable valeur de Ia constante. Si l 'on faisait x = �tz on aurait 1 1 1 1 C - ro = 1 - 3 + 5 - 7 + . . .

4

ou C = � w comme precedemment. L 'objection se reduit done a celle-ci : l 'equation w-x . 1 . 1 . 1 . (b) -- = sm x + - sm 2x + - sm 3x + - sm 4x + · · · 2 2 4 3 que foumit le calcul de l 'auteur n'est point vraie. En differentiant on a 1 - 2 = cos x + cos 2x + cos 3x + cos4x + . . .

multipliant par dx et integrant et determinant la constante pour que l 'equation (b) ait l·ieu pour x = 0 on a 1 1 1 . - 2'7: = sin X + 2 Hin 2x + 3 Slll 3 X + ( C) ·

·

·

qui ne peut s 'accorder avec la precedente. La reponse consiste a remarquer que l 'equation (b) a lieu pour toutes les 'oaleurs de x qui sont > 0 et < 2w (ici la page est coupee) et que l 'on ne satisfait pas a cette condition des limites en determinant la constante de maniere que l'equation ait lieu lorsque x = 0. Ainsi on ferait precisement la meme objection si l 'on se n�duisait a dire : [ 'equation 1 . w-x 1 . . -- = sm x + - o;m 2x + - sm 3x + · · 2 2 3 ne peut pas subsister car elle n'a pas lieu lorsque x = 0. (puis en tout petit, au bas de la page) En general on ne peut point separer l 'usage d 'une equation de ce genre de la consideration des limites entre lesquelles les valeurs de la variable doivent etre considerees. ·

Here is an abbreviated translation of the last and mo;;t important part : Here is the objection (of Lagrange) : (b) is not true, because differentiating, then integrating in such a way that the equation holds when x = 0 one gets (c) , which is not compatible with (b). The answer (of Fourier) is that (b) holds when 0 < x < 2ro and this condition is not satisfied when x = 0 The same objection (of Lagrange) could be ma.d.c by simply considering (b) and saying that it does not hold when x 0. It is a general fact that an equation of this type cannot be used without speci fying the limits between which the values of the variable have to be considered. =

The whole page is rather lenghty, but it is a perfect explanation of the La grange's mistake [10] . 2.3. Now we are faced with a puzzle. Why is this draft of Fourier enclosed in the handwritten papers of Lagrange ? Why was it endorsed by an academic committee 7 When and why was it written 7 Is there a relation between these two papers and the "entschiedenste" opposition of Lagrange 7 I shall first provide the reader with some documents and then give my inter pretation of a possible order of the events.

198

JEAN-PIERRE KAHANE

2.3.1. First, the series under consideration, (a) in the paper of Lagrange, is the matter of n° 182 of the Analytical Heat Theory [5] . Fourier checks that the partial sums of even order can be written x 2

+ �) dx J cos(mx 2 cos �

and shows that the integral tends to 0 using an integration by parts. It is a perfect proof of the formula. Precisions come later, after considering other examples, in n° 184 : Il faut observer, a l 'egard de toutes ces series, que les equations qui en sont formees n'ont lieu que lorsque la variable est comprise entre certaines limites. (This is almost the :-;arne as the end of the draft.) Moreover, for the series (a), the calculation "donne Ia valeur x/2 toutes Jes foi:-; que X est plus grand que 0 et moindrc que w. Mais elle n'equivaut plus a x/2 si r arc depasse w . " (Again, this is expressed in the draft.) 2.3.2. According to Fourier, (a) is not new ; it. was known before (end of n° 182). Anyway, it was not accepted by Lagrange. There is a long letter of Fourier to Lagrange, published by Herivel (He), likely 1808 or 1809, the draft of which is kept in the manuscript fund of the French National Library [6] . The draft is rough and looks like the paper in the files of Lagrange ; but it is organized in order to be as convincing as possible. About (a) he insists : Je vous prie de jeter les yeu.'C sur cette derniere note qui etablit clairement la convergence de la serie et dont la partie essentielle etait dans le memoire (art.). Vous reconnaitrez facilement que cette matiere n'est pas du domaine de la foi mais de celui de la geometric, ce qui est b'ien different, et il me semble que si de pareilles demonstrations ]Jeuvent etre contestees, i[ jaut renoncer a ecrire quelque chose d 'exact en mathematiques. (I urge you to look at the last note, which proves the convergence of the series and which was contained essentially in the memoire (art). You will agree that this is not a matter of faith but of geometry, which is quite different, and that one should give up writing anything exact in mathematics if such a demonstration was not accepted.) Then he tells how he found the general formula by different methods ; he mentions that he had communicated his results to Diot and Poisson ; he discusses the possible priority of d'Alembert and Euler for the method of integration ; and he insists that nobody before him had tried to develop a constant into a series of cosine, and that one should be cautious about the limits between which the development holds. A note explains that he was not able to consult mathematical books in Grenoble, and that he is willing to quote those who contributed to the subject before him. There is no mention of Lagrange in the letter, but Herivel gives very good raisons to believe that it was addressed to him. At the same time, Fourier wrote to Laplace, again on the subject of trigono metric series. These letters were also published by Herivel. 2.3,3. Then came the subject proposed for the Prize : donner la theorie mathematique des lois de propagation de la chaleur, et comparer le resultat de cette theorie a des experiences exactes (to give the mathematical theory of the propagation of heat, and compare the results to exact experiments).

PARTIAL DIFFERENTIAL EQUATIONS

199

The memoir of Fourier, Theorie du mouvernent de la chaleur dans les corps solides, with the subtitle et ignem regunt numeri (Plato), was received on September 28, 181 1 and given the number 2, as certified by "le comte Lagrange, le comte Laplace, Malus, Legendre." The Prize was announced on January 6, 1812 : La classe a deceme le prix, d 'une valeur de 3000 F, au memoire enregistre sous le n° 2, portant cette epitaphe "et ignum regunt nurneri (Plato) ". Cette piece renferrne les veritables equations differentielles de la transmission de la chaleur, soit a l 'interieur des corps, soit a leur surface j et la nouveaute du sujet, jointe a son importance, a determine la classe a couronner cet Ouvrage, en observant cependant que la maniere dont / 'Auteur parvient a ses equation n'est pas exempte de difficultes, et que son analyse, pour les integrer, lai.�se encore q·uelque chose a desirer, soit relativement a la generalite, soit meme du cote de la rigueur [17] . Clearly i t was not a complete approval of Fourier, and even his analysis (what we now call Fourier analysis) was condemned
The minister of the Interior was the mathematician Lazare Carnot. The com mittee worked until the end of 1815 and succeeded in classifying Lagrange's hand written papers [19] . Fourier was elected as member of the Academy on May 27, 1816, he was turned down by Louis XVIII, elected again and confirmed on May 12, 1817. He was elected as secretaire perpetuel on November 18, 1822 , and he died on May 16, 1830.

2.4. Let me try now to tell the story as I see it . Fourier had fought as much as he could after 1807 in order to let his proofs be known and the objections discussed. The way he obtained the Prize in 1812 was a disappointment for him. He had a high respect for Lagrange, and Lagrange had ignored Fourier's arguments. Lagrange died soon afterwards . He left an enormous amount of unpublished works, and for some reason Fourier had access to the handwritten papers of Lagrange. He saw the page on equation (a), he felt angry, he immediately wrote his comments and joined them to Lagranges's paper. When Lacroix, Legendre, Poisson and Prony found these two sheets of paper together, Prony wrote "deux feuilles" and the others

200

JEAN-PIERRE KAHANE

agreed. Fourier never told the story. To Dirichlet and other young people he liked to speak on different matters. These young people knew the works of Fourier and appreciated his treatment of trigonometric series. Fourier had to explain why it was badly received. Then he spoke on vibrating strings, Daniel Bernoulli, Lagrange, and declared that Lagrange strongly opposed his own conclusions and that there was a piece in the Archives proving this opposition. Dirichlet repeated the statement of Fourier to Riemann, and Riemann wrote what we saw. The second part of this paper , on Lagrange and Fourier, received the constant help of several collaborators of the Archives de l'Academie des sciences and Bib liotheque de l'Institut. I am particularly thankful to Mesdames Florence Greffe, director, and Claudine Pouret at the Archives, and Mireille Pastoureau, director, and Annie Chassagne, conservateur en chef, at the Library. I also thank Robert Ryan for his careful reading and the many linguistic im provements he made. The history of mathematics is a pleasant opportunity to go to libraries and read (or try to read ) old papers. Shall we leave such opportunities to future math ematicians about what we do now ? This question is my conclusion.

PARTIAL DIFFERENTIAL EQUATIONS

Lagrange Joseph-Louis Copyright Academie des sciences de l'Institut de France

201

202

JEAN-PIERRE KAHANE

1•' .0 "'RU•: R , ('.l...,.pl) 1

rl

l�·;n/,�. ,(,. t,' if 7uw tf,:,,,n�lfl� . 'o.

II;(' ' { 1.- 'U •1(. �� '� .(' �I* .,.,.. ft..�... , 1;;.� ._,... 1 ....-.<1 .. .._ ._ _ _ _ _ _ ...._ _ _ _ _ _ _ __.

L

/.

Fourier Joseph Copyright Academie des sciences de l'Institut de France

203

PARTIAL DIFFERENTIAL EQUATIONS

Excerpts from the "Schriftstuck"

Courtesy of B'bl' 1 1otheque de l'Institut de France

' t§,£-r •'-f. _

1.

c

",

,.,. -� •·· u - • ,.. _ , . _ J . --. � � -..,.. "1 o.u •..,._, 1: ....,; , ... J u d <>b-4 v . v to' u-.1......._ (,;;ra.( � -...L4AM.. J ... .,· � Q.. ?e- ]<4!. "" �u 4-.L
""...1 �

..

"

-\-<•

.

4- bO {, - -

..;.

..

..

:r,W+....._u.:..;.,. � "-f'fA4

_,.,__, � .,....,.�. """"� �-.... ,..,_ "" v:..J;. .. o. r. � - a lif -y _

� %'

� � ., -� .

'(

� :J�

p,..,. ,,� _,-.._ � ' c. '- 7.1 �.:J = - w' ,d-;> <> < � /r"'"< {,J.. f (u "" '..;'71 "' .,J ......V <;;J� ,:;; '-"..:� ;t' c...... e. 4 ..,J€u..f � "fla. ...;),._ t'- o iif"· u...-i �u.. v tr b.� -- ·-z ai'. � � t...� � "-\P... c:.,..,... .. JLc OM. ci ....., '1-
·

- •

Courtesy of B 1'bliotheque de l'Institut de France

204

JEAN-PIERRE KAHANE

Courtesy of Bibliotheque de l'lnstitut de France

References [1] D. Bernoulli,

Sur le melange de plusieurs especes de vibrations simples isochrones, qui peuvent

exister dans un meme systeme de corps,

Bernoulli, Basel, Birkhiiuser 1982 . . [2] H. Burkhardt, Entwicklungen nach

Mem. Acad. Berlin 55 (1753) , in Die Werke von Daniel

oscillirenden Functionen und integration der Differen

tialgleichungen der mathematischen Physik,

Erster Hauptteil : die Ausbildung der Methode der Reihenentwicklungen an physikalischen und astronomischen Problemen, Jahresbericht der Deutschen Mathematiker Vereinigung X 2, 1908, 1-1804. [3] G. Cantor, Uber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen, Math. Annalen Leipzig 5, 123-132, in Georg Cantor gesammelte Abhandlungen, Berlin 1932. [4] A. Denjoy, Ler,:ons sur le calcul des coefficients d'une serie trigonometrique I, II, III, IV, Paris, Gauthier-Villars 1941-1949. [5] J. Fourier, Theorie analytique de la chaleur, Paris, Firmin-Didot, 1822 (also in CEuvres, Paris, Gauthier-Villars 1888-1890). [6] J. Fourier, Minutes de lettres, service des manuscrits, Bibliotheque Nationale de France, Fr 22501, fol. 72-74. [7] J. Herivel, Joseph Fourier, Lettres inedites, 1808-1816, Paris 1980. [8] G. Jouve, Imprevus et pieges des cordes vibrantes chez d 'Alembert {1 755-1 783), doutes et certitudes sur sur les equations aux derivees partielles, les series et les fonctions, These de doctorat de l'Universite Claude Bernard Lyon 1 (10 juillet 2007), I These principale, 166p. I I Annexes 398 p. [9] J. L. Lagrange, Recherches sur l a nature e t la propagation du son, Miscellanea Taurinensia I, 1759, Nouvelles recherches sur la nature e t l a propagation du son, Miscellanea Taurinensia II, 1760, in CEuvres de Lagrange , Tome I , Paris, Gauthier-Villars 1867, 37-150 and 15 1-332. [10] J. L. Lagrange, manuscrits, Bibliotheque de l'Institut de France, vol. 906, 102-103.

205

PARTIAL DIFFERENTIAL EQUATIONS

[11] V. M az ya and T. Shaposhnikova, Jacques Hadamard, A Universal Mathematician, History of Mathematics vol. 14, American Mathematical Society, London M athemat ical Society, 1998. [12] B. Riemann, Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe, (Habilitation dissertation, Gottingen 1854), in besammelte mathematische Werke, Leipzig 1876. [13] L. Schwartz, Gener-alisation de La notion de fonction, de derivation, de transformation de Fourier, et applications mathematiques et physiques, Annales de l'Universite de G reno ble, 1946. [14] C. Truesdell, The rational mechanics of flexible and ela.stic bodies 1 638-1 788, in L . Euleri Opera Omnia, II lln sectio altera, Zurich 1960. [15] A. P. Youschkevitch, About the historrJ of the debate on vibrating strings (d'Alembert and the use of "discontinuous " functions), (in Russian) Istoriko-matematitcheskie iss ledovania 20 1975, 221-231 ( French translation in Jouve II). [16] A. Zygmund, Trigonometric series /, II, Cambridge Universi ty Press 1959. [17] Seance publique de I 'Institut Imperial de France du 6 janvier 1812, classe des sciences mathematiques et physiques, proclamation des prix, Archives de I' Academie des sciences . [18] Livre des seances de Ia premiere classe de l'Institut Imperial de France 1812-1815, Archives de l' Academie des sciences. [19] Prod�s verbaux des travaux de La commission chargee de mett re en ordre les papiers de Lagrange, 181 5- 1817, dossier Lagrange ( paquet n° 3, p iece n° 40, 8 novembre 1815), Archives de l Academie des sciences. '

'

DEPARTMENT OF MATHEMATICS, UNIVERSITY PARIS-SUD

E-mail addres., Jean-Pierre . Kahane�ath . u-psud . fr

9 1 406

0RSAY CEDEX

Proceedings of Symposia in Pure Mathematics Volume 79, 2008

Quantitative Unique Continuation, Logarithmic Convexity of Gaussian Means and Hardy's Uncertainty Principle Carlos E. Kenig

In this paper we describe some recent works on quantitative unique continuation for elliptic, parabolic and dispersive equations. We also discuss recent works on the logarithmic convexity of Gaussian means of solutions to Schrodinger evolutions and the connection with a well-known version of the uncertainty principle, due to Hardy. The elliptic results are joint work with J. Bourgain [BK] , while the remainder of the works discussed here are joint works with L. Escauriaza, G. Ponce and L. Vega ( [EKPVl], [EKPV2] , [EKPV3] , [EKPV4] , [EKPV5] ) . The paper is based on lectures presented at WHAPDE 2008, Merida, Mexico. I am grateful to the organizers of WHAPDE 2008 and to the participants in the workshop for the invitation and the very friendly atmosphere of the workshop. For further references and background on the problems discusses here, see [BK], [K1], [K2] , [EKPV 1] , [EKPV2] , [EKPV3] , [EKPV4] , [EKPV5] and the references therein. 1 . Some recent quantitative unique continuation theorems

Here I will discuss some quantitative llllique continuation theorems for elliptic, parabolic, and dispersive equations. I will start by describing the elliptic situation. This arose as a key step in the work of [BK] which proved Anderson localization at the bottom of the spectrum for the continuous Bernoulli model in higher dimensions, a question originating in Anderson's paper [A] . Briefly, this says the following: consider a random Schrodinger operator on JR.n , of the form H. = -/::;, + V.,, where V., (x ) =

L Ej r/J (x - j), ¢ E C0 (B(O, 1 / 10)) ,

j E'Z!'

0� ¢� 1

and Ej E {0, 1} are independent. It is not difficult to see that inf spec H. = 0 a.s. In this context, Anderson localization means that for energies E near the bottom of the spectrum (i.e. 0 < E < o) H, has pure point spectrum, with exponentially decaying eigenfunctiontl, a.s . . When V., ha�:> a continuum; site distribution (Ej E [0, 1] ) this has been understood for some time ( [GMP] n = 1, [FS] n > 1). For the Anderson-Bernoulli model this was known for n = 1 ([CK.M] ; [SVW] ), but not in higher dimensions. We now have: 2000 Mathematics Subject Classification. Primary 35Q53; Secondary 35G25, 35D99. Partially supported by NSF Grant #DMS-0456583. @2008 American Mathematical Society

201

208

CARLOS E. KENIG

THEOREM 1 . 1 ( [BK] ) . There exists 0 > 0 s.t. for- 0 < E < !5, H, displays Anderson localization a.s. , n � 1 .

In establishing this result we were lead to the following deterministic quantita tive unique continuation theorem: S upp os e that u is a solution to 6u + Vu = 0 in �n , where l V I � 1, and lui � Co , u(O) = 1. For R large, define M(R)

=

Note t h at by unique continuation, Tl!EOH.BM

inf

sup

!xo i= R B (xo , l )

lu(x) l .

SUPB (xo ,l ) i u(x) l > 0.

How

small can M(R) be?

1 . 2 ( [BK] ) .

M(R) � C cxp( -CR413 tog R).

REMARK 1. 3. In ord er for our arr;ument to give the desired application to Anderson localization for the Bernoulli model, we would need an estimate of the form 1\I(R) � C exp( -CR8), wiLh f3 < � 1.35. Note that 4/3 = 1 . 333 . . . .

1 +2v'3

As it turns out, this is a quantitative version of a conjecture of E.M. Landis. He conjectured (late 60's) that if 6u + Vu = 0 in Rn , where l V I � 1 , lui � Co , and iu(x) l � C exp ( -Cix l l + ) , then u = 0. This conjecture of Landis was disproved by Meshkov ([M] ) , who constructed such a V, u =/= 0, with lu(x) l � C exp (-Ci x l 4/3). This example also shows the sharpness of our lower bound o n M ( R) . One should note however that in Meshkov 's example u, V are complex valued. Our proof uses a rescaling procedure, combined with well-known Carleman estimates. Q:. Can 4/3 be improved to 1 in our lower bound for M (R) for real valued

u, V'?

to

Let us now turn our attention to parabolic equatimlfi. Thm;, consider solutiom; OtU -

6u + TV (x, t) · Vu + V(x, t)u = 0

Then, as is well-known, the following x (0, 1] , with IWI � N, l V I � M. backward uniqueness result holds: If iu(x, t)i � C0 and u(x, 1) = 0, then u = 0 ( see [LO] ) . This result has been extended by Escauriaza-Seregin-Sverak ( [ESS] ) who showed that it is enough to as sume that u is a solution on R� x ( 0 , 1 ] , where R+ = {x = (x ', x n ) ; Xn > 0}, without any assumption on u i aiR'j: x (O,l} · This was a crucial ingredient in their proof that weak (Leray-Hopf) solutions of the Navier Stokes system in �3 x [0, 1 ) , which have uniformly bounded L� norm are regular and unique. In 1974, Landis-Oleinik, [LO] , in parallel to Landis' conjecture for elliptic equations mentioned earlier, formulated the following conjecture: Let u be as in the backward uniqueness situation mentioned above. Assume t.hat, instead of u(x, 1) = 0, we assume that iu(x, 1)1 � C exp( - C i x l 2+ e ) , for some «:: > 0. Is then n = 0'? Clearly, t he exponent 2 is optimal here. in JR.n

THEOREM 1 .4 ( [EKPVl] ) . The Landis-Oleinik conjecture holds. More pre cisely, if l lu(-, l) I IP (B(O . l)) � 6, there exists Ro = Ro(c5, M, N, n) > 0 s.t. for I YI � Ro , we have l l u( ·, 1 ) I I P CB(O,t)) � C exp ( - Ciy l 2 log i y i ) .

Moreover, a n analogous result

holds for n only defined in IR +

x

( 0 , 1] .

209

QUANTITATIVE UNIQUE CONTINUATION . . .

The proof of this result uses space-time rescalings and parabolic Carleman estimates, in t.be spirit of the elliptic case. It holds for both real and complex solutions. We hope that this result will prove useful in control theory. We now turn our attention to dispersive equations. Ler us consider non-linear Schrodinger equations of the form in ]Rn

i8tU + i":.u + F(u, u)u = 0,

X

[0, 1),

for suitable non-linearity F, and let us try to understand what (if any) is the analog of the parabolic result we have just explained. The first obstacle is that the Schrodinger equations are time reversible and so "backward" makes no sense here. As is usual for uniqueness questions, we consider linear Schrodinger equations of the form i8tU + 6u + Vu = 0, in IR n x [0, 1 ] , and deal with suitable V(x , t ) so that we can, i n the end, set V(x, t )

=

F( u(x , t) , u(x, t)) .

In order to motivate our work, I will first recall the following version of Heisenberg's uncertainty principle, due to Hardy, [SS] : if f : lR -+ 0, if A · B > 1 , then f = 0. For instance, if l f (x ) l � C, exp( -C. Ixl2+c ) , If({) I � C, exp( -C.I{I2+'), then f = 0 . This can easily b e translated into an equivalent formulation for solu tions to the free Schrodinger equation. For, if v solves

iOtV + a; v

with v(x, 0)

=

v0(x) , then

v (x , 1 )

=

0

in IR

X

[0, 1] ,

� J eilx-ylf4tvo (y)dy, Ceilxl2 14 j e -ixyf2ei l yl214v0 (y) dy .

v(x , t) so that

=

=

If we then apply the corollary to Hardy's uncertainty principle to f (y) we se that if l v(x , O) I

s; C, exp(-C. I x l2+')

=

eiY2 14v0 (y),

lv(x, 1 ) 1 � C, exp(-C. I xl 2+' ) ,

and

we must have v = 0. Thus, for time-reversible equations, the analog o f backward uniqueness should be "uniqueneHs from behavior at two different times" . Thus, we are interested in such results with "data eventually 0" or even with "decaying very fast data" . ThiH kind of uniqueness question for "data eventually 0" has been studied for some time. For the 1 -d cubic Schri:idinger equation

iOtU + a;u =f lul2u

=

0 in

IR

X

[0, 1],

B.Y. Zhang ( [Z2]) showed t hat if u = 0 on (-oo, a] x {0, 1 } or on [a, +oo) x {0, 1}, a E IR, then u = 0 on IR x [0, 1]. His proof used inverse scattering, a non-linear

Fourier transform, and analyticity. In 2002, [KPV3] did away with scattering and analyticity, proving corresponding results for solutiom; to

i8tu + 6u + V(x, t)u

=

0

in

IR"

x

:o, 1 ] ,

n

� 1.

210

CARLOS E. KENIG

THEOREM

1. 5 ( [KPV3] ) . If V E Li L':' n Lfo'c and

0 J J V J J L1L00 (Ixi>R) -+ R�oo and there exists a strictly convex cone r C !Rn and a y0 1

supp u ( - , 0)

then we must have u

=

C

supp u ( ·, 1 )

Yo + f,

0 on !Rn

x

E

C

!Rn such that

Yo + r,

[0, 1 ] .

Clearly, taking V (x t) = J u J 2 (x, t) , we recover Zhang's result mentioned above. This was extended by [IK] who considered more general potentials V and the case when r !Rf- . For instance, if v E L � (R.n X [0, 1] ) or even v E Lf LH!Rn X (0, 1]) with 2/p + njq :::; 2, 1 < p < oo (n = 1 , 1 < p < 2) or V E C([O, 1 ] ; Lnf 2 (R.n)) n � 3, the result holds with r a half-plane. Our extension of Hardy's uncertainty principle, to this context, now is: ,

=

THEOREM

1 .6 ( [EKPV2] ) . Let u be a solution of iEltu + 6u + Vu 0, in lR.n X [0, 1]. =

Assume that V E L00(lR.n

x

[0, 1] ) , 'Vx V

E

LU(O, 1]; L;' (IR., )) and

}l� J J V I I L! L=(i xi> R) = 0.

If there exists a > 2, a > 0, such that u ( ·, 0), u(·, 1) E H1 (ealxl"' dx), then u = 0.

It is conjectured that Theorem 1.6 remains valid assuming only that u, 'Vu at timeH 0 , 1 are in £2 ((Yo + f), ealx l "' dx) , with Yo + r as in Theorem 1 .5. This extension of Theorem 1 . 6 would clearly imply Theorem 1.5. Let me sketch the prof of this result. Our t:>tartiug point is: LEMMA 1 .7 ( [KPV3]) .

3E >

0 s.t. if I J V I IL1 L;;o :::;

iEltU + 6u + Vu = H,

and uo (x) = u (x, 0) , u1 (x)

X

and u solves (0, 1],

u(x, 1) belong to L2 (e2f3x1 dx) n £2(dx) and

II E Li{L2 ( e 2f3x1 dx} =

in lR.n

t"

n

L2(dx)),

then and

S C { 1 J uo J I £2(e2i3zidx) + J l u1 J l£2( e213"' dx) + [ [ H I I L!£ 2 (e2il"I dx) } with C independent of {3.

This is a delicate lemma. If we a priori knew that u E C([O, 1]; L2(e2f:3x1 dx) ) , a variant of the energy method, splitting frequencies into 6 > 0, 6 < 0 , gives the result. But, since we are not free to prescribe both u0, u1 , we cannot use a priori estimates. This is instead accomplished by "truncating" the weight 2{3x1 and introducing an extra parameter. Or next step is to deduce, from Lemma 1. 7, further weighted estimates:

2 11

QUANTITATIVE UNIQUE CONTINCATION . . .

COROLLARY

1. . Assume that we ar-e under the hypothesis of Lemma 'l and a > 0, a >8 1, u0 , u1 E L2(eal x l "' dx) , H E dx) . Then 3Ca > 0, b > 0 s.t. sup e x l " i u {x, tWdx < Idea for the proof of Corollary 1. 8 : Multiply u by ?1R (x) fJ(x/R), 0 for l x l � 1, 17 = 1 for l x l > 2. We apply Lemma 1.7 to un(x, t) (x/R)u {x , t) , with (3 rRcx_-l, for suitable and the corollary follows. 2 1.

for some

Li,L; (ea lxl

O
1

l x i >Ca

bl

"

oo .

17 =

=

=

= rJ

r

The next step of the proof is to dedu�e lower bounds for

L space-time integrals, in analogy with the elliptic and parabolic arguments. These are "quantitative" . THEOREM

1. 9 . Let u solve iOtU + 6.u + Vu 0, x E !Rn, t E [0, 1]. Assume that i u l 2dxdt 2:: 1, 1[o 1Rn lu l2 + IY'u l 2 � A, and that 1�-k Then there exists Ro Ro ( A, n) > 0 and with II V II oo � s.t. if R > Ro 2:: C e-c nR2 • t5(R) 1o( 1{ ( i u l 2 + i 'Vu i 2 )dxdt n Clearly, Corollary 1. 8 applied to u, 'Vu, combined with Theorem 1 .9 yield our version of Hardy's uncertainty principle. 1

=

r! + � 1 )

r

lxl
=

L.

=

(

Cn

L,

R - l � lx i � R

1

2

In order to prove Thmrem 1.9, we use a Carleman estimate which is a variant of one due to V. Isakov [I

].

LEMMA ( [EKPV2] ) . Assume that real function. Then, theTe exists C = C ( n,

1.10 R > 0 and ¢> : [0 , 1 ] JR. is a smooth ll¢'lloo, ll¢" ll oo ) > 0 s.t. a;2 l lea:liH(t)<71 12 91 1 2 � Cn llea l fEHCtled2 (i8t + .6.)g t2 ' £ for all a > CnR2, g E C0 (JR.n+1) s.t. suppg {(x, t) : 1 -H + ¢(t)e1 ! 2:: 1}. The proof of Lemma 1.10 follows by conjugating the operator (i8t + .6. ) with the weight exp (a 1 -H + ¢>( t )e1 1 2) , and splitting the resulting operator into a Her mitian and an anti-Hermitian part. Then, the commutator between the two parts is positive, for g with the support property above and 2:: CnR2• In order to use Lemma 1.10 to prove Theorem 1 .9, we choose (}R, (} E C00 (!Rn), ¢ E C0([0, 1]) so that BR (x) 1 if lxl < R - 1, On(x) 0, lx l 2:: R; O(x) 0 if lx l < 1 , O(x) = 1, when l x l 2:: 2; 0 � ¢> � 3, with ¢> = 3 on [� - �� + �] and ¢ = 0 on [0, 1 /4]U[3/4, 1]. We apply Lemma 1.10 to g (x, t ) = (}n(x ) · O ("H + ¢ (t)e1 ) u(x , t), a R2, to obtain, after some manipulations, the desired result. We next turn our attention to corresponding results for the KdV equations. In [Z1] it is proved that if + a�u + 0, in JR. [0, 1], and uo(x) u(x, O), u 1 (x) u(x , ) are supported in (a, +oo) or in (-oo, a), then u = 0. This was later extended by [KPV l ] , [KPV2] , who also showed that if v1 , v 2 -4

C

a

=

=

�

OtU

=

UOx U =

=

l

X

==:

212

CARLOS E. KENlG

are solutions of

OtV + 8�v + VkO V = 0, k 2: 1 , x and u 0 = vi (x, 0) - v2 (x, 0) , u i = V I (x, 1) - v2 (x, 1) are supported in ( a, +oo) or in ( - oo , a), then VI = v2 . Further results are due to L . Robbiano ( [R) ) . He considered n a solution to

( 1 .1)

OtU + o;u + a2 (x, t)o;u + ar (x, t)8xu + ao(x, t)u

=

0

with coefficients aj in suitable function spaces. He showed that, if u (x , 0) x E (b, oo) some b, and 3 CI , C2 > 0 s . t . J� u (x , t) J :s; CI exp ( - C2 x " ) ,

0,

(x, t) E (b, oo) x [0 , 1]

for some a > 9/4, then u = 0. On the other hand, the Airy function

Ai (x) is the fundamental solution for

IAi (x ) J =s;

=

j e2"ixHeid�

OtU + 8�u = 0,

and verifies 2 C(l + x_ ) - I/4 exp(- Cx � ).

We now have

1 . 1 1 ( [EKPV3]) . !f u is a solution of ( 1 . 1 ) on JR. x [0, 1] such that 3/2 u(x, O) , u (x, 1) E HI (e0x+ dx) for any a > 0, and aj belong to suitable function spaces, then u = 0 THEOREM

3/2

This is clearly optimal for OtU + 8�u = 0. The same result holds for eux_ dx . The proof of this theorem also has two steps, one consisting of upper bounds, the other of lower bounds. The second step follows closely that used for Schrodinger operators, but the upper bounds can no longer be obtained by any variant of the energy estimates. These are now replaced by suitable "dispersive Carleman esti mates" . A typical application of Theorem 1 . 1 1 is:

THEOREM 1 . 12 ( [EKPV3] ) . Let

Ul J U2 E C( [O , 1]; H3(IR)) n L2 (JxJ 2 dx) ,

solve Assume that

u i ( · , 0) - u2(·, 0) , u1 ( ·, 1 ) - u2 ( - , 1 ) E H 1 (e ax+ dx) 3/2

for any a > 0 . Then ui

=

u2 .

Finally, we end with a result that shows that this result is sharp, even for the non-linear problem.

THEOREM 1.13 ( [EKPV3] ) . There exists u ¢ 0, a solution of s.t.

Ot U + 8� u + Uk Ox U

=

0

in IR

X

[0, 1]

2 J u (x, O) J + J u (x, 1) 1 =s; C exp( - Cx � ) .

QUANTITATIVE UNIQUE CONTINUATION . . .

213

2. Convexity properties of Gaussian means of solutions to Schrodinger equations

As mentioned before, (EKPV2] proved that if u E C((O, 1 ] ; H1 (1Rn)) solves in !Rn (0, 1] iBtu + 6.u + V (x, t)u = 0

{

and ·ui

x

u(O) = uo u(1) = u1

E L2 (ea l "' l 8 dx)

for some a > 0, (} > 1, then 3Ce > 0, b > 0 s.t. sup f

O Ce

e*l 9 l u(x, tWdx < oo

when the (complex) potential verifies I I V I I q L� � e, E = E1, . We will next re examine this result and precise it, in the case (} = 2. \Ve will first deal with potentials V V (x), V real valued; I I V I Ioo ::; M1 . We will consider u E C([O, 1] ; L 2 (1R11 } ) which verifies 8t u i(l:,.u + Vu) in !Rn [0, 1] . We will assume that there exist positive numbers o: and ;3 such that ! l e l x l 2/�7u(O) I I , l l e lxl2 / a 2 u(1) II are finite. Here and in the sequel I I II denotes the L2 norm in Then =

=

X

l e:xl2 /(at+( l -t)/:l)2 u(t) l

x.

·

at+( l -t) f:l

is "logarithmically convex" in [0, 1] , i.e. THEOREM 2.1 ( (EKPV5] ).

have

There exists N =

N(o:, ;J)

so that for O <

s

< 1

·w e

l elxl2/(a.•+(l- s){3)2 ( ) I I CN(M! +!vq) I l e l x l• ;a2 u(O) 1 /:l(l-s)/(as+(l-s){:j) I I e ',·" l 2 /"'2 u(l) l as/(as+( l - s)/3) . �

v. s

x

,_,

Moreover ( "smoothing effect ")

I Jt(l - t)elxl2/(at+( l -t)i3)2 vu(t) l £2 (JR"

�

(0,1}) 2 NeN(Ml+Mf) 1 1el :r. I 2 1�\,.(0) I I + i l e l x l 2 / o: u(1 ) 1 1 . X

[

]

Note that when o: = ;3, we have o:t + (1 - t)j:J = o: and this gives the precise version (for this case) of the [EKPV2] result . We start with the sketch of the proof iu the case = ;3. It turns out that a formal argument giving the proof is not too dificult, but a rigurous justification is tricky. This is imp ort nt fact, since, as we will see, the formal arguments actually can lead to false results. To justify the interest of the case o: =f. ;3, consider the case V 0, i.e. the free particle. Then, if o:

uo

= u(O),

an

=

a

214 so

that, with Ct = (2it) n/ 2 ,

CARLOS E. KENIG

Ct e -i l xl2 /4Lu(x , t ) = ( eil· l 2 /4tuot(x j2t) .

In this context the Hardy uncertainty principle says that if

u( O ) E L2 (e 2 1xl, /.82 dx),

with a/3 � 4, then uo

=

u ( l) E L2 (e21xl2/o? dx ) ,

0 and 4 is sharp.

KEY CoNVEXITY LEMMA 2.2 (abstract) . Let S be a symmetric operator, A an anti-symmetric one (possibly both depending on t), F a positive function, f(x , t) a "reasonable function". Let H(t) = (!, f), D(t) = ( Sf, f) , 8t 8 = St and N(t ) = D(t)j H(t) (the "frequency function"). Then i) of H = 28t Re ( 8tf - Sf - Af, f) + + 2 ( Stf + [S, AJ J, f) + ! lot ! - Af - Sf W - l l 8d - Af - Sf ! l 2 and ii) N(t) :;::: ( St f + [S, A]f, f) /H - ! ! 8d - Af - Sf l !2 /(2H) iii) Moreover, if !od - Af - Sf! � Mt !f! + F and

in Rn

x

[0, 1],

St + [S, AJ � -Mo

M2 = SUPO$t:9 1 ! F (t ) l ! / l ! f (t) l ! < oo , then H(t) is "logarithmically convex" in [0, 1] and H(t ) � e N(Mo+Ml ·I M�+M,+M;)H (O) l- t H (1) t .

PROOF.

H(t) = 2 Re (8t f, f) = 2 Re (8d - Sf - Af, f) + 2 (Sf, f) ,

so

H(t ) = 2 Re (8d - Sf - Af, f) + 2D(t) .

(2. 1)

AlHo,

H(t) = Re (ot f + Sf, f ) + Re (otf - Sf, f) ,

D(t) = 2 Re (8tf + Sf, f) - '2 Re (utf - Sf, f) .

1

1

Multiplying

H(t) D(t) = Re (8t f + Sf, J) 2 - 21 Re (8tf - Sf, f) 2 . 2 Adding an anti-symmetric part does not change the real parts, so

(2.2)

(2.3)

.

1

H. (t) D( t )

=

1 , 2 2 1 2 Re (8t f + Sf - Af, f) - 2 Re (8tf - Sf - Af, J) .

Differentiating D(t), we get (2.4)

D(t) = ( Stf, f) + ( Sotf, f) + (Sf, ot f) = ( Stf, f) + 2 Re (od, Sf) = = ( Sd + [S, A]f, f) + 2 Re (8tf - AJ, Sf) =

= ( Stf, [S, Al f, f) + 21 1 ! 8t ! - Af + Sf ! ! 2 - 2 i l 8d - Af - Sf ! !

1

2

QUANTITATIVE U N IQ CE CONTINUATION . . .

by polarization. This and

2 15

(2.1) gives i). Next,

iv(t) = (Stf + [S, A]f, f) /H+

+

� [ilotf - Af + Sf l l2 [ [ f [ [ 2 - ( Re (otf - Af + Sf, f)) 2] /H2+ � [( Re (otf - Af - Sf, !))2 - ilatf - Af - Sf WI I f l l2 ] IH2 +

follows from (2.3) . Now, the second line is non-negative (Cauchy-Schwartz),

(Re (otf - Af - Sf, !))2 ;::: 0, so ii) follows. When we are in the situation of iii),

.l'l' (t) ;::: -Mo + M12 + M 2 , 2 so that (2 . 1 ) now gives Ot [log H(t)] = 0 ( 1 ) + 2N( t ). If G' (t ) = 0(1), G(O) = 0, we get 8t [log H( t) - G(t)] = 2N(t), so that ot[log H(t ) - G(t)] ;::: -(Mo + M� + Mi ), •

so that

at [log H(t) - G(t) + (Mo + M� + MJ ) t2 /2) ;::: 0 D

which gives the desired "log convexity" .

Sketch of Proof (a = {3 = 1) . Let us now indicate how the "formal argument" 2.1 would follow, when a = {3 = "Y· Suppose now (for later use) that for t.be fu·st part of Theorem

OtU = (a + ib) (6u + V(x, t)u + F( x, t)) in �n

x

[0, 1],

a � 0, [ [e1'1xl2 u (O) [ j < SUP[o,J] j j e1'1xl2 F(x, t ) [ [ / [ [ u (t) [ i = M2 , [ i e1' 1 xl 2 u(1 ) [ i < V is complex valued, I I V I I oo � M1. Let f = e u, where ¢ (x, t) is to be chosen. Then, f verifies otf = Sf + Af + (a + ib) (Vf + e1'¢ F), oo ,

oo ,

where

'Y

S = a(6 + 12 [\i'¢[2)

- ib"Y(2\7¢ · \7 + 6¢) + /Ot
so that

and the Lemma "gives" the (formal) "log convexity" result. We need to have an argument which gives us the required smoothness and decay to justify the formal argument. Before doing that, we give the "formal" argument for the smoothing estimate: first note that integration by parts shows that

J [V'!12

+ 4'Y2 1x l 2 1 f l 2

=

J

e2, Jx l2 (j Y'u [2

- 2nf' [ u [2 ) dx

CARLOS E. KENIG

216 where f give

=

e' lxl 2 u. Also, since

Adding we obtain (2 .5)

2

Recall

(/

n =

V' · x , integration by parts and Cauchy-Schwartz

I V'f l 2 + 4'"'? 1 xl2 1fl 2

2:

) f

e2,[xi, I Y'u l 2dx .

28t Re (8tf - Sf - Af, f) + 2 ( Stf + [S, A] f, f) + + l l 8tf + Sf - Af l l2 - I l ot ! - Af - Sf W 2: 2: 28t Re ( 8t f - Sf - Af, f ) - l l8tf - Af - Sfl l 2 + 2( Sd + [S, A]f, f ) . Multiply by t(l - t ) and integrate by parts to obtain ot H(t)

=

1a Af, f) + 11

1

2 1 t(l - t) ( St f + [S, Al f, f )dt + 2 1 H(t )dt � H ( ) + H ( O) + 1 t(l - t) ll8tf - Sf - Afl l2 dt. + 2 (1 - 2t) Re (8tf - Sf -

1a

1a

We now use

St + [S, A ] = - l'(a2 + b2) [86 - 321'2 lxn lot! - Sf - Afl � Ja2 + b2 (M1 I f l + e1 1x 2 1 FI) ,

to obtain:

16')'(a2 + b2)

Finally, \1 f

11 j t(1 - t) [ {

{ I V'fl2 + 4')'2 l x l2 lf l 2 } �

2

� ( N M + 1) su� ll e' l xl 2 u(t) 1 1 + sup ll ellxl 2 F l 1

[0, 1J

[0,1]

]

2 (a2 + b2 ) .

e"l x l 2 ( 'Vu + 2xu')') , and (2.5) gives the bound: (!' > 0) l l vt(l - t)e' 1 x 1 ,'Vu i i £Z(Rn x[o,1]) + 1 1 Jt (1 - t) lxl e'"� l x l 2 u i i£Z CJR:" x [o, ]) � 1 5_ N + v'"M; + 1"-h ) sup ll e'Yix l 2 l u(t) 1 11 + sup ll e"r l x l 2 F l l £2 (1R" x [O,l ] ) · [0,1] [0,1] =

]

[(1

How to justify the formal arguments? We first change i(L-.+V) by (a+i) (L-.+V), we change l x l 2 by lx l2 -<, a > 0, E > 0 and then pass to the limit. This can be justified when V = V( x ), real, bounded. This is how we proceed: LEMMA

satisfies

2 . 3 (Energy method) . A ssume that u E L00 ([0, 1]; L2)

= (a + ib) (L-.u + V(x, t )u) + F( x, t) in Rn a > 0, b E R . Then, for 0 5:. T � 1 , OtU

e-Mr lle'a l x l 2 / (a+41 (a,+ b2 )T) u(T) I I �

x

n

1

£2 ([0, 1]; H1 )

[0, ] ,

�

ll e' l x l 2 u(O ) I I + J� b2 ll e'a l x 1 2 /(a+41(a2 +b2 ) T) F l l +__ a2_

Ll ([O,T];£2) '

QUANTITATIVE UNIQ1JE CONTINUATION . . .

I J a Re V - blm VII Ll([O,TJ;L"") · PROOF. For ¢ real, to be chosen, v = e<�> u , v verifies OtV = Sv + Av + (a + ib)eF in IR.n (0, 1], where S =sym, A =anti-sym, S = a(6 + I Y' ¢1 2 ) - ib(2V'¢ · V' + 6¢) + (at¢+ aRe V - blm V) and A = ib(6 + IY'¢1 2 ) - a(2V'¢ V' + 6¢) + i(bRe V - aim V). Dtll v W = 2 Re (Sv, v) + 2 Re ((a + ib) eri>F, v) (formally) . where Mr

=

21

7

x

·

A (formal) integration by parts gives

Re ( Sv, v) = -a j I Y'v l2 j (ai Y'¢12 8t ¢)1 v l 2 + + 2b m I v\7¢ . V'v + I (aRe b m V)l v J 2 . +

+

v- l

l

Cauchy-Schwartz gives

Otll v (t)W :S 2JJaRe V - blm VJiooll v (t)JI 2 + 2Va2 + b2 J JecP F(t)JI II v (t)ll when :s o.

(a + �) IY'¢12 + Dt

h(t) = "'(aj (a + 4"'( (a2 + b2)t). To formalize the calculations, given R > 0, ¢R(x) { �J2 JJ xx Jl :S RR ' �

=

choose a radial mollifier ()P and set

r/Jp, R (X, t)

= h(t)Bp 0, Gaussian decay at t = 0 is preserved, with a D Next, we prove that if u L 00 ( [0, 1 ] ; £2) £2 ([0, 1] �H 1 ) verifies OtU = (a+ ib)(!::, u + V(x , t)u + F(x , t)), p

�.

*

*

*

p

E

n

-->

-->

oo,

2 18

CARLOS E. KENIG

where I I V I I u "' � M1 , SUP [o , 1J I I e-r l x l 2 F(t) l l / l lu(t) l l = M < oo, and l l e'lxl2 u(O) I I , 2 l l e'lx l 2 u(1) 1 1 are finite, we have a "log convex" estimate, uniformly in a > 0, smalL In fact, we now repeat the formal argument, but replace ¢(x) = l x l 2 by lx l � 1

l x l 2: 1

and then by ¢ , p (x) = eP * if;€ , where eP E C'Q is radiaL We then have: ¢€,p E C1 , 1 , it is convex and grows at infinity slower that l x l 2-' and 0 , E > 0, p > 0, our argument applies rigurously, since u(O)e'lxl2 E L2 => 0 < t < 1 , u(t)e'Yixl z _, E L 2 , and for a t independent if; ,

€

St + [S, A] = -'Y(a2 + b2) (4\7 (D2¢\7) - 41'2 D2¢\7¢ . \7¢ + .t:-2¢) .

One can see that l l .t:. 2 ¢e,p l l oo ::::; C(n, p)t:, which gives the desired log convexity when 0, then p ____, 0, for a > 0. Once the log convexity holds, for a > 0 again, the "local smoothing" argument applies. The conclusion of these considerations is: ·

€ ____,

LEMMA 2.4. Assume that u E L 00 ([0 , 1]; L2(1Rn)) n L2( [0, 1]; H 1 ) verifies Btu = (a + ib) (.t:.u + V(x, t)u + F (x, t)),

'Y > 0 where a > 0, b E IR, I I V I I oo ::::; M1 . Then, 3N-y s.t.

in lRn

x

[0, 1] ,

sup l le1'1xl2 u(t) l l ::::; [0, 1]

::::; e

N"� [(az +bz)[M�+M�]+ �(M, +Mz )]

l l e-y lxl z u(O) I I l -t l l e'Y i xl2 u(1) W ,

l l vt( l - t) e'Y i xl 2 u iiL (Rn x [O , l] ) ::::; N-y (1 + M1 + Mz ) 2 where Mz

=

SUP[o,1] l l e'Yi xl2 F (t) l l/ l l u(t) l l < oo.

{

sup l l e-r l x l 2 u(t) l l [0.. 1]

}

,

Conclusion of the argument when V ( x, t) = V (x), real. We now consider the Schrodinger operator H = .t:. + V, which is self-adjoint. We consider u E C([O, 1]; L2) solving Bt u = i((L. + V)u) in IRn X [0, 1] and assume that l l e'lxl2 u(O) I I < oo, l l e1'1x l2u( l ) l l < oo. From spectral theory, u(t) = eiHt u(O). Moreover, for a > 0, consider the solution of BtUa = (a + i)((6 + V)ua )

in IRn

X

[0, 1], Ua (O)

=

u(O).

We now have Clearly

= l l e'Y i xl 2 u(O) I I · a Also, ua(1) = e Hu(l). Recall, from the "energy method" that if

l l e'Y i xl2 Ua(O) I I

{

Btv = a(L. + V)v v(O) = vo

'

V real,

l l e'Yalxlz /(a+4-yaz) v (1) I I � exp(Ml ) l l e-rlxl z vo l l '

219

QUANTITATIVE UNIQUE CONTINUATION . . .

= u(1), then v(1) = eaHvo

where that

M1

Let Ia obtain

= 1/(1 + 41a) and apply now our log-convexity result for ua, Ia · We then

=

l l aVI I Ll([o,l];L=)·

Now, if vo

l e'Yixl2 /(1+4rya) Ua (1) I

::=;

=

ua (1), so

exp(Ml) j je'Yi x12 u(1) I I ·

l l e'Ya lx l 2 Ua(s) l l :S eNMl l le'Yalx l z Ua(1) 1 1 1 - s l le'Y" Ixl2 Ua (O)W :S

:::; eNMl exp(Ml ) l l ellxl2 u(1) w-s l l e"Yixl 2 u(O) W .

We then let a - 0 and obtain the "log convexity" bound. To obtain the "local smoothing" bound, we again use the ua , let a ! 0. This establishes Theorem 2.1 when a = (3.

REMARK 2.5. Solutions so that e'Yix l 2 u(O), el l x l 2 u(1) E L2 certainly exist for some I· In fact, if h E L2 (ef l x l2 dx) and u0 = e6Cli.+V) h, our "energy method" gives this for u(t) = ei t (li.+V) u0 , (V = V(x)) . (We are indebted to R. Killip for this remark. ) When V = 0, this characterizes such u! (see [EKPV4]) .

A misleading convexity argument : Consider now f = ea(t) J xl 2 u , where u

solves the free Schrodinger equation

OtU = i6u in IR x [-1 , 1]. Then,

f verifies

otf = Sf + Af, S = -4ia(x8x +

In this case we have

St + [S, A]

=

�)

+ a'x2,

a' 2 --;; S - 8ao; +

(

A = i(o; + 4a2x2 ) 32a3

+ a" - 2

-)

.

(a1 ) 2 - x2 • a

If a is positive, even, and a solution of (a') 2 32a3 + a" - 2 = 0 in [-1, 1], a then our formal calculations show that 8t (a 1 8t log Ha (t) ) � 0 in [-1, 1 ] . Hence, for s < t we have

-

--

a(t)8t log Ha (s) :S a(s)8t log Ha (t) . Integrating between [- 1 , 0] and [0, 1] and using the evenness of a, we conclude Ha( O) :S Ha ( - 1 ) 112 Ha (1) 1 12 .

Now, if a solves

{

32a3 + a11

-

2 (a ) 2

�

a (O ) = l, a'(1) = 0

=

0

a is positive, even, and limR-""' Ra(R) = 0. Also, aR(t) equation. If the formal calculation holds for HaR '

= Ra(Rt) also solves the

CARLOS E. KENIG

220 In particular, u = 0. But

u(x, t) = (t - i) -112 eilxl2 f4(t -i) is a non-zero free solution,

which decays as a quadratic exponential at t

=

±1.

3 . The case a -:;6 (3 ; the conformal or Appel transformation

Assume u(y , ) verifies 85u = (a + ib) (6u + V ( y , s) u + F(y, s)) in !Rn [0, 1] , a + ib -:;6 0, a > 0, (3 > 0, IR and set ..j(ifJ X fjt u(- x, t) = a(l Jcif3 u - t) + fjt a(l - t) + fjt ' a (l - t) + f]t (a - f3)1xl2 x exp 4(a + ib) (o(l - t) + {3t) Then u verifies OtU = (a + ib)(6u V ( x , t)u + F( x , t)) in !Rn [0, 1], ( o/3 fjt x V o ( l..j(ifJ V x t) = (o(l - t) + /3t)2 , - t) + ,6't ' o ( l - t) + ,6't ' X fjt F( x t) = ..j(ifJ F o(l..;c;p , (o( l - t) + f3t ) � +2 - t) + ,6't ' o(l - t) + f3t · Moreover, if = ,6'tj (o(l - t) /3t), [ 7"'/3 �a -,:3)4 ] 1 12 u (s ) I e'"Yixl u(t) II e [ a/3 I I eil x l " F(t) I I = (o(l - t) + (3t) 2 e (<>•+ll(l-s))2 4(a2+b2)(c.s+;J(l ] 1 y12 F( s) . s

LEMMA 3.1.

x

'Y E

) (

(

)

n/2

(

+

+

s

=

2

I

•

(

x

(

(o.s+ll(L- s))2 + 4(a2+0 )(o.a+il(L-s))

I

)

7nil

+

y

(u-{l)a

•))

I

)

.

)

I

The proof is by change of variableH. Conclusion of the proof of Theorem 2.1: \Ve can assume o -:;6 /3. We can also assume o < (3 (change for u( l - t)). (This gives (a - fj)a < 0.) As before ,

u

Ua e(a +i)tHu(O) eatHu(t) ,

= H = (6 + V), By the "energy estimate" we now have

=

a > 0.

ll e l xl2 /a2 ua ( l) l l s; eaiiVIIoo ll el xl,/a, u ( l)l l and I Jelx l 2 /�2 Ua (O) I I s; ll e ! xl 2 N2 u(O) II · We now have also OtUa = (a + i) ( 6·ua + Vua), so when we do the Appel transform, we have, with 'Ya = l foa f3a, a Otiia = (a + i)((6 + v )ua ) , where

ya (

)

aaf3a

(

V (aa (l� -

)

X x , t = ( a a( l - t ) + f3a t) 2 t) + f3a t . Now, fo r a > 0 we have "log convexity" in this last problem. Moreover, by the Appel Lemma and our definitions, we have

ll elalx l \ia(O)II s; llelxl 21!3\t (O) II ' ll e-ralxl2 ua(l)ll s; eai!VIIoo llel xl 2/a2 u( l) ll ·

22 1

QUANTITATIVE UNIQUE CONTINUATION . . .

Thus,

I e"1a lx l 2

Ua (t)

I

:S:: e N( l +M, +Mf) ea i i V I I "'

II

e lxl l /132 u (O )

1 1 -t ll e lx l 2 /a? u ( 1) w

and the corresponding "local smoothing" eHtimate. But now, letting a -+ 0 and changing variables our result follows. Time dependent, complex potentials: We will consider complex potentials V(x, t) , I I V I I oo :S:: 1'\tfo . We will also assume

N!:!!o I I V I I £l ((O, lj, L oc (lxi>R)) = 0.

We first recall a result in [KPV3] . LEMMA

3.2. There exists N

=

N ( n) , t:o

=

t:o (n) > 0 so that, if X E !Rn,

V E L 1 ( [0, 1]; L00), I I V I I £l ((O.l);Loc) :S:: Eo , then if u E 0([0, 1]; L2 ) satisfies OtU = i(6.u + V(x, t)u + F(x, t )) in !Rn x [0, 1], then

THEOREM

3.3. Let V

0([0, 1 ] ; L2) solve

E

L}L�,

limR-u I I V I I u ( [O, lJ,Loc(l x i>R))

Ot U = i{6u + V (x, t)u)

Assume in addition that V E

L

00

(!Rn -.- l ) , and that

ll el x l 2 /i32 u {O ) II < oo ,

Then, 3N = N (a, {3) s. t.

sup ll e lxl2 / (at+(l-t).6)2 u (t ) l l [0,1 )

+

in JR.n

x

[

0. Le t u E

[0, 1].

l l e lx l2/<> 2 u (l ) ll

< oo.

ll vt( 1 - t ) elx!2 /(ot+( l -t) i3) 2 V'u(t) l l

C" N ,NI I VII�

=

l l e l • l ' 1•' u(O) II + II e l • l ' I•' u ( l )

PROOF. We start out by using the Appel transform, 1/a{), (a + ib) = ·i . We now have ii E 0([0, 1 ] ; L2 ),

I

£2(JRn x [0,1J )

+

f,�� l l u (t) I l

:::::;

l

·

ii(x, t) an
=

=

i (6.ii + V(x, t)ii), and it is easy to check that the potential V verifies Otii

l l if l loo :S::

and limR--.o l l if l l u ( [o,l] , L=(lR"\Bn) l l ii{t ) l l

=

l l u(s) l l ,

IWJJC

=

{ �' �}

0. Also, we

I I V I I oo have

8 =

Choose now R > 0 such that 1 1 �7 1 1 v ( [o.t) ,L""(lR"'\B R ) ) :::::; Eo , Eo Then,

a::;

,6t --:-: ---;- +-{3 t)7t · a ( 1--

in Lemma 3.2.

222

CARLOS E. KENIG

where VR(x ,

t) =

XIR"\BR V(

x , t) , FR

=

X Bn V

u

.

By the Lemma we have:

e -1 -XI2 1 2 and integrate both

-f

-f

Now, replace ..\ by 2..\vf'Y, square both sides, multiply be sides with respect to 5: in !Rn . Using this and the identity

.....

J e2.JYXxe- i XI2/2d); (27r )nf2 e2-y[x l2 , =

we obtain the inequality tE[O,l)

sup

l e-rlxl\ i(t) I I ::; ::; N

[llel x i2/.C12 u( O) II + l el x l 2 /<>2 u ( l ) I I + sup IIV(t) II sup ll u(t) I l

[0,1]

To prove the regularity of u , gives

For 0

< a < 1 , set

F

we

a(t)

[0,1]

l

proceed as follows: the standard Duhamel formula

=

2· .

a+z

eat/::, (v(t)u(t))

,

Clearly, We now have, from the "energy estimates" , with

'Ya

=

�,

[0,1] l elal xl 2 ua(t)l l ::; sup [0,1] l e' l"'l2 u(t) l sup l le1'a.l x l 2 Fa (t) l ::; elrVIIoo sup l e-rl x l 2 u (t) l · sup

�.�

�.�

But then, our formal "smoothing effect" argument applies and gives: (using the first step) (key Lemma)

I I Jt(l - t)Y'ii.ae1'a.lx l2 11 L2 (1R" x[O,l] ) ::; N eNIIVIIoo [llel x l2 //32 u (O) I + I el x 12 /<>2 u(l) I + sup II V( t ) II sup ll u(t) Il We now let

a

[0,1]

-+

0.

[0 ,1]

l

·

0

QU ANTITATIVE UNIQUE COKTINUATION

223

_

4. The Hardy uncertainty principle

Recall that for free evolution, OtU = i6u, Hardy's uncertainty principle says that if u(O) E £
THEOREM 4. 1 . Let V

I I V I I oc

< oo , limR_,o

solution of

V(x), V

real, [jV[[oo < oo, or V = V (x, t), V complex,

I I V I Iu([lt,l},T�oo( l xi>R)) = 0 . Assume that u E C ( [O, 1] ; L 2 ) is a =

OtU =

i(.6u + V(x, t)u) in IR.n [0 , 1], such that elx l 2 u(O ) E L2 , e l x l 2 /a2 u( 1 ) E L2, and, Ct,B < 2. Then //3 2

x

u =

0.

Preliminaries: Let 'Y = 1 /a.B . Using the Appel transform and our convexity and "smoothing" estimates we can assume, without loss of generality, that the following holds for 'Y > 1 / 2 : (4 . 1 )

sup ll e�' l xl 2 u(t) ll (0, 1 ]

I 0,1]

+ sup yft( 1 - t)e'Yi x '

£2 [

2 V'u(t ) l

l

£ 2 (R" X [0, 1 ])

<

oo.

Let me first give a formal argument, in the spirit of our "log convexity" in equalities. If e 1 = (1, 0, . . . , 0), R > 0, set f = el-'lx+ Re,t ( I-t) l 2 u , where 0 < J1 < ., , and H (t) = (!, f). At the formal level, it is easy to show (for the free evolution) that oz log H (t) ;:::: -R2 /4J1, so that H(t)e-R2 t(l - t)/&J-L is log convex in [0, 1] and so Letting J1 i

'Y

H ( 1/2) S JI {O ) l /2 H ( 1 ) lf2 e R2 /32�-' .

we see that

J

e 2,[x+ .!!p- [2 [u( 1/ 2) 1 2 S ll er-l x l 2 u( O ) ll ll e' l x l 2 u( 1 )

Thus,

r la(• R/ 4)

I eR2 /32�r .

[u(1/2) 1 2 S ll e/JxJ2 u( o) l l l l e-yJxJ2 u(1) 1 1 e [R2 (l - 412 (1 - E) 2

)Jj:32·r ,

0 < f. < 1 , which implies u(1/2) = 0 as R -+ oo, b > 1/2). The path from the formal argument to the rigorous one is not easy. Vl/e will do it instead with the Carleman inequality: [0,

LEMMA 4. 2 . Let ¢>(t) , 1./J( t) be smooth functions on [0, 1 ] , g (x, t) 1]) , e1 = ( 1 , 0, . . . , 0) . Then, fur· J1 > 0, we have (for R » 0},

jj [·tj/'(t) - 3�: [¢"(tWJe2
S

Then

S

2

E

Cgo (JRn

e2'1/.>(t) e2�tl � -¢(t)e, l 1 (i Ot + 6)gl2.

e�-' l � +tt>(t )e1 [ 2+w (t) (i8t + 6)g = 51-' f + A�-' f'

x

224

CARLOS E. KENIG

where S1, = s; , AJJ. inner product ) , and

- A� {the adjoints are now with respect to the L2 ( dxdt)

=

41-L !:._ + ¢e 1 A11 = vVe then have:

( R R

Jl e2w{t) e21' i }4: -¢(t)e, l 2 j ( i8t

)

+

·

V' -

21-Ln

R2

- 2iw.t/

( xR1 + ¢e 1 ) - ·i'lj/.

6 )g j2 =

= ( ( SJJ. + AJJ. ) J, ( SJJ. + AJJ. ) f) = {SJJ.j, SJJ. J) + (AJJ.j, AJJ.f) + + (S11 f, AJJ. f) + (A�' f' SJJ. f ) � { [S�' , A�']! , f ) .

We now compute [Sf! , AIL] and obtain: ' '13Jt [Sf! , A IL] = - R'2 !::,

+

32 /.L3

R4

Thus,

1

1 R + 2/.L ( � X

2 + ¢el + +

¢ e 1 ¢11 + 2t.L (¢1 ) 2 - Bi q/ Bx , + 'lj/1•

�

)

and the Lemma follows. Next, choose ¢(t)

=

't//' (t) -

t ( l - t), lj.•(t) = - ( 1 + E)

R4

( /') 2 (t) = 321-L q

(1

1�: t(l - t). Then

+ t: ) 4 - R4 = .!__ 4 R R

8{.l

and so our inequality reads, for g E C0 (JR.n

x

8{.1

[0, 1] ) ,

8!-L

0

22!j

QUANTITATIVE UNIQUE CONTINUATION . . .

+

We next fix R > 0, recall that u solves i8t'u t:::. u = Vu, and that the estimates (4.1) bold. Choose then rJ ( t), 0 :::; rJ :::; 1 , rJ = 1 where t ( 1 - t) � 1 / R, 1) = 0 near t = 1 , 0, so that supp rJ' c {t(1 - t ) � 1/R }, lrJ' I � C R.

g(x , + (

Choose also M » R, e E COC (IRn), and now set t) rJ(t)(}(x/Af)u(x , t), which is compactly supported in IR" x (0, 1 ) , so that our estimate holds.

( t·at -t-· u. /\ )g Finally,

let

J.L =

+

vg

=

I

Zr]' ( t)e

·

+

(x)

II

M

u

=

x)u 1 " () ( TI

TJ'iu.

+

+ 2vO(x/M)·vu)

III.

M

(1 + t) - 3rR2 • Our inequality then gives:

� ( 1 �t) 3 Rz if e2•i>(tJ ez,, J J'i H( tJed2 JgJ2 :::; �

The

contribution

JJ e21i,(t) e2'' 1 -TI H(t)c t j2 {I + II + TIT} .

of I to the right hand side is bounded hy I I V I I ::>a

jj ez..!(t)e2., j i;"H(t)P, I ! Igl2, =

so that, if R is very large, we can hide it in the left side, t o see that we only have � to deal with II and III. Recall that '1/J(t) (1 + t) 1�>(1 - t) 0, so e2•b(tJ � 1. On the support of 1)1 , we have t( l - t) � 1/ R, so that 0 � ¢(t) � 1/ R. We 11ow estimate 2 J.L

I � + ¢(t)e112

=

(�:��:J { I� 1 +

� ¢(t ) ¢(t)2 } ( 1 �'Yt) :J l xl 2 + , 2, 2"y 2 z 2 ,/x R t) + ( 1 t 3 R ¢ (t ) :::; 21 1:r l + ) ( 1 + E) 3 1 q;( 2

+

2

=

+

C,

+

on supp rJ', where ¢(t) :::; 1 /R. Thus, because of {4. 1 ) , the contribution of II is bounded by C€R. The contribution of II I is controlled by (recalling that rJ = 0 � when t ( 1 - t) � � )

C4 /r{

M

ji:r. IS2M

x

e, j , + i u ( , t) J 2 e21/:( t) e21• 1 1i: H (t) +

c M2

j"Jlx'r �2M I V'tt(x, t ) i 2 e2.;•(t) e2J.t l "fi +cl>(t)c, J 2 t ( l - t )R .

If we use (4. 1 ) , 7/J � 0, the bound above for 2tt j:I;j R + ¢ (t)e1 1 2 beeome::;

�

2r lxl2

( 1 + !) 3

Thus, letting .1t1 -.

+

2"Yi x t l R ( 1 + t) 3 4

R2

21 2 (l + E) 3 16 � 2rl xl + CcR·

we see that, for fixed R, II I -. 0, so that, since

t ( 1 - t) � 1/ R, we obtain: oc ,

+

:_ (1 + t)" 2 R 8 1

/r{

lt(t-t)'?.l/R

e2�·(t) e2�J "ff H( th i, J'tlf � C R .p .

17

=

1

011

22 6

CARLOS E. KENIG

We are now going to restrict to integration over the region where 8 is small, to be chosen. Then,

so that

I � I S o, I t - 1/21 S 8 ,

I.:_ + ¢(t)e1 j 2 - 2._16 - 26 (�2 - 28) . R

>

so that, in our region of integration,

smce J.l = _l_ (l+e)3 R2 . But, if 1 > 1/2, .

"Y

(1 + e) 3

-

(1 + E)4

4�r

>

0'

for some E small, and so, for /j smaller than that we get a lower bound of C,,,sR2 • We thus have But then, since

I

f

ll t - 1/21 � 6 11 7\' l>o

iu1 2 =

e2'Yixl2 e - 2')'lxl2 1 u l 2 < f f llt- 1/21�6 Jl i l 9 s

e- 2'Y82 R2

r llt-1/21 9

by (4.1), we see that , for appropriate c, ,,,t5 we have

(lr

l t- 1/2 1 �6

Letting R ---+ oo , we see that u

=

I iui 2 ) ec, ,,,6 R2

J e2'Yixl2 1ul2

s

s

C'Y e -2'Y!j2 R2

C-.,.,e,t5·

0 on { (x, t) : I t - 1/21 S 8 } , therefore u = 0. References

[A] [BK]

P. Anderson, A bsence

1492-1505. J. Bourgain and

of diffusion in certain mndom lattices, Phys. Review .

C . Kenig

dimensions, Invent. Math

[C KM]

,

109 ( 1 958),

On localization in the Anderson-Bernoulli model in higher

161 (2005), 38�426.

R. Carmona, A. Klein, and F. Martinelli, Anderson localization for Bernoulli and other

singular potentials, Commun. Math. Phys. 108

(1987) , 41-66.

QUANTITATIVE UNIQUE CONTINUATION

22 7

[EKPV1] L. Escauriaza, C. Kenig, G. Ponce, and L. Vega , Decay at infinity of calonc func tio ns within characte1·istic hyperplanes, Math . Res. Lett. 13 (2006 ) , 441-453. [EKPV2] , On uniqueness properties of solutions of SchTo ding er equations, Commun. in PDE 31 (2006), 181 1-1823. [EKPV3] --- , On uniqueness properti es of solutions of the k-generalized KdV equation.�, Jour. Ftmct. Anal. 244 (2007}, 504-535. [EKPV4] , Convexity properties of solutions to the free Schroding er equation with Gauss ian decay, to appear, MRL. [EKPV5] , Hardy 's uncertainty prin cip l e, convexity and Schrodinger evolutions, to appear, JEMS. L. Escauriaza, G. Seregin, and V. Sverak, £3·00 solution.� to the Navier-Stokes equations [ESSJ and backward uniqueness, Russ. Math. Surv. 58:2 (2003) , 2 1 1-250. .J. Frolich and T. Spencer , Absence of diffusion with Anderson tight binding model for· [FS] large dis order or low en ergy, Commun. M ath . Phys. 88 (1983), 1 5 1 -1 84 . [G MP] Y. Goldsheid, S. Molchanov, and L. Pastur, Pure point spectrum of st ochasti c one dime11.� ional Schrodinger operators, Funct. Anal. Appl. 11 (1977), 1-10. A. lonescu and C. Kenig, LP Carleman inP.quali tie s and uniqueness of solutions of [IK] non-linear Schrodinger equations, Acta Math. 193 (2004) , 193-239. [I] V. Isakov, Carleman type estimates in anisotropic case and applications, J. Diff. Eqs. 105 (1993), 217-238. C. K enig , Some recent quantitative unique continuation theorems, Rend. Accad. ::-laz. [K 1 ] Sci. XL Mem. Mat. Appl. 29 (2005) , 231-242. , Some recent applir.ations of unique continuation, Contemp. Math. 439 (2007), [K2] 25-56. [ K PV1] C. Kenig, G. Ponce, and L. Vega, On the support of solutions to generalized Kdll equatio n, Annates de l'Institut H. Poincare 19 (2002), 1 9 1-208 . [K PV2] , On th e unique continuation of s o lu ti o ns to the ge neralized Kd V equati on , Math. Res. Lett 10 (2003), 833-846. [KPV3] , On uniquP. continuation for nonlinear Schro dinge r equatwns, Commun. Pure Appl. Math. 60 (2002), 1247-1262. E. M. Landis and 0. A. Oleinik, Ge ne 1·alized analyticity and some related properties of [LO] solutions of elliptic and parabolic equations, Russ. Math. Surv. 29 (1974), 195-212. V. z. Meshkov, On the poss ib le rate of decay at infinity of s o lut i o ns of second ord fr [M] partial differen ti al equations, Math. USSR S obornik 72 (1992) , 343-360. L. Robbiano, Unicite forte a l 'infini pour KdV, ESAIM Control Optim. Calc. Var. 8 [R] (2002), 933-939. [SVW) C. Shubin, R. Vakilian, and T. Wolff, Some ham�onic analysis questions suggested by Anderson-Bernoulli models, GAFA 8 (1998), 932-964. [SS] E. M. Stein and R. Shakarchi, Complex an alysis, Princeton Lectures in Analysis, II, Princeton University Press, Princeton, NJ, 2003. B . Y. Zhang, Uniqu e continuation for the Korteweg -de Vries equation, SIAM J. Math. (Z1] Anal. 23 (1992), 55-7 1 . [Z2] , Unique continuation for the nonlinear Schrodinger equation, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 191-205. ___

___

___

___

___

___

___

DEPATMENT

OF

MATHEMATICS, UNIVF.RSITY OF CHICAGO, CHICAGO, IL 6063 7 , USA

E-mail address: ce k�ath . uchicago . edu

Proceedings of Symposia in Pure Mathematics Volume 79, 2008

Boundary Harnack Inequalities for Operators of p-Laplace Type in Reifenberg Flat Domains John L . Lewis , Niklas Lundstrom, and Kaj Nystrom In this paper we highlight a set of techniques that recently have been used to e stablish bou nd ary Harnack inequalities for p-harmonic functions vani shing on a portion of the boundary of a domain whi ch is 'fiA.t' in the sense that its boundary is well-approximated by hyperplane!;. Moreover, we use these techniques to establis h new results concerning boundary Harnack inequalities and the Martin boundary problem for operators of p-Laplace type with variable coefficients in Reifenberg flat domains. ABSTRACT.

1. Introduction and statement of main results In [LN] , [LN1], [LN2], see also [LN3] for a survey of these results, a number of results concerning the boundary behaviour of positive p-harmonic functions, 1 < p < oo, in a bounded Lipschitz domain D c R,. were proved. In particular, the boundary Harnack inequality, as well as Holder continuity for ratios of positive p-harmonic functions, 1 < p < oo, vani�:;hing on a portion of 8D were established. FUrthermore, the p-Martin boundary problem at w E 8D was resolved under the assumption that D is either convex, C 1 -regular or a Lipschitz domain with small constant. Also, in [LN4] these questions were resolved for p-harmonic functions vanishing on a portion of certain Reifenberg flat and Ahlfors regular NTA-do.tnains. From a technological perspective the toolbox developed in [LN, LN1-L�4] can be divided into (i) techniques which can be used to establish boundary Harnack inequalities for p-harmonic functions vanishing on a portion of the boundary of a domain which is 'flat' in the sense that its boundary is well-approximated by hyperplanes and ( ii) techniques which can be used to establish boundary Harnack inequalities for p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain or on a portion of the boundary of a domain which ean be well approximated by Lipschitz graph domains. Domains in category (i) are called Reifenberg flat domains with small constant or just Reifenberg fiat domains. They indudc domains with small Lipschitz constant, C1-domains and certain quasi-balls. Domains in category (ii) include Lipschitz domains with large Lipschitz constant and certain Ahlfors regular NTA-domains, which can be well approximated by Lipschitz graph domains in the Hausdorff distance sense. The purpose of this paper is to highlight the techniques labeled as category (i) in the above discussion and to use these techniques to establish boundary Harnack inequalitie!:i as well as to 2000

Primary 35J25, 35J70 . Keywords and phrases: boundary Harnack inequality, p-harmonic function, A-harmonic function, variable coefficients, Reifenberg fiat domain, Martin boundary. Lewis was partially support ed by NSF DMS-0139748. Nystrom was partially supported by grant 70768001 from the Swedish Research Council. Mathematics 81;.bject Classification.

Key words and phrases.

229

©2008 American Mathematical Society

230

JOHN L.

LEWIS . NIKLAS LUNDSTROM, AND KAJ NYSTROM

resolve the Martin boum.lary problem for operators of p-Laplace type with variable coefficients in Reifenberg fiat domains. To state our results we need to introduce some notation. Points in Euclidean n sp ace Rn a.re denoted by x = (xb . . . , Xn ) or (x', Xn ) where x' = (x1 , . . . , x,__I ) E R"- 1 . Let E, 8E, diam E, be the closure, boundary, diameter, of the set E C R" and let d(y, E) equal the distance from y E R" to E. ( , ·) denotes the standard inner product on Rn and lxl = (x, :r:)112 is the Euclidean norm of x. Put B ( x r ) = {y E R" : l x - Y l < r} whenever x E Rn , r > 0, and let dx be Lebesgue n-measurc on Rn . We let -

-

,

h(E , F)

=

max(sup{d(y, E) : y E F} , sup{d(y, F) : y E E} )

be the Hausdorff distance between the sets E, F C R". If 0 c R" is open and 1 :::; q :::; oo, then by w·l,q (O) we denote the space of equivalence clas::;e:; of functions f with distributional gradient 9 f Ux , , . , fxJ , bo th of which are q th power integrable on 0. Let. IIJII1.q = llfllq + ll l 9f l ll q be the norm in Wl. q (O) where ll · ll q denotes the usual Lebesgue q-norm in 0. Next let C0 (0) be the t:et o f infinitely q differentiable functions with compact support in 0 and let Wci'' ( 0) be the closure of C0 (0) in the norm of W 1 ·" (0). By 9· we denote the divergence operator. We first introduce the operators of p-Laplace type which we consider in this paper. =

.

.

Definition 1.1. Let p, ;3, a E (1, x ) and 1 E (0, 1). Let A = (A1 , . . . , A,. ) : Rn X an __, Rn , assume that A = A (x, 17) is continuous in R" X (R" \ {0 } ) and that A(x, 1J), for fixed x E Rn, is continuously differentiable in 17k , for every k E { 1 , . . . , n}, whenever 17 E Rn \ { 0}. We say that the function A belongs to the class A1p(a, 8, -y) if the following conditions are satisfied whP:n.ever x, y, e E Rn and 17 E Rn \ {0} :

(i) (i1: ) (iii) (iv)

a - 1 11JIP - 2 1�1 2 :::;

n 8A

L

� (x , 17)�i�j , i,j=l 17J

(x , 17 ) :::; a i1J I P- 2 , 1 :::; i , j :::; n,

I �� 1

I A ( x 17) - A ( y , 17) j :::; tJi x - Yl7lrJJ P - I , A (x , 77) = l11lp-l A ( x , 11/ l771 ) . ,

For short, we write l'vfp (a) for the class Mp (a , 0, 1) .

Definition 1.2. Let p E (l, oo) and let A E Mp (c., f3, �r) for some (a, (3, -y) . Given a bounded domain G we say that u is A-harmonic in G provided u E W1 ·P(G) and (1 .3)

j (A(x, 9u(x) ) , 98(x)) dx

=

0

whenever () E Wci- ·P(G) . If A(x, 17) = I77I P -2 (171 , . . . , 17 ), then u is said to be p n harmonic in G. As a short notation for (1 . 3) we write 9 · (A(x, 9u)) = 0 in G.

The relevance and importance of the conditions impo�ed through the assump tion A E Mp (a, (3, -y) will be discussed below. Initially we just note that the class Mp (a, (3, -y) is, �ee Lemma 2. 15, closed under translations, rotations and under di lations x __, rx, r E (0, 1] . Moreover, we note that an important class of equations

BOUNDARY HARNACK I�EQUALITIES FOR OPERATORS OF p-LAPLACE 'l'YPE

which is covered by Definition 1 . 1 and \7

( 1 .4)

·

[

2 31

1.2 is the class of equations of the type

]

(A(x)\7u, \7u)PI2-1 A(x)\7u

=

0 in G

where A = A(x) = {a;,1 (x)} is such that the conditions in Definition 1.1 (i) - (iv) are fulfilled. Next we introduce the geometric notions used in this paper. \Ve define, Definition 1 .5. A bounded domain n is called non-tangentially accessible {NTA) if there ex·ist M 2:: 2 and ro > 0 such that the following are fulfilled: (i)

(ii) (iii)

corkscrew condition: for any w E an, 0 < T < To, there exi�-;ts ar(w) E n n B(w, r/2) , satisfying M- 1 r < d(ar (w) , an), R11 \ f2 satisfies the corkscrew condition, uniform condition: if W E an, 0 < r < ro , and Wt , W2 E B(w, r) (1 n, then there exists a rectifiable curve 1 : (0, 1 ]-}n with ! (0 ) = w , �r(1) = w2 , 1 and such that (a) H1 (!) :::; M l w 1 - w2 l , (b) min{H 1 ( 1 ( (0, t] ) ) , H 1 (! ( [t , 1 ] ) ) } :::; M d (! (t) , 00) .

In Definition 1 .5, H1 denotes length or the one-dimensional Hausdorff measure. We note that (iii) is different but equivalent t.o the usual Harnack chain condition given in (JK] (see (BL], Lemma 2.5). M will be called the NTA-constant of n. Definition 1.6. Let n c R11 be a bo unded domain, w E an, and 0 < r < To . Then an is said to be uniformly (J, r0) -approximable by hyperplanes, provided there exists, whenever w E an and 0 < r < ro , a hyperpla.ne A containing w such that h(aO n B(w, r), A n B(w, r)) :::; 8r.

We let :F(8, r0 ) denote the class of all domains n which satisfy Definition 1.6. Let n E :F(J, r0), w E &0., 0 < r < ro , and let A be as in Definition 1.6. We say that &0. separates B(w, r), if ( 1 . 7)

{ x E n n B(w, r) : d(x, an) 2:: 28r} c one component of Rn \ A.

Definition 1.8. Let n C R11 be a bounded domain. Then 0 and &0. a1·e said to be (8, ro) -Reifenberg fiat provided n E F(J, ro ) and {1. 7) hold whenever 0 < r <

ro , W E a0.

For short we say that n and an are c5-Reifenberg flat whenever 0 and an are (8, r0)-Reifenberg flat for some r0 > 0. We note that an equivalent definition of a

Reifenberg flat domain is given in [KT] . As in [KT] one can show that a 8- Reifenberg flat domain is an NTA-domain with constant M = M ( n), provided 0 < 8 < J and J is small enough. In this paper we first prove the following theorem.

232

JOHN

L.

LEWIS,

NIKLAS LUNDSTRO M , AND KAJ NYSTR O M

Theorem 1 . Let n c Rn be a (o, r0) -Reifenberg fiat domain. Let p, 1 < p < oo, be given and assume that A E Mp (a., f3, /) for some (a, f3, /) . Let w E an, o < r < r0, and suppose that u, v are positive A -harmonic functions in n n B(w, 4r), continuous in O n B(w, 4r), and u = O = v on an n B(w, 4r) . There exists J < J, a > O, and c1 � 1 , all depending only on p, n, a, (3, /, such that ·if 0 < a < J, then (T u(y2 ) 1 < 11 u(yl) ( 1 YI Y2 i) og -- - l og c1 r v (y2 ) v(y i ) --

whenever Yl , Y2

E n n B(w, r/cl ) ·

We note that in [LN] we obtained for p-harmonic functions u, v , in a bounded Lipschitz domain n, whenever w E an, and Y 1 , Y2 E n n B(w, rfc) . Here c depends only on p, n, and the Lipschitz constant for n. Moreover, using this result, we showed, in [LNl], that the conclusion of Theorem 1 holds whenever u, v, are p-harmonic, and n is Lipschitz. Constants again depend only on p, n, and the Lipschitz constant for n. In this paper we also prove the following theorem.

Theorem 2. Let n c Rn, /), To, p, a, (3, , and A be as in the statement of Theorem 1 . Then there exists o* = /j* (p, n, a, f3, /) < J, such that the following is ,

true. Let w E an and suppose that u, v are positive A-harmonic functions in n with u = 0 = v continuously on an \ { w}. If 0 < a < o* , then u(y) = Av(y) for all y E n and for some constant >..

We remark, using terminology of the Martin boundary problem, that if u is W> in Theorem 2, then u is called a minimal positive A-harmonic function in n, relative to w E on . Moreover, the A-Martin boundary of n is the set of equivalence classes of positive minimal A-harmonic functions relative to all boundary points of n. Two minimal positive A-harmonic functions are in the same equivalence class if they correspond to the same boundary point and one is a constant multiple of the other. Note that the conclusion of Theorem 2 implies that u is unique up to constant multiples. Thus, llince w E an is arbitrary, one can say that the A-Martin boundary of n is identifiable with an. We remark that in (LNl] the Martin boundary problem for p-harmonic func tions was resolved in domains which are either convex, C1-regular or Lipschitz with sufficiently small constant. Also, in [LN4] the Martin boundary problem was resolved, again for p-harmonic functions, in Reifenberg flat domains and certain Ahlfors regular NTA-domains. Theorem 2 is new in the cW>e of operators of p Laplace type with variable coefficients. Recall that n is said to be a bounded Lipschitz domain if there exists a finite set of balls { B(xi , ri ) } , with X i E an and Ti > 0, such that { B ( xi ri ) } constitutes a covering of an open neighbourhood of an and such that, for each i, ,

(1 .9)

n n B( x ) an n B(x; , r;) i , r·i

{x {x

=

=

( x' , x.,. ) E Rn : Xn > ¢; (x')} n B(x; , ri ) , ( x' , Xn ) E Rn : Xn = ¢;(x')} n B(xi , ri ) ,

BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE

233

in an appropriate coordinate system and for a Lipschitz function c/Ji. The Lipschitz constant of n is defined to be M = maxi IIIV'¢i l lloo · If n is Lipschitz then n is NTA with ro = min rife, where c = c(p, n , M) 2:: 1. Moreover, if each ¢i : Rn - 1 -tR can be chosen to be either C1- or C1·a -regular, then n is a bounded C1- or C1•a -domain. We say that n is a quasi-ball provided n = f(B(O, 1 ) ) , where f = (/I , h , . , fn) : Rn -t Rn is a }( > 1 quasi-conformal mapping of Rn onto R n. That is, fi E W 1 •n(B(O, p) ), 0 < p < oo , 1 ::; i ::; n, and for almost every x E Rn with respect to Lebesgue n-measure the following hold,

.

(i) (ii)

lDf( x) l n

=

sup

l h l=l

I DJ( x )hln :S K lJt (x) l ,

J, (x) � 0 or J, (x) ::; 0.

for the Jacobian matrix of f and In this display we have written D f( x) = (�) 1 J, (x) for the Jacobian determinant of f at x .

Remark 1.10. Let n c Rn be a bounded Lipschitz domain with constant M. If M is small enough then S1 is (6, ro) -Reifenberg fiat for some 6 = o(M), r·o > 0 with 6(M) -t 0 as M -t 0. Hence, Theorems 1-2 apply to any bounded Lipschitz domain with sufficiently small Dipschitz constant. A lso, if n = f(B(O, 1)) where f is a K quasi-conformal mapping of Rn onto Rn, then one can show that an is 6-Reifenberg fiat, with r0 = L where 8-tO as K-t 1 (see [R, Theorems 12.5 -12. 7}). Thus Theorems 1, 2, apply when n is a quasi-ball and if K = K (p , n) ·is close enough to 1 . To state corollaries t o Theorems 1-2 we next introduce the notion of Reifenberg flat domains with vanishing constant. Definition 1 . 1 1 . Let S1 C Rn be a (o, ro) -Reifenberg fiat domain for some 0 < 6 < J, ro > 0, and let w E an, 0 < r < ro. We say that an n B(w , r) is Reifenberg fiat with vanishing constant, if for each E > 0, there exists r = r( E) > 0 with the following property. If X E an n B (w, r) and 0 < p < r, then there ·is a plane P' = P'(x, p) containing x such that h(Gn n B (x , p), P' n B ( x , p)) ::; f.p .

The following corollaries are immediate consequences of Theorems 1-2. Corollary 1. Let n c R" be a domain which is Reifenberg fiat with vanishing constant. Let p, 1 < p < oc , be given and assume that A E Mp(a, {3, ry ) for some (a, {3, ry) . Let w E an, 0 < r < To . Assume that u, v are positive A -harmonic functions in n n B (w, 4r) , u , v are continuous in D n B (w, 4r) and u = 0 = v on an n B(w, 4r) . There exist ri = ri (p, n, o:, {3, ry) < r and C2 = C2 (p , n, o: , ,B, ry) :::: 1 such that if w' E an n B(w, r) and 0 < r' < ri , then < C2

_

whenever

YI , Y2 E n n B( w', r') .

( IYI - Y2 l ) a r1

Corollary 2. Let [! c Rn , p, a, 13, ry and A be as in the statement of Corollary 1 . Let w E an and suppose that u, v are positi ve A-harmonic functions in n with

234

JOHN L. LEWIS, NIKLAS LUNDSTROM, AND KAJ NYSTROM

u = 0 = v continuously on on \ {w}. Then u (y ) = >.v ( y) for all y E n and for some constant >.. Remark 1 .12. We note that if n is a bmmded C1 -domain in the sense of {1.9) then n is also Reifenberg fiat with vanishing constant. Hence Corollaries 1-2 apply

to any bounded C 1 -domain.

Concerning proofs, we here outline the proof of Theorem 1.

Step 0. As a starting point we establish the conclusion of Theorem 1 , see Lemma 2.8, when A E .l'vfp(a), n is equal to a truncated cylinder and w is the center on the bottom of n .

Step A . (Uniform non-degeneracy of IV'ul - the 'fundamental inequality') . There exist <51 = <51 (p , n , a , (3, 1) , \ i = c1 (p , n , a , (3, 1 ) and :\ = :\ (p , n , a , (3, 1), such that if 0 < 6 < 8 , then 1

:s;

:s; >.

- u(y) whenever y E n n B( w, r I f:l ) . d(y , an) I f ( 1 . 13) holds then we say that I V' ul satisfies the '.fundamental inequality' in x- l

( 1 . 13)

u(y) d(y , an)

IV'u(y) I

n n B(w, rjc! ) .

Step B . (Extension of IV'uiP-2 t o an A2-weight). There exist 02 = o2 (p , n, a , f3, "f) and c2 = c2 (p, n, a, (3, "f) such that if O < 6 < o2 , then I VuiP- 2 extends to an A2 (B(w, rj(c1c2))-weight with constant depending only on p, n, a, (3, ,. For the definition of an A 2 -weight., see section 4. The 'fundamental inequal ity' established in Step A is crucial to our arguments and section 3 is devoted to its proof. Armed with the results established in Step A and Step B we introduce certain deformations of A-harmonic functions. In particular, to describe the con structions we let 0 c Rn , o, r0, p, a, (3, -y, A, w, r, u and v be as in the statement of Theorem L Let J = min{ <51 , 82 } where 81 and 82 are given in Step A and Step B respectively. We extend n and v to B(w, 4r) by defining u = 0 = v on B(w, 4r) \D.

Step C. (Deformation of A-harmonic functions) . Let r* = rjc and assume that (a)

(b) ( 1 . 14) (c)

o :s;

u :s; v / 2 in n n B(w, 4r* ),

c- 1 :S: u(ar• (w)) , v(a,.• (w)) :S: c , c-1h(ar· (w)) :s; gJ.ax h :s; ch(ar• (w)) whenever h =

nnB(w,4r•) _

u or v.

Here c � 1 depends only on p, n, a, ;3 , I· At the end of section 4 we then show that the assumptions in ( 1 . 14) can be easily removed. Hence, to prove Theorem 1 we can without loss of generality assume that ( 1 . 14) holds. We let u(·, T) , 0 :s; T :::;: 1, be the A-harmonic function in 0 n B(w, 4r*) with continuous boundary values, ( 1 . 15) u(y, T) = Tv(y) + (1 - T)u(y) whenever

Using ( 1 . 14), (1. 15), we ;;ee that if t, T u(· , t) - u(- , T) 0< ( 1 . 16) t-T

E =

y E 8(n n B(w, 4r* )) and T E [0, 1] .

[0, 1], then

< c(p' n ' a ' ·8 ' -y) v-u-

BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE

235

on a(nn B(w, 4r* ) ) . From the maximum principle for A-harmonic functions it then follows that the inequality in (1.16) also holds in n n B (w, 4r*). Therefore, using ( 1 . 16) we see that T --'>u(y, T) , r E [0, 1], for fixed y E n n B(w, 4r* ) , is Lipschitz continuous with Lipschitz norm :S c. Thus uT(y, · ) exists, for fixed y E nnE( w, 4r* ) , almost everywhere in (0,1]. Let {Yv } b e a dense sequence o f 0 n B(w, 4r*) and let W be the set of all r E [0, 1] for which uT ( Ym, ) exists, in the sense of difference quotients, whenever Ym E {Yv } · We note that H 1 ( [0 , 1] \ W) = 0 where H 1 is one-dimensional Hausdorff measure. Next, applying the 'fundamental inequality', established in Step A, to u(-, r) , r E [0, 1], we see that there exist constants c and :X, which depend on p, n, a, /3 , 'Y, but are independent of r, r E [0, I], such that if y E n n B(w, 16r'), r' = r* /c and r E (0, lj, then ·

A

-_1

( 1 . 1 7)

u(y , r) < I ( I < - u(y, r) d(y , an) - Vu y , T) - A d(y , an) . _

One can then deduce, using the fundamental theorem of calculus and arguing as in [LN4, displays ( 1 . 15)-(1 .23)] , that

(1.18)

log

( ) v (ym. ) u (ym )

=

log

(

u(ym , l ) 1i (ym , O)

)

=

f (ym , r ) d !1 u(ym r ,r )

0

whenever Ym E {Yv }, Ym E OnB(w, r' ) , and for a function f which has the following important properties,

( 1 . 19)

=

f 2: 0 is continuous in B(w, r') wit h f f (ym , T) UT (yrn, T)

(i) (ii)

=

0 on B(w, T') \ n,

whenever Ym E {yv } , Ym E n n B(w, r'), r E W. Moreover, f is locally a weak solution in 0 n B (w, r') to the equation

( 1 .20) where

(1.21) whenever y E n n B(w, r') and 1 :S i, j :S we see that

a- 1 � ( y, r)l�l 2

n.

Also, using Definition 1.1 (i ) and (ii)

:S

L ;;ij (y , r)�iej ::; a� cv. r)lel 2 i ,j whenever y E n n B(w, r') and where � (y, r) = I Vu(y , r) I P- 2 • Finally, a key obser vation in this step is that ( = u(-, r ) is also a weak solution to L in n n B(w, r'). Indeed, using the homogeneity in Definition 1 . 1 (iv) we see that (1 .22)

(1.23)

L bii (y , r)u11j (y , r) j

We conclude from ( 1 . 23) that (

=

�A'T]J (y, Vu(y, r) ) u113 (y, r)

=

L

=

(p - l)Ai (y, V u(y, r) ) .

j

u( · , r ) is also a weak solution to L.

236

JOHN L. LEWIS, NIKLAS LUNDSTROM. AND KA.J NYSTROM

Step D . (Boundary Harnack inequalities for degenerate elliptic equations) . Using the deformations introduced in Step C the proof of Theorem 1 therefore boils down to proving boundary Harnack inequalities for the operator L. The idea here is to make use of Step B to conclude that 5,(-, 7), 7 E [0, 1] , can be extended to A2 -weights in B(w , 4r"), r" = r'/(4c2) · Then the operator L can be considered as a degenerate elliptic operator in the sense of [FKS] , [FJK] , [FJK 1 ] , and we can apply results of these authors. In particular, to do this we first observe that the sequence {y,_. }, introduced below (1 . 16), is a dense sequence in n n B (w, r' ) , and V } 0 ! (· , 7) , v2 (·) = u(·, 7), are positive solutions to L, see ( 1 .20)-( 1.22) , vanishing continuously on n n B(w , r'). Second, we observe from Step B that 5, (y , 7 ) = I 'V ii.(y, 7) 1P- 2 can be extended to an A2 ( B(w , 4r") )-weight. Hence, from [FKSJ, [FJK] and [FJK1]: we can conclude that there exist a constant c = c(p, n , a, {3, 1) , 1 :s; c < oo, and (j = (j(p, n, a, ,8, 1 ) , (j E (0, 1), such that if r"' = r" jc, then =

u VI (a,.m (w)) Vt (Yl ) IY1 vl (Y2) Y21 c < ( 1 . 24 ) v2 (yi ) v2 (y2) - v2 (ar"' (w) ) ( r " ) whenever y1 , Y2 E 0 n B(w, ." ) . Hence, assuming ( 1 . 14) we see that Theorem 1 now follows from ( 1 . 18), ( 1 . 24 ) as (1 . 25 ) 0 :s; f(a,.u-('w ), -r) :s; u(ar"' (w) , -r) � c- 1 , whenever 7 E (0, 1]. ( 1 . 25 ) is a consequence of (1. 16) and ( 1 . 14) (b) .

I

1

,

' ,

c,

The proof of Theorem 2 can also be decomposed into steps similar to steps A-D stated above. Still in this case details are more involved and we refer to section 5 for details. The rest of the paper is organized as follows. In section 2 we �>tate a number of basic estimates for A-harmonic functions in NTA-domains and we obtain the conclusion of Theorem 1 when A E Mp (a) , f2 is equal to a truncated cylinder (see (2.7) and Lemma 2.8), and w is the center of the bottom of fl (Step 0) . section 3 we establish the 'fundamental inequality' for A-harmonic functions, u, vanishing on a portion of a Reifenberg flat domain (Step A). In section 4 we first state a number of results for degenerate elliptic equations tailored to our situation and we then extend IY'u i P- 2 to an A2-weight (Step B). In this section we also complete the proof of Theorem 1 by showing that the technical assumption in ( 1 . 14) can be removed. In section 5 we prove Theorem 2. Finally in an Appendix to this paper (section 6), we point out an alternative argument to Step C based on an idea in [W] .

In

2. Basic estimates for A-harmonic functions and boundary Harnack inequalities in a prototype case In this section we first state and prove some basic estimates for non-negative A-harmonic functions in a bounded NTA domain n c R11• We then prove the boundary Harnack inequality for non-negative A-harmonic functions, A E Mv (a) , vanishing on a portion of a hyperplane. Throughout this section we will assume that A E Mp(a, {3, 1) or A E Mp (a) for some (o:, ,8 , 'Y) and 1 < p < oo. Also in this paper, unless otherwise stated, c will denote a positive constant ;:::: 1 , not nec essarily the same at each occunence, depending only on p, n, M, a, (3, 1 where l'vf denotes the NTA-constant for 0 c Rn . In general , c(a 1 , . . . , am ) denotes a positive

BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE

237

constant 2: 1, which may depend only on p, n, A1, a, {3, "( and a1, . . . , am , not neces sarily the same at each occmrence. If A � B then A/ B is bounded from above and below by constants which, unless otherwise stated, only depend on p, n, M, a, /3, ; . Moreover, we let max u , min u be the essential supremum and infimum of u on B(z.s) B(z,s) B(z, s) whenever B(z, s) C an and whenever 7L is defined on B(z, s). We put .D.(w, r) = an n B(w, r) whenever w E an , 0 < r. Finally, ei, 1 :::; i :::; n, denotes the point in an with one in the i th coordinate position and zeroes elsewhere.

Lemma 2 . 1 . G-iven p, 1 < p < oo, assume that A E Mp (a, /3, ;) for some (a, /3, ;) . Let u be a positive A-harmonic function in B(w, 2r) . Then (i)

there

f B(w,r/ 2)

IV'njP dx :::;

c

( max u)P, B(w,r)

< c min u. - B(w,r) exists iT = iT(p, n, a, ,B ,;) E (0, 1) such that if x , y E B(w, r),

(ii)

Furthermore, then

rp - n

(iii)

B(w,r)

max

u

c(1x;ul)ii

iu(x) - u (y ) l :::;

max u . B(w,2r)

Proof: Lemma 2 . 1 (i), (ii) are standard Caccioppoli and Harnack inequalities while (iii) is a standard Holder estimate (see [S]). 0

Lemma 2.2. Let n c an be a bounded NTA-domain, suppose that p, 1 < p < oo , is given and that A E Mp (a, {3, -y) joT some (o:, /3, ;) . Let w E an, 0 < r < ro , and suppose that u is a non-negative continuous A-harmonic function in n n B (w, 2r) and that u = 0 on .D.(w, 2r) . Then

( i) Furthermore, there exists B(w, r), then ( ii)

i V' iP d f QnB(w,r'/2) u

iT =

x

:::; c ( max u)P. OnB(w,r)

a(p, n, M, a, {3, ;) E (0, 1) such that if x, y E n

( )

iu(x) - u(y) l :::; c lx�yi ii

OnB(w,2r)

max

n

u.

Proof: Lemma 2.2 (i) is a standard subsolution inequality while (ii) follows from a Wiener criteria first proved in [M] and later generalized in [GZ) . 0 Lemma 2.3. Let D. c Rn be a bounded NTA -domain, suppose that p, 1 < p < oo, is given and that A E Mp (a, /3, "() for some (a, /3, ;) . Let w E an, 0 < r < r0, and suppose that u is a non-negative continuous A-harmonic function in fi n B(w, 2r) and that u = 0 on .D. (w, 2r) . There exists c = c(p, n, M, a, /3, "(), 1 :::; c < oo, such that if r = TIc, then max u ::::; c u ( ar (w )) . SlnB(w,•' ) Proof: A proof of Lemma 2.3 for linear elliptic PDE can be found in [ CFMS] . The proof uses only analogues of Lemmas 2 . 1 , 2.2 for linear PDE and Definition 1 .5. In

JOHN L. LEWIS, NlKLAS LUNDSTROM, AND KAJ NYSTRO::vt

238

particular, the proof also applies in our situation.

0

Lemma 2.4. Let n c Rn be a bounded NTA-domain, suppose that p, 1 < p < oo, is given and that A E Mp (a., /3, ''!) for some (a. , /3, ')') . Let w E an, 0 < r < A r0 , and suppose that u is a non-negative continuous -harmonic function in n n B (w , 4r ) and that u = 0 on u(w, 4r ) . Extend u to B(w, 4r) by defining u = 0 on B ( w , 4r) \ n. Then u has a representative in W1·P (B ( w, 4r)) with Holder continuous partial derivatives of first order in n n B(w, 4r) . In particular, there exists & E (0, 1], depending only on p, n, a., /3, 1' such that if x , y E B(w, f/2), B(w, 4r) c n n B(w, 4r), then c - 1 1'\?u(x ) - '\i' u (y ) i :::;

Proof: Given (2.5)

E

(lx - Yl/r)& B(w,r) In§\� l'\?ul

:::; c f-1 (lx - Yi/f)&

rn_ax_ u .

B(w, 2r)

> 0 and small, let

.4 (y, 7J, E )

=

J A (y , 'T) - x)BE(x)dx

whenever

(y, ·r1) E Rn x Rn,

R"

Bdx = 1 and e. (x) = cne(x/e) whenever X E Rn . where e E COO (B(O, 1)) with From Definition 1.1 and standard properties of approximations to the identity, we deduce for some c = c(p, n) 2: 1 that

JRn

(i)

(2 .6)

( ii)

(iii)

(ca:) -l ( t: + I7JI )p-2 1�12 ::;

� ��;

l

Ai

:t � i,j=l

'r/;

(y , .,,, t: ) �i�j ,

(y, 'T), t:) :::; ca:( t + I1J I )P-2, 1 ::; i , j ::; n ,

IA (x , ·q, e) - A (y , 1J, t) l :::; c.Bi x - y i'Y(e + l rJI ) P-l

whenever x, y, 1J E Rn . Moreover, A(y, · , f ) is, for fixed (y, t:) , infinitely different iable . To prove L emma 2.4 we choose u ( · , E), a weak solution to the PDE with struc ture as in (2.6), in SUCh a way that u ( · , E) is continuous in f!nB(w, 3r) and u(·, c) = U on 8[!1nB(w, 3r)]. Existence of u(-, f) follows from the Wiener criteria in [G Z] men tioned in the proof of Lenuua 2.2, the maximum principle for A-harmonic functions, and the fact that the W1·P-Dirichlet problem for these funct ions, in n n B(w, 3r) , always has a unique solution (see [HKM, Appendix I]) . Moreover, from [T] , [Tl] , it follows that u( - , E ) is in C1·& (n n B(w, 2r) ) for some a > 0 with constants inde pendent of t. Letting t---+0 one can show, using Definition 1 . 1 , that subsequences of { u(-, €) } , {Vu(-, t) } , converge pointwise to u, Vu. In view of Lemma 2 . 1 and the result in [T] it follows that this convergence is uniform on compact subsets of n n B(w, 3r ) . Using this fact we get the last display in Lemma 2.4. Finally we note that in [T] a stronger a.o:;sumption, compared t o (2.6) (iii), is imposed. However, other authors later obtained the results in [T] under assumption (2.6) (see [Li] for references) . D Next we show that the conclusion of Theorem 1 holds in the case of a truncated cylinder with w the center on the bottom of the cylinder (Step 0). To this end we

BOUNDARY HARNACK INEQUALITIES FOR OPERATOIL<; OF p-LAPLACE TYPE

2.'39

w = (w1, ... , wn) E Rn , the truncated cylinders , ' Qa,b (w) = {y = (y', Yn ) IY - w' l < a , IYn - Wn l < b}, Q !,b (w) = {y = (y', Yn ) : IY' - w' l < a , 0 < Yn - Wn < b}, Q ;;,b ( w) = {y = (y', y.,. ) : IY' - w' l < a , - b < Yn - Wn < 0}. (2.7) Furthermore, if a = b then we let Q a ( w ) = Qa,a (w), Qt (w) = Qt a ( w ) Q;:;- (w) =

introduce, for

a,

b E R+

and

:

,

Q;, a (w).

,

p , 1 < p < oo , is given and that A E Mp (a) for some a. Assume also, that u , v are non-negative A-harmonic functions in Q{(O), continuous on the closure of Q{ (O), and with u = 0 = v on oQi(O) n {Yn = 0}. Then there exist c = c(p, n, a) , 1 :S c < and u = u(p, n, a) E (0, 1] such that

Lemma 2.8. Suppose that

oo,

l ( ��D ( ��:D I log

- log

:S cly1 - y2 la

whenever Y1, Y2 E Q{14 (0) .

Proof. Let A = A(ry) be as in Lemma 2.8 and let p be fixed, 1 < p < oo. Note that Yn is A-harmonic and that it suffices to prove Lemma 2.8 when v = Yn · Define A(1J, � ) as in (2.5) relative to A and let u ( - , t:) be the solution to (A(V'u(y , t: ) , t:)) =

0 with

V' u � ( t: + IV'u (y, t:) l) 2 -v [��;( V'u(y, t:) ,E)+ ��:("vu(y,t: ) ,t: ) J

continuous boundary values equal to

A;i (y , t:) =

on

oQi (0).

·

Let

y E Q{12 (0) and 1 :S i , j :S n. From (2.6) (ii) and Schauder type estimates we see that u(·, t:), y.,, are classical solutions to the non-divergence form uniformly whenever

elliptic equation,

L*( =

(2.9)

n

L

i,j =l

A;j (y,

t:)(y;yj = 0,

t:))

y E Qi12 (0). Note also from (2.6) that the ellipticity constant for (Ai1 (y , and Ai1( y , t:) , 1 :S i,j :S n, in Qi1 2(0), depend only on a , p , n . From this note we see that if z = (z', Zn ) E Qi12(0) and 1 0 - 3 < Pt < P2 < 1 0 ::1, then e -Ni y -zl2 - e-Np� (2 . 10 ) ·lj; ( y) = e - Np21 - e -Np22 for

the L00-norm for

is a subsolution to L* in Qt (o) n [B(z, p2) \ B(z, pi )], if N = N(a, p, n ) is sufficiently large, and 1/J = 1 on oB(z, Pt) while '1/J = 0 on oB(z, P2) · Using this fact, with z = (z' , 1/ 1 6) , lz' l < 1/2, p1 = 1/64 , p = 1/16 and Harnack's inequality for L* (see [GT, Corollary 9.25] ) we get

2

(2. 1 1 ) whenever

y E Qj1 4(0).

c-1 Yn u(en/4, E ) :S u( y , t:)

Moreover, using

1 - 1/J, z = (z', -e .,. /64) , lz'l < 1/2 , Pl =

1/64, P2 = 1/16, in a similar argument it follows that (2.12)

u(y, c) $ cyn

m a.x

Q�/4 (U)

u( · , t:) :S C Yn u(en/4 , ) E

\

240

JOHN L. LEWIS, NIKLAS LUNDSTROM, AND KAJ NYSTROM

in Q{11 (0 ) . In partic.ular, the right-hand inequality in (2.12) follows from the ana logue of Lemma 2.3 for L * . Fix x E 8Q{12(o) n { y : Yn = 0}. From (2. 1 1 ) , (2.12), and linearity of L* one can deduce (see for example [LN, Lemma 3.27]) that there exists B, O < (} < 1 , such that osc (p/4) :::; B osc (p) (2. 13) when 0 < p :::; 1/4, where osc (t) M(t) - m(t) and we have put -u(y,•) , m (t)

=

M(t)

=

Q i (x)

max

Yn

=

min u Qi(x)

(y,e) . Yn

one can simply apply the same argument as in (2. 1 1 ) , (2.12) to u - m(p)yn , Yn and M(p) yn - u, Yn in Qt (x). Iterating (2 .13), we obtain for some .\ > 0, c > 1, depending on o:, p, n, that

To get (2.13)

osc (s) :::; c (s/t) :>.. osc (t) , 0 < s < t :::; 1/4.

(2.1 4)

Letting f--70 it follows as in the proof of Lemma 2.4 that u( · , c) converges uniformly to ·u on compact subsets of Q{12 (0). Thus (2 .11), (2 . 12) and (2.14) also hold for u. Moreover, (2. 1 1 ), (2. 12), (2. 14) , arbitrariness of x, and interior Harnack - Holder continuity of u are easily shown to be equivalent to the conclusion of Lemma 2.8 when v(y) = Yn · 0 Vve note that boundary Harnack inequalities for non-divergence form linear symmetric operators in Lipschitz domains can be found in either [B] or [FGMS] . We end this section by proving the following lemma. Lemma 2.15. Let G C Rn be an open set, suppose that p, 1 < p < oo, is given and let A E Mv (er., /3, /) for some (er., /3, /) · Let F : Rn --7 Rn be the composition of a tmw;lat-ion, a rotation and a dilation z --7 rz, r E (0, 1] . Suppose that u is A-harmonic in G and define u( z ) u(F(z)) whenever F(z) E G. Then ft is A. harmonic ·in F 1 (G) and A E Mp(o:, /3, 1). =

for some w E R''-, i.e., F is a translation. In this case the conclusion follows immediately with A(z, ry) = A(z + w, rJ) and A E Mp(o:, /3, !) · Suppose that F(z) = rz, where r it> an orthogonal matrix w i th det. r 1 . In thit:> case the conclusion follows with A(z, 17) = A(rz, rry) and A E Mp (er., /3, /) · Finally, suppose that F(z) rz for some r E (0, 1]. Then u is A-harmonic in F-1 (G) with A(z, 17) rP-l A(rz, r - 1 ry) . Moreover, property (i), (ii) and ('iv) in Definition l. l follow readily. To prove (iii) in Definition 1 . 1 we see that j A(z , ry) - A (y , ry)j :::; (3r�' j z - yj�' i11ip - l :::; /3iz - yj�' jrylp-l whenever r E (0, 1]. This completes the proof of Lemma 2.15. 0 Proof. Suppose that F(z)

=

z+

w

=

=

=

3. Non-degeneracy of j\7uj

In t.his sP-etion we establish the 'fundamental inequality' referred to as Step A in the introduction. To do this we first prove a few technical results.

BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p- LAPLACE TYPE

241

Lemma 3.1. Let 1 < p < oo, and assume that A 1 , A2 E Mp ( o:, {3, 'Y) with IA1 (y, r1) - A2 (y , ry) I S:: EjryJP-l whenever y E

Qi (0)

for some 0 < E < 1 /2 . Let u2 be a non-negative A2 -harmonic function in Qi (O), continuous o n the closure Qi (O ) , and with u2 = 0 on BQi (O) n { yn = 0 } . Moreover, let u 1 be the A1 -harmonic function in Qi12 (0) which is continuous on the closure of Qi 2(0) and which coincides with u2 on 8Qi12 (0 ) . Then there exist, given 1 ( 0 , 1/16) , c, c, 0, and T, all depending only on p, n, 0:, {3, 'Y, SUCh that J u 2(y) - 'Ul (y) j S

cf'' u2 ( en /2 )

S c/J P-7U2(y) whene v er y E Q i14( 0)

p

E

\ Q i;4,p (0 ) .

To begin the proof of Lemma 3.1 we note that the existence of u1 in Lemma 3 . 1 follows from the Wiener criteria in [GZ] , see the discussion after Lemma 2.2, the 1 maximum principle for A-harmonic functions, and the fact that the W ·P-Dirichlet problem for these functions in Q 2 ( 0) always has a unique solution (see (HKM, 1 Appendix I] ) . Observe for x E Rn, A. E R't, � E Rn \ {0}, and A E Mp (o: , {3, ')' ) , that

Proof.

i

{ 3 2)

.

for i

E

(3.3)

{1, . , n}. Using

.

(3.2) and Definition

1.1 (i), (ii),

we see that

2 c- 1 (JA.J + JWP- J A. -�j 2 s; (A( x , .X) - A(x , �) , .X-0 :S c ( j .X J + JWP- 2 j .X- � j 2 .

Moreover, from (3.3) we deduce that if

I=

then, {3.4 )

I

:S cJ, J : =

whenever p ;:::

(A1 (y, \?u 1 (y)) - A1 (y, \?u2 (y)), \?u2(y) - \?u 1 (y) ) dy,

2.

Also, if

1 < p < 2,

I :::; cJPI 2 J is

whenever

as

y E

definition of

(3. 6) J =

J

Qi;2(0)

j \?u2 - \? u1 /Pdy ,

Qi/2(0)

(3 .5 )

where

I

I

(I

we see from (3.3) and Holder's inequality that

Qi/ 2 (0)

1 p/2

j\?u1 J P + j \?u2 jPdx

)-

\7 · (A1 (y, \7u1 (y))) = 0 = \7 · (A2(y, \? u 2(y))) u 1 E W�'P (Q;t"12 (0) ) , we see from the 0 = u2

defined in (3.4 ) . As

Qi12 (0)

and

in (3.4) that

J

Qi/ 2 (0)

as

-

{A2(y, \?u2 (y)) - A 1 (y, \?u2 (y) ) , \? u2 (y) - "Vut (Y) ) dy.

242

JOHN L. LEWIS, NIKLAS LUNDSTROM, AND KAJ NYSTROM

Hence, using (3.4) , (3.6), the assumption on the difference IAt (y, ry) - A2 (Y, 17) 1 stated in the lemma and Holder's inequality we can conclude, for p 2: 2, that

I S C€

(3.7) Also, for 1

<

p

<

I

(I V'ut lv + IY'u2 IP)dx.

Q{/ 2 (0)

2, we can use (3.5) to find that I :5 cEP/2

(3.8)

I

(I V'ul lr + IY'u2 IP)dx.

Q{;,( O)

Now from the observation above (3.6), (3.3) with � = 0, and Holder's inequality we see that

I

IY'u1 IPdx

:5

c

J

(A 1 (x, V'u 1 (x) ), \i'u 2 (x))dx

Q{/2 (0)

Q{/2 (0)

::; ( 1 /2)

J

I V'·u1 IPd:r +

e

Qi/2 (0)

Thus,

J

(3.9)

i'Vu 2 IPdx.

Qt. 2 (0)

IY'ut iPdx :5 c

Q{/2(0 )

j

J

IY'u2 IP dx.

Q{/2 (0)

Let a = min{ 1 , p/2} . Using (3.9) in (3.8), (3.7) , and Lemmas 2 . 1 - 2.3 for u2 we obtain (3.10 )

Next using the Poincare inequality for functions in w� ·P ( Qi/2 ( 0 )) we deduce from (3. 10) that

(3.11)

J

ln2 - Ut lr dx S

Q{/2 (0)

In the following we let (3.12)

E = {y

E

11 =

c

J

IY'uz - V'u t iP dx S CE a (u2 ( e,. /2))P.

Q{/2 (0)

aj(p + 2) and we introduce the sets

Q"t12 (0 ) : l u 2 ( y) - Ut (Y) I S E17 u2 (e n /2 ) } , F

=

Q"t12 (0 ) \ E.

Moreover, for a measurable function f defined on Qi1 2 (o) we introduce, whenever y E Q"t12 ( 0) , the Hardy-Littlewood maximal function (3. 13)

Let

M ( f) (y) : =

j

1 lf(z) i dz . ( Y) I Q I {r>O , Q r(y) c Q {1 2 (0) } Q . ( y)

sup

r

,

(3.14)

where XF is the indicator function for the set F. Then using weak ( 1 , 1)-estimates for the Hardy-Littlewood maximal function, (3 .11) and (3.12) we see that 1Qi;2 (0 ) \ Gl :5 CE- 17 1 F I :5 CE - ?JE-pryEa = CE?J (3.15)

BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE

by our choice for ry. Also, using continuity of

lu2 (y) - u l ( Y) I =

(3.16)

If

�, r) l J

�� I B (

B(y,r)

243

u2 (y) - u1 (y) we find for y E G that lu2 (z) - u1 (z) l dz :::; ce1lu2(en/2) .

y E Q{14(0) \ G, then from (3 . 15) we see there exists fj E G such that I Y - t/ 1 :::;

c(n)e7Jin.

Using Lemmas

�

!u2 (y) - u1 (y) j

2 . 1 , 2.2,

!u2 (fj ) - u1 ( Y) I + ! u2 (y) - u2 (fj) l + l u l ( Y) - u 1 (Y ) I c(e1J + ei11J/n ) u2 (en /2) .

�

( 3.17)

we hence get that

This completes the proof of the first inequality stated in Lerruna 3.1. Finally, using the Harnack inequality we see that there exists T = T(p, n, a, {3, �f ) 2: 1 such that

u2 (en/2) :::; cp-ru2 ( Y)

whenever

y E Qi14(0) \ Q{14,p (O).

D

We continue by proving the following important technical lemma.

R" be an open set, suppose 1 < p < oo, and that A1, A2 E Mv(a, {3, "f). Also, suppose that u 1 , u2 are non-negative functions in 0, that ·ih ·is A1 -harmonic in 0 and that u2 is A2 -harmonic in 0. Let ii 2: 1 , y E 0 and assume that 1 u 1 (y) u1 (y ) � a � d( y , 80) � I V u l (Y)i d(y, 80) ' Let £-1 = (cii)( l +iY) /& , where &- i.e; as in Lemma 2.4. If

Lemma 3.18. Let 0

C

•

_

u2 1 ( 1 - f:) L � -;-- � ( 1 + f:)L in B(y, 100 d(y , 80) ) •

for some L, 0 < L <

oo,

•

Ul

then for c = c(p, n, a, {3, "!) suitably large,

1 u2 (y) < I " . ) I < u2 (y) v u2 ( y - ca cii d(y, 80) d(y, 80) · _

Proof. Let ii 2: 1, y E 0 be as in the statement of the lemma. Using Lerruna and the Harnack inequality i n Lemma 2.1 (ii) we sec that, IVu2 (zl ) - Vu2 (z2 ) i :::; ct17

(3. 19)

B(y,td(y,80)) max

2.4

I Y'u2 (·) i :::; c2 tu- u2 (y)jd(y, 80)

whenever z1 , z2 E B(y, td(y, 80)) and 0 < t � 1 0 - 3 . Here c depends only on p, n, a, [3, 'Y· Using (3.19) we see that we only have to prove bounds from below for the gradient of u2 at y. To achieve this we suppose that,

( 3 . 20)

for some small ( > then deduce that

0 to be chosen.

From (3. 19) with

z

=

z1 , y

=

z2 and (3.20) we

(3.21 ) whenever

IY - :0 1

(3.22)

=

z E B(y, td(y, 80) ) . Integrating, it follows that if fj td(y, 80) , t = ( 1 /&-, then iu 2(y) - u2 (Y ) i � c'( 1 +1 /& u2 (y).

The constants in

(3.21 ), (3.22)

depend only o n p , n , a , {3, "f ·

E

8B(y, td(y, 80) ),

244

JOHN L. LEWIS,

NIKLAS LUNDSTR OM , AND KAJ NYSTRO M

��j1 .

(3.19)

Next we note that also holds with u2 replaced by u1 . Let A = Then from (:U9) for 11 1 and the non-degeneracy assumption on l \7iL 1 1 in Lemma we find that

3. 18,

1 d(y, 80 ) ), { \i'u 1 ( z ) , A} � (1 - c ii() l\i'u1 (y) l whenever z E for some c = c(p, n, o., fJ, I)· If � ( 2cii) - 1 , where c is the constant in the last

display,

B(y, ( /&

(

then we get from integration that

1& (3 .23) c• (u1 (Y) - iLt (Y) ) � ii- 1 ( / u l (Y) with y = y + ( 1/& d(y, 80)A and where the constant c• depends only on (3, From (3. 2 3), (3. 22), that if f. is in Lemma 3.18, then u2 (y) c'(1+1/" ) u (y ) ( (1 f.)t 1"l1 (fj) � 1 + (1 /&f (iic• ) ul (Y ) - ( 1++ c'(lj(ac• +1/& ) A (3.24) (1 + ) 1 (1 /& ) L < ( 1 - c)L provided 1/ (iic) 1/& ( 1 /& � iic € for some large c = c(p, o., (3, 1). This inequal 1 ity and (3. 2 3) are satisfied if €- = ( cii ) Cl+&)/& and (- 1 Moreover, if the hypotheses of Lemma 3.18 hold for this then in order to avoid the contradiction in ( 3.24) it must be true that (3.20) is false for this choice of (. Hence Lemma 3 . 18 is true. Armed with Lemma 3.1 and Lemma 3 . 18 we prove the 'fundamental inequal ity' for A-harmonic functions, A E Mp ( o., f3, /') for some (o., ,B , 1), vanishing on a portion of {y : Yn = 0}. we see

a.'l

<

_

�

<

p, n, a.,

1+

I·

2

_

e:

n,

=

f.,

,

Cii .

D

< p < oo , and A E Mp(o., f3, 1) for some (o., f3, 1). Suppose that 7L is a positive A -harmonic function in Qi (0) , continuous on the closure of Qi(O), and that u = 0 on 8Q i (O) n = 0}. Then ther-e exist c = c(p, n, a, (J, I) and X = X(p, n, a, {3, �f) , such that u( y) X-1 u (y) � whenever E Qi;c (O). �X

Lemma 3.25. Let 1

{Yn

y Yn l\i'u(y) i Yn Let A E Mp(a, 1), A A(y, 17), be given. Put A 2 (y, 17) A (y, 17) , A(O, rJ) . Clearly, A1 , A2 E Mp (o., /3, /). We decompose the proof into the {3,

Proof. A1 (17) = following steps.

=

=

Step 1. Lemma 3.25 holds for the operator To see this we note once again = 0}. Let 11 2 = u . is A1 -harmonic and u 1 = 0 on 8Qi (O n that u t (y) = Applying Lemma to the pair u 1 , u2 we see that

A1 .

Yn

(3 . 26) whenever inequality (3.27)

I

2.8

log

) {Yn

( log �:��D � cly1 - Y2 l a � I ( ��:D -

y1 , y2 E Qi14(0) . Exponentiation of this inequality yields the equivalent

245

BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE

whenever obviously

Yt . Y2 E Qi14(0).

Let 0 =

Q i14 (0)

and note that if

E

Yz

Qi18 (0)

then

(3.28)

a = a(n) . Let r be defined through the 3.18. Using (3.27) we then see that

for some Lemma

(1

(3.29)

f./2)

Lemma

Step 2 .

J E (0, 1/8)

3.25

c'ra

=

�f. where f. iH aH in

(y2 ) � �1 (y1 ) � ( 1 + f./2) �1 ( y2 ) �1 u u ) 2 ( Y2)

u2(Y1 whenever y 1 E B ( y2 , r ) . From (3.28) , (3.29), Lemma 3.25 holds for the operator A 1 . _

relation

z (Y"2)

and Lemma 3.18 we conclude that

We let p

is valid for the operator A2•

E

(0, 1 / 16)

aud

be degrees of freedom to be chosen below. Let ih be the A1-harmonic

Qt1 2 (0) which is continuous on the closure of Qt12 (0) and which satisfies u 1 u on oQt12 (0) . Using Step 1 we see there exist ..\1 = ..\1 (p, n, a) , c1 = c1 (p, n, a) � 1 , such that function in =

1 u1 (y) ..\ 1 -Yn

(3.30)

� 1 'V1t_ 1 ( y ) I :=::; ..\1 u1(y)

Moreover, using Definition 1 . 1

(3.31) Let

u2

=

we

� E l17 1

whenever

p, n, a, {3, �r,

Ju"2 (y) - u 1 (Y) J

y E Q+ J/f:t ( 0) .

have

p- 2

with F.. =

From Lenm1a 2 . 15 and Lemma

depending only on

(3.32)

(iii)

IA2 (y, ry) - A1 (y, ry)l

u.

Yn

--

such that

3.1

2(38"�

whenever

y E Qt (O).

we see there exist c , 0 , '

a replaced by ..\ 1 1/ (32c1) . Fix i5 subject to c'Elip-T = c' (2{3i5"�)0 p-T = min{f./2, 10 - 8 }. we note that J = J(p, n, a, {3, I ) · Then from (3.32) we see that

Using

( 3.34)

l -E� _

u2 (y) --;:-� 1 + E whenever y E QJ+;4 ( 0) \ Q3+14 1( )

ll y (3.30) , (3.33), and 1 u2 ( y) ..\2- -Yn

for some ..\"2 =

�

each

� c'/J p- T u2 (y ) whenever y E Qt14 (0) \ Qt14,pJ (O) .

Let f. be as in the statement of Lemma 3.18 with

( 3.33 )

r,

_

Lemma

"' )J I v u2 ( y

3.18

, p0

and put p =

In particular,

( 0) .

we therefore conclude that

uz (y) r; whenever y E Q+ � ..\2 --

Yn

u

Ct

( 0) \ Q+ •; - ' 2 poF ( 0) , " Ct

..\ 2 (p, n, a, {3, I ) · Moreover, if y E Qt;c1 ,2pJ( O) , then we can also prove that (3.34) is valid at y by iterating the previous argument and by making use of the invariance of the class Mp (a, {3, ·-y ) with respect to translations and dilations, see Lemma 2.15. This completes the proof of Lemma 3.25. D Finally we use Lemma

3.25

to establish the main result of this section.

Lemma 3.35. Let n c R" he a (8, r0 ) -Reifenberg flat domain, w E 80, and 0 < r < min{"r0, 1 } . Let p, 1 < p < oo, be given and ass7L1nP- Uwt A E A1p(a, i3, 1) joT some (a, (1, 1 ) . Suppose that u is a positive A-harmonic function in OnB(w, 4r ), that n is continuous in n n B(w, 4r) , and that u = 0 on .6.(w, 41·). There exist 6 =

246

JOHN L.

LEWIS,

NIKLAS LUNDSTROM, AND KAJ NYSTROM

b (p, n, o:, /3 , /) , c = c(p, n, o: , /3 , /) an d � = �(p, n, o:, /3 , /), such that if O < fJ th en

::::;

J,

r

u (y) - u(y) . d(y , 80.) ::::; I V'u(y) l :::; ), d(y, 80.) whenever y E 0. n B(w, r /c) Proof. Let A E Mp(o:, /3, /), A = A(y, ry) be given. Let w E 80., 0 < r < r0, suppose that u is a positive A-harmonic function in 0. n B(w, 4r), that is continuous in Q n B(w, 4r), and that u = 0 on .6-(w , 4r). We intend to use Lemma 3.25 and Lemma 3.1 to prove Lemma 3.35. Let u = 0 in B(w, 4r) \ 0.. Then u E W 1 ·P (B(w, 2r)) and is continuous in B(w, 4r). Let c1 c be as in Lemma 3.25 and choose c' 2: 100c1 so that if f) E 0. n B(w, rjc'), s = 4c1 d(y, 80.), and z E 80. with I Y - z l = d(f), an), then max ::::; cu(f)) (3.36 ) B(z,4s) - 1

u

=

v,

u

for some c = c(p , n , o:, ,8, 1) . Using Definition 1 .6 with w , r replaced by that there exists a hyp erplane A such that

z, 4s, we see

h(80. n B(z, 4s), A n B(z, 4s)) ::::; 4os.

(3.37)

For the moment we allow J in Lemma 3.35 to vary but shall later fix it satisfying several conditions. Using ( 1 . 7) we deduce that

{y E 0. 11 B(z, 4s) : d(y, 80.) 2: 88s} C

as

a number

one component of Rn \ A.

Moreover, using Lemma 2.15 we see that we may without loss of generality that A = {(y',Yn) : y' E Rn -t, Yn = 0} and

assume

{y E 0. n B( z, 4s) : d(y, 80.) 2: 8os } C {y E R" : Yn > 0} .

(3.38)

From (3.38)

find that

if we define

A' = {(y' , 0) + 20ose , y' E Rn - l }, 0.' {y E Rn : Yn > 20os }, we

n

=

then

0.' n B(z , 2 s) c 0. n B(z, 2s). Let v be a A-harmonic function in 0.' n B(z, 2s) with continuous boundary values on 8(0.' n B(z, 2s)) and such that v ::::; u on 8(0.' n B(z , 2s)). Moreover, we choose v so that v(y) = u(y) whenever y E 8(0.' n B(z, 2s)] and Yn > 408s, v(y) = 0 whenever y E 8[0.' n B(z, 2s) ] and Yn < 305s. (3.39)

of v follows once again from the Wiener criteria of [GZ] , the maximum principle for A-harmonic functions, and the fact that the W 1 ·P-Dirichlet problem for these functions in 0' n B(z, 2s) always has a solution. By construction and the maximum principle for A-harmonic functions we have v ::; u in 0' n B(z, 2s). Also, since each point of 8 [0.' n B(z, 2s)] where u 1- v lies within 808s of a point where u is zero, it follows from (3.36) and Lemmas 2.2, 2.3 that u ::; v + c8"' u(y) on 8[0' n B(z, 2s)] . In particular, again using the maximum principle for p-harmonic functions we conclude that Existence

v ::::; u S

v

+ c85u(y) in n' n B(z, 2s) .

BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE

Thus, using the last inequality and

(3.36)

1 ::; u (y) ::; ( 1 - co,_)_1 v (y

(3.40)

we see that

whenever

provided J is small enough. Using Lemma

( ) A• - 1 d vf) yaD) ( ,

(3.41)

::;

247

3.25

,

IV'v (y) l

y E D' n B(y, � d( y, Gn' ))

and the construction we also have

::; A v ( y ) d( y ,l )D) " •

for some j, = j, (p, n ) . In particular, from (3.40) , (3.41) we see for 0 < o < J, and J = J (p, n, a, /3, "! ) suitably small, that the hypotheses of Lemma 3.18 are satisfied with 0 = D' n B(z, 2s) and We now fix J and from Lemma 3.18 we conclude

-

that

for some

Lemma

5. 1

3.35

a = �. (y)

A t d (uY, aD)

=

-

1

:S I V'u

5. 1 (p, n, a, /3, 1 ) . Since f)

is complete.

D

( ') I

-

u (y )

y :S At d( f) , 00 )

E D n B(w, rjc')

is arbitrary, the proof of

4. Degenerate elliptic equations and extension of IV'uiP- 2 to an A2-weight Let w E Rn , 0 < r and let A(x) be a real valued Lebesgue measurable func tion defined almost everywhere on B(w, 2r). A(x) is said to belong to the class

A2 (B(w, r)) (4. 1)

if there exists a con::;taut r such that r:- 2n

J

B(w,f)

A dx .

J

B(w,f)

A-1dx ::; r

w E B(w, r) and 0 < f ::; r. If A(x) belongs to the class A2(B(w, ·r )) then A is referred to as an A2 (B(w, r) )-weight. The smallest r such that (4.1 ) holds is referred to as the A2-constant of A.

whenever

In the following we let D C Rn be a bounded (t5, r0)-Reifenberg flat domain with NTA-constant 1'vf. We let w E aD, 0 < r < ro , and we consider the operator

,

(4.2 )

L

n

=

a

(·

a

L - bij (x) OX · i,j =l ox •

J

)

in D n B (w, 2r). We assume t h at the coefficients {bij ( x)} are bounded, Lebesgue measurable functions defined almost everywhere on B(w, 2r). Moreover, n

c-1 A(x) l�l2 :S L bij (x)�i�j :S cl�l2 A(x) i,j =l for almost every x E B(w , 2r) , where A E A2 (B(w, )) By definition L is a degen erate elliptic operator (in divergence form) in B( w, 2r) with ellipticity measured by the function ,\. If 0 c B(w, 2r) is open then we let W1•2(0) be the weighted Sobolev space of equivalence classes of functions with distributional gradient \i' v (4.3)

r

v

and norm

ll v flt2

=

.

j v2 Adx + j IV'vi2 Adx < oo.

0

0

248

JOHN L. LEWIS, NIKLAS LUNDSTROM, AND KAJ NYSTROM

2 Let W� ' 2 (0) be the closure of C0 (0) in the norm of W1• (0). We say that v is a weak solution to Lv = 0 in 0 provided v E W1•2 (0) and

1 �::)ij Vxi
(4.4)

0

t,J

whenever ¢ E C0 (0) . The following three lemmas, Lemmas 4.5-4. 7, are tailored to our situation and based on the results in [FKS] , [FJK] and [FJK 1]. We note that these authors as sumed L was symmetric, i.e., bij = bji, 1 � i,j � n, but this assumption was not needed in the proof of these lemmas. Essentially one can say 'ditto' to the discus sion in [KKPT, section 1] for nonsymmetric uniformly elliptic divergence form PDE. Lemma 4.5. Let n c an be a NTA-domain with constant M' w E an, 0 < r < To, and let A be an A2 (B(w, r))-weight with constant r. Suppose that v is a positive weak sol1£tion to Lv = 0 in nn B(w, 2r) . Then there exists a constant c, 1 � c < oo , depending only on n, M and r, such that if w E 0, 0 < r, B(w, 2r) C 0 n B(w, r), then (i)

(ii)

c-1r2

I

J V'v J 2 Adx �

B(w.r/2 ) max v < c min v. B (w,r) - B (w,r)

c( I

D(w,r)

Ad'J: ( m� v ) 2 � c

)

B (w;r)

1,

B (w 2 f)

Jvl 2 .Adx,

Furthermore, there e:Lists a = a(n, M, r) E (0, 1) such that if x , y E B(w, r) then

(iii )

J v(x) - v(y) J �

m_ax_ v. c(l x�yl) ii B(w,2r)

Lemma 4.6. Let n c Rn be a NTA-domain with constant lv!, w E aD., 0 < T < ro , and let .A be an A2(B(w, 1')) -weight with constant r. Suppose that v is a positive weak solution to Lv = 0 in n n B ( w, 2r) and that v = 0 on .6. ( w, 2r) in the weighted Sobolev sense. Then there exists c = c(n, M, r), 1 � c < oc , such that the following holds with f r /c. =

( i)

( ii )

1

OnB(w,r/2) � cv(a;:(w)).

max v onB(w,f)

nnB(w,r)

Furthermore, there exists a = a(n, M, f) E (0, 1) such that if x, y then (-iii)

)

Jv(x) - v(y) J � c lx�yl a

(

E

D. n H(w, f),

max _ v . OnB(w,2r)

Lemma 4.7. Let n c Rn be a NTA-domain with constant M, w E an, 0 < r < 1·o , and let A be an A2 (B(w, 1·)) -weight W'ith constant r. Suppose that v1 and v2 are two positive weak sol?Llions to Lv = 0 in n n B( w, 2r) and Vt = 0 = V2 on .6.( w, 2r) in the weighted Sobolev sense. Then there exist c = c(n, J\..1, I') , 1 � c < oc, and

UOUN DARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE

cr = cr(n, M, r) E {0, 1) such that if f = rjc, v1 ( a.;: (w) ) in fl n B(w, rjc) and if Yb Y2 E fl n B(w, r jc), then

l vl(Yt) Vz (yt)

_

=

249

v2 (ar ( w) ) , then vtfv2 ::; c

V t {Y2) vz(Yz)

-

r

To continue the proof of Theorem 1 , we have the following lemmas.

<

Lemma 4.8. Let fl C Rn be a bounded (o, ro) -Reifenberg fiat domain. Let p, 1 p < oo, be given and assume that A E Mp (o:, f3, 'Y) for some (o:, (3, '"y ). Let w E 80, 0 < r < ro and suppose that u is a positive A-harmonic Junction in fl n B(w, 4r·), u is continuous in D n B(w, 4r) , and u = 0 on �(w, 4r) . Then there exist, for t:* > 0 given, J = J(p, n , o:, fJ, 'Y, t:* ) > 0 and c = c(p, n, o:, fJ, 'Y, t:*), 1 ::; c < oo, such that C- 1

('!:") 1+£'

< -

whenever 0 < o ::; J and 0 < f < r/4 .

U Uf ( W) ) < u( ar ( w) )

(

c

()

f l - £. r-

Let n c Rn be a bounded (o, r0) -Reifenberg fiat domain. Let p, 1 p < oo, be given and assume that A E Mp(o:, fJ, 'Y) for some (o:, {3, '"y) . Let w E an, 0 < r < min { ro 1 } , and suppose that u is a positive A-harmonic function in fl n B(w, 2r) , u is continuous in n n B(w, 2r), and u = 0 on � (w, 2r) . There exist {/ = o'(p, n, o:, fJ, 'Y) , and c c(p, n, o:, (3, 1) ;:::: 1 such that if O t5 < 6', and f = r/c, then J V'u JP-2 extends to an A2 (B( w, f) )-weight with constant depending only on p, n, a, (3, 'Y ·

<

Lemma 4.9.

,

<

=

Proof of Lemma 4.8: A2 (y, 7J) = A(y, 7J), A1 (77)

Let A E

Mv (a, (3, 1), A

A(w, 7J) . Then A1 , A2 E

=

A (y , 77)

be given and set

Let u be a Az harmonic function as in the statement of the lemma. We extend u to B (w, 4r) \ fl by putting u = 0 in this set and then note that u is continuous in B(w, 4r) . We also =

Mv (a, fJ, 'Y) ·

observe from Definition 1 .8 that it suffices to prove Lemma 4.9 for {J = J. Also, as discussed after Definition 1 . 8 we may assume that n is a NTA-domain. Moreover, using Lemma 2.15 and Definition 1 . 1 (iv), we can without loss of generality assume that r = 4, w = 0 and ·u (a 1 (0)) = 1 . In the following we let � b e a small constant t o b e chosen below. I n particular, { will be fixed to depend only on p, n, a, (J, 'Y · For � fixed we can, again using Lemma

2.15, without loss of generality also assume that

h(P n B(0, 4�) , an n B(0 , 4{)) ::; 4J�, where P = { E R" : = 0}. Furthermore, if may assume, as in ( 3.38 ) , that '

(4. 10)

y

Yn

3

=

43 is small enough, then we

B ( 0, 4) n { (y', Yn) : Yn 2 2 b{} C n B(0, 4) n { (y', yn) : Yn ::; -26�} c R" \ fl .

Moreover, we see that to prove Lemma

4.8 it suffices to show that

(4.11) In the following we will use the notation introduced in (2. 7) .

250

JOHN L.

LEWIS, NIKLAS LUNDSTROM, AND KA.T NYSTROM

To begin the proof of (4. 11) we introduce two auxiliary functions u+ and u- . In particular, we define u+ to be A2-harmonic in Q7,(l sil')€ (85�en) with continuous boundary values on aQ 7, (1 88 ) /88�en) defined as follows,

-

� , - SS)€(8<5�en) (1 {y : 16<5 � Yn } if y E oQt,(l - s;5)€(8J�en) n {y : 8J� < Yn < 1660, u + (y) u+ (y) = 0 if y E aQt,(l SJ)e (8J�en) n {y : Yn 8J�}. Similarly, we define u- to he the A2-harmonic function in Q7, ( 1+ S il') e ( -88�en ) which satisfies u- = u on aQ7,( HsJ)� ( -8J�en)· From the maximum principle for A harmonic functions and (4.10) we see, by construction, that u+ (y) � u (y) � u - (y) whenever y E Q7,(l Si5) �(88�en)· (4.12) =

u+ (y) = u (y) (Yn u (y)

�:8"0

I. f y E aQ{+,(l

=

Using Definition 1 . 1 (iii) we next note that

clrJip - 1 whenever y E Q7,( l +SJ)€ (-8J�en), E = 2(3(1 . To proceed we let v,+ be the Arharmonic function in Qt12 ,(1 / 2 _8J)e (8 J�en ) which is continuous on the closure of Qt1 2,( 1 /2 _ 86)�(8J�en) and which coincides with u+ on aQt12 , ( 1 /2 sJ)€ (8J�en). Similarly, we let u- be the A1 -harmonic function in Qt12 , ( l / 2+Sil')e ( -8J�en) which is continuouH on the closure of Qt12 , ( 1 /2+Bil' ) { ( - 8J�e11 ) and coincides with uaQt/2 , ( 1 /2+86 ){ (-8J�en. ) · Finally, we define v + (y ) Yn - 8J�, v-(y) := Yn + 8J� whenever y E Rn . Hence v+ and are A 1-harmonic functions and grow linearly in the en-direction. We first focus on the right hand inequality in (4. 11). Using (4. 1 3), Lemma 2.15, (4. 13) f A 2 (Y, rJ) - A l (Y , 11) 1 �

Oil

v

:=

-

and Lemma 3.1 we see that

(4.14) for y

Q€+/4,(l/4 +B il')� ( -8<5�en.) n { -4<5� < Yn < �/2} and for a constant c- = c(p, n, o:, {3, "f ). Moreover, sing (4. 12), the maximum principle and the Harnack inequality for A-harmonic functions, (4.14) , as well Lemma 2. 8 applied to the functions ii- , v- we see that there exists a constant c = c(p, n, o:) , 1 � c < oo, such E

u

as

that

(4. 15)

u (y) � u- (y) � (1 - cc8J-.,.)-1u - (y) � c( 1 - ci1 J -.,. ) - 1 u- (a{;s (O))

v-?)

y E !1 n B(O, �/c) . From (4.15) we conclude that u (y) � c(l - cc8J-.,. ) - 1 u- ( ae;s ( O)) (Yn /�) (4. 16) whenever y E !1 n B (O, �/c). Let J < 1/(16c) and let � be defined though the relation 1/2 = cf.8J _.,. = c(2f3C)8 J-... Then � �(p, n, (3, ')', J) and from Lemmas 3.1, 2.2, 2.3, as well as the maximum principle for A harmonic functions, we observe that max u( ao 8 (0) ) u- (ao s (O) ) u - ( a{;s (O)) )u Q <. ! > . (1/2+85)<. ( - S o �en whenever

=

a,

�

�

�

+

_

BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE

25 1

where proportionality constants depend only on p, n, a, (3, I· Using these displays in (4. 16), we get u (aJ� (O)) $ cJ u(a� ;8(0)) . Moreover, suppose by way of induction that we have shown, for some k E {1, 2, ... }, (4.17) u(aJk( ( O)) $ (c6) k u (ae;8 (0)) where c depends only on p, n, o:, {3, I· Then, from Reifenberg flatness we see there exists a plane P' containing 0 such that h(P' n B ( O, 4Jk E), an n B(O, 48k 0 ) $ 4J8k E · �

V\Te can now repeat the above argument with P replaced by P' and 4 replaced by Jk E, since we can 46'k E . Here however we use a cylinder with radius and height � already apply Lemma 3.1. We get

u ( a.5k+ 1 e (D) ) $ c J u(aJk e (O) ) $ (cJ) k+1u (ae ;8 (0)). Thus by induction the inequality in (4.17) is true for all positive integers k. Next we fix 8 so that 8-(· = c where c is the constant in the above display. Then 8 and E both depend only on p, n, a, {3, 1 and t:* . Given 0 < f < E, let k be the smallest integer such that 8k E $ f. Then from (4. 1 7) and our choice of 8 we see that u(ar (O )) $ cf 1 - (· , for some c = c(p, n, o:, {3 , {, E*). Here we have also used the fact that u(ae ;8(0) ) $ c* = c.(p, n, o:, /3, ')'), which follows from Lemmas 2. 2 , 2.3, and fact that u (a1 (0) ) = 1. This completes the proof of the right-hand side inequality in (4.1 1 ) . Second we focus on the left-hand inequality in (4. 1 1 ) . In this case we first apply Lemma 2.8 to the functions u+ , v+ in Q7;z ( l / Z - sc5)( (88Een ) · Indeed, using Lemma , 2. 8 and the Harnack inequality we see, provided 8 is small enough, that u+ (a 32Je ( O )) u+ (ae; 8(0) ) u+ ( ae; 8(0)) (4. 18 ) E v+ (a3 J( ( O)) � v + (ae/4(0)) � 2 Here A � B means that A/ B is bounded from above and below by constants which only depend on p, n, o:, {3, 1·. From (4 . 18) we get (4. 19) u+ (a32J�( O)) � c- 1 8u+ (ae;B(O)) for some c = c(p, n, o:, O, 1 $ c < oo. Moreover, using Lemma 3.1 we also see that (4.20) u+ (y) $ (1 - c £86--r)-1 u+ (y) for y E Q+e/ Z ( l /Z - Sc5)( (86E<5e n ) n {1 68E < Yn < E/ 2 } and for a constant c- = , c(p, n, o:, /3, 1)· Using (4.19), (4.20), the fact that the class Mp(o:, /3, 1) is closed under translations, rotations, suitable dilations, and multiplication by constants (see Lemma 2.1 5 and Definition 1 . 1 ( iv)) , we can argue as in the proof of the right-hand inequality in (4. 1 1 ) . Thus by induction we obtain (4.21) u (a c32s) • e ( O) ) � ( c- 1 8) k u (au8 ( 0) ) for k = 1, 2, ... �

�

-

To complete the proof we let 8 be so small that c 1 8 � (32J) l +e* and assume that f E [(32J) k +1 e, (32J) k �] . With 8(p, n, o:, {3, 1, £*) now fixed, it follows from Harnack's inequality for A-harmonic functions that (4.22) u(ar (O) ) � c- 1u ( ac 3zJ) k � ( O) ) � c-1 (32J)k ( l +e*lu(ae ;8 (0)) � c- 1 f(He·) . for some c c(p, n, o:, /3, 1, €* ) . In (4.22) we have also used the fact that u(ae;s ( 0)) � 1/c+(p, n, o:, ,6 , 1, €* ) , for some c+ � 1 , which follows from the definition of 3 in =

.TOHN

252

L.

LEWIS,

NIKLAS LUNDSTRO M , AND KAJ NYSTRO M

terms of € "' , Harnack's inequality, and the fact that u( a1 (0)) the proof of (4. 1 1 ) and hence the proof of Lemma 4.8. D

=

1 . ( 4.22) completes

Proof of Lemma 4.9. Lemma 4.9 follows from Lemma 4.8, in exactly the same way as Lemma 3.30 in [LN4] followed from Lemma 3.15 in [LN4] . For the readers convenience we include the details of the proof. Let Qj = Q(:rj , Tj ), j = 1 , 2, ... be a Whitney decomposition of Rn \ n into open cubes with center at Xj and sidelength Tj. Then U/ J(xj , Tj) = Rn \ n and Q(xj , rj ) n Q(xi , ri) = 0 when i =1- j. We furthermore construct the Whitney cubes in such a way that w - 4"d(Q1, an) � Tj � 10- 2nd(Qj , an). Let r = r j c2 , where c = c(p, n, a, j3, ,) , 1 :::::; c < oo, is so large that the 'fundamental inequality' in Lemma 3.35 holds in n n B(w, r jc) . From the NTA property oH ! we may also suppose c is so large that if Qj n B(w, 50·?) =f. 0, then there is a Wj E n n B ( w, cr) for which d( Wj , an) d( Wj , Xj) d(xj ,< :Jn) . Here A ,...., B means that A/ B is hounded from above and below by constants which only depend on n. Next we define >.(x) = J'Vu(x)JP- 2 whenever x E n n B(w, 50r) and we let r be the set of all j such that if j E f t.hen Qj n B(w, 50.?) -=f. 0. Moreover, if j E f then we choose w1 E n n B(w, cr) as above and define >.(x) = >.(wj) when x E Q1 . This defines >. almost everywhere on B (w, 50r) with respect to Lebesgue n measure, since it follows from ( 4.27) that for 8 small enough, an n B(w, r) has Lebesgue n measure zero. From the definition of >., Lemma 3.35, and the Harnack inequality for A-harmonic functions we see that rv

(4.23)

>.(x) = >.(wi)

>. (z) whenever x

E

rv

Qj and z

E

B(wj , d(wj , DD) /2) .

Let .\ = >. if p � 2 and .\ 1/ >. if 1 < p � 2. If w E B (w, r) and d( w, an) /2 < f � r, then from Lemmas 2 . 1 - 2.3, (4.23), and Holder's inequality it follows that �

=

j

(4.24)

Adx :::::; cu ( ar (w ) ) IP - 21 fn- I P - 2 1 .

B(w,r)

Here w E an with Jw - wJ = d(w, an) . Also, from Lemma 4.8 we get for J small enough and y E n n B(w, cf) , that cu(y) � u(ar (w))

(4.25)

(

d(

y�an)

)

l+£"

Here €* > 0 is a small positive number which will be fixed after the display following (4.27). From (4.25) and Lemma 3.35, we see that if d(w, an)/2 < f � r, then (4.26)

j

B(>v,r)

>.- 1dx :::::; cf (l+£") 1P- 21 u(ar (w ) ) - IP- 21

j

nnB(w,ci'J

d(y, OO) _,. I P- 21 dy.

To complete the estimate in (4.26) we need to estimate the integral involving the distance function. To do this we define I ( z, s) =

f

nnB(z, .. )

whenever z E an n B(w, r) , 0 <

s

Ek = D n B(z, s) n {y

d(y, an) -c*lp-2l dy

< r. Let :

d(y , an) :::; ok s} for k

=

0, 1, 2, . . .

BOUNDARY HARNACK INEQLALITIES FOR O PERATORS OF p-LAPLACE TYPE

253

Then since an is 8-Reifenberg fiat we deduce that

(4.27) e 1 = e+ (p, n) .

where

Indeed, from 5-Reifenberg flatness it is easily seen that this

cj8n- l balls an n B(z, s ). We can then repeat the argument in each ball to get that (4.27) holds for E2 . Continuing in this way we get (4.27) for all positive integers k. Using (4.27) and writing I (z, s ) as a sum over Ek \ Ek+l • k = 0, 1 , 2, . . . we get statement holds for E0 , E1 . Moreover, E1 t:a.n be covered by at most

lOOos

of radius

I( z, s) ::;

for some

with centers in

[1

csn-," lp-21

+ 5-€"lp-2l

f c� 8k(l-e" lp-21)]

<

k= l

c_ 8n-,* lp-2l ,

c_ = c_ (p, n) � 1 , provided 4�:* 1P - 21 ::; 1 and o' > 0 is small enough. z = w, s = cr, we can continue our calculation in (4.26)

Using this estimate with and conclude that

f

(4.28)

A- ldx ::; cu(ar (w)) -IP- 21 -rn + IP - 2 1 .

B(w,r)

Combining

(4.24) , (4.28),

we get

j

B(w,r)

when

d(w, a0)/2 ::; f ::; f.

easily from Lemma that Lemma

4.9

3.35.

Adx ::; cf2n

B(w,f)

Tliis inequality is also valid if

r ::; d(w, a0)/2,

as follows

We conclude from this inequality and arbitrariness of ii.J , ·r,

is true. D

4.1. Proof of Theorem 1 . 3, 4, we see that Steps 0, A, ll,

completed.

j

A-1dx

From the results proved or stated in section

2,

C and D outlined in the introduction are now

Hence, to prove Theorem

1

it only remains to remove assumption

1 . 1 (iv) that Theorem 1 is invariant u, v by constants. Using this note and Lemma 2.3 we see if r* = Tjc, for c = c(p, n, a, (3, /) large enough, then we may assume that

(1 . 14) .

To do this

we

first note from Definition

under multiplication of that

(4.29)

fln B(w,4r•) max

h

�

h(ar• (w) ) = 1

whenever

h

=

u or

Let u, ii b e the A-harmonic functions in

v.

0 n B(w, 4r*) with boundary values ii. = min(u, v) and ·u = 2 max (u , v) respectively on a(On B(w, 4r*) ) . From the maximum principle for A-harmonic functions we then see that 11 ::; u, v ::; ii/2 in n n B(w, 4r* ) . Using this inequality and applying Theorem 1 t o ii. , ii with r replaced by r • , we get max(ujv, vju) ::; vjii. ::; c in 0 n B (w, f).

ujv that (1. 14) ( a) can be achieved in n n v by a suitably large constant. wh ich can be chosen to on p, n, a, /3, I · Thus Theorem 1 is true. D

Finally we note from bonndedncss of

B(w, 4r* )

by multiplying

depend depend only

254

JOH::'-1

L.

LEWIS,

NIKLAS

LUNDSTROM,

AND KAJ

NYSTROM

5. The Martin boundary problem: preliminary reductions

Let n c Rn , r5, r0, p, a, /3, -y, and A be as in the statement of Theorem 2. Moreover, let w E an and let 0 < r' « r0 , where ro = min{ro, 1 } . Assume that u is an A-harmonic function i n n \ B(w, r') and that u = 0 continuously on an \ B(w, r'). We can apply Lemma 3.35 to conclude that there exist o* , 0 < o* < 1 , c) . 2: 1 , depending only on p, n , a, /3, -y, such that if 0 < o :::; rS* , then, for each f) E an \ B(w, er') , the 'fundamental inequality', x- 1

(5.1)

u(y) >- u( y) y) d( y, an) :::; IV' u ( l :::; d(y, an) -

A

holds whenever y E 80 n B(f}, lfi - w i fe) n B(w, r0). Using this fact we see that if 0 < 0 ::::: o* then there exists r;, depending only on p, n, a, /3, ,, such that if we define a non-tangential approach region at w E an, denoted fl(w, fj), by 0(w, fj) { y E n : d (y 80) 2: ill Y - wl}, then u satisfies (5. 1 ) in [n \ O(w, ij)] n (B(w, r0) \ B(w, cr')). (5 .2)

,

=

We observe that the above argument applies for any small r' > 0 if u is a minimal positive A-harmonic function with respect to w. We note, in analogy with the proof of Theorem 1, that if we apriori knew that (5 .1) held. in n n B(w, r) for some r > 0, then we could apply the argument in Steps C, D of the introduction to get an analogue of Theorem 1 in n n B(w, r) \ B(w, cr'). Letting r'-tO we would then get Theorem 2. Unfortunately though we do not know this apriori and we do not see how to 'deduce' this inequality from simpler functions as in the proof of Lemma 3.35. Still, if (5. 1) holds in nn B(w, r), whenever A E Mp (a) , then we can make use of appropriate versions of Lemmas 3.1 and 3. 18, as well as Definition 1.1 (iii), to conclude that (5.1) holds in n n B (w, s) , for some s < r, whenever A E Mp (a, /3, -y) . Thus to prove Theorem 2 we first prove Theorem 2 under the assumption that (5.3)

In particular, we start by showing that if one such A-harmonic function satisfies the 'fundamental inequality ' then all such functions, relative to the given A, have this property. More specifically we prove,

Lemma 5.4. Let n be a bounded (o, r0) -Reifenberg fiat domain and let w E an. Let A E Mv (a) for some a and 1 0 be A-harmonic in n \ B(w, r'), continuous in Rn \ B(w, r') , with u = v = 0 on Rn \ [n U B(w, r')]. Suppose for some r1, r' < r1 < ro , and b 2: 1 , that b- 1

u (y) < 0, >., c 2: 1 , depending on p, n, a, b, su ch that if 0 < o < J+ < J (8 as in Theorem 1), then r1

whenever y

E

v (y ) d(y, an) :::;

I V'v(y) I A

v ( y)

:::; >. d( y , an)

n n [B(w, rtfc) \ B(w, cr')] . .Moreover,

r' ( z ) - lo u (y) g( l
)

a

BOUNDARY HARNACK INEQUA LITIES FOR OPERATORS OF p-LAPLACE TYPE

2 55

whenever z, y E f2 \ B(w, cr') .

Proof: We note that to prove the last statement of Lemma 5.4 we can assume that since otherwise there is nothing to prove. Let r = cr' . If C. = c(p , n, a)

r' /r1 < < 1 ,

is large enough,

we may assume

u :S v /2 :S cu in f2 \ B(w, r),

(5.5)

,

as we see from Theorem 1, Harnack's inequality, and the maximum principle for A-harmonic functions. As in (1. 15) let u ( · , t), t E [0, 1 ] be A-harmonic in f2 \ B (w, r), with continuous boundary values, we

(5.6)

u( -, t)

=

(1 - t) fL(· ) + tf {)

on 8 [0 \ B ( w , f)].

Extend u ( · , t), t E [0, 1] , to be continuous on Rn \ [n U .8(w, f)] by setting u(- , t) 0 on this set. Next we note from Lemma 3.18 that there exists Eu = t:o(P, n, a, b) such that if s 1 and p1 satisfy f :S s1 < pJ /4 5 rt/16, t E [0, 1] , and =

(1 - t:o )L :S u(- , t ) /it( · ) :::::; (1 + co)L,

(5.7)

in f2 n [B(w, 2pl ) \ B(w, sl ) ] for some L, then

�-1

(5.8)

u (y ,

t) < I Vu( y t) l < ,\ u (y, t) an) d(y, ' - d(y , ao)

-

whenever y E 0 n [B(w, pl ) \ B(w, 2s 1 ) ] where ,\ = _\(p, n, (5.6), that if t1, t2 E [0, 1], then 1 c- u(· , h )

a,

b) . Observe from (5.5),

:

u( · , t ) - �( · , t1 ) 2- 1 v (·) - it(·) ::::: c u ( · , t l )

<

U ( - , t1 , t2 )

(5.9)

=

on 8[0\B(w, f) ] . Moreover, from the maximum principle we see ihat this inequality also holds in 0 \ B (w , f) . Thus for Eo as in (5.7), there exists fo, 0 < Eo :::; with the same dependence as Eo, such that if lt2 - t1 l :S E0, then 1 - Eo/2 :::;

(5 . 10)

.

u ( · , t2)

u ( tl ) ·,

fl) ,

S 1

+ Eo/2 in f2 \ H(w, f) .

Divide [0, 1 ] into closed intervals, disjoint except for endpoints, of length f00/2 except possibly for the interval containing 1 which is of length s E0/2. Let 6 0 < < . < em = 1 be the endpoints of these intervals. Thus [0 , 1] is divided into 6 { [�k , �k+1 ] }J." . Next suppose for some l, 1 :::; l :::; m - 1 , that (5.8) is valid whenever t E [e! , 6+1] and y E 0 n [B(w, p1 ) \ B(w, 2sl)] . Under this assumption claim for some i\ , c2, a, depending only on p, n, a, b, that

.

=

(5. 1 1)

I

Iog

u ( z , �l + l )

(

we

u , ) < , s1 - log (y �! +1 c1 min(lz - wl, IY - wl) u (y el ) u( z, et)

, I

)

0

whenever y E n n (B(w, Pl/f':-2) \ B(w, c2 s 1 ) ] . I ndeed we can retrace the argument in Step C of the introduction to get, for z, y E 0 n [B (w, pl /c) \ B ( w, cs1 ) ] , that z,

256

JOHN L. LEWIS, NIKLAS LUNDSTROM, AND KAJ NYSTROM

there exists f as in ( 1 . 19) and u > 0 as in ( 1 .24) such that <

-

(5 . 12)

S

e1+1

fI

e1

c

f (z t) , u(z, t)

_

1

f (y, t ) dt u (y, t )

( min z - �1, (l

IY - wl)

)

a

To get the last inequality in (5 . 12) we have used a slightly more general version of Lemma 4.7. We now proceed by induction. Observe from (5. 10) and u (· , 6) = 11. , that (5. 7) (5.8) hold whenever t E [6 , 6]. Thus (5. 1 1 ) is true for l 1 with s 1 = r, P l = r l /4. Let s2 = c2 s1, P2 = P1/c2. By induction, suppose for some 2 :<; k < m, =

(5. 13)

I

log

u(z, �k ) u(z)

- log

u(y, �k) u(y)

l :<; (k - ( , 1)cl

sk min( l z - w l , IY - wl)

)a

whenever z, y E n n [B(w, Pk ) \ B (w, sk)] , where u, i\ are the constants in (5 . 1 1 ) . For ry > 0 given and small we choose s � � 2sk, so that

I

u ( z , �k) u(z)

_

u(y , �k) u(y)

I

<

-

u (z, � k) T/ u(z)

whenever z, y E 0 n [B(w, Pk) \ B (w , sk)]. Moreover, choose r1 > 0 so small that (1 _

(5. 14)

fix z as in the last

display and

u (z, �k) < u (y, t) < ( I + u (z , �k) ) E Q u(z) , fQ u(z) u(y) )

whenever y E n n [B(w, Pk) \ B (w, sUJ and t E [�k . �k+ l l · To estimate the size of ry observe, for t E [�k. �k+l ] , that u(y, t) u(y)

=

u(z, �k) u(y, t) . u(y, �k) < (1 + E /2)(I + ) O f/ u(y) u(z) . u(y, �k)

Thus if ry = Eo/4 (€0 small) , then the right hand inequality in (5.14) is valid. A similar argument gives the left hand inequality in (5. 14) when ry = Eo/4. Abo since k :<; 2/E� , and E� , u depend only on p, n, a, b, we deduce from (5. 13) that one can take s� = c3sk for c3 c(p n, a, b) large enough. From (5. 14) we first find that u�,;)) in O n [B(w, pk) \ B(w, s�)] and thereupon that (5.8) (5.7) holds with i also holds. From (5.8) we now get, as in (5.12), that (5. 11) is valid for l k in O n [B(w, �) \ B(w, 2c2s}. )] . Let sk +l 2cac2sk and Pk +l = "ft · Using (5.11) and the induction hypothesis we have =

,

=

=

=

l log

(5.15)

u(z, �k+ l ) u(z)

u y k ) - 1og ( , � +l u(y)

I I log llog <

-

+ <

-

kc 1

u(z , �k+ 1 ) u (z, �k)

- log u(y , �k+ I ) u(y, �k)

u (z, �k) _ 1 u(y , �k ) og u(y) u(z)

(

sk+I min ( l z - w l , IY - w l )

l )a

I

BOUNDARY HARNACK INEQUALITIES

FOR OPERATORS OF

p-LAPLACE TYPE

257

whenever z, y E f! n [B(w, Pk+l) \ B(w, sk+t)]. From ( 5 . 1 5) and induction we get (5. 13) with k = m. Since u(-, �m ) = v and Sm :S cr', Pm � rt fc, for some brp;c c = c(p, n, a) , we can now argue as in (5. 14) to first get (5.7) with u(· , t) replaced by v and then (5.8) for v. We conclude that Lemma 5.4 is valid for z , y E f! n [B (w, rtf c) \ B(w, cr')] provided c is large enough. Using the maximum principle for A-harmonic functions it follows that the last display in Lemma 5.4 is also valid for z, y E n \ B(w, ri /c) . D

5.1. Proof of Theorem 2 when A E Mp (a). Let n c Rn , w E 8n, c5, p, r0, as in Theorem 2. Let A E Mp(a), and suppose that u , v, are minimal positive A-harmonic functions relative to w E on. If (5. 1) holds for u in f! n B (w, r1 ) , then we can apply Lemma 5.4 to u, v and let r'---+0. We then get that u/v equals a constant, which is the conclusion of Theorem 2. Thus to complete the proof of Theorem 2 for A E Mp (a) , it suffices to show the existence of a minimal positive A-harmonic function u relative to w E an and 0 < T} < To for which the 'fundamental inequality' in (5 . 1 ) holds in n n B(w, rl ) . Moreover , it suffices to show that ( 5. 1 ) holds for some r1 = r1 (p, n, a), 0 < r1 < r0 , >. = 5.(p, n, a ) � 1 , in O(w, ii) n B(w, r 1 ) where fj = ij(p, n, a ) is as i n (5. 2 ) . To this end we show there exists c = c(p, n, a) � 1 such that if c2r' < r < r0/n, and p = rfc, then (5.1) holds for u on O(w, fj) n 8B(w, p) . Here u > 0 is A-harmonic in n \ B(w, r') with continuous boundary values and u = o on an \ fJ(w, r'). It then follows from arbitrariness of r, r', the above discussion, and Lemma 5.4 that Theorem 2 is valid whenever A E Mp (a) and u is a minimal positive A-harmonic function relative to w E 80. With this game plan in mind, observe from Lemma 2.15 and (1.7), that we may assmne r = 1, w = 0, and a,

{3, "f, be

n, H( O , 4n) n {y : Yn :S -J.£} c Rn \ 0 , 1 where J.£ = 500nc5* , 0 < p. < 10- 00 and r' < ( o* )2. Here 8* is temporarily allowed to vary but will be fixed after the proof of Lemma 5.19. Extend fi. to be continuous on Rn \ B(O, r'), by putting u = 0 on Rn \ (0 u B(O, r')) . Using the notation in (2.7), let Q = Qt, 1 _J.I(I1en ) \ B( O , ,fii) and let v 1 be the A-harmonic function in Q with the following continuous boundary values, (5.16) B (O, 4n) n {y : Yn � 1'·}

Vt (Y)

=

vt (y)

c

u(y) , y E 8Q n {y : 2p. :S Yn} , (Yn - l1) u(y) , y E 8Q n {y : p. :S Yn < 211}· j.£

Comparing boundary values and ul:lirtp; the maximum principle for A-harmonic func tions, it follows that ( 5 . 1 7)

We now set

1-£

v1 :::; u in Q .

=

!-£(«'-) = exp(-1/«'-). We ::ihall prove,

Lemma 5. 18. Let 0 < E S €,

J.£ = p.(«'-) be as above and let ij be as in (5.2}. If E is small enough, then there exists B = O(p, n, a) , 0 < {J :::; 1/2, such that if p = J.£1 12-e, then

whenever

y

E 0(0, fi/4) n

1 :S u( y) fvt (Y) :S 1 + t:

[B(O, p) \ B(O, 2fo)].

2G8

JOHN

L. LEWIS,

NIKLAS LUNDSTRO M , AND KAJ NYSTRO M

Lemma 5. 19. Let v1 , f, f., 0 , J.L be as in Lemma 5. 18 and let ij be as in (5.2). If € is small enough, there exist 8 e(p, n, a), O < () < 0/4) . = 5-(p, n, a) > 1, such that

if P

=

J.L l / 2 - 20' a

= J.L - e ' then

=

< I vi (Y) V'v1 (y) l � A d(y, an) d(y, an)

- _ 1 v1 (y)

A

whenever y

E

0 (0, �/2)

n

-

-

[B(O, ap) \ B(O , pja)] .

Assuming Lemmas 5. 18, 5. 19, are true we complete the proof of Theorem 2 when A E lv!p (a) as follows. From these lemmas and Lemma 3.18 we de duce, for sufficiently small f. = f.(p, n, a: ) > 0, that (5.1) is valid for u and for some ). = 5. (p, n, a:) 2 1 in D. (w, ij) n aB ( O, p). With € now fixed we put 8* = p,(E)/ (500n) and conclude from (5.2), Lemma 2.15, arbitrariness of r, that (5. 1) holds in n n [B (w , ri ) \ B(w, r' )] with r1 = r0 /c, r' � r0jc', provided c, c' are large enough, depending only on p, n, a:. Thus we can apply Lemma 5.4 and proceed as in the discussion after that lemma to get Theorem 2 under the assumption A E Mp ( a: ) . Proof of Lemma 5. 18. To begin the proof of Lemma 5.18 observe from (5.17) that it suffices to prove the righthand inequality in this display. We note that if y E aQ and u (y) =!= v1 (y), then y lies within 4J.L of a point in 8Q. Also maxaB (O ,t ) u is non-increasing as a function of t 2 r', as we see from the maximum principle for A-harmonic functions. Using these facts and Lemmas 2 . 1- 2.3 we find that (5.20)

on fJQ. By the maximum principle this inequality also holds in Q. Here iJ is the exponent of HOlder continuity in Lemma 2.2. Using Harnack's inequality, we also find that there exist T = T (p, n, a: ) 2 1 and c = c(p, n, a: ) > 1 such that (5.21)

max{ V; ( z ) , lb (y) } $ c(d( z , fJQ) /d(y , 8Q)t min{ 'f/' ( z ), 1/'(y) }

whenever z E Q , y E Q n B ( z , 4d( z , 8Q)) and 1/J = u or v1 . Also from Lemmas 2.12.3 applied to v1 , we get (5.22)

Let p, {) be as in Lemma 5.18. Using (5.20) - (5.22), we see that if y E 0 (0, ij/4) n [B(O, p) \ B ( O, 2 fo)] , then (5.23)

u(y) $ vi (y ) + CJ.L&/2 u(foen ) �

( 1 + C2f.J,& /2-e-,- ) VI (Y)

$ (1 +

f)VI (Y)

provided f. is small enough and ih = G-/4. The proof of Lemma 5.18 is complete

.

D

Proof of Lemma 5.19. To prove Lemma 5.19 we let VJ , t:, f., e, f-1, be as in Lemma 5.18. Using Lemmas 2.2 - 2.3 and Harnack's inequality we see that there exists ¢> = ¢> (p , n, a:) > 0, 0 < ¢> $ 1/2 , and c = c (p, n, a:) > 1 with

u (y) $ c(sjt) ri>u(sen ) provided y E Rn \ B(O, t) , t � s � 2r'. Using (5.24) with t Lemmas 2.1 - 2.3 we see that (5.24)

(5.25)

=

1 , s = 2fo, and

BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE

2ri9

where c depends only on p, n, a. Let ii be the A-harmonic function in Q with continuous boundary values ii = 0 on 8Q \ B(O, JJi) , and v = v 1 on 8B(O, fo). Then from (5.25) and the maximum principle, we see that ii S v 1 S v +

( 5 .26)

Let p = J.-L 1 12 - 28 , e small, and a = to '¢ = v we find v

(5.27)

CJ.-L12u( ..fiien ) in Q.

J.-L-8

be as in Lemma 5 . 19. Using (5.21) applied

2 c- 1 (J.-L112jap fu(.Jiie n) = c- 1 !138" u( Jiie n)

on 0(0, ij/8)n[B(O, 2ap)\B(O, pj (2a))], where T is aB in (5.21) and the nontangential approach region 0 was defined above (5.2) relative to w, ij. Abo, since ij depends only on p, n, a, it follows that c = c(p, n, a) in (5.27) . If we define () by () = min{¢/(12T) , B/4}, then from (5 .26) , (5.27) we get

V1 < 1 + € <1ii -

(5.28)

in 0 ( 0, r//8) n B(O, 2ap) \ B(O, pj(2a)), whenever 0 S small. Next let v be the A-harmonic function in

E

S f., provided f. is sufficiently

Q' = QL_t<(J.-Len ) \ B ( 2 J/iem J/i) with continuous boundary values v = 0 on 8Q' \ B(2..(iien , .jji) and v = 1 on aB(2foen, fo). We claim that (5.29) v (y) S c(2foen - y , Vv(y)) when y E Q'. Assuming claim (5.29) we can complete the proof of Lemma 5.19 in the following manner. First observe that (5.29) implies there exists c = c(p, n, TJ) 2: 1, for given ry , 0 <

·TJ

S 1/2, with

v(y) v(y) _ I Vv(y) l < c d (y &Q' ) d(y BQ' ) < , , in Q(O, ry) \ B (O , 10fo), where Q(O, ry) is the non-tangential approach region defined relative to 0, ry, Q, aB above (5.2) . From the observation in (5.2) with u, n, replaced by v, Q and (5.30) for suitable 7J = TJ(p, n, a) we deduce that (5.30) in fact holds in Q \ B(O, lOJJi) . We can now use Lemma 5.4 in Q \ B(O, 10fo) with v, v playing the role of u, v, respectively. In particular, we get for some large c = c(p, n, a) that

(5. 30 )

(5. 31)

c

_1

c- 1 d

_

v (y) (y , an)

s

I VTv(y) I

<

_

v (y ) c d (y, an)

in Q n B(O, 1/c* ) \ B(O, c* y'"[i) for some c* = c* (p, n, a ) . Finally, note that if 0 S E S f. and if f. is sufficiently small, then 1/c* > 2ap > pf(2a) > c* fo· From this fact, (5.31), (5.28), and Lemma 3.18 applied to v, vb we deduce that Lemma 5.19 is valid subject to claim (5.29). To prove claim (5.29) we first observe from Lemma..<.; 2.2, 2.3 that v(z) S 1/2 in Q' n B(O, 10fo) for some z whose distance from {)Q is at least c - 1 fo where c = c(p, n, a). Using Harnack's inequality it follows for some c' > 1 that v S: 1 - 1/ c' on {)B(2foen, 3fo/2) . If y E B( 2foen , 3fo/2) \ B(2foen, y'fi) , set (5.32)

((y) =

eNiy - zl2 I�' - eN

e9Nf4 _ eN

260

JOHN L. LEWIS,

NIKLAS LUNDSTR OM , AND KAJ NYSTRO M

where z = 2.,fiie n . Then ( = 0 on 8B (2.Jiien , .,fii) , and ( = 1 on 8B(2.Jiien, 3ffi/2). Also, if N = N(p, n, a) is large enough in (5.32), then from direct calculation and Definition 1 . 1 , we find V' · A ( V'( ) 2: 0 in B (2.Jiie n, 3ffi/2) \ B(2Jiien, .,fii) . More over, using these facts and the maximum principle we deduce (5.33)

in B(2ffien , 3 ffi/2) \ B(2foen , /Ji) provided c+ Next for fixed t > 1 put

=

c + (p, n, a) is large enough.

2 foe n + t ( y - 2 foen ) E Q '}, v(y) - v(2 Jiien t( y - 2 Jiie n )) ( whenever y E O . F y, t ) = t-1

{y E

0

F (y )

Q'

:

+

From (5.33) for t > 1 fixed, t near 1 , and basic geometry it follows that (5.34 )

We note that (iv) of Definition 1.1 and A E Mv (a) imply that an A-harmonic function remains A-harmonic under scaling, translation, and multiplication by a constant. From this fact we see that F is the difference of two A-harmonic func tions in 0 and one of them is a constant multiple of v. Using this fact, (5.34), and the maximum principle for A-harmonic functions, it follows that F 2: c - 1v in 0. Letting t---- 1, using Lemma 2.4 and the chain rule, we get claim (5.29). The proof of Lemma 5.19 is finished. D As mentioned earlier, Lemmas 5.18, 5 . 19 together with Lemma 5.4 imply The orem 2 when A E Mv (a) .

5.2. Proof of Theorem 2 . We are now ready to prove Theorem 2 in the general case. Lemma 5.35. Let n be a bounded (b, ro) -Reifenberg fiat domain and let w E 80.. Let A E Mp (a, ;3, "'!) for some ( a , ;3, "'!) and 1 0 be A-harmonic in 0. \ B(w, r') , continuous in Rn \ B (w, r'), with u := v := 0 on Rn \ [0. u B(w, r') ] . Then there exists 8* , CJ > 0 , c+ � 1, depending o n p, n , a , /3, 'Y , such that 4 0 < 8 < 6. < J (J as in Theorem 1} and r1 = ro fc+ , then

I

log (

wheneveT z, y

E

' r u(z ) u (y) ) - log ( )I < c+ ( min ( rl , lz - w l , v( y ) v(z )

0. \ B (w, c+ r') .

I Y - wl )

)

a

Proof: Once again we assume that r' fr1 < < 1, since otherwise there is nothing to prove. As in (5.5) we may assmne for some c = c(p, n, a, /3, "'!) that u :S: v/2 :S: cu in 0. \ B(w, 2r') .

(5.36)

Let u(·, t ) , t E [0, 1], b e A-harmonic in 0. \ B(w , 2r'), with continuous boundary values, (5.37)

·u (·, t)

=

( 1 - t ) u(·) + t v(·) on 8 [0. \ B(w, 2r') ] .

261

BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE

We claim there exists c, ). 2: 1 depending only on p, n, a, /3, 'Y such that if t E [0, 1], and y E n n [B(w, fo/c) \ B(w, cr')], then - _1

.A

(5.38 )

u(y , t )

d (y, an) < I Vu ( y , t) l _

<

_

-

u(y , t )

:.. d (y , an ) "

Indeed let A1 (y, ry) = A(w, ry) whenever y E Rn and "' E Rn \ {0}. Let 1 < a < b and suppose that p is such that 2r' :::; p/a < bp :::; f0/2. Let v (· , t ) , for t E [0, 1], he A1-harmonic in n \ B(w, pja) with continuous boundary values equal to u ( · , t). Then, from Lemmas 5.4, 5.18, 5.19 we see that if a a(p, n, a) is large enough, then =

(5.39)

IVv( · , t) l

�

v ( ·, t) /d( · , an )

on n n 8B (w, p). Here � means with constants depending only on p, n, a. Let h (·, t) he the A1-ha.rmonic function in n1 = n n [B(w, bp) \ B (w, pja)] with continuous boundary values equal to u(·, t) . We claim that if b = b(p, n, a, /3, 'Y) is large enough then (5.39) is also valid for h . In fact (5.24) holds with u replaced by u(-, t ) for t E [0, 1], where now rf> rf>(p, n, a, /3, 'Y) and s 2: 2r'. Ut;ing (5.24) for u(-, t) we get =

v (·, t) :::; h ( - , t ) :::; v (· , t ) + c (ab) -if> u(penfa, t ) on anl ·

From the maximum principle this inequality also holds in nl . Moreover' for T (5.2 1 ) we deduce, v (· , t) 2: c-1 a -Tu(penfa, t) on O(w, fJ/2) (5.40)

n

Clli

in

(B(w, 2p) \ B(w, p/2) ). Thus, for some c' = c'(p, n , a, (3 , 'Y) 2: 1 , v ( · , t ) :::; h ( · , t) :::; ( 1 + c'aT-t/> b-if>)v(·, t)

on O (w, �/2) n (B(w, 2p) \ B (w, p/2)). Choosing b = b(p, n, a, (3, 'YHa.rge enough in (5.40), using (5.39), Lemma 3.18, it follows that (5.41)

X+ 1 h (y, t)/d( y , an) :::; IVh ( y, t ) l :::; A+h ( y , t)jd (y , an )

whenever y E 0( w, fJ) n 8B( w , p) for some )\+ = >.+ (p, n, a, (3, 'Y) 2: 1 . From (5.2) we see that (5.41) holds on n n 8B(w, p) provided -A-r (p, n, a, p, 'Y) is large enough. With a , b, now fixed, depending only on p, n, a, /3, 'Y, we can use Lemma 2 . 15 and argue as in Lemma 3.1 to conclude for given f. > 0, the existence < of r·1 = ·r1 (p, n, a, (3, 'Y, t) so small that if bp :::; r1 r0 , then 1 - E :::; u (- , t)jh(· , t) :::; 1 + t

on O(w, fJ/2) n (B(w, 2p) \ B(w, p/2)) . In view of this inequality, (5.41 ) , and Lemma 3 . 18, we see that if t = t(p, n, a, (3, 'Y) is small enough, then (5.4 2)

I Vu( ·, t ) l

�

u( ·, t) /d ( - , an)

on fl(w, fj) n 8B(w, p) , where proportionality constants depend only on p, n, a, (3, 'Y In view of (5.2), this inequality holds on n n 8B(w, p) . With r 1 , a, b fixed we see from arbitrariness of p that (5.38) is true. We can now argue as in Lemma 5.4 or just repeat the argument in ( 1 . 1 8) - ( 1 .25) to conclude Lemma 5.35. 0 As pointed out earlier in this section, if u, v are minimal A-harmonic functions relative to w E an, then we can apply Lemma 5.35 and Let r' -tO to get Theorem 2. The proof of Theorem 2 is now complete. D

2

62

JOHN L. LEWIS, NIKLAS LUNDSTROM, AND KAJ NYSTROM

6. Appendix : an alternative approach to deformations

In this section we show that Step C in Theorem 1 can be replaced by a some what different argument based on ideas in [WJ. The first author would like to thank Mikhail Feldman for making him aware of the ideas in [WJ . In the following all con stants will depend only on p, n, a, (3, --y and we suppose that u, v are A-harmonic in n n B(w, 4r) and continuous in B(w, 4r) with u = v = 0 on B(w, 4r) \ n. From Lemma 3.35 we see that if o is small enough, f = rjc, and c is large enough, then for some p, ;::: 1 , (6. 1)

j],

-1

h(y) h(y) 'Vh(y) l d(y, an) :::: I :::: P- d(y, an )

whenever y E 0 n B(w, 4f), h E {u, v }. Also from Lemma 4.8 exists J1,. ;::: 1 , for t:* > 0 fixed, such that ( 6.2)

p, ;

1

() s

f

1 +<.

::; h( a. ( w) ) h(ar (w) ) � 11-•

see that there

()

whenever y E 0 n B (w, f), h E { u, v}, where 0 < x, >. E Rn , � E Rn \ {0}, that

j:

we

-<· s 1 f s

< 4f. Observe again, for

1

A; (x, >.) - Ai(x, 0

t

0

(6.3)

A, (x , t>. + ( 1 - t)�)dt

n

L ( Aj - �j )

·-t J-

1

'

aA ( JOTJJ 0

x,

t). + ( 1 - t )� )dt

for i E { 1 , ... , n}. In view of (6.3), (6 .1), and A-harmonicity of u, v, we deduce that u - v is a weak solution to L( = o in n n B(w, f) , where

-

L((x)

(6.4)

and aij (x)

=

n

a · (a;j (x)(,j ) i,j=l ax t

L

1

=

J �:: (tV'u(x) + (1 - t)'Vv(x) )dt, 0

for 1 � i, j ::; n. Moreover, from the strueture assumptions on A, see Definition 1 . 1 , we find that (6.5) whenever

c:;:1 ( I V'u (x) l + I'Vv(x) I )P-2 1 � 12

X

n

L

a;j (x ) �i�j i,j=l < c+ ( I'V u(x) l + I'Vv(x) l)p-2 1 �12 <

E n n B (w, f). Next we prove the following lemma.

Lemma 6.6. There exists c ;::: 1, oo > 0 , such that if r* = r· jc, and 0 < o < oo, then (I'V u l + I 'Vv i ) P- 2 extends to an A2 -weight ·in B(w, r*) with Az -constant depending only on p, n, a, (3, --y.

BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE

263

Proof: The proof is essentially the same as the proof of Lemma 4.9. That is , we use a Whitney cube decomposition of Rn \ D to extend ( I'V'ul + I 'V'v i ) P- 2 to a function >.. on B(w, 4r* ) . Let w E B (w, r*) and 0 < f < r* . Let w E 80 with lw - w l = d(w, 80) and suppose that l w - wl/2 < f < r•. We assume, as we may, that

(6.7)

Let 5. = >.. when p �

max{ u (ar(w )) , v (ar(w ))} = u (ar ( w )) . 2 and 5. = 1/>.. for 1 0, small enough, that

j

(6 .8)

B('w ,r)

and

5.dx

:::;

r.n (ar (7i1 )) 1P-2 1 r:n- IP-2 1

J

onB(w,50f') B(w,r ) 2 < cu(ar ( w )) -IP -21 ;-:n+lp- 1 . (6.9) These inequalities em ain true if r :::; lw - wl/2, follows e ily from (6. 1). Com bining (6.8), (6.9) , and using arbitrariness of w, f, we get Lemma 6.6. D r

as

as

Using the ideas in [W] we continue by proving the following.

oo, w E 80, 0 < r < rn, suppose that u and v are non-negative A-harmonic Junctions in n n B (w , 2r) with v :::; ft. Assume also that u, v, are continuous in B (w , 2r ) with u = 0 = v on B(w, 2r) \ D. Lpt r* be as in Lemma 6. 6. There exists c � 1 such that if f = r* /c, then r( c- 1 u(af'(w)) - v(af'(w)) < u (y) - v(y) < c u(af'(w)) - v(a w)) v(y ) v (a;;(w)) v(ar(w)) whenever y E D () B (w , f) .

Lemma 6.10. Given p, 1 < p <

Proof: We first prove the lefthand inequality in Lemma 6.10. To do so we show the existence of A, 1 :::; A < oo, and c � 1 , such t hat if r' = r* jc and if v(y) u( y) - v( y) (5.11) e ( y) = A u(ar• (w)) - v(ar• (w)) v(ar• (w)) for y E D () B(w , r* ) , then e (y ) � 0 whenever y E D () B(w, 2r ) . ( 6. 1 2) To do this, we initially allow A, c � 1 to vary in (6. 11). A, c, a e then fixed near the end of the argument. Put A u (y ) u' ( y) - u (ar• (w)) - v(ar• (w)) , A v( y) v( y ) ' ( y) + - ft(ar• (w)) - v(ar• (w)) v(ar• (w)) . Observe from (6.11) that e = u' Using Definition 1 . 1 (iv) we see that u', v' are A-harmonic functions. Let L be defined in (6 .4) using u' , v', instead of u, v,

)

(

_

1

v

_

' v .

as

r

:.!64

JOHN L. LEWIS, NIKLAS LUNDSTROM, AND KAJ NYSTROM

and let e 1 , e2 be the solutions to Lei boundary values:

=

0, i = 1, 2, in n n B(w, r* ), with continuous

(6.13) whenever y E 8( n n B (w, r* )). From Lemma 6.6 we see that Lemma 4 . 7 can be applied and we get, for some c+ 2: 1 and r+ = r� jc+ , that

e l (Y) _ 1 el (ar+ (w)) + e2 (ar+ (w)) :::; e2 ( y) :::; whenever y E !1 n B ( w, 2r+ ) · We now put

( 6·14 )

c

, 1 c = c+' r

and observe from (6 . 14) that

=

c+

e1 (ar+ (w))

e2 ( ar_ (w))

e (ar' (w) ) r+ ' A = c 2

e 1 ( ar' (w)) '

Aet (Y) - e2 (y) 2: 0 whenever y E !1 n B(w, 2r').

(6 . 15)

Let e = A e1 e2 and note from linearity of L that e, e , both satisfy the �arne linear locally uniformly elliptic pde in O n B ( w, r*) and also that these functions have the same continuous boundary values on 8(!1 n B ( w, r*)). Hence, using the maximum principle for the operator L it follows that e = e and then by (6.15) that e (y) ;::: o in On B(w, 2r' ) . To complete the proof of the left-hand inequality in Lemma 6. 10 with f = 2r', we observe from Lemmas 4.5, 4.6, that A :::; c . The proof of the right-hand inequality in Lemma 6.11 is similar. We omit the details. D -

We note that in [LN5] Lemma 6.10 was proved under the assumptions that u and v are non-negative p-harmonic functions in !1 n B( w, 2r) and that !1 c Rn is a Lipschitz domain. In this case the constants in Lemma 6.10 depend only on p, n and the Lipschitz constant of !1. Moreover, in [LN5] this result is used to prove regularity of a Lipschitz free boundary in a general two-phase free boundary problem for the p-Laplace operator.

Proof of Theorem 1. Let u, v, A, 0, w, r be as in Theorem 1 and let u, v be the A-harmonic function� in !1 n H(w, 2r) with u = max{u , v} and v = rnin{ u , v } on 8[!1 n B(w, 2r) ] .

From the maximum principle for A-harmonic function� we have u 2: v and hence we can apply Lemma 6.10 to conclude that c- 1

u (ar (w)) v (ar (w ) )

<

-

u(y) v(y)

<

-

c u (ar (w)) v (ar(w ) )

whenever y E OnB(w, f). Moreover, using the definition of u , v , and the inequalities in the last display we can conclude that (6.16) Next if X

u (z ) u (y ) whenever y, z E O n B(w, r) . $c v (z) v (y) E an n B(w, r/8), then we let u u M(p) = sup - and m(p) = inf R(:-c,p) 'V B(x,p) V -

-

_

BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE

265

when 0 < p < r. If p is fixed we can apply Lemma 6.10 with u = u, v = m(p)v, and 2r replaced by p to conclude the existence of c* , c", such that if p = p/ c* , then (6. 17)

M(p) - m(p)

�

c* (m(jj) - m(p) ) .

Likewise, we can apply Lemma 6.10 with u (M(p) v - u)/u

�

=

M (p) v and iJ = u to conclude

constant on n n B (w, jj) .

Using this inequality together with (6.16) it follows that

(M(p)v - u)/v � constant on n n B (w, p).

Here we have used heavily the fact that A-harmonic functions remain A-harmonic after multiplication by a constant as follows from Definition 1 .1 (i v) . Thus if c* is large enough, then

M(p) - m( p) � c* (M(p) - M(jj)) . If osc (t) = M(t) - m(t), then we can add (6. 1 7) , (6. 18) and we get , after some arithmetic, that c -1 (6. 19) osc (jj) � _ *_ osc (p) c* + 1 We can now use (6.19), since c* is independent of p. m an iterative argument. Doing this we can conclude that (6. 18)

0

(6.20)

osc (s) � c(s/t)0 osc (t) whenever 0 < s < t

� r/2

for some (} > 0, c � 1. (6.20), (6.16) , along with arbitrarin�ss of x E 80 n B(w, r/8) and interior Holder continuity - Harnack inequalities for ·u , ·u, are easily seen to imply Theorem 1. D

References [B] P. Bauman, Positive solutions of elliptic equations in non - divergen ce form and their adjoints, Ark. Mat. 22 (1984), no.2, 153 - 173 . [BL] B. Bennewitz and J. Lewis, On the dimensi o n of p-harmonic measure , Ann. Acad. Sci. Fenn. 30 (2005), 459-505. [CFMS] L. Caffarelli, E. Fabes, S. Mortola, S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana J. Math. 30 (4) (1981) 62 1-640. [FGMS] E. Fabes, N. Garofalo, M. Malave, S. Salsa, Fatou theorems for some non-linear elliptic equations, Rev. Mat. lberoamericana 4 (1988), no. 2, 227 - 251. [FKS] E . Fabes, C. Kenig, and R. Serapioni, The local regularity of solutions t o degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), no. 1, 77 - 116. [FJK] E . Fabes, D. Jerison, and C. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, 151-182. [F J Kl ] E. Fabes, D . Jtlrison, and C. Kenig, Boundary behavior of solutions to degenerate elliptic equations. Conference on harmonicn analysis in honor of Antonio Zygmund, Vol I, II Chicago, Ill, 1981, 577-589, Wadsworth Math. Ser, Wadsworth Belmont CA, 1983. [GT] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, second edition, Springer-Verlag, 1983. [GZ] R. Gariepy and W. Ziemer, A regularity condition at the boundary for solutions of qaasilinear elliptic equations, Arch. Rat. Mech. Anal. 67 (1977), no. 1, 25-39. [JK] D. Jerison and C. Kenig, Boundary behaviour of harmonic functions in nontangentially accessible domains, Advances in Math. 46 (1982), 80-147. [KKPT] C.E Kenig, H. Koch, J. Pipher, T. Taro, A new approach to absolute continuity of elliptic measure with applications to non-symmetric equations, Adv. in Math 153 (2000), 231-298. [KT] C. Kenig and T. Toro, Harmonic meas ure on locally fi at domains, Duke Math J. 87 (1997), 501-551.

266

JOHN L. LEWIS

,

NIKLAS LUNDSTROM. AND KAJ NYSTROM

[LN] J. Lewis and K. Nystrom, Boundary behaviour for p-harrnonic functions in Lipschitz and starlike Lipschitz ring domains, Ann. Sc. Ecole Norm. S up. (4) 40 (2007), no. 4, 765-813. [LN1] J. Lewis and K. Nystrom, Boundary behaviour and the Martin boundary problem for p harmonic functions in Lipschitz domains, submitted. [LN2) J . Lewis and K. Nystrom, Regularity and free bo und ary regul arity for the p-Laplacian in Lipschitz and C 1 - dom ains , Ann. Acad. Sci. Fenn. 33 (2008) , 1 - 26. [LN3] J . Lewis and K. Nystrom, New results for p-harmonic functi ons, to appear in Pure and Applied Math Quarterly. [LN4] J. Lewis and K. Nystrom, Boundary behavionr ofp-harrnonic functions in domains beyond Lipschitz domains, Advances in the Calculus of Variations 1 (2008), 1 - 38. [LNS] J. Lewis and K. Nystrom, Regularity of Lipschitz free bounda,·ies in two-phase problems for the p - Laplace operato r, subm it ted . [Li] G. Lieb erman , Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), no. 1 1 , 1203-1219. [M) V.G. M az'ya , The continuity at a boundary point of the solutions of quasilinear elliptic equations (Russian), Vcstnik Lenin grad. Univ. 25 (1970), no. 13, 42-55. [R] Y.G Reshetnyak Y.G., Space mappings with bounded distortion, Translations of mathematical monographs , 73, American Mathematical Society, 1989. [S] J. Serrin, Local behavior of solutions of quasilinear elliptic equations, Acta Math. 111 (1964), 247-302. [T] P. 'lblksdorf, Regula1·iy for a more general class of quasilinear elliptic equations, J. Differential Equatio ns, 51 (1984), no. 1, 126-150. [Tl] P. Tolksdorf, Everywhere regularity for some quasilinMr sy.�tems with a lack of ellip ticity, Ann. Mat. Pura Appl. (4) 134 (1983), 241-266. [W] P. Wang, Regularity of free boundaries of two-phase problems for fully non-linear elliptic

of second order. Part 1: Lipschitz free boundari es are C 1 • Q , Communications on Applied Mathe mat ics . 53 (2000), 799-810. Current address: Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA

equations Pure and

E-mail address: j obn(!)ms . uky . edu

Current address: Department of Mathematics , Umea University, E-mail address: email : niklas . lunds tromillm at h . umu . se

Current address: De part ment of Mathematics, Umeii. University, E-mail address: kaj . nystromtDmat h . umu . se

S-90187 Umca, Sweden

S-90187 Umeii., Sweden

Proceedings of Symposia

Volum� '!'9, 2008

in Pure Mathematics

Waves on a steady stream with vorticity M. Lilli and J. F . Toland

ABSTRACT. The existence question for two-dimensional p erio dic water waves on the surface of a flow with vorticity is one of finding a region upon which the solution of a semi-linear elliptic equation simultaneously satisfies two inde pendent boundary c onditions. Here we reduce this problem to a quasi-linear elliptic equation on a fixed domain with one nonlinear boundary condition and study the existence of non-trivial solutions using bifurcation theory. Although our reduction is a very slight variant of the classical one due to Dubreil-Jacotin, it significantly simplifies some of the analysis and extends the scope of the the ory. For example, for a large class of laminar flow profiles, we find bifurcating k-modal waves with negative speeds for all k E N, and with positive speeds for a finite family of k.

1.

Introduction

When the wavelength is normalized to be 2tr, the existence question for two dimensional periodic water waves on the surface of a flow with vorticity is one of finding a domain upon which the solution of a semi-linear elliptic equation simul taneously satisfies two independent boundary conditions ( 1.1 d) and ( 1 . 1e) below: ( 1 . 1a)

(l .lb) ( l . lc) ( 1 . 1d)

-

6.1/J (x , y )

=

1/J(-, y) is 27r-periodic in x,

1(1/J(x, y)),

-d < y

<

1J(X ) ,

1/J(x, -d) = 0,

1/J (x, 17(x)) = c1 , 1 ) ( l . le) 2 J'V'f/;(x, 1J(x) ) J 2 + 917(x = c2 , where c1 (the volume flow rate) and c2 (the Bernoulli constant, also known as the total head) are constants. Here 'ljJ denotes the stream function and the vorticity in the flow has been assumed to be a given function 1 of 1/J. The impermeable bottom of the channel is located at y = -d, the acceleration due to gravity is g and the 1991 Mathematic.� Subject Clas!lijication . Primary 35R.35, 74J15; Secondary 74J30, 76D27. Key words and phrases. Water waves, vorticity, l*lrni-linear elliptic, free-boundary problems M . Li lli acknowledges the German Science Foundation which supported his work at the University of Bath. J. F. Toland holds a Royal Society/Wolfson Merit Award. .

@2008 American Mathematical Society

257

268

M.

LILLI AND

J. F.

TOLAND

curve { (x, TJ(x)) ; x E R}, where 17 is 27r-periodic, is the unknown free boundary upon which the stream function (1. 1d) and the pressure (1.1e) must be constants. In a study of the existence of small amplitude waves, Dubreil-Jacotin [12] defined a new function h(x, p) on the fixed domain R = R x [0, 1] as (1.2) 'if; (x , h(x , p) ) = c1p , x E R , p E [0 , 1] , and observed that the free-boundary problem (1.1) is equivalent to; h� hxx + (1 + h; )hpp - 2hp hx hxp - c11'Y(CJ P)h� = 0 on R, (1.3a) (1.3b) h (x , O) = -d, 1 + h� + 2c1 2 (gh - cz)h; 0, p = 1, (1.3c) h(, p ) is 27r-periodic in x. (1.3d) This system has become a cornerstone of the growing literature on large-amplitude water waves with vorticity that began with the work of Constantin & Strauss [5, 6] , in which the vorticity-stream-function 1' E Cl,a is prel:lcribed. In other work on waves with vorticity [1, 3, 9, 14, 15] , 1' is not fixed. Instead, the vorticity of a bifurcating wave is presumed to originate in the parametrized family of laminar streams from which it bifurcates. This is our point of view. To pursue it we introduce variables different from those of [12] . As with (1.3), the new system, (2.4) below, involves nonlinear operators that are real-analytic functions of the unknown function � . and there is an obvious variational structure. Moreover, it has a trivial solution, � (z) = z, independent of the laminar flow to which it corresponds, and the linearization (2.8) about that solution leads to transparent bifurcation criteria in a large number of situations, see Theorem 5.4. If it is required, this method can readily be adapted to yield an alternative approach to the theory in [5, 6, 15]. =

2. Formulation of the Problem

Consider a laminar running stream for which the vertical distribution of hori zontal velocity is given by U(y) , y E [-·d, 0] . Suppose U E C2 ( -d, 0) n C1•19 [-d, 0] , ,J E (0, 1), and that U(y) =/= 0 on [-d, 0] . The corresponding stream function is then (2 .1)

w(y)

(2. 2)

"f(W(y))

=

[yd U(z) dz,

and the dependence of vorticity on the stream function is given by =

-U' ( y),

y E [-d, O] ,

which is the definition of a C'9-function 1' on the interval {w(y) ; y E [-d , OJ}. With 1 Ct = \li (O) and c2 = 2 u(0)2, w is a solution of (1 .1) and the question is whether there are non-laminar solutions of ( 1.1) for the same vorticity function "f . REMARK 2.1. Although these hypotheses, and later (5.5) and (5.9), are quite unrestrictive, neither this formulation nor that in [5] cover all cm:;es of practical interest. For example, [3, page 102] and [14] considers waves bifurcating from running streams of the form U(y) = c + u(y) , where u(y) = (d + y) 117 and c is a parameter. Her e U E C117 and the vorticity at the bottom is infinite. This

WAVES 0:-1 A STEADY STREAJ\1 WITH VORTICI'l'Y

269

leads to a version of problem (2.4) below with singular coefficients to which we will 0

��-

-d, 0 'lfJ (x, �(x, z)) = \ll (z), x E IR., z E [-d, Oj . (2.3) Since w'(z) = U(z) and w"(z) = U'( z), z E ( -d, 0), the system to be sathified by � is U'(z ) { 1 - 1 ;��; } + U(z) { (�:) � C;�� ) J = 0, (2.4a) (2.4b) � ( x , -d) = -d, U (z) 2 (1 + �;) + 2(g� - � U(0) 2 ) �; = 0, z = 0, (2.4c) f)(-,z) is 21!'-periodic in x. (2.4d) The trivial solution of (2.4). Since we seek non-trivial solutions of ( 1 .1), it is important to note that f)(z) = z is the solution of (2.4) that corresponds to the laminar-flow solution of ( 1 . 1) , no matter what the given function U may be. This free-boundary problem can be transformed into a problem on a fixed domain by defining � : lR. x [ ] -> lR. as

x -

Variational Structure of (2 .4) . At this point we make the formal observation

that the transformed system (2.4) has variational structure. This is no surprise since the original free-boundary problem ( 1 . 1) has the variational structure discussed in [2] (see also [10]) and our change of variables (2.3) leads from there to the functional J below. (An analogous variational formulation [7, § 4.1] of the Dubreil-Jacotin equation (1.3) follows similarly. ) For functions � which are periodic in on the semi-infinite strip S = lR. x in the )-plane, with = let

x [-d, 0] f) (x, -d) -d, 2 1 J ( �) � J Ish U2 { ;zf); + f)z } dxdz - � 1 7!' f)(x, 0)2dx, (2.5) where S2 = ( ) ( -d, 0). Then critical points of (2.5) satisfy the system (2.4). Moreover, f)(z) = z is a critical point of J. So let f)(x, z) = z+��: ( x, z) in the formula for J. Then the first term has the form � J /' U 2 { 2 + /\.; + K; } dxdz = C + � J r U2 { II:� + ��:; } dxdz , 2 Js2-. 2 Js2.. 1 + ;;,z 1 + ;;,z (x, z

=

rr

- 1r , 1r

x

where C is independent of ;;,. Therefore where

we

are interested in critical points of J,

= -2l/ 1 ( "'12 ++ll:z"'2z ) U(z)2dxdz - -2g 11r ��:(x, oY.! dx. Critical points of J satisfy the system (2. 7a) ( 1U2 ;;,ll:z ) ( 1u+2 "' ) 2 ( U2(1(;;,+2 ll:+z)11:22 ) ) 0, (2.7b) ��:(x, -d) = 0, (2.7c) U(0):-! ( 1 + K�(x, O)) + (2g��:(x, O) - U(0) 2 )(1 + ��:z (x,0)) 2 = 0, (2.7d) ��:(·, z) is 21r-periodic in x. (2.6)

J(;;,)

S2-.r

+

:r:

x

+

X

Z

ll:z

z

1

_,.

X

Z

z

•

270

M. LILLI AND J.

(2.8a)

(U21ix)x + (U2�z)z = 0,

F.

TOLAND

Linearization of (2.4) . The functional J has a critical point K = 0 , irrespective of U and the linearization of the Euler-Lagrange system (2.7) , with respect to K, about this zero solution is li(x, -d) = 0, 2 g'K(x, 0 ) - U(0) 'Kz (x, 0 ) = 0 , 'K( · , z) is 27r-periodic in x.

(2.8b)

(2.8c) (2 .8d)

We will see that this linear problem is easy to analyze using separation of variables.

3. Parametrized Families of Laminar Streams

Now we consider a parametrized family of laminar running streams, U(y; c) , y E [-d, 0),

cEI

C JR, where, for c E I (an open interval) , U(· ; c) E C2(-d, O) , U(y; c) i= 0 on [-d, O] and c �---+ U(· ; c) E C2 (I; C1•19 [-d, Ol) . Here no physical meaning is assigned to the parameter c, the dependence of U on c being quite general. Let the corresponding

stream function be denoted by

\II (y; c) =

jy

-d

U(z; c) dz,

and the dependence of vorticity on the stream function by 1(\II (y; c) ) = -U' (y; c) ,

With

c1 (c) = \II (O; c) and c (c)

y E [- d, 0] . 1

2 u(O; c) 2, U(· ; c) and 1 = 1 ( · ; c) . The corresponding 2

=

lit ( · ; r:) is a solution of ( 1 . 1 ) when U solution of (2.7) is "' = 0 for all c E I. The question is whether there are other (non-laminar) solutions of ( 1 . 1 ) for the same vorticity function 1(· ; c) for certain values of c. This is a global question, but here we regard it as a question of finding bifurcation points on the line of trivial solutions {K 0 , c E I} of system (2.7) . =

=

4. Bifurcation Theory

We now consider basic bifurcation theory [8] for the nonlinear problem ( U(z; c)2 Kx ) + ( U (z; c ) 2Kz ) 1 + Kz x 1 + Kz z

(4_ 1a)

( 4.1b) (4. 1 c)

-

� U (z; c) 2 (K; + K�) 2

r;, (x, -d) = 0 ;

(

( 1 + Kz) 2

)

z

=

O,

U(O, c) 2( 1 + l'i:; (x , 0)) + (2gl'i: ( X, 0 ) - U( O , c) 2) ( 1 + l'i:z (x, 0) ) 2 = 0 , K ( - , z ) is 27r-periodic in x ,

(4.1d)

regarding c as the bifurcation parameter. To simplify matters we will seek solutions K that are even in x. To this end let X=

{"' E C2·1J (S) : K is even and 21r-periodic in x and K( x , -d) = 0} , 0 y = {,. E C ·1J (S) : /'i, is even and 2tr-periodic in X} , Z = { w E C1•19(JR) : w is even and 27r-pcriodic},

2 71

WAVES ON A STEADY STREAM WITH VORTICITY

which are Banach spaces when endowed with the usual Holder-space norms. Let B denote the open ball of radius 1 about the origin in X x Y and define F B x JR y

X

:

Z by

F (c, K)

-+

=

(

( U(z; c)2i'i:z ) ( U (z; c)2 Kx ) + 1 + "-z z x 1 + Kz

_

� ( U(x; c)2 (K; + K; ) ) z 2 (1 + Kz) 2

)

U(O; c)2(1 + l'i:�(x, 0)) + (2gl'i: (X, 0) - U(c; 0)2) (1 + Kz (x , 0) )2

It is clear that F is twice continuously differentiable from B x I into Y x Z and that F (O, c) = 0 E Y x Z for all c E I. In order to show that a particular c* is a bifurcation point for the problem F ( K, c) = 0 , it will suffice to show that, for some � E X \ {0} , ker d�
(4.2a)

�c (d��:F[(O, c)]�) l c=c• � Range d�< F[(O, c* )] ,

(4.2b)

range d�
(4.2c)

x

Z.

REMARK 4. 1 . It is interesting to note that the parameter c occurs nonlinearly in the linearized problem (2.8). Nevertheless, the system (4.2) coincides with the hypotheses in [8] that ensure that c* is a bifurcation point for (4. 1 ) . If U depends real-analytically on c, as it does under the hypotheses of Theorem 5.4, the operat or in (4. 1 ) is real-analytic on B x JR. In t h t case the theory of [4] is available t o extend the local real-analytic curve that bifurcates from the s imple eigenvalue to a uniquely defined global curve which has, in a neighbourhood of each of its points, a 0 local real-analytic parametrization. Stipulations (4.2a) and (4.2b ) mean that the solutions � of the linear problem a

(4.3a) (4.3b)

(U(z; c*)2Kx)x + (U(z; c*)2Kz)z = 0, K(x , -d) = 0 ,

(4.3c)

g�(x, 0) - U(O; c* )2Kz (x, 0) = 0 , �( · , z) is even and 21r-periodic in x ,

( 4 . 3d )

form on dimensional subspace of X, and a

(4.4)

e-

(

(2U(z; c*)

�� (z; c* )Kx )x + (2U(z; c* ) �� (z; c* )�z ) -2U(O; c*) �� (0; c*)�z(x, 0) #

(

z

)

(U(z; c* ?ii:x)x + (U(z; c* )2t'iz )z gti(x, 0) - U(O; c* )2tiz (x, 0)

for ny ti E X. The meaning of (4.2c) is that the set a

K (x , -d) = 0, K ( -, z ) even and 21r-periodic,

)

}

has codimension 1 in Y x Z. This will follow by s t andard arguments if we can show that there is a unique solution of (4.3), up to scalar multiplication, because (4.3) is

272

M. LILLI AND J. F.

TOLAND

a self-adjoint eigenvalue problem in an L2 setting. (See the last paragraph of [13, § 6.7] . ) In fact, in that ca..<-;e (j, h) E Y x Z is in the range of d,.;F[ (O , c*)] if and only if

j [2"

�(x, z )f(x, z) dxdx

+ I: �(x , O)h(x) dx

=

0.

Thus c* will be shown to be a bifurcation point for ( 4.1 ) if we can show � is unique up Lo normalisation and that ( 4.4) holds. \Ve will study the uniqueness question presently, but first here is an observation that will be useful in checking that ( 4.4) holds. Suppose that (4.4) does not hold. More precisely, suppose that K, f. 0 satisfies (4.3) and that equality in (4.4) holds for some K. E X . A multiplication of the first component of the equality in ( 4.4) by K, and integration by parts over 82., , using the periodicity in the x direction, yields

J [2 " U (z; c* ) �� (z; c*) I Y''KI 2 dxdz =

1"

-11"

U(O; c* )

0U ( 0 ; c*)K.(x, O) 'Kz (x, 0) dx 0c

+ � I: U(O; c* )2 ('Kz(x, O)K.(x, 0) - ;:.:(x , O)K.z(x, 0) ) dx = 0,

from equality in the second components of (4.4). Therefore, if (4.4) is false, then

(4.5)

j [2"

U(z; c*)

�� (z; c* )j\7'Kj 2 dxdz

=

0.

If, for example, if (oU joc) (z; c*) is not zero on [-d, 0] , this cannot happen. Separation of variables. VIe seek values of c for which there exists a non-trivial solution of (4.3 ) . It is easy to see, by separation of variables and completeness of the eigenvalues in an L 2 setting, that if such a K, exists then it must be in the form ;:.:(x, z) = a(x) b(z) , where a is 2n-periodic. This means that there exists k such that (4.6)

(4. 7a)

a" (x ) + k2 a (x)

=

(U(z; c) 2 b')'

(4.7b )

b ( - d) = 0,

(4. 8a)

v" - (k2

0 where a is 2n-periodic,

-

k2 U( z; c)2b = 0,

gb(O) - U(O; c) 2bz (O)

=

0.

The equation for a has constant eoeffi.cients and may be solved explicitly if and only if k is an integer. Its only even solution is a multiple of cos kx . In general, we cannot solve the equation for b explicitly and we will study it in greater detail later. However, in one case at least, all its solutions are known in closed form. To see how this is so, suppose that b satisfies (4 . 7) and let v ( z) = U( z; c) b(z). Then v satisfies (4.8b)

+

U" ( z ; c) ) v = 0, U( z; c)

k E N,

(g + U(O; c) U'(O; c)) v(O) = U(O; c)2 v'(O ).

WAVES ON A STEADY STREAM WITH VORTICITY

273

running stream with constant vorticity, including irrotational flows. An im U(y; c) : = c woy -=f. 0 on [-d, 0] . Here the vorticity -wo, (4.9a) v" - k2 v = 0, (4.9b) (g +woe) v(O) = c2 v'(O). There is a solution for certain values of . (k(d + z) ) where (g +wk oc) tanh(kd) = c2 . v(z) = smh (4.9c) The fact that 0 does not lie between c and c-w0d, equivalently Lhat U(- ; c) does not vanish, is a further restriction, but this problem can be analyzed completely. Let g > 0 and k E N be2 given. When wo = 0, (4.9c) says that c is uniquely determined up to its sign by c jg = k - 1 tanb (dk), which is the classical value of c for the kth point of irrotatinnal waves from streams of depth d. However, when wobifurcation -=f. 0, for each k E N there exists cJ; < 0 < ct satitifying (4.9c). If w0 > 0, then 0 > cj; + woy fc 0 on [-d, 0] for any k. Hence c;; is an admissible solution of (4.9c) for all k E N. However ct + woy fc 0 on [-d, OJ means that ct > wod > 0. Therefore ct is admissible solution of (4.9c) only for k E N with d2 2 (1 - t khd(kd) ) < f!.k tanh(kd) . (4. 10) The cases wo > 0 and < 0 are symmetrical, with the superscripts + and 0 inter hanged.

A portant case of (4.8) is when + 1 ::::::: a constant, and (4.8) bas the form

c:

E lR

an

an

Wo

w0

c

5. General Linear Theory of a Running Stream As an illustration of these general mnsiderations we look at the important case when for some E C 1 ,19 • Here we may astiume that 0, without loss of generality, because the value of can be absorbed in the parameter In this set up, represents the horizontal-velocity profile of a running stream in a fixed frame of reference and, relative to a frame moving with velocity the horizontal velocity profile becomes c Thus non-zero solutions of (4.1), for some corresponds to waves travelling with velocity -c on this stream. Our general theory bas reduced the question of bifurcation points to proving that, for a certain value of c, the linear problem

u(O) =

U(y; c) = c + u(y) c. u -c, c

u

u(O)

+ u.

(5.la) ((u +c)2b1)1 = >.(u +c)2b, b(-d) 0, gb(O) = c2bz (O) , (5.1b) has a simple ige lue = k 2 , ome k E N, where c is such that u(y) +c =f=. 0 on [problem -d, 0] . When u(y) c =f=. 0 on [-d, 0] the problem for >. is a regular Sturm-Liouvlle + for which th are given by a classical Rayleigh-Ritz minim e

nva

>.

=

for o

e eigenvalues .,\

ax

principle for the quotient formula

(u(y) + c)2v' (y) 2 dy - gv(0)2 ° 1 d Q(v; c) = -

(5.2)

in which

g

and

·u

are fixed.

0

1 (u(y) + c)2v(y)2 dy -

d

,

274

M. LILLI

AND J. F. TOLAND

LEMMA 5 .1 . Suppose that g is fixed and that u + c #- 0 on [-d, 0) . Then there exists an increasing sequence of eigenvalues Aj (c), j E N, of (5.1), characterised as follows: Aj(c) =

in£

dim(E)=j

Moreover, >.1 (c)

is

a

{sup Q(v, c} : v E E C W 1 • 2 [-d, O], v =f:- 0 , v(-d) = O } .

simple eigenvalue .

PROOF. This minimax characterization of Aj (c) is part of the classical theory, see [11, §4.5) , for example. Moreover, solutions of the eigenvalue problem (5. 1} attain these minimax values. In particular, .At (c} is attained at a certain function v. Since Q(lvl; c) � Q(v; c), we may assume that ..\1 (c) is also attained at lvl . Now suppose that >.1 (c) is attained at v1 and v2 , and therefore that lv 1 l and lv2 l are eigenfunctions of (5.1) for the eigenvalue >.1 (c). If lv1 l and lv2 l are not linearly dependent, it follows that lv1 l l v2 l (u + c)2dz = 0. Since this is false, lvd is a

jo

-d

scalar multiple of lv2 l· Since both satisfy (5.1), it follows that v1 is a multiple of v2 , as required for ..\ 1 ( c) to be a simple eigenvalue. D

o j Pc =

Let

-d

dy >0 (u (y ) + c )2

and con�:>ider the eigenvalue problem (5.3a} !" = J.Lf, f( O ) = 0, gf(Pc ) = j' ( Pc), (5.3b)

f ¢ 0.

It is easy to see that there exists a solution with JL = v2 > 0 if and only if gPc > 1 , in which case f(z) = a sinh vz for some a #- 0, where v is uniquely determined by tanh vPc 1 vPc

g Pc .

When gPc > 1 all the other eigenvalues J.L of (5.3) (there are infinitely many) are negative and determined by tan vPc 1 . J.L = -v 2 and f ( z) = sm vz where P. . v

P.

c

=

g

c

By a similar calculation, every eigenvalue of (5.3) is negative when gPc < 1 , and when gPc = 1 all its eigenvalues are non-positive, exactly one (counting multiplicity) being zero with eigenfunction f(z) = z. As with Q and (5.1), these eigenvalues correspond to minimax values of q (f; c) =

J:c f' (z)2 dz - gf(Pc ) 2 J:c f(z)2 dz

over the class of non-zero functions f E W1· 2 (0, Pc) with f(O) = 0. From the above observations we infer that when gPc ::::; 1, inf { q( f, c) : f E W 1 • 2 [0 , Pc) , f #- 0, f (O ) ) = 0 } 2: 0. (5.4a) However, when gPc > 1, ( 5.4b)

inf { q(f, c) : f E W1. 2 [0, PcJ , f #- 0, f(O)) = 0 }

<

0

275

WAVRS ON A STEADY STREAM WITH VORTICITY

and, for all j 2:: 2, . { sup q ( f, e) : f E F c W1 •2 [0, PcJ, f cf 0, f (O)

dtm(F)=J

(5.4c)

. inf

=

0} > 0.

We return now to our study of (5 .1). In addition our basic assumption that u E C2 ( - d, 0) n C1•'9 [ - d, 0] with u(O) = 0, we now as;,ume that u (y) < 0 , y E [- d, 0) .

(5.5)

When f : [0, P,] --) IR is smooth and f(O)

v (y ) =

jy

-d

=

0, let

dt (u(t) + c)2

Then v ( - d) = 0 and, when substituted in (5.2) , we infer from (5.4a) thaL >.1 (c) > 0 when gPc < 1 .

(5.6)

Because (4.8) and ( 5 . 1 ) are equivalent, our main results on the eigenvalue problem (5. 1) represent a significant simplification and extension of [15, Lemma 2.5] . LEMMA 5.2. Suppose that (5.5) holds and that c < 0. Then >.1 (c) --) -oo as c / 0 and >-t (e) > 0 for all c < 0 with lei sufficiently large. Hence, for each k E N, there ex-ists c; < 0 such that - k2 = >.1 (c;) . PROOF. Note first that

Pc --)

(5.7) since u(O) substitute

=

0 and lu'(O)I <

dy j-od 2 = oo as e / 0, u (y )

oo.

v ( -d)

=

jy

:

[0, Pc] --) IR be such that f(O)

=

0 and

dt v ( ) = -d (u(t) + e) 2 . y

Then

Let f

0 and, when substituted in (5.2), we find that

). 1 (c) < -

I:

- 1°

(5.8)

-

Now, 1 1 2:: p

c

Pc - gP; (u( y) + c) 2

_}u( y) + c)2

ly

-d

(Lyd t�: r ) ( jy (u (

pc- 1 - g

1

Pc

-d

c) 2

dy

dt 2 dy 2 t) + e) (u(

dt --+ 0 for y E [-d, I) as c / 0. (u (t ) + c) 2

by (5.7), and hence >.1 (c) --) -oo as c / 0, by (5.7) and the dominated convergence theorem. It follows from (5.4) that >.1 (e) ::; 0 if and only if Peg 2:: 1 , which is true for all c < 0 with lei sufficiently small, by (5.7) . Finally note from (5.6) that >.1 (c) > 0 for all lei sufficiently large, since gPc --) 0 as lei --+ oo. Since >.1 (c) obviously depends continuously on c < 0, the result 0 follows.

M. LILLI AND J.

276

F. TOLAND

To consider the behaviour of >. 1 (c) for positive c suppose that (5.9) u( -d) < u(y) for all y E ( -d, 0] and let y -u( -d) > 0. Note that (5. 10) Pc ---> oo as c \, y. We now restrict attention to c E (�, oo) . =

LEMMA 5.3. Suppose that (5.9) holds. (a)

liminf >. 1 (c) :::;

(5 . 1 1 )

c "..:g

-g

2 iiu + � II P( -d, o)

{b) >.1 (c) > 0 for all c > 0 sufficiently large. (c) Suppose that -k2 > lim infc� >.1 (c) , k E

-k2 . PROOF. Since P, ---> oo as c \, �,

>.r (ct)

.

N. There exists ct > y, su ch that

=

1 1 2: p c

jy

-d

dt

·

( 'IJ, (t ) + c) 2

--->

1 as c \, y for all y E ( - d, 0] ,

and ( 5 . 1 1 ) follow.s from (5.8) . As in the preceding proof, >.1 (c) > 0 for all c suffi 0 ciently large.

REMARK. An example of Lemma 5.3 (a) arises in the problem of bifurcation of waves on flows of constant vorticity with d = 1, in which case u(y) Wo Y , Wo E R When w0 > 0 the hypotheses of Lemma 5.2 are satisfied. Moreover, in Lemma 5.3, y, wo and il u + Y i ll2(-r,o) = w5/3. Hence -k2 = >.r ( c) for some c > y, > 0 is an eigenvalue if w5 k 2 < 3g. In fact, for this example we have seen from an explicit calculation that -k2 = >.1 (c) for some c > y, if and only if w5 (1 - k- 1 tanh k) < g k - 1 tanh k. (In particular, -1 >.1 (c) for some c > w0 if and only if w5 < 3. 194g (which may be compared 0 with w5 < 3g , the criterion in the Lemma) . =

=

=

THEOREM 5.4. The parameter values ct in the preceding two lemmas are furcation points for problem (2.7) when U(z ; c) = u(z) + c.

bi

PROOF. We have shown that (4.2a) is satisfied when c ct , and (4.2c) follows by the self-adjointness of (4 . 3 ) in an L2 setting. Since 8Uj8c(O; ct ) 1, it follows from ( 4 .5 ) that K, 0 if ( 4.2b) is false. Hence hypothesis (4.2) is satisfied, and it follows that the eigenvalues ct are bifurcation points for the problem of waves on 0 a running stream. =

=

=

References [1]

A.

J.

Abdullah, Wave motion

at the

surface of a current which has an exponential distribution

of vorticity, Annals New York Acad. Sci. 51

( 1 949), 425 - 441. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. , 325 (1981), 105-144. [3] T. B. Benjamin, The solitary wave on a stream with an arbitrary distribution of vorticity, J. Fluid Mech. 12 (1962), 97 -1Hi. [4] B. Buffoni and J. F. Toland, Analytic Theory of Global Bifurcation - An Introduction. Prince ton University P ress , Princeton, N. J., 2003.

[2]

H. W.

WAVES ON A STEADY STREAM WITH VORTICITY

277

[5] A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., Vol. LVII (2004), 481-527. (6] A. Constantin and W. Strauss, Exact periodic traveling water waves with vorticity, C. R. Math. Acad. Sci. Paris 335 (10) (2002), 797-800. (7] A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math. , Vol. LX (2007) , 911-950. [8] M. G. Crandall and P. H. Rabinowitz, B ifurcati on from a simple eigenvalue, J. Funct. Anal. 8 (1971), 321-340 . (9] R. A. Dalrymple and J. C. Cox, Symmetric finite-amplitude r ot ational water waves, J. Physical Oceanography, 6 (1976), 847-852. [10] I. I. Danil iuk, On i ntegral functionals with a variable domain of integration, Proc. Steklov Inst. Math. 118 (1972). In English, Am er. Math. Soc. (1976). [11] E. B. Davies, Sp�ctral Theory and Differential Operators. Cambridge University Press, Cam bridge, 1995. (12] M.-L. Dubreil-Jacotin, Sur Ia determination r:igoureuse des ondes permanentes periodiques dampleur finie. J. Math. Pures Appl. 13 (1934), 217291. [ 13] D. Gilbarg and N. S. Trudinger , Elliptic Partial Differential Equations of Second Order. 2nd Edition. Springer, New York, 1983. (14] J. N. Hunt, Gravity waves in flowing water, Proc. R. Soc. London A, 231 (1955) , 496-504. [15] V. M. Hur and M. Lin, Unstable surface waves in running water, ArXiv 0708:0541VI [Math. AP] 3rd Aug 2007, to appear. UNIVER.SITii.T AucsBURG, lNSTITUT FUR

MATHF:MATIK,

U!-�IVERSITTSSTR.ASSE 14,

86159 Aves

BURG, GERMANY

Current address : Department of Mathematical Sciences, U niversity of Bath, Claverton Down, Bath BA2 7AY, UK E-mail address: lilli/Dmath .uni-augsburg .de DEPARTMENT OF MATHEMATICAL SCIENCES, UNlVERSlTY OF

BA2 7AY, UK

E-mail address:

jftiDmaths . bath . ac . uk

BATH,

CLAVRR1'0N DOWN, BATH

Proceedings of Symposia in Pure Mathematics Volume 79, 2008

On analytic capacity of portions of continuum and a question of T. Murai Fedor Nazarov and Alexander Volberg This paper is dedicated to Vladimir Maz'ya

We give an answer to an old question of T. Murai concerning the characterization of the boundednesss of the Cauchy integral operator on arbitrary set� of finite Hausdorff length. If the set is a continuum, we got a new proof to a theorem of Guy David characterizing the rectifiable curves on the plane for which the Cauchy integral operator is bounded on L2(ds). In doing that we use also a nonhomogeneous version of a certain Tb theorem first proved by M . Christ in homogeneous spaces. We are going to "compute" in metric terms the analytic capacity of the intersection of an arbitrary continuum and a half-plane ( or a disc, or any domain with piecewise smooth boundary) . ABSTRACT.

1. Introduction Takafumi Murai asked in [Mu] the following question: given a compact set E C C su ch that its Hausdorff !-dimensional measure sat isfies 0 < H1 (E) < oo, is that true that Cauchy integral operator is bounded in L2(E, H1 I E) if and only if H1 (E n Q) S C ')'(E n Q) .

Here

/'

stands for analytic capacity defined in the next paragraph.

We give a positive answer to this question here.

THEOREM 1 . 1 . Let E be a compact on the complex plane such that 0 < H1 (E) < oo. Then the Cauchy integral operator T (and also r•) is hounded on L2 (E, H1) if and only if there exists a constant C such that for every square Q on

the plane

(1. 1 ) 2000

Mathematics Subject Classification.

Key words and phrases.

Primary 47B36; Secondary 42C05. Analytic capacity, Hausdorff content, nonhomogeneous harmonic

analysis, accretive functions. The first author was supported in part by NSF Grant 0501067. The second author was supported in part by NSF Grant 0501067.

279

F. Nazarov, A. Volberg

280

Definition. Let K be a compact set in C.

"f( K) : = sup{ lim l z f (z) l : J E Hol(C \ K) , lf( z) i :::; 1 Vz E C \ K, J(oo) = 0} , Z-+00

"f+ (K) : = sup{}�.� lz f(z)l : f(z) = By definition,

I �f.L�(I , f.L E M+ (K) , lf( z) l :::; 1 Vz E C \ K} .

.

(1.2)

In [T4] Tolsa proved that the opposite inequality also holds with absolute constant. It is a very tough theorem. We discuss its relations with results of this paper in the last section. A natural question arises: how verifiable is this criterion? Strangely enough it is sometimes verifiable, and this is the second main topic of this article. In Theorem 1.3 we compute (up to the absolute constant) the analytic capacity of certain class of sets. This allows us to observe in Section 3 that a famous theorem of Guy David is a one-line consequence of the above criterion ( 1 . 1). David 's theorem we are referring to is the one that characterizes all rectifiable curves on the plane on which the Cauchy integral operator is bounded. Let nH recall that there is another criterion of the boundedness of Cauchy integral. It is obtained in [NTV2] , [Tl] and we want to formulate it now. To do that we need to recall the reader the notion of Menger's curvature of a measure. Given three pointH z1 , z2 , z3 E C we call R(z 1 , z2 , za ) the radius of the circle ( may be oo) passing through those points. Then Menger's curvature of a positive measure f.L is by definition c2(/L) :=

)

1 2

R2 (z1 , z2 , z::�) H 1 1E for a certain compact E, O < H 1 (E) < oo, we will use the following

If f.L notations: =

(.I J .I

df.L ( zl ) df.L(z2 ) df.L(z3 )

cz(E) := c2(H1 IE) . We are ready to quote the criterion proved by Nazarov-Treil-Volherg in [NTV2] and Tolsa in [Tl] . THEOREM 1.2. 1) Cauchy integral operator is bounded in £2 (/L) if and only if there exists a finite constant C such that for every square Q (1.3) f.L(Q ) :::; c f( Q) ' where £(Q) is the length of the side of Q, and ( 1 .4)

C2 (f.LIQ ) 2 :::; c J-L( Q) .

2) In particular, if J-L = H 1 IE, for a certain compact E, O < H 1 (E) < oo, then the boundedness of the Cauchy integral operator in L2 ( E, H1 ) is equivalent to (1.5) and (1.6)

On analytic capacity of portions of continuum

28

1

Remark. Actually one can sometimes Rlcip assumption (1.3) as Tolsa has shown in Lemma 5.2 of [T5] . This is the case for measures J.t such that their upper density lim sup J.t(B(x, r) ) /r r--+0

is uniformly bounded. We are grateful to the referee for this remark. We will ::;how below that Theorem 1.1 implies easily part 2) of Theorem 1 .2. On the other hand, one can deduce Theorem 1 . 1 from Theorem 1 .2, but this requires a much more efl"orts. This deduction is based on already mentioned very tough result of Tolsa [T4] . This deduction is briefly discussed in the last section of this article. Another interesting feature of our criterion (1.1) is that one can prove its multi dimensional analogs, however, the multi-dimensional analogs of criterion (1.6) from [NTV2] , [Tl] are not known now (because they involve the notion of Menger's curvature that did not yet get multi-dimensional understanding) .

1 . 1 . Theorem 1 . 1 implies easily the second part of Theorem 1.2. The difficult part is to prove that ( 1 .5), (1 .6) imply the boundedness in L2 (E, H11E) of the Cauchy integral operator. \Ve want to use Theorem 1 . 1 . So for our goals it is sufficient to prove the following implication ( Q is always a square ) (1.7) VQ c2 (E n Q)2 :S C1 H1 (E n Q) :S C2f(Q) * H1 ( E n Q) :S C21(E n Q) . To prove (1 .7) we need the trivial inequality 1 � I+ and the following charac terization of I+ due to Melnikov (see e.g. [Tl] , it can be found in [Vo] also ) :

(1.8)

I+ ( K)

�

sup

p,: p,(B(x,r ) ) � r Vr 'Vx

11�112

c2 (J.t) +

II J.tl l

Obviously the second inequality in the left hand !:>ide of (1. 7) implie!:l that measure H 1 I E satisfies the growth condition: H 1 (E n B(x, r)) :::; C1 r .

Hence, m>ing (1 .8) for test measure

p, : =

C}1 H1 I E n Q we obtain

Hl (E n Q)2 1 I ( E n Q) � I+ ( E n Q) � a c2(E n Q) 2 + H l (E n Q) � a H (E n Q) .

We got the right hand side of (1.7) , which is the reduction we wanted.

1.2. David's characterization of bounded Cauchy integral operator on rectifiable curves: Ahlfors-David curves. If the set E is a rectifiable curve

r, then a theorem of Guy David describe::; all !:>uch curvet-> on the plane for which the Cauchy integral operator is bounded on L 2 (r, d H1) . This is the class of curves I' (called Ahlfors-David curves) satisfying

( 1 .9)

H1 (D(x, R)

n

r)) s C R

for a.ny disc D(x, R). Lipschitz curves and Lavrentiev (chord-arc) curves give us the examples satisfying (1 .9) . It is slightly strange that for Lipschitz curves (and even for chord-arc curves ) there exists a purely analytic proof of the boundedness of the Cauchy operator on £2 ( r, d H1) (see [CJS] , [Chr] ) , but all the proofs of the theorem of David are the mixtures of analytic and geometric arguments, [Dal] ,

282

F.

Nazarov,

A.

Volberg

[DaJ] . In the present paper we are going to show, in particular, that Guy David's characterization follows from two ingredients: a) Theorem 1.1, h) a simple computation in geometric terms of 1(f n Q) , where Q is a square and r is an arbitrary continuum, Theorem 1.3 belows. In the present paper our main idea is to use a local Tb theorem of M. Christ [Chr] (and not the usual Tb theorem or Tl theorem). The difference with [Chr] is that we have to use a nonhomogeneous version of a local Tb theorem. This nonhomogeneous version of Christ 's local Tb theorem leads us naturally to the "computation" of the analytic capacity of the intersection of our E with a square (or a disc) . It is quite well understood now that the analytic capacity of an arbitrary compact cannot he measured in terms of simple geometric characteristics of the compact. The result of Tolsa [T4] (see also the exposition of this result in [Vo] together with its multi-dimensional analogs) only confirms this because the analytic capacity is proved to he computable in metric terms, but only in quite complicated ones. See also, for example, [JM] , where it is shown that the Buffon needle probability cannot serve as a metric equivalent of analytic capacity in general. Surely, one can derive that on compact subsets of an Ahlfors-David curve the analytic capacity is equivalent to just H 1 measure. We will discuss this later, but now let us notice that the equivalence constants are not absolute. They depend on the geometry of the ambient curve. Secondly, to establish this equivalence one needs heavy tools: either the theorem of David or geometric arguments of Jones [J] and Melnikov [Me2] . However, to our surprise, there exists a class of compacts for which one can get the simple geometric measurement equivalent to the analytic capacity up to absolute constants. And this class is large enough to enable us to use our version of Tb theorem resulting in a new proof of the theorem of David. This class consists of intersections of any continuum with any closed half-plane {or any closed disc, or any closed square, ... ) . Let us introduce some notations. In what follows, a, a' , a" , A denote various positive finite absolute constants. Letters II, Q and D stand for various half planes, squares and discs respectively. \Ve will use the symbol h1 (E) to denote !-dimensional Hausdorff content of E, namely ,

h1 (E)

:=

inf{

L

r1

:

E C Uj D(xj , rj ) } .

It is clear that H1 is larger than h 1 , and they vanish simultaneously. For any continuum r the Hausdorff content is equivalent with the diameter. But the same is true for the analytic capacity 1(r). Thus, for a continuum r

(1. 10) We are going to prove {1.10) for sets ourselves to the the case of half-planes. THEOREM

Then (1.11)

r n II, r n Q, and r n D. We restrict

1 3 . Let II be a closed half-plane, and let r be a continuum. .

a

h 1 (r n ll)

:::; 1(r n II) :::; A h 1 (r

n II) .

283

On analytic capacity of portions of continuum

We prove this result below. Our proof was superseded by the proof by John Garnett [JG] that is considerably easier than ours. But we still decided to present our proof for the possible future generalizations to higher dimension. The simpler Garnett's proof relies on complex analysis observations.

2. The Cauchy integral operator on sets E, 0 < H 1 (E) < oo , and nonhomogeneous accretive system Tb theorem Let us remind that the function K(x, y) , x, y E IR.n is called a Calder6n Zygmund kernel of dimension m if there exist finite and positive constants C, c: such that

\ K(x, y) \ :::; Cdist (x , y ) -m ,

' ( ) I K x, y) - K ( x, y ) \ + \ K ( y, X

Cdist(y,) y'+) <:

K ( y ' X ) \ :::; ' ,

-

d1St ( X, y m <:

whenever dist (y, y') :::; l / 2dist (x, y) . Constants C, e are called Calderon-Zygmimd constants of the kernel K. A spe cial class of Calderon-Zygmund kernels are antisymmetric Calder6n-Zygnmnd ker nels, namely, such that K(x, y) = -K ( y, x) . For a given antisymmetric Calder6n Zygmund kernel K and a positive measure J.L with compact support such that �.t ( D (x , r)) :"::: C rm one can define an operator T with Calder6n-Zygmund kernel K as in [DaJ]. One can also define a maximal singular operator as follows T* f (x ) := sup e>O

Ir

}y:jy- x i>e:

K(x , y)f (y) d�.t (Y )

I

THEOREM 2.1 . Let J.L be a measure with compact support in JR.n . Let K be a Calder6n-Zygmund kernel of dimension m, and let for any disc D(x, r)

�.t(D(x, r )) :":::

C rm .

Let K(x, y) = -K(y, x ) . We denote by T a Calder6n-Zygmund operator with kernel K and assume that there exist finite positive constants B, o such that for every cube Q in IR.n with J..L ( Q) > 0, we have a function bq such that supp bq

\ (bq ) q \

:=

C

Q , \\bq \\co :::; 1 ,

�.t(�)l k

bq d�.t\ 2: o ,

\\T*bq \\oo :::; B

Then T* is a bounded operator on L2(J..L ) , and its norm depends only on B, o, n and the Calder6n-Zygmund constants of the kernel K. The system of functions { bQ} as above is called local accretive system. The statement imitates the statement of the theorem of M. Christ from [Chr] , but the proof requires considerations from the nonhomogeneous harmonic analysis (see

[NTV3] ).

\Ve are ready t o prove Theorem

1.1.

F. Nazarov, A. Volberg

284

PROOF. First, we are going to reformulate (3.1) in terms of the Cauchy trans form with respect to measure H1 on E. So we call

dHl := Je{ f(() (-z

Tf

in the sense of Calder6n-Zygmund operators, see [Dal] or [DaJS] . We also denote e: > O

T* f( z ) := sup

1 j(EE / ,I(-zi>c - l f(() dHl ( Z

This is called maximal Cauchy operator.

LEMMA 2.2. Let E satisfy (3.1). Then there exist positive finite constants K, fJ such that for any square Q with H1 ( Q n E) > 0 there exists a function bq so that supp bq c Q n E , ll bo lloo :<S; 1 , bq dHl l ::::: fJ , l ( bq ) o l : = I ( n E) H IJ T* bq lloo � K .

� I lone

The constants K, fJ depend only on the constant in (3. 1).

PROOF. Fix a square Q with H1(Q n E) > 0. From (3.1) we conclude that there exists a holomorphic function fq E H00(C \ (E n Q)), such that ll fq lloo � 1 and such that fq (oo) = 0, l f� (oo)) l ::::: k H1 (E n Q). Clearly (because E n Q has a finite H 1 measure), f q can be represented as a Cauchy integral of a measure bq dH1 on E n Q , where bq E L00 (dH1 ) , IJ bq ll oo :<S; 1. We also know that f; (oo) = bq (()dH1 ((), and thus ·

fenQ

I lre Q bq dH1 l ::::: k · H1 (E n Q)

(2.1)

n

Notice that the first two claims of our theorem are already proved (the second claim follows from (2. 1)). To prove the third claim we use the following standard estimates. Fix a point z E C and e > 0 . Consider

1(EE.I(-zl>< { !Q,e:,z (w) = --; lvcz,p) { JQ(w) dA(w) - � / 1 � JD(z,p) JD(z,p) (EEnD(z,<) JQ,e:,z (w ) =

bq (()dHl (() (-w

This function is holomorphic in D (z, c) . In particular, putting p = c:/2, we get 1rp

=

1rp

1rp

/q , e,z( w)dA( w)

bq (()dHl (() dA( w) ( W = tl + t2 .

The first term is good because it is an average of a bounded function: lt1 l :<S;

� { l fo ( w)l dA(w ) � ll fq lloo � 1 . 1rp D( z,p)

J

In the second term we put the absolute value inside and interchange the order of integration:

On analytic capacity

it2 i �

f � 7r J � f PJ P

�

EnD(z,t:)

EnD(z,e)

of

portions of

dH1 (() {

JD(z,p)

285

continuum

1r

1 Wl

-

'>

-

dA(w)

dH 1 (() ::; c::_ H1(E n D( z, p)) P

n Q(z, p) ) � 9:.H1 (E n Q(z, p)) ::; !!_'"'f(E cp P

a

::; -"((Q(z, p)) ::; A C a , cp where c denotes the constant. from (3.1), C = c-1 , and Q(z, p) denotes the smallest square with the same center as the disc D(z, p) and containing thiti disc. We used here (3.1) and an obvious estimate that the analytic capacity is bounded by the diameter of a set. Lemma 2.2 is proved. 0

p=

Now one can apply this Theorem 2.1 to the kernel K((, z) H1 IE. Of course m = 1 now, and the assumption

=

<:�z and mea.'lure

p ( B(z , r)) = H1 (B (x, r) n E) ::; C ·r follows from H1(B(x, r) nE) � C-y(B(x, r)nE) � C'Y(B(x, r)) � Cr by the obvious estimate that the analytic capacity is bounded by the diameter of a set. In Lemma 2.2 we just proved the existence of a local of functions required in the assumptions of Theorem 2.1. To finish the proof of our main Theorem 1.1 we just need to apply Theorem 2.1. Theorem 1 . 1 is completely proved. 0 3. Application: the Cauchy integral operator on Ahlfors-David curves and nonhomogeneous accretive system Tb theorem THEOREM 3.1. Let C be an Ahlfor&-David curve. There exist positive finite constants b, B such that

b H 1 (C n Q) s 'Y(C n Q) s B H 1 (C n Q)

(3.1)

PROOF. The right inequality is well known. The proof of the left inequality is the corollary of Theorem 1.3. In fact, let {D(xj , rJ)} be a covering of C n Q such that I:; rj � 2 h 1 (C n Q). By the Ahlfors-David property of C, one has

H1(C n Q) � L H1(C n D(xj , rj)) � B L rj � 2B h 1 (C n Q) . j

We combine this with (1.11) (for squares Q instead of half-planes II) and we get (3. 1).

0 THEOREM 3.2. If C is an Ahlfors-David curve, then T* is bounded in Conversely, if T* is a bounded operator in L2 (C, dH1 ), where C is a reetifiable curve, then C is an Ahlfors-David curve.

L2(C, dH1).

PROOF. In Theorem 3.1 we checked the assumption (3 1 ). We are now in a 0 position to apply Theorem 1 . 1 and to get the result. .

286

F.

)!aze.rov, A.

Volberg

Notice that one can replace "if' by "if and only if" in the statement above. But "only if'' part is simple and standard. The original proof of this result [Dal] was the mixture of analysis and geometry. Notice that in [Chr] another proof of David's theorem has been already given. The proof of [Chr] involves a hard geometric coru;truction of "dyadic cubes" on arbitrary space of homogeneous type. We avoid this by working with the usual dyadic squares on the plane, we do not build special "squares" adapted to a problem. Another proof can be derived from the ideas of Jones [J] in conjunction with the Menger's curvature characterization of L2 (J.L) boundedness of the Cauchy operator from [NTV2] , [Tl]. It seems to us that our just given proof of David's theorem is somewhat less involved than the other ones we mentioned. It is built on computation of analytic capacity of the portion of a continuum inside an arbitrary square (Theorem 1 .3) and on local Tb theorem of Christ [Chr] (only the nonhomogeneous version is needed-Theorem 2.1). We are left to prove Theorem 1.3. 4. Analytic capacity of portions of continuum. The proof of Theorem 1.3

All our considerations will be made for the intersection of a continuum with a half-plane. This is done for the sake of brevity, but one can replace the half-plane by a square or a disc. Of course this will change the absolute constants involved in the estimate, but this will be the only change. The right inequality in (1.11) is standard. In fact, let {D1 } be a finite family of discs of radii r-1 covering continuum r and such that I: Tj :s; 2 h1 (f) . Let K = Uj i5j . We use

(4.1)

and the monotonicity of �r to conclude 'Y(f) :s; 'Y(K) :s; A H 1 (8K) :s; A I: ri < 2A h1 (r). In the rest of this !:lection we prove the left inequality of (1.11). Let r be a continuum, and IT = {z : Imz 2 0} be the upper half-plane. All our constants will be absolute. So we can assume without the loss of generality that r is a real analytic curve. Then the intersection r n TI consists of f1 , . .. , rn pieces, and r1 n IT # 0, j 1, . .. , n. Let Yi E rj n IR, Tj = diam ( rj ) , j = 1, . . . , n . Obviously h l (r n TI) :s; I: Tj Considcr {D(y1, ,.i)} and choose a subfamily {D(yj, rj )}jE.1' such that =

(4 . 2)

(4.3) Clearly, (4.4)

On analytic capacity of portions of continuum

287

Let '"Yi := "Y(ri)- For every i E :F let fi denote a function from H00(C \ ri) such that llfi lloo S 1 and such that (4.5) Clearly 1/4 ri S '"'!i S 2 ri. Let us fix an interval such that -y(Ji) = '"Yi · Obviously, '"Yi S Jt . Thus,

Ji

:= [Yi - 1/2 Li , Yi + 1/2 Li]

(4 .6) Let L = L, i E .:F Li. For every i E :F let such that ll9i lloo S 1 and such that

9i

denote a function from H00 (C \ Ii )

(4.7) Writing 9i (z) =

1; g�(��x

we can assume from symmetry that 9i (x) is a real valued function. We also know that J gi (x)d.r. = '"'li · Let C denote C U oo and let 0 : = C \ UiE.rD(yi, 1 0 ri)- This is a compact set on the Riemann sphere. We consider CR (O) , the class of real-valued continuous function on n. Denote V1 : = {f

E CIR (D) : llfll

�

S 1} ,

cni 2:: 6 L , c = {ci } i E .:F E l� , lielloo ::; 1 } . { L cilmfi(z) : 1 i E.:F i E:F Suppose that

v2

:=

L:

(4.8) Then we will show that the left inequality in ( 1 . 1 1 ) holds. In fact, if we have ( 4.8) then there exists a collection c = { Ci h E .r such that the function f(z) := L, cdi (z) satisfies I Imf l S 1 in n. But f is a holomorphic ftmction in C \ UiE.:FC- Fix i E :F. On OD (yi, 10ri) we see that icdi I is bounded by 1 . Thus IIm L,jE .:F # i ci!J l is bounded by 2 on this circle, and, hence, on the whole disc , D(yi , 10ri), where J cdi l is bounded by 1 in its turn. Therefore, lim L,i E .:F ciiJ I = Jimfl is bounded by 3 in D(y.; , 10ri)· And this is so for every i E :F. Thus IImf l is bounded by 3 everywhere in C \ uiE:Fri. On the other hand, l (oo) = L,J E :F cnJ 2:: �e L. Vl/e conclude

1

(4.9) In particular, we obtain the left inequality of { 1 . 1 1) :

-y(r n II) 2:: -y(uiE.:F ri ) 2:: a' L 2:: a" h 1 (r n II) .

We are left to prove that ( 4.8) holds.

288

F.

Nazarov,

A.

Volberg

5. Maximal functions, weak type inequalities, and the proof of

(4.8)

Suppose (4.8) does not hold. We are going to come to a contradiction. There exists a real measure on 0 such that

v

II

but

vi S: 1 ,

[ (L cd (z)) ln iEF i

dv(z) > 1 for all collections c = {ci} iE F E lR', l c l oo 1 , l:iEF Cili � L . Now we are going to find such a collection for which (5 .1) fails. This collection will consist only of 1 's and O's. Define for x (Mtv)(x) := sup I vi (D (x, r)) . Let us define another maximal function on = Ui EF Namely, if x E Ji , i v*(x) sup l v i (D(x , r)) . Im

(5. 1 )

S:

E JR. I

r>O

:=

�

c,

Ii .

T

E :F

The segment Ii is well inside D(yi, lO ri) and v is outside of it, therefore sup

xEI,.

r

r>3r;

v*(x) ::; inf (M1 v)(x) . xEl;

Consider :Fo d� {i E :F : E h v* (x ) � 1�0 } . On Io := UiEFo Ji we have ( this is just

:lx

(4.2)) 100 M1 v L (5.2) On the other hand, 100 } I S: 5L II v i L . l {x : (M1 v) (x) � L 100 20 This is a usual weak type result. Consider \ :Fo {i E :F : Vx Ji , v* (x) 1�0 }. We conclude from :F1

:=

:F

�

=

E

=

::;

(5.2) and the last inequality that ( J1 �f uiEFJi) :

(5.3)

l ·h l

�

20 £ , and , \lx 19

E J1 ,

100 . v (x) ::; L •

Without the loss of generality we can think that 1 . v�

E F : l . 'Yi ::;

2oooo L . In fact, if we have the opposite inequality for a certain io , then V1 =f because we choose just ci 1 if i = io ci = 0 otherwise, and see that f = 2: cdi = fio belongs to V1 n V2. But we assumed V1 V2 Let us call a subset :F' of :F1 admissible if the following holds 1 1 L. << L 20000 iEF' - 5000 (5.4)

,

=

--

-

n

"" 'V· L... "

=

0.

n

V2

0

On analytic capacity of portions

of

289

coutinuum

By (5.3), (5.4) there are plenty of admissible subsets. We call c = {ci}iE.r admissible if <; = 1 on a certain admissible subt>et :F' of indices and ci = otherwise. In particular, = 0 for all i in F0. Let F' be admissible. Consider Im

Ci Jn L (

iE

J'"'

0

fi (z)) dv(z)

=

Im

I: 1 (fi (z) - 9i (z) ) dv(z) +

iEF' n

Clearly,

8 1'i "Yi I z - Yi l 2 · and (5.3), we get

1 /i (z) - 9i (z) l Thus, using the fact that

fn i !i -

�

i E :F' C F1 9i l dl vl

� 8 v* (yi) 'Yi �

8�0

"Yi

•

In particular, combining this with the admissibility of :F', we get

< 1 0"1 1 -

(5.5) To estimate

4 L 800 800 <-.-< -. """' 'V· � '' - L 5000 - 25 L iEF'

-

0"2 we will write it down 0"2 = -Im

as

jL

i EJ'"'

follows:

9i(x)v(x) dx ,

where v(x) := In d;_i:). The great advantage now is that all gi (x) are real valued . In particular, as ll 9i l l oc � 1, we get la2 l = i

(5.6)

f� i

'

gi (x) lm D(x) dx l

�

� h' IIm D(x) l dx .

i

'

Im;�d/J(z) = P(z, x) dl vi (z), where P(z, x) de Immediately, jim D (x) l � In In

notes the Poisson kernel. Consider F2 C F1 such that F2 = {i segment Ji lies well inside D( yi , lOr,), and i

inf E F2 � xEl;

And so

L

(5.7)

Jj

iEF2 I, On the other hand,

P(z, x) dlvi (z) � lO . L

P(z, x) d l vi (z)dx �

� L lli l · iEF2

� J, j P(z, x) dl v l (z)dx j (iu::F2I; P(z, x) dx) dlvl (z) ,

i and, hence,

J

supx E T; I P(z, x) dlv l (z) � 1�0 }. The v is supported outside of it, therefore, :

=

F. Nazarov, A. Volberg

290

(5 . 8)

L /,. J P(z, x) dlvl(z)dx = J cj P(z, x)xu, E r-21, dx) dlvl (z) :5 J 1 dlv l (z) :5 1 .

iEF2

I,

Now (5.7) and (5.8) give us that

L

"""' < L II' I 10

iE F2

(5.9)

Let :F3 = F1 \ F2 . Let Ja := U; EF3h From (5.3) and (5 .9) we get

0

(5.10) But also we have for every

x

E J3

r P(z , x) dlvl (z) :5 Jn

(5. 1 1)

wo

L

.

Let us consider any admissible F' such that F' C F3. From (5.4) and (5. 10) it follows that there are plenty of such subsets. Now we can finish the estimate of a2: (5.12)

l u2 1 =

1

u i e .r' J,..

dx

100 L J P(z, x)dl v l (z) S: L · 5000

:5 50 ·

1

Notice that we really used admissibility of F' here. Combining (5.5) and (5.12), we obtain that for an admissible c = { ci} , namely for ci = 1, i E F' , ci = 0 otherwise one has

lim J L cdi(z)dv(z)l :5 254 + 501 = 509 < 51

But this contradicts (5 .1). Theorem 1.3 is completely proved. 6. Discussion. 1) Theorem 1.1 can be derived from Menger's curvature criterion (1.6) if one com bines this criterion with a very difficult result of Tolsa (-y ::=: 'Y+ , see [T4] ). In fact , one can prove ( 6 .1) TIQ c2 (E n Q) 2 :5 C1 H1(E n Q)
This is the inversion of (1 .7). While (1. 7) was based on characterization ( 1.8), the direct proof of the reverse implication is more subtle. Still suppose that the reverse to (1 .7) is already proved. Still to derive (1. 6 ) from ( 1 .1) ( that i:o to derive Theorem 1 . 1 from Theorem 1.2) we, however, need to use that 'Y <::::: A 'Y+ of [T4) , which is several orders of magnitude harder than anything proved here.

2) We cannot help but quote another very subtle result related to Theorem 1.1.

On analytic capacity of portions of continuum

29 1

THEOREM 6. 1 . Let v be a complex measure with compact support K C
c v (z ) Then there exi::;t::; a po::;itive

f.1

:=

dv( (

}K z - J.., , r

ICV (z) l s 1 .

on K ::;uch that

ICIL (z) l S 1 ,

(

and such that

l v(K) I f.l (K) 2: a lv {K ) I l v i (K I ) where a > 0 is an absolute constant.

)64 ,

We can apply Theorem 6 . 1 to the proof of Theorem 1 . 1 . We will meet the situation, where II v i i = H 1 (E n Q), lv(K)I = 'J'(E n Q), and the assumption gives us the following inequality that, in its turn, brings the proof of Theorem 1 . 1 (after the use of the curvature criterion from [NTV2] , [Tl]): (6.2)

I+ (E n Q) 2: a1 (E n Q) , a > 0 ,

(which is true in general by the abovementioned Tolsa's solution of Vitushkin's problem, [T4] ). In {6.2) we obtained Tolsa's conclusion without using difficult result of [T4] . We just used Theorem 6.1 proved in [NTV5] , [Vo] and assumption ( 1 . 1 ) . But Theorem 6. 1 is itself a pretty difficult one. Actually, iL is one of two main ingredients in Tolsa's [T4] . So we are back to a very lengthy and difficult proof of our first main result. 3) In this paper we show how to avoid using difficult stuff from [T4] , [NTV4] , [NTV5], [Vo] in the proof of Theorem 1.1. We avoided curvature criterion from (NTV2] , [Tl] as well.

4) This, in particular, shows that their exists also a multi-dimensional analog of our Theorem 1.1. Let u s b e i n !Rn now. Let E be a compact subset of !Rn such that 0 < Hn-l (E) < oo. Let T denote the vector Riesz transform operator with kernel R(x - y) , where

R( x) : = (

1:tn' . . . , l�r)

·

We recall the reader that there exists in !Rn a full analog of analytic capacity. It is called Lipschitz harmonic capacity, and it was introduced by Mattila and Paramonov. We will call it 1 as before, the reader can get acquainted with it by reading [Vo] . \Ve have THEOREM 6.2. Operator T is bounded in L2 (E, Hn-1 IE) if and only if there exists a finite constant C such that for every cube Q in JRn we have

(6.3)

292

F. Nazarov, A.

Volberg

References [Chr] M. CHRIST, A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), 601-628. [CW] R. R. COIFMAN, G. WEISS, "Analyse harmonique non-commutative sur certaines espaces homog(mes" , Lecture Notes in Math. 242, Springer-Verlag, Berlin, 1971. [CJ S) R. R. COIFMAN, P . W. JONES, ST. SEMMES Two elementary proofs of the L2 boundedness of Cauchy integrals on Lipschitz curves, J. of Amer. Math. Soc., 2 1989, No. 3, 553-564. [Da1] G . DAVID , Operateurs integraux singuliers sur certaines courbes du plan complexe . Ann. Sci. Ecole Norm. Sup . , 17 ( 1984), 157-189. [Da2] G. DAVID, Completely unrectifiable 1 -sets on the plane have vanishing analytic capacity, Revist a Mat. Iberoamericana, v. 14 (1998), 369-479. [DaJ] G. DAV!U, J .-L. J O V RN E , A boundedness criterion for generalized Calder6n-Zygmund op erators, Ann. of Math., 120 ( 1984) , 371-397. [DaJS] G. DAVID, J .-L. JOVRNE, S. SEMMES, Operateurs de Calder6n-Zyqmund, fonctions paraaccretive et interpolation, Rev. M at. Ibcroamer . , 1 (1985), 1-56. [Fa] K . FALKONER "The Geometry of Fractal Sets" , Cambridge Univ. Press, 1985. [Gam] T. GAMELIN "Uniform Algebras" , Prentice-Hall, Englewood Cliffs, N.J. 1969. [JG] . J. GARNETT Personal communication. [J) P. W. JONES Rectifiable sets and the traveling salesman problem, Invent. Math., 102 (1990), 1-15. [JM] P . W. JONES AND T. MURAl, Positive analytic capacity but zero Buffon needle probability, Pacific J. Math., 133 ( 1988), no. 1 , 99 1 14. [Ma] P . MATTILA "Geometry of Sets and Measures in Euclidean Spaces" . Cambridge Univ. Press, 1995. [Mel] M. MELNJKOV A n estimate of the Ca1Lchy integral over an analytic arc. Sbornik: Mathe matics, 71 ( 1966), NO 4, 5030514. [Me2] M . MELNIKOV Analytic capacity: discrete approach and curvat·ure of a measur·e. Sbornik: Mathematics, 186 (1995), 827-846. [Mu] T. M u RAl "A Real Variable Method for the Cauchy Transform, and Analytic Capacity" , Lecture Notes in Math., 1307, Springer Verlag, Berlin-Heidelberg-New York, 1988. [MMVj P . MATTILA, M. MELI'IKOV, J . VERDERA, The Cauchy integral, analytic capacity, and uniform rectifiability, Ann. of Math . , 144 (1996), 127-136. [NT] F. NAZAROV, S. TREIL, The hunt for Bellman function: applications to estimates of singular integral operators and to other classical problems in harmonic analysis, Algebra i Analysis, 8 (1997), no. 5, 32-162. [NTV1) F. NA ZAROV , S . TREIL, A. VOLBERG, Weak type estimates and Cotlar inequalities for Calder6n-Zygmund operators on nonhomogeneous spaces. IMRN International Math. Res. Notices, 1998, no. 9, 463-487. [NTV2] F. NAZAROV, 8. TREIL, A. VOLBERG, Cauchy integral and Calder6n-Zygmund operators on nonhomogeneous spaces. IMRN International Math. Res. Notices, 1997, no. 15, 703-726. [NTV3] F. NAZAROV, S. TREIL, A. VOLBERG, Accretive system Tb theorem of M. Christ for nonhomogeneous spaces, Duke Math. J . , 113 (2002), pp. 259-312 [NTV4) F. NAZAROV, S. TREIL, A . VOLBERG, The Tb-theorem on nonhomogeneous spaces, Acta Math., 190 (2003), 151-239. [NTV5] ___ , Nonhomogeneous Tb theorem which proves Vitushkin's conjecture, P reprint No. 519, CRM, Barcelona, 2002, 1-84. [St] E. STEIN, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Inte grals" , with the assistance of Timothy S. Murphy, Princeton Math. Ser. 43, Monographs in Harmonic analysis, iii, Princeton Univ. Press, Princeton, 1993. [T1] X. TOLSA , L 2 -boundedness of the Cauchy integral operator for continuous measures, D uke Math. J . , 98 ( 1 999), no. 2, 269-304. [T2] X. TOLSA, Cottar's inequality and the existence of principal values for the Cauchy integral without doubling condition, J. Reine Angew. Math. 502 ( 1998), 199-235. [T3] X. TOLSA, Curvature of measures, Cauchy singular integral, and analytic capacity, Thesis , Dept. Math. Univ. Auton. de Barcelona, 1998. [T4] X TOLSA, Pain/eve's problem and the semiadditivity of analytic capacity, Acta Math., 190 (2003) , no. 1, 105-149.

On

[T5] X. [Vo] A .

293

a.na.lytic ca.pa.city of portions of continuum

TOLSA, On the analytic capacity I'+ , Indiana Univ. Math. J . , 51

(2), (2002), 31 7-344.

VOLBERG, "Calder6n-Zygmund Capacities and Operators on Nonhomogeneous Spaces" ,

CBMS Regional Conference Series in Mathematics, v.

100, 2003,

pp.

1-167.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF WISCONSIN, MADISON, WI.

E-mail address:

53706

ne.zarov
DEPARTMENT OF MATHEMATICS, MICHIGAN STATE UNIVERSITY, EAST LANSING, MICHIGAN

48823, USA

AND SCHOOL OF MATHEMATICS, EDINBURGH UNIVERSITY,

E-mail address: volberg
EH9 3JZ,

UK

Proceedings of Symposia. in Pure Mathematics Volum� '79, 2008

The Christoffel-Darboux Kernel Barry Simon* AHSTRACT. A review

of the uses

of the CD kernel in the spectral theory of

orthogonal polynomials, concentrating on recent results.

COl\ TENTS

1 . Introduction 2. The ABC Theorem 3. The Christoffel-Darboux Formula 4. Zeros of OPRL: Basics Via CD 5 . The C D Kernel and Formula for MOPs 6. Gaussian Quadrature 7. Markov-Stieltjes Inequalities 8. Mixed CD Kernels 9. Variational Principle: Basics 10. The Nevai Class: An Aside 1 1 . Delta Function Limits of Trial Polynomials 12. Regularity: An Aside 13. Weak Limits 14. Variational Principle: Mate-Nevai Upper Bounds 15. Criteria for A.C. Spectrum 16. Variational Principle: Nevai Trial Polynomial 17. Variational Principle: Mate-Nevai-Totik Lower Bound 18. Variational Principle: Polynomial Maps 19. Floquet-Jost Solutions for Periodic Jacobi Matrices 20. Lubinsky's Inequality and Bulk Universality 21. Derivatives of CD Kernels 22. Lubinsky's Second Approach 23. Zeros: The Freud-Levin-Lubinsky Argument 24. Adding Point Masses References

296 298 299 302 303 304 306 308 309 311 312 315 316 317 319 320 32 1 322 323 323 324 325 328 329 331

2000 Mathematics Subject Classification. 34L40, 47-02, 42C05. Key words and ph-rases. Orthogonal polynomials, spectral theory.

This work was supported in part by NSF grant DMS-0652919 and U.S.-Israel Binational

Science Foundation (BSF) Grant No. 2002068.

295

@2008 Barry Simon

296

B.

SIMON

1. Introduction

This article reviews a particular tool of the spectral theory of orthogonal poly nomials. Let 1-l be a measure on C with finite moments, that is, (1 . 1 )

for all n = 0, 1 , 2 , . . . and which is nontrivial in the sense that it is not supported on a finite set of points. Thus, {z n }�=O are independent in U (C, dJ1), so by Gram Schmidt, one can define monic orthogonal polynomials, Xn(z; df..t) , and orthonormal polynomials, Xn = Xn / IIXn ll£2 · Thus,

j z1 Xn(z; dt-t) dl-l(z) = 0

j = 0, . . . , n - 1

Xn (z) = zn + lower order

j Xn (z) Xm(z) df..t = Onm

(1.2) (1.3 )

( 1.4)

V'.fe will often be interested in the special cases where J1 is supported on � (especially with support compact), in which case we use Pn, Pn rather than Xn, Xn , and where 1-l is supported on 8]])) ( lDl = { z I l z l < 1 } ) , in which ease we use �Pn , 'Pn· We call these OPRL and OPUC (for "real line" and "unit circle" ). OPRL and OPUC are spectral theoretic because there are Jacobi parameters {an, bn }�= l and Verblunsky coefficients { an }�=O with recursion relations (P - 1 = 0;

Po = o = 1) :

ZPn( z) = an+IPn+ l (z) + bn+lPn (z) + anPn- I ( z) �Pn+ l (z) = z n (z) - anc�»;, (z) n �(z) = z 1Pn( 1 /z)

(1.5) ( 1 .6) ( 1 . 7)

We will sometimes need the monic OPRL and normalized OPUC recursion relations: zP,. (z) = Pn+I (z) + bn+IP,. ( z) + a� Pn - I (z ) Z<,On (z) = Pn <.On+I (z } + Cin
(1.8) (1.9) ( 1 . 10 )

Of course, the use of Pn implies I an i < 1 and all sets of { an}�0 obeying this occur. Similarly, bn E �. an E (0, oo ) and all such sets occur. In the OPUC case, {an }�=O determine d/.l, while in the OPRL case, they do if sup (lan l + l bn l ) < oo, and may or may not in the unbounded case. For basics of OPRL, see [93, 22, 34, 89] ; and for basics of OPUC, see [93, 37, 34, 80, 81, 79] . vVe will use �n (or �n (d!-l) ) for the leading coefficient of Xn , Pn , or 'Pn. so �n = I I Xn ii "Zi(dfL)

The

(1. 1 1 )

Christoffel-Darboux kernel (named after [23, 28] ) is defined by Kn (z , () =

n

L Xj (z) Xj (()

j =O

( 1.1 2)

297

THE CHRISTOFFEL-DARBOUX KERNEL

We sometimes use Kn (z, ( ; Jl) if we need to make the measure explicit. Note that if c > 0, Kn (z, (; cp,)

=

(1. 13)

c- 1 Kn (z, (; p,)

since Xn (z; cdJl) = c- 1 12 x n (z; dp,) . By the Schwarz inequality, we have

(1.14) There are three variations of convention. Some only sum to n 1 ; this is the more common convention but (1. 12) is used by Szego [93] , Atkinson [5] , and in [80, 81]. As we will note shortly, it would be more natural to put the complex conjugate on Xn (() , not Xn ( z )-and a very few authors do that. For OPRL with z , ( real, the complex conjugate is irrelevant-and some authors leave it off even for complex z and (. As a tool in spectral analysis, convergence of OP expansions, and other aspects of analysis, the use of the CD kernel has been especially exploited by Freud and Nevai, and summarized in Nevai's paper on the subject [68] . A series of recent papers by Lubinsky ( of which [60, 61] are most spectacular) has caused heightened interest in the subject and motivated me to write this comprehensive review. Without realizing they were dealing with OPRL CD kernels, these objects have been used extensively in the spectral theory community, especially the diagonal kernel n Kn (x, x) = 1Pj (x) l 2 { 1 . 15) j=O -

2:::

Continuum analogs of the ratios of this object for the first and second kind poly nomials appeared in the work of Gilbert-Pearson [38] and the discrete analog in Khan-Pearson [49) and then in Jitomirskaya-Last [45) . Last-Simon [55) studied � Kn (x, x) as n ___. oo. Variation of parameters played a role in all these works and it exploits what is essentially mixed CD kernels (see Section 8). One of our goals here is to emphasize the operator theoretic point of view, which is often underemphasized in the OP literature. In particular, in describing p,, we think of the operator Mz on L2 (C, dp,) of multiplication by z: (Mz/) ( z)

=

zf (z)

(1.16)

If supp (dJl ) is compact, Mz is a bounded operator defined on all of L2 (C, dJl) . If it is not compact, there are issues of domain, essential selfadjointness, etc. that will not concern us here, except to note that in the OPRL case, they are connected to uniqueness of the solution of the moment problem (see [77] ) . \Vith this in mind, we use a(dJl) for the spectrum of Mz , that is, the support of dp,, and aess (dJl) for the essential spectrum . \Vhen dealing with OPRL of compact support ( where Mz is bounded selfadjoint ) or OPUC (where Mz is unitary) , we will sometimes usc aac (dJ.L) , asc(dJl) , app(dJ.L) for the spectral theory components. (We will discuss aess(dp,) only in the OPUC / OPRL case where it is unambiguous, but for general operators, there are multiple definitions; see [31].) The basis of operator theoretic approaches to the study of the CD kernel de pends on its interpretation as the integral kernel of a projection. In L2 (C, dp,) , the set of polynomials of degree at most n is an n + !-dimensional space. We will use

B. SIMON

298 7rn

for the operator of orthogonal projection onto this space. Note that (rrnf)(() =

j Kn (z, () f(z) dJ.L(z)

(1.17)

The order of z and ( iH the opposite of the usual for integral kernels and why we mentioned that putting complex conjugation on Xn (() might be more natural in (1.12).

In particular, deg ( f) � n =? f (( )

=

j Kn (z, ()f(z) d�J(z)

( 1 .18)

In particular, since Kn is a polynomial in ( of degree n, we have Kn (z, w )

=

j Kn(z, () Kn((, w) dp.(()

( 1 . 19)

often called the reproducing property. One major theme here is the frequent use of operator theory, for example, proving the CD formula as a statement about operator commut ators . Another theme, motivated by Lubinsky [60, 61] , is the study of asymptotics of � Kn (x, y ) on diagonal (x y ) and slightly off diagonal (( x - y) 0 ( �)). Sections 2, 3, and 6 discuss very basic formulae, and Sections 4 and 7 simple applications. Sections 5 and 8 discuss extenHicm'3 of the context of CD kernels. Section 9 starts a long riff on the use of the Christoffel variational principle which runs through Section 23. Section 24 is a final simple application. Vladimir Maz'ya has been an important figure in the spectral analysis of partial differential operators. \Vhile difference equations are somewhat further from his opus, they are related. It is a pleasure to dedicate this article with best wishes on his 70th birthday. I would like to thank J. Christiansen for producing Figure 1 (in Section 7) in Maple, and C. Berg, F. Gesztesy, L. Golinskii, D. Lubinsky, F. Marcellan, E. Saff, and V. Totik for useful discussions. =

=

2. The ABC Theorem

We begin with a result that is an aside which we include because it deserves to be better known. It was rediscovered and popularized by Berg [11], who found it earliest in a 1939 paper of Collar [24] , who attributes it to his teacher, Aitken--so we dub it the ABC theorem. Given that it is essentially a result about Gram Schmidt, as we shall see, it is likely it really goes back to the nineteenth century. For applications of this theorem, see [13, 47] . Kn is a polynomial of degree n in z and (, so we can define an (n + 1 ) x (n + 1) square matrix, k( n) , with entries k;::, 0 � j, m � n, by n L k):: zm(j (2.1) Kn (z , () =

j,rn =O

One also has the moment matrix

m)�)

=

{zi , zk)

=

j zj zk d11(z)

(2 . 2)

THE C HRISTOFFEL-DARBOUX KERNEL

299

0 � j, k � n. For OPRL, this is a function of j + k, so mCn) is a Hankel matrix. For OPUC, this is a function of j - k , so m( n) is a Toeplitz matrix.

2.1 (ABC Theorem) . ( m ( n) ) - 1 = k ( n) PROOF. By (1.18) for .e = 0, . . . , n , Kn(z , ( ) zf dp(z) = (£ THEOREM

J

Plugging (2. 1) in for K , using

which is

(2.4)

(2.2) to do the integrals leads to (2.5)

j, q =O which says that

(2.3)

'""" n (n) > L kj( q ) mq.e = Uj£

(2.6)

q

(2.3).

D

Here is a second way to see this result in a more general context : \Vrite j

so we can define an (n + 1)

x

Xj (z ) = L aj k Zk k =O (n + 1) triangular matrix a l n) by (n) = aj ajk k

Then (the Cholesky factorization of k)

(2. 7) (2.8)

k ( n) = a Cnl ( a( n) ) *

(2.9)

(xi , xc) = �J!.

(2 . 1 0 )

with * Hermitean adjoint. The condition

says that

(2 . 1 1) (a( n) ) * m(nl (a( n ) ) = 1 n 1 ( l ) the identity matrix. Multiplyi ng by (a ) * on the right and [ (a Cn )*] on the left yields (2.3). This has a clear extension to a general Gram-Schmidt setting. 3. The Christoffel-Darboux Formula The Christoffel-Darboux formula for OPRL says that

Kn ( Z, -,�') - an+ 1 _

(Pn+l (z) p,.((z) -- Pn(Z} p,.+ I (( ) ) ., ;

(3.1)

and for OPUC that

l (() - c,on + I (z ) c,on + I (( ) l ) Kn ( z, �"., ) = 'P�+ (z cp�+

1 - z(

(3.2)

The conventional wisdom is that there is no CD formula for general OPs, but we will see, in a sense, that is only half true. The usual proofs are inductive. Our proofs here will be direct operator theoretic calculations.

300

B. SIMON

We focus first on (3.1 ) . From the operator point of view, the key is to note that, by (1 . 1 7) , (g , [Mz , Trn]f> =

J g(() (( - z)Kn ( z, ()f(z) dJ.L(()dfl (Z )

(3.3)

where [A, B] = AB - BA. For OPRL, in (3.3) , ( and z are real, so ( 3.1) for z, ( E a(dJ.L) is equivalent to (3.4)

While (3.4 ) only proves (3.1) for such z, ( by the fact that both sides are polynomials in z and (, it is actually equivalent. Here is the general result:

THEOREM 3.1 (General Half CD Formula) . Let J1. be a measure on C with finite moments. Then:

( 1 - 1rn) [Mz , 1rn] (1 - 1rn) = 0 1fn [Mz , 1fn]1fn = 0

(3.5)

(3.6)

Xn (3.7) ( 1 - 1rn ) [Mz , 1fn l 1fn = II +lll 1\Xn , · ) Xn+ I JJ Xn ll REMARK. If J1. has compact support, these are formulae involving bounded operators on L2 (C, dJ.L). If not, regard 7rn and Mz as maps of polynomials to polynomial�:>. PROOF. (3.5)

follows from expanding [Mz , 7rn] and using

If we note that [Mz, 1fn] (3.8) again,

1fn (1 - 1fn) = (1 - 1fn)1fn = 0

=

(3.8)

- [Mz , (1 - 1fn)], (3.6) similarly follows from (3.8). By

(3. 9) On ran(1fn- d , Trn is the identity, and multiplication by z leaves one in Trn , that is, (3. 10) (1 - 1rn)Mz 1rn I ran(1rn-d = 0

On the other hand, for the monic OPs, since Mz1fn Xn

Xn+l · Since

=

(3. 11) ( 1 - 1fn) Mz1fnXn = Xn+l n+ l + lower order and (1 - 7rn) takes any such polynomial to z

JI Xn+l ll (Xn 1 Xn)Xn+I = Xn+l II Xn ll we see (3.4) holds on ran(l - 7rn ) + ran(7rn _1) + [XnJ , and so on all of £ 2 .

0

From this point of view, we can understand what is missing for a CD formula for general OP. The missing piece is ( 3. 1 2)

The operator on the left of (3. 7) is proven to be rank one, but (1 - 1fn)M;7rn is, in general, rank n. For
30 1

THE CHRISTOFFEL--DARBOUX KERNEL

(a.e. z E O"(dJ-L)). In the first case, zcp E ran(7rn ) if deg(cp) � n - 1 , and in the second case, if cp (O) = 0. Thus, only for theHe two cases do we expect a simple formula for [Mz , 7r] . THEOREM 3.2 (CD Formula for OPRL) . For OPRL, we have (3.13) [Mz , 7rn] = an+ l [(pn, · )Pn+l - (Pn+ l > · )Pn] and (3. 1 ) holds joT z # ( . PROOF. Inductively, one has that Pn (x) (a 1 . . . an ) - l xn + . . . 1 (3. 14) IIPnll = a 1 · · . an J-l (� ) 12 and thus, , so

=

IIPn+t ll (3. 15) = an+l IIPn ll Moreover, since M; = Mz for OPRL and [A, B ] * = - [A*, B*] , we get from (3.12) that (3.16) 11'n [Mz , 7rn] ( 1 - 1fn) -an+ l (Pn+l > · lPn (3.5)-(3.7), (3. 14}, and (3. 16) imply (3.13) which , as noted, implies (3. 1). 0 =

For OPUC, the natural objeet is (note MzM; = M; Mz = 1) (3. 1 7) 3.3 (CD Formula for OPUC) . For OP UC, we have

THEOREM

11'n - Mz7rn M; = (cp �+l> · )cp�+ l - ('Pn+l > l'Pn+l ·

(3.18)

and (3.2) holds. PROOF. Bn is selfadjoint so ran(Bn) = ker (Bn ).l.. Clearly, ran(Bn) C ran(7rn) + Mz [ a ( )] ran(7rn+l) and Bn ze = 0 for £ = 1, . . . so ran(Bn) = { z, z2, . . . , zn }l. n ran( 7rn+l) is spanned by 'Pn+l and cp�+ l · Thus, both En and the right side of (3. 18) are rank two selfadjoint operators with the same range and both have trace 0. Thus, it suffices to find a single vector 1J in the span of 'Pn+ l and cp�+ l with Bn1J = (RHS of (3. 18))1J, since a rank at most one selfadjoint operator with zero trace is zero! We will take 1J zcpn , which lies in the span since, by (1 .9) and its * , r n 7rn

=

, n,

=

(3.19)

By (3. 16) , (3. 1 7) , and II
we have that Bn(Zcpn) = [7rn 1 Mz]'Pn = - ( 1 - 11'n )Mz71'n'Pn = - Pn'Pn+l On the other hand, cp�+l j_ { z, . . . , zn+ l }, so

(cp�+l, Zcpn ) = 0

and, by (3.19),

(3 . 20)

(3.21)

302

B. SIMON =

so

Pn

[LHS of (3.18)]z
=

-PniPn +l

Note that ( 3.2) implies Kn+l (z, ()

=

Kn (z, () + IPn+l (z) IPnH ( ()

(3.22)

0

C=;�)

IP�+ l ( z) IP�+ l ( () - z( <.On+! (z) IPn+l ( ( ) � � � � � � � � � � � � � 1 --z� ( so changing index, we get the "other form" of the CD formula for OPUC, =

Kn (z, ()

�
(3.23 ) 1 - Zip We also note that Szego [93] derived the recursion relation from the CD formula, so the lack of a CD formula for general OPs explains the lack of a recursion relation in general. =

4. Zeros of OPRL: Basics Via CD

In this section, we will u::>e the CD formula to derive the basic facts about the zeros of OPRL. In the vast literature on OPRL, we suspect this is known but we don't know where. Vve were motivated to look for this by a paper of Wong [105] , who derived the basics for zeros of POPUC (paraorthogonal polynomials on the unit circle) using the CD formula (for other approaches to zeros of POPUCs, see [19, 86] ) . We begin with the CD formula on diagonal: THEOREM 4 . 1 . For OPRL and x real, n.

2:::: 1Pj (x) l 2 = an + IfP�+ l (x )pn (x) - P�(x)Pn+ l ( x)]

j =O

(4. 1)

PROOF. In ( 3.1 ) with z = x, ( = y both real, subtract Pn+ l (Y)Pn ( Y) from both products on the left and take the limit a..'l y -+ x. 0 COROLLARY 4.2. If Pn(xo ) = 0 for Xo re.al, then

Pn+ I ( x o)P;Jxo) < 0

PROOF. The left-hand side of (4. 1 ) is strictly positive since p0(x) = 1.

(4.2)

0

THEOREM 4.3. All the zeros ofpn ( x ) are real and ,g imple and the zeros ofPn +l strictly interlace those of Pn . That is, between any successive zeros of Pn+ I lies exactly one zero of Pn and it is strictly between, and Pn+ l has one zero bet·ween each s-uccessive zero of Pn and it has one zem nhove the top zero of Pn and one below the bottom zero of Pn .

PROOF. By (4.2) , p.,. ( x0 ) = 0 ::::;. p� (x0 ) � 0, so zeros are simple, which then implies that the sign of p� changes between its siH:cesHive :.�eros. By (4.2), the sign of Pn +l thus changes between zeros of Pn , so Pn+l has an odd number of zeros between zeros of Pn .

303

THE CH RISTOFFEL-DARB OUX KERNEL

P1 is a real polynomial, so it has one real zero. For x large, Pn (x) > 0 since the leading coefficient is positive. Thus, p� (x0) > 0 at the top zero. From (4.2), Pn+l (xo) < 0 and thus, since Pn + l (x) > 0 for x large, Pn +l has a zero above the top zero of Pn · Similarly, it has a zero below the bottom zero. We thus see inductively, starting with Pt , that Pn has n real zeros and they interlace those of Pn- 1 . 0

We note that Ambroladze [3] and then Denisov-Simon [29] used properties of the CD kernel to prove results about zeros (see Wong [105] for the OPUC analog) ; the latter paper includes: THEOREM 4. 4. Suppose a00 = upn an < oo and Xo E IR has d = dist (xo , u( dp,)) > 0 . Let o = J2 /(d + v'2 a00 ) . Then at least one ofpn and Pn- 1 has no zeros in ( xo

-

s

o, Xo + o) .

They also have results about zeros near isolated points of u(dp,) . 5. The C D Kernel and Formula for MOPs

Given an e e matrix-valued measure, there is a rich structure of matrix O Ps and MOPUC). A huge literature is surveyed and extended in [27] . In particular, the CD kernel and CD formula for MORL are discussed in Sections 2.6 and 2. 7, and for MOPUC in Section 3.4. There are two "inner prod cts maps from £2 matrix-valued functions to rna trices, (( · )) and (( · ))L . The R for right comes from the form of scalar homo X

(MOPRL

geneity,

, ·

for

example, H.

·

,

u

,"

(5.1)

but ((!, Ag)) R i s not. related to ((!, g )) R· There are two normalized OPs, pf(x) and pj(x), orthonormal in (( · · )) R and (( , )) L , respectively, but a single CD kernel (for z, w real and t is matrix adjoint), ·

·

,

n Kn (z, w) = L P� (z)p� ( w)t k=O

n = L Pt ( z) tphw)

One has that

k=O

((Kn ( · , z) , f ( · ) ))R

=

( 7rn f) ( z )

(5.2) (5.3) (5.4)

7rn is the projection in the Tr( (( )) R ) inner product to polynomials of degree n. In [27] , the CD formula is proven using Wronskian calculations. We note here that the commutator proof we give in Section 3 extends to this matrix case. Within the Toeplitz matrix literature community, a result equivalent to the CD formula is called the Gohbcrg-Scmcncul f rmul ; see [10, 35, 39, 40, 48, 100, where

101].

· ,

·

o

a

304

B. SIMON

6. Gaussian Quadrature

Orthogonal polynomials allow one to approximate integrals over a measure dp, on R by certain discrete measures. The weights in these discrete measures depend on Kn (x, x). Here we present an operator theoretic way of understanding this. Fix n and, for b E R, let ln;F (b) be the n x n matrix ln;F (b)

=

b1 a1 0

0

a1 b2 az

a2 b3

(6. 1 ) bn + b

(i.e., we truncate the infinite Jacobi matrix and change only the corner matrix

element bn t o bn + b) . Let iJn) (b) , j = l, . . . , n,

< x2 <

b e the eigenvalues of ln;F (b) labelled by X 1 ( We shall shortly see these eigenvalues are all simple.) Let r.pj"l be the nor malized eigenvectors with components [rp�n) (b)],, £ = 1 , . . . , n, and define

....

x;n) (b)

=

( 6 . 2)

l [rf>Jn) (b)h l2

so that if el is the vector (1 0 . . . oy ' then

(6.3)

is the spectral measure for ln;F (b) and e 1 , that is, {el , Jn;F (b) eel) =

'II

2.': 5-)n)(b)iJn)(b/

j =l

{6.4)

for all f.. We are going to begin by proving an intermediate quadrature formula: THEOREM 6 . 1 . Let p, be a probability measure.

0, 1, . . . , 2n - 2,

For any b and any e =

t >-)nl (b) x)n) (b) e j =l If b = 0, this holds also for e = 2 n - 1 .

j xe dp,

PROOF.

=

(6.5)

For any measure, { aj , bj }j�1 determine {Pj }j��, and moreover, (6.6)

If a measure has finite support with at least n points, one can still define {Pj } j��, Jacobi parameters { aj , bj }j;:f , and bn by (6.6). n dp, and the measure, call it ap,i ) , of (6.3) have the same Jacobi parameters {aj , bj }j;:f , so the same {PJ } j�J , and thus by k = 0, 1 , . . . , j - 1 ; j = 1, . . . , n - 1

(6.7)

THE CHRISTOFFEL-DAR.BOUX

305

KERNEL

we inductively get (6.5) for f = 0, 1 , 2, . . . , 2n - 3. Moreover, determines inductively (fi.fi) £ = 2n - 1.

for

/ Pn-1 (X) 2 df..l = 1

f. = 2n - 2. Finally,

(6.8 ) if b

= 0, (6.6) yields (6.5) for

0

As the second step, we want to determine the x)nl (b) and ).)n) (b) .

THEOREM 6.2. Let Kn ; F = 1r11 - t A1z 1fn -1 r ran( rrn- t) for a general finite mo ment measure, ft, on tC . Then (6.9)

PROOF. Suppose Xn. (z) has a zero of order £ at z0. Let r.p = X11 (z)/(z - z0 ) e. Then, in ran(n11 ) , (6. 10) (Kn ; F - zo) j r.p =J 0 j = o, 1, . . . , e - 1 \ (6. 1 1 ) (Kn ; F - zo) ? = 0

=

since (J1,1z - zo) �' cp X (z) and 1fn- 1 Xn = 0. Thus, zo is an eigenvalue of Kn :F of n algebraic multiplicity at least €. Since Xn ( z) has n zeros counting multiplicity, this accounts for all the roots, so (6.9) holds because both sides of monic polynomials 0 of degree n with the same roots. COROLLARY

6.3. We have for OPRL det (z - .ln;F (b))

=

Pn (z) - bPn- l (z)

The eigenvalues x)n) (b) are all simple and obey for 0 < b < ( with X11.q (O) = oo) , n n n x(J ) (0) < a;(J ) (b) < x(J +)l (0) and for

-oo

< b < 0 and j

=

1, . . . , n ( with Xn - t (O) =

x)�\ (0) < xjn) (b) < xjnl (O)

- oo ) ,

(6. 12) oo

and j

=

1, . . . , n (6. 1 3) (6. 14)

PROOF. (6.12) for b = 0 is just (6.9). Expanding in minors shows the determi nant of ( z - .ln;F(I!)) is ju::;t the value at b = 0 minus b times the (n - 1) x (n - 1) determinant, proving (6. 12) in general. The inequalities in (6 . 13)/(6. 1 4) follow either by eigenvalue perturbation theory 0 or by using the arguments in Section 4. In fact, our analysis below proves that for 0 < b < oo , 1 iJn) (O) < XJ11) (b) < x;n- ) (0)

(6.15)

The recursion formula for monic OPs proves that p1 ( xj (b) ) is the unnormalized eigenvector for l ; F (b). Kn_ 1 (xj (b) , Xj (b)) 112 is the normalization constant, so n since Po = 1 (if f..L (lR) = 1 ) : P ROPOSITION

6.4. lf f..L (R.) = 1 , then

>-.)n) (b)

= (Kn-t (X�n) (b) , XJ11) (b) )) - 1

(6. 16)

B.

306

SIMON

Now fix n and xu E R. Define

Pn (xo ) Pn - l (xo ) with the convention b = oo if P,,_l (xo) = 0. Define for b -=f. oo , ) j = 1, . . . , n x)n ( xo ) = x)n) ( b (x o ) )

(6.1 7)

b (xo ) =

and if b(xo ) = :xJ,

- (n - l) ( O) xj(n ) (xo ) - x1

j = 1, . . . , n - 1

(6.19)

AJn) (xo) = (Kn-1 (xJn) (xo) , xjnl (xo))) - 1

(6.20)

_

and

(6.18)

Then Theorem 6.1 becomes THEOREM

6.5 (Gaussian Quadrature) . Fix n, x0 . Then

j Q(x ) d�-t = j=l :t >-Jn) (xo)Q(x,;n) (xo)) for all polynomials Q of degr·ee up to: ( 1) 2n - 1 if Pn ( xo ) = 0 (2 ) 2n - 2 if Pn (xo) =/- 0 =/- Pn-l (xo ) (3) 2n - 3 if Pn -1 (xo) = 0. REMARKS. 1. The sum goes to n - 1 if Pn -l ( xo ) 2 . We can define x)n ) to be the solutions of

=

(6.21)

0.

Pn-l (xo )Pn (x) - Pn (xo)Pn-l (x) = 0 which has degree n if Pn -l (xo) -=f. 0 and n - 1 if Pn- l (xo) = 0.

(6.22)

3. (6.20) makes sense even if �-t(IR) =f- 1 and dividing by p,(IR) changes J Q(x) dp, and >.)nl by the same amount, so (6.21) holds for all positive J-t (with finite mo ments) , not just the normalized ones. n 4. The weights, AJ ) (x0), in Gaussian quadrature are called Cotes numbers. 7. Markov-Stieltjes Inequalities

The ideas of this section go back to Markov [6 3] and Stieltjes [92] based on conjectures of Chebyshev [21] (see Freud [34] ) . LEMMA 7. 1 . Fix X l < . . . < Xn in IR distin ct and 1 :::; e < n. Then there is a polynomial, Q, of degree 2n - 2 so that (i) 1 j = l, . . . , £ Q(xj ) = ( 7.1) 0 1 = £ + 1, . . . , n (ii) For all x E IR, (7.2 ) Q(x) � X( - oo,xeJ (x)

{

REMARK.

Figure 1 has a graph of Q and X(- oo,xe ] for n = 5, C = 3,

Xj

=

j - 1.

THE CHRISTOFFEL-DARBOUX KERNEL

-1

0

3

FIGURE

4

307

5

1 . An interpolation polynomial

PROOF . By standard interpolation theory, there exists a unique polynomial of degree k with k + 1 conditions of the form Q(yj ) = Q'(yj ) = . . . = Q(nj)(yj) = 0

2:j nj = k + 1 . Let Q be the polynomial of degree 2n - 2 with the n conditions in (7 .1) and the n - 1 conditionti ( 7.3) Q'(xj ) = O j = l , . . . , £ - 1, £ + 1, . . . , n Clearly, Q' has at most 2n- 3 zeros. n - 1 are given by (7.3) and, by Snell's the orem, each of the n - 2 intervals (xr , x2 ) , . . . , (xt- 1 , x.e ), (xH l > XH 2 ) , . . . , (xn -1 , Xn ) must have a zero. Since Q' is nonvanishing on (xe, xe+d and Q (xe) = 1 > Q (xHI ) = 0, Q'(y) < 0 on (xe, X£+1 ) . Tracking where Q' changes sign, one sees

that (7.2) holds.

THEOREM 7.2. Suppose df.l is a measure on JR. with finite moments . Then 1 � f.l(( -oo , xo] ) L (n) ( n)

,X {jlx)nl (xo)�xo } Kn- 1 (xj (xo) j (xo)) �

f.l((-oo, xo )) �

L

{ J.1 x j(n) ( xo ) <xo }

1

0

(7.4)

( ) ( ) Kn -l(xjn (x o) , xjn (xo ))

REMARKS. 1. The two bounds differ by Kn -1 (xo , xo)-1 . 2 . These imply

f.l ({xo }) ::::; Kn- 1 ( xo , xo ) - 1 In fact, one knows (see (9.2 1) below)

f.l ( {x o } ) = lim Kn- l (xo , xo) - 1

If f.k( {xo}) = 0, then the bounds are exact as n ........, oo. n - oo

(7.5) (7.6)

308

B.

PROOF. Suppose

Pn - 1 (xo)

SIMON

=/:. 0. Let ( be such that

the polynomial of Lemma 7.1. By (7.2),

t-t( ( -oo, xo]) $

x�n) (xo) = xo. Let Q be

j Q(x) dt-t

and, by (7.1) and Theorem 6.5, the integral is the sum on the left of (7.4). Clearly, this implies 1

t-t((xo, oo)) :::::

(x 0 )> xo } {). 1 x (nJ j

2:::

Kn - 1 (xj(n) (xo) , xj( n) (xo))

which, by x ---> -x symmetry, implies the last inequality in (7.4). COROLLARY 7. 3 �

k- 1 L.

j=H 1

.

If ( $ k - 1, then

< 11.([x(£n) (x 0 ) - ,_., ' (n) (n ) K (xj ( xo ) , xj (x 0 )) 1

PROOF. Note if X1 =

'

x (kn) (x 0 )] )

D

(7. 7)

x�n\ xo) for some e, then x}n) (xo) = x;n) (x1), so we get D

(7.7) by subtracting values of (7.4).

Notice that this corollary gives effective lower bounds only if k - 1 2: £+ 1, that is, only on at least three consecutive zeros. The following theorem of Last-Simon [57] , based on ideas of Golinskii [41], can be used on successive zeros (see [57] for the proof) . THEOREM 7 4 . If E, E'

� IE - E' l ,

then

.

IE - E' l

2:

are distinct zeros of Pn (x) ,

o2 - ( 2.! I E - E' l 2)2 3n

[

E

=

� (E + E')

Kn (E ' E) sup lv-EI:$<5 Kn ( Y, y)

]1/2

and (j > (7.8)

8. Mixed CD Kernels

Recall that given a measure JL on lR with finite moments and Jacobi parameters

{an , bn }�= 1 , the second kind polynomials are defined by the recursion relations ( 1 .5)

but with initial conditions

qo(x) = 0 so qn (x) is a polynomial of degree n - 1.

(8.1) In fact, if [l is the measure with Jacobi

parameters given by then

qn (x; dt-t) = a1 1Pn - 1 (x; d[l)

It is sometimes useful to consider

KAq) ( x, y)

n

=

L qj (x) qj (y)

j=O

(8.2) (8.3)

309

THE CHRISTOFFEL-DARBOUX KERNEL

and the mixed CD kernel

Pj (Y)

(8.4)

- - 2K y,

(8.5)

"

K,V'ql (x, y) = L qj (x) j=O

Since (8.2) implies

K(q)( n

X, y ,· d/-1)

n - 1 (X,

al

· d-) /-1

K.

there is a CD formula for K(q) which follows immediately from the one for There is also a mixed CD formula for K!fql. OPUC also have second kind polynomials, mixed CD kernels, and mixed CD formulae. These are discussed in Section 3.2 of [80] . Mixed CD kernels will enter in Section 21.

9. Variational Principle: Basics If one thing marks the OP approach to the CD kernel that has been missing from the spectral theorists' approach, it is a remarkable variational principle for the diagonal kernel. We begin with:

Fix (a1 , . . . , a,.) E em . Then l min (f 1zj l2 1 f, ajZj 1 ) = (fl aj 1 2 )J =l J=l J=l with the minimizer given uniquely by LEMMA

9.1.

(9. 1)

=

zj(0)

=

· 12

c'ij j L...- =l I Q:J "\"m

(9. 2)

REMARK. One can use Lagrange multipliers to a priori compute z.� o) and prove this result. P ROO F .

then

If

m L: aj Zj j=l

t,

!zj - z) 0l l2

from which the result is obvious.

=

=

(9.3)

1

t, l z1 1 2 - ( t, )

If Q has deg(Q) :S n and Qn(zo)

la1 l2

(9.4)

-l

0

= 1, then n

Qn (z ) = L ajXj ( z)

j= O with Xj the orthonormal polynomials for a measure dJL, then 2: O:jXj ( z0) II Qn ii 1,2(C,dJ.<) = 2:;=0 la1 l2 . Thus the lemma implies:

(9.5) =

1

and

(Christoffel Variational Principle). Let be a meas1tre on C with finiteTHEOREM moments.9.2Then for E C, min (f !Q., (z W dp, I Q ( ) 1, deg(Qn ) ) (: ) (9.6) I'·

z0

n

zo

=

:S

n

=

Kn

o , zo

310

B. STMON

and the minimizer is given by

Kn (zo , z) ( Q n z, Zo ) - K z n ( o , zo ) _

(9.7)

One immediate useful consequence is:

THEOREM 9.3. If p :5 v1 then

Kn (z, z; dv) s; Kn (z, z; dp)

(9.8)

For this reason, it is useful to have comparison models: EXAMPLE

9.4. Let dp = d()j 27r for z = re·ifJ and ( = ei'P. We have, since

1 - ,,.n + l ei(n+l)(
=

the Poisson kernel,

Pr (B, ;.p) =

For r = 1 , we have

1 - ,- 2 1 + r - 2r cos (f) -
(9. 12 )

the Fejer kernel. For T > 1, we use

(� , �)

(9. 1 3)

d

(9.14)

Kn ( z, () = Z"11(nKn

which implies, for z = ei cp, zo = rei0 , r IQn(z, zo) i

EXAMPLE

(9.11)

>

2 d

1,

__.

0

9.5. Let dp0 be the measure dpo(x) =

� V4 - x2 X[- 2' 2J (x) dx 27r

on [-2, 2] . Then Pn are the Chebyshev polynomials of the second kind, sin(n + 1)fJ e Pn (2 COS ) = . f) HID In particular, if l x l s; 2 - 8,

IPn( x + iy) l s; Ct,<5 enC2 '61Yi

(9.15)

(9. 16) (9.17)

and so (9.18) 0

The following shows the power of the variational principle:

THE CHRISTOFFEL-DARBOUX KERNEL

THEOREM

9.6. Let

dfL = w(x) dx + dfLs Suppose for some Xo , 6, we have

311 (9. 1 9)

w (x) ;:::: c > O

for X E [xo - 6, Xo + 8] . Then for any ()' < 0 and all X E [xo - o'' Xo + 8'] , we have for all a real, 1 ia ia ( 9.20) - Kn x + - , x + - � C1 ec2 1 a I

(

n

n

n

)

We can find a scaled and translated Now use Theorem 9.3 and (9. 18). O · fL ;:::: fL PROOF.

version

of the dfLo of (9. 15) with

D

The following has many proofs, but it is nice to have a variational one: THEOREM

9.7. Let fL be a measure on IR of compact support. For all x0 E IR, 1 (9.21) lim Kn (x o , xo ) = p,({xo})-

REMARK. Jf

PROOF.

n -> co

ft({x0 }) = 0, the limit

is infinite.

Clearly, if Q (xo) = 1 , J I Qn(x) l 2 df.l � p,({xo }), so

Kn (Xo , xo) � fL ( {x 0}) -1 On the other hand, pick A � diam(a(dJt)) and let (x )2 n Q 2n (x) = 1 -

��o

(

For any a,

sup I Q2n (x ) l

lx-xo l � a xEa (dll)

=

)

M2 n(a) ---" 0

so, since Q2n :S 1 on a ( dfL),

Kn (xo , xo) � [f.l((xo - a, xo + a)) + M2n (a)r1

(9.22) (9.23) (9.24)

(9.25)

so

(9.26) lim inf Kn (xo , xo) ;:::: [f.l((xo - a, xo + a))] for each a. Since lima l o f.l( (xo - a, xo + a)) = f.l({xo}) , (9.22) and (9.26) imply D (9.2 1).

10. The Nevai Class: An Aside In his monograph, Nevai [67] emphasized the extensive theory that can be developed for OPRL measures whose Jacobi parameters obey (10.1) for some b real and a > 0. He proved such measures have ratio asymptotics, that is, Pn+ 1 (z)/ Pn ( z) has a limit for all z E C\IR, and Simon [78] proved a converse: Ratio asymptotics at one point of C+ implies there are a, b, with (10. 1 ) . The essential spectrum for such a measure is [b - 2a, b + 2a] , so the Nevai class is naturally associated with a single i nterval e C JR. The question of what is the proper analog of the Nevai class for a set e of the form ( 10.2)

312

B.

SIMON

with (1 0 .3)

has been answered recently and is relevant below. The key was the realization of L6pez [8, 9] that the proper analog of an arc of a circle was !a n i -+ a and C:in+ ID:n -+ a2 for some a > 0. This is not that On approaches a fixed sequence but rather that for each k, n+ k

'"" laj - aei0 1 e'6 E8D L.....t

min

J=n

-+

0

(10.4)

as n -+ oo. Thus, a1 approaches a set of Verblunsky coefficients rather than a fixed one. For any finite gap set � of the form (10.2) /( 10.3), there is a natural torus, J. , of almost periodic Jacobi matrics with CTess (J) = e for all J E J• . This can be described in terms of minimal Herglotz functions [90, 89) or reflectionless two-sided Jacobi matrices [75] . All J E J. are periodic if and only if each [aJ > fiJ] has rational harmonic measure. In this case, we say e is periodic. DEFINITION. dm ( { an, bn } :=I ' {an , bn}:=l) =

00

L e -j ( i am+j - i'im+j I + lbm+j - bm+j i)

j =O

JEJ,

dm ({ an , bn } , J. ) = min dm ({ an , bn}, J ) DEFINITION. The

with

(10.5) ( 10.6)

Nevai class for e, N(e), is the set of all Jacobi matrices, J, (10. 7)

as m -+ oo . This definition is implicit in Simon [81) ; the metric dm is from [26] . Notice that in case of a single gap e in 8][}, the isospectra.l torus is the set of { D:n } ;;::>= 0 with O:n = aei & for all n where a is e dependent and fixed and () is arbitrary. The above definition is the Lopez class. That thit> it> the "right" definition is Heeu by the following pair of theorems: THEOREM 10.1 (Last-Simon [56] ) . If J E N(�) , then

O'ess ( J) = e THEOREM 10.2 ( [26] for periodic �'s; [75] in general) . If CTess ( J)

then J

E

( 10.8)

= O'ac (J) = e

N(e). 11. Delta Function Limits of Trial Polynomials

Intuitively, the minimizer, Qn ( x, x0) , in the Christoffel variational principle must be 1 at z0 and should try to be small on the rest of o- (dp,). As the degree gets larger and larger, one expects it can do this better a.nd better. So one might guess that for every J > 0, sup I Qn (x, xo) l -+ 0 (11.1) lx-xol>o X E (]"(dp)

THE CHRlSTOFFEL-DARBOUX KERNEL

31 3

While this happens in many cases, it is too much to hope for. If x1 E a(dJ.L) but J.L has very small weight near x 1 , then it may be a better strategy for Q n not to be small very near x 1 . Indeed, we will see (Example 1 1.3) that the sup in ( 1 1 . 1 ) can go t o infinity. What is more likely is to expect that I Qn(x , x0)1Z df.l will be concentrated ne ar x0 . We normalize this to define

dry( xo ) (x) = I Qn (X , xoW2df.L(x) JI Qn (x , xo) l dJ.L(x)

( 1 1 .2)

I Kn (X , xo W df.L (X )

(1 1.3)

n

so, by

(9.6)/ (9.7) , in the OPRL case,

dTfn(xo ) (X ) =

Kn(x, xo)

We say f.L obeys the Nevai 8-convergence criterion if and only if, in the sense of weak (aka vague ) convergence of measures,

( 1 1.4)

dry�xo) (x) --+ <>xo

the point mass at x0. In this section, we will explore when this holds. Clearly, if xo � a(df.l), (1 1 .4) cannot hold. We saw, for OPUC with df.L = d0/21r and z � alDl, the limit was a Poisson measure, and similar results should hold for suitable OPRL. But we will see below (Example 1 1 . 2) that even on a(df.l-), (11.4) can fail. The major result below is that for Nevai class on eint, it does hold. We begin with an equivalent criterion: DEFINITION. We say Nevai's lemma holds if

1Pn (Xo ) l 2 n-x· Kn (Xo , xo ) lim

=

( 1 1 .5)

O

THEOREM 1 1 . 1 . If df.l is a measure on IR with bounded support and then for any fixed xu

E

( 1 1 .6)

inf n an > 0

IR,

REMARK. That (11.5) Breuer-Last-Simon [14] .

(11.4)

=}

{:}

(11. 5 )

( 1 1 .4) is in Nevai [67] . The equivalence is a result of

PROOF. Since

1

Kn-l(xo , xo) = 1Pn(xo ) l 2 Kn (xo , xo) Kn (xo , xo) x K (11. 5 ) <=> n-t(Xo, o) 1

_

Kn (xo, xo)

so We thus conclude

(U.S) <=?

( 1 1 . 7) ( 1 1 .8)

--+

IPn ( xoW + IPn+l ( xu ) l 2

Kn (Xo, xo) By the CD formula and orthonormality of Pj (x) ,

--+

O

f ix - xoi2 1 Kn (x, xo)l 2 df.l = a�+l [p,.,_ (xo) 2 + Pn+l (xo)2]

.

( 1 1 .9) ( 1 1.10)

3 14

B.

SIMON

so, by (11 .6) a.nd (11.10),

j lx - xol2 dry�xo) (x) - 0 � ( 1 1 .5)

when a, is uniformly bounded above and away from zero. But since d1Jn have support in a fixed interval, 0

ExAMPLE 1 1 . 2 . Suppose at some point x0 , we have lim (IPn (xo W + 1Pn+ l (xo) l 2 ) 1 /n - A > 1 n -> oo

( 1 1 . 1 1)

lim sup

( 1 1 . 12)

IPn (xoW 0 > n-+= Kn (xo , xo) for if ( 1 1. 1 2 ) fails, then (11.5) holciH and, by (1 1.7), for any for n 2: No ,

We claim that

so

t: ,

we ean find N0 so ( 1 1 . 1 3)

lim Kn (Xo , Xo) l/n :S 1 So, by (1 1.5), ( 1 1 . 1 1) fails. Thus, (11.11) implies that (11.5) fails, and so (11.4) D fails. REMARK.

lim inf > 1.

As the proof shows, rather than a limit in (11.12), we can have a

The first example of this type was found by Szwarc [94] . He has a dp, with pure points at 2 - n- 1 but not at 2, and so that the Lyapunov exponent at. 2 was positive but 2 was not an eigenvalue, so ( 1 1 . 1 1) holds. The Anderson model (see [20)) provides a more dramatic example. The spectrum is an interval [a, b] and ( 1 1 . 1 1) holds for a.e. x E [a, b] . The spectral measure in this case is supported at. eigenvalues and at eigenvalues (11 .8), and so (11.4) holds. Thus (11.4) holds on a dense set in [a, b] but fails for Lebesgue a. e. x0 ! ExAMPLE 1 1 .3. A Jacobi weight has the form

with a , b > -1. In general, one can show [93] has

Pn (l) "' cna+l/ 2

(11.14) ( 1 1 . 15)

so if xo E ( - 1, 1) where IPn(xoW + IPn - l (xoW is bounded above and below, one I Kn (Xo , 1) 1 Kn(xo , xo)

rv

na+ l/2 na - 1 2 = / n

so if a > ! , 1Qn (x0, 1)1 --+ oo. Since dp,(x) is small for x near 1, one can (and, as 0 we will see, does) have (11.4) even though (11.1) fails. 'With various counterexamples in place (and more later!), we turn to the positive results:

315

THE CHRISTOFFEL-DARBOUX KERNEL

THEOREM 1 1 .4 (Nevai [67] , Nevai-Totik-Zhang [69] ) . If dp, is a measure in the classical Nevai class (i. e., for a single interval, e = [b - 2a, b + 2a] ) , then ( 1 1 .5) and so ( 1 1.4) holds uniformly on e .

THEOREM 1 1 . 5 (Zhang [108] , Breuer-Last-Simon [ 14 ] ) . Let e be a periodic finite gap set and let p, l·ie in the Nevai class for· e. 'l'hen ( 1 1 . 5) and so ( 1 1 .4) holds uniformly on e .

1 1 .6 (Breuer-Last-Simon [14] ) . Let e be a general finite gap set and let f-L lie in the Nevai class for e. Then ( 1 1. 5) and so (11 .4) holds uniformly on compact subsets of eint . THEOREM

REMARKS. 1 . Nevai [67] proved ( 10.4)/(10.5) for the classical Nevai class for i every energy in e but only uniformly on compacts of e nt . Uniformity on all of e using a beautiful lemma is from [69] . 2. Zhang [108] proved Theorem 1 1 .5 for any 11· whose Jacobi parameters ap proached a fixed periodic Jacobi matrix. Breuer-Last-Simon [14] noted that with out change, Zhang's result holds for the Nevai class. 3. It is hoped that the final version of [14] will prove the result in Theorem 11.6 on all of e, maybe even uniformly in e .

EXAMPLE 1 1 . 7 ( [14] ) . In the next section, we will discuss regular measures. They have zero Lyapunov exponent on O'ess ( M) , so one might expect Nevai's lemma could hold-and it will in many regular cases. However , [14] prove that if bn = 0 and an is alternately 1 and � on successive very long blocks (1 on blocks of size 3n2 and ! on blocks of Hize 2"\ then dfl, is regular for r:r(dp,) = [ - 2 , 2] . But for a.e. x E [-2 , 2] \ [-1 , 1], (10.4) and (10.3) fail. 0

1 1 . 8 ( [14] ) . The following is extensively discussed in [14] : For of compact support and a.e. x with respect to f-L, (10.4) and so (10.3)

CONJECTL"RE

general holds.

OPRL

12. Regularity: An Aside There is another class besides the Nevai class that enters in variational problems because it allows exponential bounds on trial polynomials. It relies on notions from potential theory; see [42, 52, 73, 102] for the general theory and [9 1, 85] for the theory in the context of orthogonal polynomials.

Let 11 be a measure with compact support and let is regular for e if and only if lim (a1 . . . an )lfn = C(e) n-oc the capacity of e. DEFINITION.

We say

f-L

e

=

O'ess (p,) .

(12.1)

For e = [ -1, 1 ] , C ( e ) = � and the class of regular measures was singled out initially by Erdos-Turan [32] and extensively studied by Ullman [103] . The general theory wa::; developed by StahJ-Totil< [91]. Recall that any set of positive capacity has an equilibrium measure, p. , and Green's function, c., defined by requiring c. is harmonic on C \ e, G. (z) = log izl + 0(1) near infinity, and for quasi-every x E e , ( 12.2) lim c. (zn) = 0 Z n --t X

3 16

B SIMON

(quasi-every means except for a set of capacity 0) . e is called regular for the Dirichlet problem if and only if { 12.2) holds for every x E e. Finite gap sets are regular for the Dirichlet problem. One major reason regularity will concern us is:

Let e C lR be compact and regular for the Dirichlet problem. Let p, be a measure regular for e. Then for any �::, there is o > 0 and Ce so that sup IPn (z, dp,) l ::; G_, ec- l n l ( 12.3) THEOREM 1 2 . 1 .

dist(z,•)
For proofs, see [91, 85] . Since Kn has sup

dist( z ,e)
n + 1 terms,

I K (z , w) l ::; (n +

n

and for the minimum (since K., (zo , zo ) � 1), sup

dist(z,e)
I Qn (z, zo) l ::;

(12.3) implies

l)G;e2"' 1 nl

(12.4)

(n + 1 ) C; e2s l n l

(12.5)

The other reason regularity enters has to do with the density of zeros. If x n) are the zeros of Pn(x, df.l) , we define the zero counting measure, dvn , to be the probability measure that gives weight to n - 1 to each x]nl _ For the following, see [91, 85] :

;

THEOREM 12.2.

Then

Let e

c lR

be compact and let J.l be a regular measure for

t.

(12.6)

the equilibrium measure for e.

In (12.6) , the convergence is weak.

13. Weak Limits

A major theme in the remainder of this review is pointwise asymptotics of n�l Kn (x, y ; df.l) and its diagonal. Therefore, it is interesting that one can say some

thing about n�l Kn (x, x; dJ.l) df.l(x) without pointwise asymptotics. Notice that

(13.1) is a probability measure. Recall the density of zeros, Vn , defined after (12. 5 ) .

THEOREM 13. 1 . Let J.l have compact support. Let vn be the density of zeros and J.ln given by (13. 1 ) . Then for any f = 0, 1 , 2, . . . ,

In particular, n(j).

df.ln(j )

if

l x dvn+l -

j xe df.ln i -> 0

(13.2)

and dvn (j ) + l have the same weak limits for any subsequence

317

THE CHRISTOFFEL · DARBOUX KERNEL

PROOF . By Theorem 6.2, the zeros of Pn + l are eigenvalues of 1rn Mx 1rn , so

1 X dvn+l n J =

£

+1

l

Tr ( (7rn Mx7rn ) )

( 13.3)

On the other hand, since {pj}j=o is a basis for ran(1rn ) ,

1 tj xt[Pj (xW dJ-L (X) J xc dJ-Ln n + j=O =

1

1 e -- Tr(7rnMx1rn ) n+1 It is easy to see that (7rnMx1rn)e - 1rn M};1r., is rank at most f., so =

goes to 0

as

n --->

LHS of ( 13 .2) 00

for e fLxed.

:::; n! 1 [[ Mx lie

(13.4)

0

REMARK. This theorem is due to Simon [88] although the basic fact goes back to Avron-Simon [7] .

See Simon [88] for an intereHting application to comparison theorems for limits of density of states. We immediately have:

Suppose that dJ-L w(x) dx + dJ-Ls with dJ-Ls Lebesg�te singular, and on some open interval dv,. dv00 and dvoo r I = Voo(x) dx and suppose that uniformly on I, lim � Kn (x, x) g(x) n and w(x) "/=- 0 on I. Then g(x) = V00w(x)(x) The theorem implies dv00 f I = w(x)g(x). COROLLARY

13.2.

=

I

(13.5)

C e =

O"ess(dJ-L) we have

--->

=

PROOF.

(1 3.6) (13.7)

0

Thus, in the regular case, we expect that "usually"

-1 Kn(x,

n

x ) --->

p.(x) w (x )

--

(13.8)

This is what we explore in much of the rest of this paper.

14. Variational Principle: Mate-Nevai Upper Bounds The Cotes numbers, An (z0), are given by (9.6), so upper bounds on .An (zo) mean lower bounds on diagonal CD kernels and there is a confusion of "upper bounds" and "lower bounds." We will present here some very general estimates that come from the use of trial functions in (9.6) so they are called Mate-Nevai upper bounds (after [65] ) , although we will write them as lower bounds on Kn . One advantage is their great generality.

B.

318

SIMON

DEFINITION. Let dfl be a measure on JR. of the form dj.L = w(x ) dx + dtts

(14.1)

where dfls is singular with respect to Lebesgue measure. We call x0 a Lebesgue point of fL if and only if

� fls([xo - �,xo + �]) - 0

n ro+f;

2 fxo- 1.

( 14.2)

lw (x) - wo(xo ) i dx -) 0

( 14.3)

n

It is a fundamental fact of harmonic analysis ([76]) that for any fL Lebesgue a.e., x0 in JR. is a Lebesgue point for It · Here is the most general version of the MN upper bound:

THEOREM 14.1. Let c C R be an arbitrary compact set which is regular for the Dir·tchlet problem. Let I C e be a closed interval. Let d�t be a measuTe with compact supporl in JR. with O'ess ( d�t) C e. Then for any Lebesgue point, x in I,

. lll Pe (X) llm . f - Kn ( X, X) � -n_,.oo n w ( x) 1

(14.4)

where dp. r I = p. (X) dx. If 'W is continuous on I (including at the endpoints as a function in a neighboThood of I) and nonvanishing, then (14.4) holds nnifoTmly on I. If Xn - x E I and A = supn nlxn - xi < oo and x is a Lebesgue, then (14.4) holds with Kn (x, x) replaced by Kn (xn, X n ) . If w is continuous and nonvanishing on I, then this extended convergence is unijoTm in x E I and X n 's with A ::; A0 < oo. REMARKS. 1. I f I c e is a nontrivial interval, the measure dp. r I is purely absolutely continuous (see, e.g., [85, 89]). 2. For OPUC, this is a result of Mate-Nevai [64] . The translation t o OPRL on [- 1, 1] is explicit in Mate-Nevai-Totik [66) . The extension to general sets via polynomial mapping and approximation (see Section 18) is due to Totik [96) . These papers also require a local Szego condition, but that is only needed for lower bounds on A n (see Section 1 7). They also don't state the Xn - x00 result, which is a refinement introduced by Lubinsky [60] who implemented it in certain [- 1 , 1] cases. 3. An alternate approach for Totik's polynomial mapping is to use trial func tions ba.sed on .Jost-Floquet solutions for periodic problems; see Section 19 (and also [87, 89]).

One can combine (14.4) with weak convergence and regularity to get THEOREM 14.2 (Simon [88]). Let c C JR. be an arbitrary compact set, regular for the Dirichlet problem. Let d�t be a measnTe with compact support in JR. with O'ess( d�t) = e and with dp, regular for e. Let I C e be an interval so w ( x ) > 0 a . e. on I . Then

(i) (ii)

Jl �

{ .!_ Kn ( X , X) d�ts ( X)

}I n

l

Kn (x , x) w ( x) - p. (x) dx -

0

0

(14.5) (14.6)

THE CHRISTOFFEL-DARBOCX KERNEL

PROOF.

3 19

By Theorems 12.2 and 13.1, 1

- Kn(x , x) dp. dpe Let v1 be a limit point of �Kn(x, x) df.l.s and dv2 = dp. - dv1 n

(1 4. 7)

--7

(14.8)

If f ?:: 0, by Fatou ' s lemma and (14.4),

(14.9) 1Jdv2 ?:: 1Pe(x)f(x) dx that is, dv2 f I ?:: Pe(x) dx f I. By (14.8), dv2 f I :S p.(x) dx. It follows dv1 f I is 0 and dv2 I I dp. f I . By compactness, �Kn(x, x) df.l.s I I 0 weakly, implying (14.6). By a simple argument [88], weak convergence of �Kn(x, x)w(x) dx p.(x) dx and (14.4) imply =

--7

--7

( 14.5) .

0

15. Criteria for A.C. Spectrum Define N so that

=

{x

lR \ N =

Theorem 14.1 implies

l lim inf � Kn(x,x) oo} {X I � Kn(x,x) oo }

E lR

E lR

lim

<

(15.1)

=

( 15 .2)

w(x) dx +df.J."

THEOREM 1 5. 1 . Let e c lR be an arbitmry compact set and df.J. = a measure with a(f.J.) = e . Let �ac = I w (x) > 0}. Then N \ �ac has Lebesgue measure zero.

{x

PROOF. If Xo E lR \ �ac and is a Lebesgue point of f.J., then Theorem 14.1 , E lR \ N. Thus,

xo

(IR \ �ac) \ (IR \ N) has Lebesgue measure zero.

=

w(xo) = 0 and, by

N \ �ac

0

REMARK. This is a direct but not explicit consequence of the Mat8-Nevai ideas [64] . Without knowing of this work, Theorem 15. 1 was rediscovered with a very different proof by Last-Simon [55] .

On the other hand, following Last-Simon [55] , we note that Fatou's lemma and

J � Kn (x , x) df.J.( X)

implies

so

THEOREM

=

1

/ lim inf � Kn(x,x) df.J.(x) :::; 1 15.2 ( [55]) . �ac \ N has Debesgue measure zero.

(15.3) (15.4)

320

B. SIMON

=

Thus, up to sets of measure zero, 2:ac N . What is interesting is that this holds, for example, when e is a positive measure Cantor set as occurs for the almost Mathieu operator (an = 1, bn = >. cos(1ro:n + B), 1>.1 < 2, >. =J- 0, a: irrational). This operator has been heavily studied; see Last [54]. 16. Variational Principle: Nevai Trial Polynomial

A basic idea is that if dp,1 and df.l2 look alike near xo , there is a good chance that Kn (xu, xu; df.l t) and Kn (x0 , xu; dp,2) are similar for n large. The expectation (1 3.8) ::;ays they better have the same support (and be regular for that support), but this is a reasonable guess. It is natural to try trial polynomials minimizing >.n (x0 , df.l1) in the Christoffel variational principle for An (x0, dJ.L2 ), but Example 1 1.3 shows this will not work in general. If df.ll has a strong zero near some other x 1 , the trial polynomial for df.l l may be large near x1 and be problematical for df.l2 if it does not have a zero there. Nevai [67] had the idea of using a locallzing factor to overcome this. Suppose e C JR., a compact set which, for now, we suppose contains a(dp, 1 ) and O'(df.l�). Pick A = diam(e) and consider (with [ · ] = integral part) ( x - xo ) 2 [en] 1(16.1) = N2 [cn j (x ) AZ

(

Then for any 0,

) -

jx-x0 j > cl xEe

sup N2[enj (X) ::;; e - c(.S ,e)n

(16.2)

so if Qn- 2 [wj (x) is the minimizer for J.ll and e is regular for the Dirichlet problem and J.ll is regular for e, then the Nevai trial function

N2 [cn] (x) Qn- 2 [cn] (x) will be exponentially small away from xo. For this to work to compare >.n (x0, df.l 1) and >.(x0 , dp,2) , we need two additional properties of >.n ( x 0 , dp1): (a) An(x0 , df.l 1) 2: C�e-m for each c < 0 . This is needed for the exponential contributions away from x0 not to matter. (b)

1. 1. 1m 1m sup e:!O n-oo

An (xo , dJ.lJ ) =1 An -2 [e :n] (x, df.li) so that the change from Qn to Qn- 2[en] does not matter. Notice that both (a) and (b) hold if

.....

nlim n>. n (Xo , df.l) = c > 0 oo

(16.3)

If one only has e = aess(df.l2 ), one can use explicit zeros in the trial polynomials to mask the eigenvalues outside e. For details of using Nevai trial functions, see [87, 89]. Below we will just refer to using Nevai trial functions.

321

THE CHRISTOFFEL-DARDOUX KERNEL

17. Variational Principle: Mate-Nevai-Totik Lower Bound In [66] , Mate-Nevai-Totik proved: THEOREM 17. 1 . Let dJ-L be a measure on 8j[)) w(O) dfJ = � dO + dJ-Ls

which obeys the Szegfi condition Then for a. e. 0=

E

g

J lo (

8ID,

w

( 17.1)

dO (B) ) 27!' > -oo

(17.2)

( 1 7. 3 ) This remains true if >-n((}oo) is replaced by >-n (Bn ) with On _. Boc obeying sup niBn Boo l < 00 . REMARKS. 1 . The proof in [66] is clever but involved ( [89] has an exposition); it would be good to find a simpler proof. 2 . [66] only has the result (),. 000• The general (}n result is due to Findley [33] . 3 . The 000 for which this is proven have to be Lebesgue points for dJ-L as well as Lebesgue points for log( w) and for its conjugate function. 4. As usual, if I is an interval with w continuous and nonvanishing, and J-Ls(I) = 0, ( 17.3) holds uniformly if 000 E I. =

By combining this lower bound with the Mate-Nevai upper bound, we get the result of Mate-Nevai-Totik [66] : THEOREM 17.2. Under the hypothesis of Theorem 1 7. 1 , for a.e. 000 E (Jj[]),

lim n >.n (Boc )

n ---+ =

=

w(()co)

( 17.4)

This remains true if >-n (B00 ) is replaced by An(Bn ) with ()n _. 000 obeying sup niBn Boc l < oo. If I is an interval with w continuous on I and J-Ls (I) = 0, then these re.mlts hold uniformly in I.

REMARK. It is possible (see remarks in Section 4.6 of [68]) that (17.4) holds if a Szcgo condition is replaced by w(B) > 0 for a.e. 0. Indeed, under that hypothesis, Simon [88] proved that 12rr dO _. 0 l w (O) (n.An (0) ) - 1 - 1 1 2 o

rr

There have been significant extensions of Theorem 1 7.2 to OPRL on fairly general sets: 1 . [66] used the idea of Nevai trial functions (Section 16) to prove the Szego condition could be replaced by regularity plus a local Szego condition. 2. [66] used the Szego mapping to get a result for [- 1, 1]. 3. l.Jsing polynomial mappings (see Section 1 8) plus approximation, Totik [96] proved a general result (see below); one can replace polynomial mappings by Floquet-Jost solutions (see Section 19) in the case of continuous weights on an interval (see [87]). Here is Totik's general result (extended from cr(dfJ) C e to aess(dtt) C e):

B. SIMON

322

THEOREM 1 7 .3 (Totik [96, 99] ) . Let c be a compact subset of !R. Let I C e be an interval. Let dfL have Uess( df..L } = e be regular for e with

Then for a. e. X00 E I,

fz log(w) dx > -oo . hm - Kn ( Xoo, Xoo ) 1

n->oo n

W

(17. 5 }

(

= ---

Pe (x oo )

X00 )

The same limit holds for �Kn(Xn , Xn) if sup n nlxn - Xoo l

(17.6}

< oo .

If fls (I)

=0

and

w is continuous and nonvanishing on I, then those limits are uniform on x00 E I and on all Xn 's with sup n n lxn - X00 I ::; A ( uniform for each fixed A) .

REMARKS. 1 . Totik [98] recently proved asymptotic results for suitable CD kernels for OPs which are neither OPUC nor OPRL. 2. The extension to general compact c without an assumption of regularity for the Dirichlet problem is in [99] .

18. Variational Principle: Polynomial Maps

Iu passing from [- 1, 1 ] to fairly general sets, one uses a three-step process. A finite gap set is an e of the form

(18.1)

where (18.2)

£1 will denote the family of finite gap sets. We write e = e 1 U · · · U el+ 1 in thiH case with the Cj closed disjoint intervalH. £p will denote the set of what we called periodic finite gap sets in Section 10-ones where each Cj has rational harmonic measure. Here are the three steps: (1) Extend to e E £p using the methods discussed briefly below. (2) Prove that given any e E £1, there is eCn) E Ep , each with the same number of bands so Cj C e;n) C e)n-l) and nne)n) = e.i · This is a result proven independently by Bogatyrev [12] , Peherstorfer [71] , and Totik [97] ; see [89] for a presentation of Totik's method. (3) Note that for any compact e, if eCm) = { x I iliHt( x , e) ::; 7k }, then e( m) is a finite gap set and e = nme(m). Step (1) is the subtle step in extending theorems: Given the Bogatyrev Peherstorfer-Totik theorem, the extensions are simple approximation. The key to e E £p is that there is a polynomial A : C -> C, so A - 1 ( [-1, 1] ) = e and so that Cj is a finite union of intervals ek with disjoint interiors so that A is a bijection from each e k to [ - 1 , 1]. That this could be useful was noted initially by Geronimo-Van Assche [36]. Totik showed how to prove Theorem 17.3 for e E £p from the results for [- 1, 1] using this polynomial mapping. For spectral theorists , the polynomial A = �b. where b. is the discriminant for the associated periodic problem (see [43, 53, 104, 9 5 , 89]). There is a direct com;trm:tion of A by Aptekarev [4] and Peherstorfer [70, 71, 72].

323

THE CHRISTOFFBI�DARBOUX KERNI!:L

19. Floquet-Jost Solutions for Periodic Jacobi Matrices

16,

As we saw in Section models with appropriate behavior are useful input for comparioon theorems. Periodic Jacobi matrices have OPs for which one can study the CD kernel and its asymptotics. The two main re�mlts concern diagonal and just off-diagonal behavior:

19.1. Let be the spectral measure associated to a periodic Jacobi matrix with essential spectrum, a finite gap set. Let dJ-1. w(x) dx on e (there can also be up to one eigenvalue in each gap). Then uniformly for x in compact subsets of x ..!:. Kn (x , x) Pe ( ) (1 9. 1 ) w(x ) and uniformly for such x and a, b in with la l A, l bl B, Kn (x + � . x + �) sin(rrp.(x ) (b - a ) ) (1 9.2) Kn (x, x) rrp. (x)(b - a) 1. (19.2) is often called bulk universality. On bounded intervals, it 1-1

THEOREM

e,

=

i nt e ,

n

�

lit

:::;

:::;

--�� --��

REMARKS.

goes back to random matrix theory. The best results using Riemann-Hilbert meth ods for OPs is due to Kuijlaars-Vanlessen [51]. A different behavior is expected at the edge of the spectrum-we will not discuss this in detail, but see Lubinsky [62] . For [- 1 , ] , Lubinsky [60] used Legendre polynomials as his model. The references for the proofs here are Simon [87, 89].

2.

1

The key to the proof of Theorem solutions of

19.1 is to use Floquet-Jost solutions, that is, (19.3)

n E Z where {an , bn} are extended periodically to all of Z. These solutions obey n +p (19.4) n For x E Un and Un are linearly independent, and so one can write in terms of u . and .. Using (19.5) Pe ( x ) p � I�� I one can prove (19.1) and (19. 2 ). The details are in [87, 89] . for

e

int

U

_

ei8(x) u

P· - l

,

u

=

20. Lubinsky's Inequality and Bulk Universality

Lubinsky [60] found a powerful tool for going from diagonal control of the kernel to slightly off-diagonal control-a simple inequality. z , (,

THEOREM

20.1. Let J-1. :::; 1-1* and let Kn, K� be their CD kernels. Then for any (20.1) I Kn (z, () - K� (z , ( ) 1 2 Kn (z, z)[I
REMARK. PROOF.

CD

Since Kn - K� is a polynomial

Kn (z, () - K� ( z , ( ) =

2::

z

of degree n:

J Kn (z , w ) [Kn (w, () - K� (w, ()] d�-L(w)

(20.2)

B. SIMON

324

By the reproducing kernel formula ( 1 . 1 9) , we get (20.1 ) from the Schwarz inequality if we show (20.3)

Expanding the square, the K';, term is Kn ((, () by (1. 19) and the Kn K� cross term is -2K�((, () by the reproducing property of Kn for df.L integrals. Thus, (20.3) is equivalent to 1K;. (w, (W drt(w) � K;.((, () (20.4)

j

This in tmn follows from {t � {l * and ( 1 . 19) for f.L* !

0

This result lets one go from diagonal control on measures to off-diagonal. Given any pair of measures, f.L and v, there is a unique measure f.L V v which is their least upper bound (see, e.g., Doob [30]). It is known (see [85]) that if f.L, v are regular for the same set, so is J.L V v. (20.1) immediately implies that (go from f.L to f.L* and then J.L* to v) :

COROLLARY 20.2. Let J.L, v be two measures and JL* = /L V v. Suppose for some -> Z00 , w.,. _, z""'" we have for ry = f.L, v, f.L* that Zn ' Kn ( Wn , Wn i 'fl ) l 1.tm Kn ( Zn, Zn ; 'fl) = lliD n -+ oo K.,. (z00, z00 ; 17) = n->oo K.,. (Zoc" Zoo i ry) and that

Then

(20.5)

X00 REMARK. It is for use with Xn = X00 + . to the various diagonal kernel results. This ''wiggle" in X00 was introduced by Lubinsky [60] , so we dub it the "Lubinsky wiggle." Given Totik's theorem (Theorem 17 .3) and bulk universality for suitable mod els, one thus gets:

THEOREM 20.3. Under the hypotheses of Theorem 1 7. 3, for a.e. X00 in I, we have unifo1mly for l al, l b l < A, 1.liD Kn(Xoo

n-> oo

+ � ' Xoo + Kn( Xx 1 X 00 )

�)

sin(1T'Pe (x00 ) (b - a)) :.....: ----=:,:�--:-'� = _....:c..,:-: 1l'p.(xoo ) ( b - a)

REMARKS. 1 . For e = [- 1, 1] , the result and method are from Lubinsky [60] . 2. For continuous weights, this is in Simon [87] and Totik [99] , and for general

weights, in Totik [99] .

21. Derivatives of CD Kernels

The ideas in this section come from a paper in preparation with Avila and Last [6] . Variation of parameters is a standard technique in ODE theory and used as an especially powerful tool in spectral theory by Gilbert-Pearson [38] and in Jacobi matrix spectral theory by Khan-Pearson [49] . It was then developed by .Jitomirskaya-Last [44, 45, 46] and Killip-Kiselev-Last [50] , from which we take Proposition 21 . 1.

THE CHRISTOFFEL-DARBOUX

325

KERNEL

PROPOSITION 2 1 . 1 . For any X , x0, we have

n-1 Pn (x) - Pn(xo) = (x - Xo) L (Pn(Xo)qm(xo) - Pm(xo)qn(xo))Pm(x)

In particular,

(21 . 1)

m=O

n-1 P�(xo) = L (Pn(xo) qrn(xo) - Prn(xo) q.,(xo))Prn(xo)

(21 .2)

m=O

Here qn are the second kind polynomials defined in Section 8. For (21.1), see [44, 45, 46, 50] . This immediately implies: COROLLARY 2 1 .2 (Avila-Last-Simon [6]) .

( + �)I ] 2 �2 �'ll [Pi (xo) ( � Pk (xo)qk(xo) ) - qj (xo)Pj (xo) ( {; Pk(xo)2 )

.!!:._ .!_ Kn xo + �n ' Xo da n =

n

a=O

j

j

(21.3)

This formula gives an indication of why (as we see in the next section is im portant) lim � Kn ( x0 + ; , x 0 + ; ) has a chance to be independent of a if one notes the following fact: LEMMA 21.3. If {an }�= l and {f1n }�=l are sequences so that lim it L:;�=l and lim it 'L:�=l ,Bn = B exist and supN [ it L �=1lan l + lf1nll < oo, then A

an =

(21 .4) This is because

j N 1 AB L i L k a f1 --+ 2 2 N j =l k=l Pi(x o ) 2 and /1j = Pi (x0 ) qj (x0 ) , one can hope to use (21 .4) 1

Setting aj = the right side of (21 .3) goes to zero.

to prove

22. Lubinsky's Second Approach Lubinsky revolutionized the study of universality in [60] , introducing the ap proach we described in Section 20. While Totik [99] and Simon [87] used those ideas to extend beyond the case of e = [- 1, 1] treated in [60], Lubinsky developed a totally different approach [61] to go beyond [60] . That approach, as abstracted in Avila-Last-Simon [6] , is discussed in this section. Here is an abstract theorem:

THEOREM 22. 1 . Let dJ..L be a measure of compact support on R. Let :.t:o be a Lebesgue point for p, and suppose that (i) For any c:, there is a Ce so that for any R, we have an N(c:, R) so that for n � N(c:, R) , z 1 (22 . 1 ) - Kn Xo + - , xo + - ::; Ce eElzJ2 n n n for all z E C with lzl < R.

(

Z)

326

B. SIMON

(ii) Uniformly for real a 's in compact subsets of IR, Kn ( xo + � , xo + � ) =1 n->oo Kn (Xo , Xo )

(22.2)

x Pn = -Kn (xo , xa ) n

(22 . 3 )

lim

Let

w( o)

Then uniformly for lim n�oo REMARKS .

z, w

in compact subsets of C,

Kn (Xo

+ � ' xo + � ) Kn(xo , xo)

1. If Pn

=

sin (1r (z

1r ( z -

w) )

w)

-

(22.4 )

p. (x o ) , the density of the equilibrium measure, then

(22.4) is the same as (1 9.2) . In every case where Theorem 22. 1 has been proven to be applicable (see below), Pn --+ p, (xo ) . But one of the interesting aspects of this is --+

that it might apply in cases where Pn dues not have a limit. For an example with a.c. spectrum but where the density of zeros has multiple limits, see Example 5.8 of [85] . 2. Lubinsky [61] worked in a situation (namely, x0 in an interval I with w (x0 ) � c > 0 on I) where (22.1) holds in the stronger form CeDizl (no square on i z i) and used arguments that rely on this. Avila-Last-Simon [6] found the result stated here; the methods seem incapable of working with (22. 1) for a fixed c rather than all c: (see Remark 1 after Theorem 22.2). Let us sketch the main ideas in the proof of Theorem 22. 1: (1) By (15. 1 ) , 1

lim inf - Kn (x o , xo) > 0

(22.5)

n

(2) By the Schwarz inequality (1.14) , (22.1), and (22.5) , and by the compact ness of normal families, we can find subsequences n(j) so Kn(j) (xo + � , x o + � ) Kn (j ) (Xo , xo )

--+

F

(

z,

w)

(22.6)

and F is analytic in w and anti- analyt ic in z. (3) Note that by (22.2) and the Schwarz inequality (1.14), we have for a, b E IR, F(a, a) = 1

IF( a, b) i � 1

(22 . 7)

By compactness, if we show any such limiting F is sin (1r ( z - w ))/ ( z - w ) , we have (22.4). By analyticity, it suffices to prove this for z = a real, and we will give details when z = 0, that is, we consider z_ ) Kn (J· ) (x o ' Xo + _p,. n Kn(j) ( xo, xo)

--+

f (z )

(22 .8)

(22 . 7) becomes f(O) = 1

i f ( x ) l � 1 for

x

real

(4) By (1.19) , (22 .9)

THE CHRISTOFFEL-DARBOUX KERNEL

32 7

which, by using the fact that x0 is Lebesgue point, can be used to show

L:

l f (x) l 2 dx :S: 1

X- 1

<0<

(22. 10)

(5) By properties of K, (see Section 6) and Hurwitz's theorem, f has zeros { xj }�-oo,i#O only on IR, which we label by <

X1

<

<

X

2 · and define x0 = 0. By Theorem 7.2, using (22.2) , we have for any j, k that ·

·

·

l xi - x k l 2 IJ - k l -

1

· ·

( 22 . 1 1 )

(22.12)

(6) Given these facts, the theorem is reduced to

(1)

THEOREM 22.2. Let f be an entire function obeying

(2)

f (O) = 1

l f (x ) l �

1

for x real

(22.13) (22.14)

(3) f is real on IR, has only real zeros, and if they are labelled by (22 . 1 1 ) , then (22. 12) holds. (4) For any e, there is a C: so (22.15) Then f(z) =

sin 7rz

(22. 16)

7rZ

REMARKS. 1 . There exist examples ( [6]) e-az2+bz sin 1rZj1rz that obey (1)-(3) and (22. 15) for some but not all c. 2. We sketch the proof of this in case one has D (22. 17) l f(z) l � Ce i z l instead of (22.15); see [6] for the general case. LEMMA 22.3. If ( 1)-(3) hold and (22.17) holds, then for any c > 0, there is Ds with mz (22. 18) l f(z) l � D:e(7r+e) I I l Sketch. By the Hadamard product formula [2] , f (z) =

eDz IT

#0

(1 - : )

ezx1

J

where D is real since f is real on l!l. Thus, for y real, l f (iy) l 2 =

By (22. 1 2) , lxi l 2 j - 1 , so l f(iy) l2 �

n

j¥.0

(� ��) +

J

( �;) ( :;J [£ ( �:) r 1+

1+

1+

3 28

B.

SlMON

which, given Euler's formula for sin 1rzjz, implies (22. 18) for z = iy. By a Phragmen-Lindelof argument, (22.18) for z real and for z pure imaginary and (22.17) implies (22.18) for all z. Thus, Theorem 22.2 (under hypothesis (22.17)) is implied by; LEMMA 22.4. If f is an entire function that obeys (22. 13), (22. 14),

then (22.16) holds.

and (22.18),

PROOF. Let j be the Fourier transform of f, that is,

](k) = (27r) -l /2

J e-ikxf (x) dx

(22.19)

(in L 2 limit sense). By the Paley-Wiener Theorem [74] , (22.18) implies j is sup ported on [- 1r , 1r) . By (22.14), 1/ 2 11 ! 11£2 = ll(27r ) - X[-,-,,.-J II£ 2 = 1 �

and, by (22.13) and support property of } , (! , (21r) - 1/2X [ - ,-,,-J } = 1 We thus have equality in the Schwarz inequality, so •

f = (21r) -1/2 X[- 1r,,.] �

D

which implies (22.16) .

This theorem has been applied in two ways; (a) Lubinsky [61) noted that one can recover Theorem 20.3 from just Totik's result Theorem 1 7.3 without using the Lubinsky wiggle or Lubinsky's inequality. (b) Avila-Last-Simon [6] have used this result to prove universali ty for ergodic Jacobi matrices wit h a.c. spectrwn where e can be a positive measure Cantor set . 23. Zeros: The Freud--Levin-Lubinsky Argument

In the final section of his book [34) , Freud proved bulk universality under fairly strong hypotheses on measures on [- 1 , 1) and noticed that it implied a strong result on local equal spacings of zeros. \Vithout knowing of Freud's work, Simon, in a series of papers (one joint with Last) [82, 83, 84, 57], focused on this behavior, called it clock spacing, and proved it in a variety of situations (not using universality or the CD kernel). After Lubinsky'�:> work on universality, Levin [59] rediscovered Freud's argument and Levin-Lubinsky [59] used tllis to obtain clock behavior in a very general context. Here is an abstract version of their result;

THEOREM 23. 1 . Let J.L be a measure uf compact support on JR.; let x0 E O' (f..t ) be such that for each A, for some en, sin( 1r (b a)) Kn- 1 ( xo + n�... , xu + n�J (23.l) 7r(b - a) Kn - 1 (xo, Xo) uniformly for real a, b with Ia!, lbl ::; A. Let x;n) (xo ) denote the zeros of Pn (x; dJ.L) labelled so -

--+

Then

(23.2)

329

THE CHRISTOFFEL-DARBOUX KERNEL

(1 )

(23.3) nen(x&n) - xo) S 1 n (ii) For any J, for large n, there are zeros x) ) for allj E {-J, - J+l, . . . , J- l , J} . (iii) . (n (n for· each j (23.4) 1m (x1. +)l - x1. ) ) ncn = 1 nl-+oo REMARKS. 1 . The meaning of x)n) has changed slightly from Section 6. 2. Only nen enters, so the "n" could be suppressed; we include it because lim sup

one expects Cn as defined to be hounded above and below. Indeed, in all known cases, Cn ---t p(xo), the derivative of the density of states. But see Remark 1 after Theorem 22.1 for cases where en might not have a limit. 3. See [58] for the OPUC case.

iJn) (xo) be the zeros of Pn(X)Pn-l(xo) - Pn (xo)Pn-l(x) labelled n 1 since as in (23.2) (with x� ) (xo) = :1:o). By (23.1), have x�� (xo)ncn sin ( 1ra) /a is nonvanishing on ( - 1 , 1) and vanishes at ± 1. The same argument shows Kn(x;;{ , x<;{ + b/ncn) is nonvanishing for lb l < � ' and so there is at most one zero near xr{ on 1/ncn scale. It follows by repeating this argument that (23.5) PROOF. Let

we

---t

for all j . Since we have (see Section 6) that

Xo S X�n ) (xo) S xin) (xo)

by interlacing, which implies (i) and similar interlacing gives (ii). Finally, follows from the same argument that led to (23.5).

(23.4) 0

24. Adding Point Masses We end with a final result involving CD kernels- · a formula of Geronimus [37, formula (3.30)]. While he states it only for OPUC, his proof works for any measure on C with finite moments. Let J.L be such a measure, let z0 E C, and let ll

= J.L

+ ADz0

(24.1)

real and bigger than or equal to - p( {zo} ). Since Xn(z; dv) and Xn(z; dp) are both monic, their difference is a polynomial of degree n - 1 , so for

A

Xn(x; dv)

= Xn( z; dp)

n- 1 + L CjXj ( z; dp) j =O

(24.2)

where Cj

= =

=

J Xj (z ; dp) [Xn (z; dv ) - Xn(z ; dp) ] dp

J Xj (z; dp) Xn(z; dv)[dv - Aaz0] -A xj ( zo ; dp) X.. ( zo ; dv)

(24.3) (24.4) (24.5)

330

B. SIMON

where (24.4) follows from Xj ( - , dJL) l_ Xn ( - , df..l) in L2 (df..l) and (24.5) from Xj ( · , df..l) l_ Xn( , dv) in L2 (dv) . Thus, (24.6) Xn (z; dv) = Xn(z; df..l) - >..X n(zo ; dv)Kn-l (zo , z; df..l) Set z = z0 and solve for Xn (z0 ; dv) to get: THEOREM 24.1 (Geronimus [37] ) . Let f..l, v be related by (24. 1). Then ·

Xn ( z,· dV) _ Xn ( z,· dfJ) -

_

>..X n(zo ; dJ.L )Kn - I (zo , z ; dp, ) 1 + >..Kn - 1 ( zo , zo ; dp,)

( 24. 7)

This formula was rediscovered by Nevai [67] for OPRL, by Cachafeiro- Marcellan [15, 16] , Simon [81] (in a weak form) , and Wong [105, 106] for OPUC. For general measures on C, the formula is from Cachafeiro--Marce1lan [17, 18]. In particular, in the context of OPUC, Wong [106] noted that one can use the CD formula to obtain: THEOREM 24 .2 (Wong [105, 106]). Let df..l be a probability measure on 81Dl and

let dv be given by

dv =

for zo E 8ID and >.. � -tL({zo } ) . Then O:n (dv)

=

G'n (df..l) +

PROOF. Let

df..l + >..8zo I + >..

(24.8)

(1 - lan (dJ.LW)l/2 . l.f'n+I (zo) l.f'n (zo )

>,.- l + Kn ( Zo , zo , df..l)

*

(24.9) (24. 10)

Since (24. 11) and (24.12) (24. 7) becomes an (dv) - o:n(dJL) = Q;; 1 n +I (zo ) Kn (zo, 0 ; df..l )

By the CD formula in the form (3.23), Kn ( zo , O)

= =

since <1>� (0)

=

( 1 - lan l 2)112 .

1 and 11�� 1 1

=

(24.13 )

(24.14) (24. 15)

ll�n ll · (24.9) then follows from ll� n+ I II / I I �n ll

0

To see a typical application: CoROLLARY 24. 3 . Let z0 be an isolated pure point of a measure df..l on 81Dl. Let di/ be given by (24.8) where >. > -tL({z0}) (so z0 is also a pure point of dv). Then for some D, C > 0, IG'n (dD) - G'n(df..l) l ::; De - Cn (24.16) PROOF. By Theorem 10.14.2 of [81], l l.f'n (zo ; df..l) i ::; D 1 e - � C

This plus (24.9) implies (24. 16).

n

(24. 17)

0

331

THE CHRISTOFFEL-DARBOUX KERNEL

This is not only true for OPUC but also for OPRL: 24.4. Let z0 be an isolated pure point of a measure of compact support d�J- on Let dv be given by (24.8) where ..\ > -J.L( {zo }) so z is also a pure D point of dv. Then for some , C > 0, (24 . 18) (i) I "-n"-n (d(dp,v)) - ( 1 + ..\)1/2 1 De- Cn (24 . 19) dv) - (1 + ..\) 1 12 ( dJ.L) [[P(dv) De - Cn (ii) n -C (24.20) (iii) [ an(dv) - an (d�J-) [ De n [ bn(dv) - bn(dJ.L)[ De -C (24. 21) SKETCH . Isolated points in the spectrum of Jacobi matrices obey (24.22) [ zo ) f ::5 D1e-C1 n for suitable Cr , D 1 (see [I, 25] ) . (24.7) can be rewritten for OPRL ) ( dj.L) (x; ) = ( x; ) (xo; dlt) 1 +K/\n-\K1(xo,x;d�J(n o , . . d ) (24.23) COROLLARY

R

::5

[[Pn (

· ,

::5

Pn

::5

· ,

:S:

Pn (

"- n

Pn

dv

Pn

d�J-

- APn

X

Since and

n l

Xo ,

M

J! Kn- l (Xo,xo; d�J-W dOx0 K - ( xo xo d ) 2 and Kn -l(xo ,xo ) is bounded (by (24.22)), we see that ff K ( xo · ; dp,) f [ P(dv) is bounded. Thus, by (24.22) and (24.23) , =

"-n ( dM) "-n (dv) - 1

;

,

M

n- l

;

= ( 1 + ..\)-1 1 2 + O (e -C1 n )

which leads to (24.18) . This in turn leads to (ii) , and that to (iii) via (24.22) , and, for example, an (dJ.L) an (dv)

=

J XPn (x ; dJ.L)Pn -l (x; dM) =J (x; d XP n (x; dv)P n - 1

dv)

dJ.l

(24.24)

v

(24.25) 0

This shows what happens if the weight of au isolated eigenvalue changes. 'What happens if an isolated eigenvalue is totally removed is much more subtle-sometimes it is exponentially small, sometimes not. This is studied by Wong [107] .

References [1] S. Agmon, Lectures on Exponential Decay of Solutions of Second- Order Elliptic Equa tions:

Boundt!

on Eigenfunctions of

N- body

Schrodinger Operators,

Mathematical Notes,

1982. [2] L. V. Ahlfors, Complex Analysis. An Introduction to the Theory of A nalytic Functions of One Complex Variable, McGraw-Hill, New York, 1978. [3] A. Ambroladze, On exceptional sets of asymptotic relations for general orthogonal polyno mials, J. Approx. Theory 82 (1995), 257-273. 29, Princeton Univ. Press, Princeton, NJ; Univ. of Tokyo Press, Tokyo,

B. SIMON

332

[4] A. I. Aptekarev,

[5] [6] [7] [8] [9]

[1 0] [11] [12]

[13] [14] [15]

[16]

[17]

[18]

[19] [20] [21]

Math. USSR Sb. 53 (1986), 233-260; Russian original in Mat. Sb. (N.S.) 125(167) ( 1 984) , 231-258. F. V. Atkinson, Disc1-ete and Continuous Boundary Problems, Academic Press, New York, 1964. A. Avila, Y. Last, and B. Simon, Rulk universality and clock spacing of zeros for ergodic Jacobi matrices with a.c. spectrum, in preparation. J. Avron and B. Simon, Almost periodic Schr·odinger· operators, II. The integrated density of states, Duke Math. J. 50 ( 1983), 369-391. D. Barrios Rolan!a and G. L6pez Lagomasino, Ratio asymptotic.� for polynomials orthogonal on arcs of the unit circle, Constr. Approx. 15 (1999), 1-31. M. Bello Hernandez and G. LOpez Lagomasino, Ratio and relative asymptotics of polyno mials orthogonal on an arc of the unit circle, J. Approx. Theory 92 ( 1 998), 216-244. A. Ben Artzi and T. Shalom, On inversion of block Toeplitz matrices, Integral Equations and Operator Theory 8 ( 1985), 751-779. C. Berg, Fibonar.ci numbers and orthogonal polynomials, to appear in J. Comput. Appl. Math. A. B. Bogatyrev, On the efficient computation of Chebyshev polynomials for several inter vals, Sb. Math. 190 (1999), 1571-1605; Russian original in Mat. Sb. 190 (1999) , no. 11, 15-50. A. Borodin, Biorthogonal ensembles, Nuclear Phys. B 536 (1999), 704-732. J. Breuer, Y. Last, and B. Simon, in preparation. A. Cachafeiro and F. Marcellan, Orthogonal polynomials and jump modifications, in "Or thogonal Polynomials and Their Applications," (Segovia, 1986), pp. 236· 240, Lect.nre Notes in Math., 1329, Springer, Berlin 1988. A. Cachafeiro and F. Marcellan, Asymptotics for the ratio of the leading coefficients of orthogonal polynomials associated with a jump modification, in "Approximation and Opti mization," (Havana, 1987), pp. 1 1 1-1 17, Lecture Notes in Math., 1354, Springer, Berlin, 1988. A. Cachafeiro and F. Marcellan, PP-rl.urbations in Toeplitz matrices, in "Orthogonal Poly nomials and Their Applications," ( Laredo, 1987), pp. 139-146, Lecture Notes in Pure and Applied Math., 117, Marcel Dekker, New York, 1989. A. Cachafeiro and F. Marcellan, Perturbations in Toeplitz matrices: Asymptotic properties, J. Math. Anal. Appl. 156 (1991) 44-5 1 . M. J . Cantero, L . Moral, and L . Velazquez, Measures and para-orthogonal polynomials on the unit circle, East J. Approx. 8 {2002) , 447-464. R. Carmona and J. Lacroix, Spectral Theory of Random Schrodinger Operators. Probability and Its Applications, Birkhauser, Boston, 1990. M. P. Chebyshev, Su•· les valeurs limites des integmles, J. Math. Pures Appl. 19 (1874), 1 57 1 60. T. S. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and Its Applica tions, 13, Gordon and Breach, New York-London-Paris, 1978. E. B. Christoffel, Uber die Gaussische Quadratur und eine Vemllgemeinerung derselben, J. Reine Angew. Math. 55 (1858), 5 1-82. A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195-206. J. M. Combes and L. Thomas , Asymptotic behaviour of eigenfunctions .for multiparticle Schrodinger operators, Comm. Math. Phys. 34 (1973), 251 --270. D. Damanik, R. Killip, and B. Simon, Perturbations of orthogonal polynomials with periodic recursion coefficients, preprint. D. Damanik, A. Pushnitski, and B. Simon, The analytic theory of matrix orthogonal poly nomials, Surveys in Approximation Theory 4 (2008), 1-85. G. Darboux, Memoire sur / 'approximation des fonctions de t1·es-grands nombres, et sur une classe etendue de developpements en serie, Liouville J. (3) 4 ( 1878), 5-56; 377-416. S. Denisov and B. Simon, Zeros of orthogonal polynomial.� on the real line, J. Approx. Theory 121 (2003), 357-364. J. L. Doob, Measure Theory, Graduate Texts in Math. , 143, Springer-Verlag, New York, 1994. -

[22] [23]

[24] [25] [26] [27] [28] [29]

(30]

Asymptotic properties of polynomials orthogonal on a system of contours,

and periodic motions of Toda chains,

THE CHRISTOFF�L-DARBOUX KERNEL

[31] [32] [33] [3·1] [35] [36] [37]

[38] [39] [40]

[41] [42) [13]

[44] [45)

[46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56]

[57]

[58] [59]

333

D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1989. P. Erdos and P. Tunl.n, On interpolation. III. Interpolatory theory of po lynomials, Ann. of Math. (2) 41 ( 1940), Slo-553. E. Findley, Universality for locally Szego measures, to appear in J. Approx. Theory. G. l''reud, Orthogonal Polynomials, Pcrg�mon Press, Oxford-New York, 1971. P. A. Fuhrmann, On a partial realization p1"0blem and the recursive inversion of Hankel and Toeplitz matrices, Contemp. Math. 47 (1985) , 149-161. J . S. Geronimo and W. Van Assche, Orthogonal polynomials on several intervals via a polynomial mapping, Trans. Amer. Math. Soc. 308 (1988 ) , 559-581. Ya. L. Geronimus, Orthogonal Polynomials: Estimates , Asymptotic Formulas, and Series of Polynomials Orthogonal on the Unit Circle and on an Interval, Consultants Bureau, New York, 1 96 1 . D . J . Gilbert and D. B. Pearson, O n subordinacy and analysis of the spectrum of one dimensional Schrodinger operators, J. Math. Anal. Appl. 128 (1987), 30-56. L C. Gohberg and N. Ya. Krupnik, A for"Tnula for the inversion of finite Toeplitz matrices, Mat. Issled 7{2) { 1972) , 272-283. [Russian] I. C. Gohberg and A. A. Semencul , On the inversion of finite Toeplitz matrices and their continuous analogs, Mat. Issled 7(2) (1972) 201-223. [Russian] L. Golinskii, Quadrature formula and zeros of para-orthogonal polynomials on the unit circle, Acta Math. Hungar. 96 (2002) , 169-186. L. L. Helms , Introduction t o Potential Theory, Pure and Applied Math. , 22, Wiley Tnterscience, New York, 1969. H. Hochstadt, On the theory of Hil l 's matric.es and related inve rs e spec tra l problems, Linear Algebra and Appl. 1 1 (1975), 41-52. S. Jitomirska.ya and Y. Last, Dimensional Hausdo1ff pmperties of s ·i ngular continuous spec tra, Phys. Rev. Lett. 76 ( 1996), 1765-1769. S. Jitomirska.ya and Y. Last, Po wer-law subordinacy and singular spectra, I. Half-line op erators, Acta Math . 183 (1999), 171-189. S. Jitomirska.ya and Y. Last, Power law subordinacy and singular spectra, II. L ine operators, Comm. Math. Phys. 2 1 1 (2000), 643-658. K. Johansson, Random growth and random matrices, European Congress of Mathematics, Vol. I ( Barcelona, 2000) , pp. 445-456, Progr. Math., 201, Birkhauser , Basel, 2001. T. Kailath, A. Vieira, and M . .Morf, Inverses of Tocplitz, innovation$ and orthogonal poly nomials, SIAM Review 20 (1978), 106-1 19. S. Khan and D . B. Pearson, Subordinacy and spectral theory for infinite matrices, Helv. Phys. Acta 65 (1992), 505-527. R Killip, A. Kiselev, and Y. Last, Dynamical ·u.ppeT bounds on wavepacket spreading, Am. J. Math. 125 (2003), 1165-1198. A. B. Kuijlaars and M. Vanlessen, Universality for eigenvalue correlations from the modified Jacobi unitary ensemble, lnt. Math. Res. Not. 30 (2002), 1575-1600. N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin-New York, 1972. Y. Last, On the measure of gaps and spectra for discrete 1 D Schrodinger operators, Comm. Math. Phys. 149 (1992), 347-360. Y. Last, Spectral theory of Sturm-Liouville operator.9 on infinite intervals: a review of recent developments, in "Sturm-Liouville Theory, " pp. 99-120, Birkhauser, Basel, 2005. Y. Last and B. S im on , Eigenfunctions, transfer matrices, and absolutely continuous spec tr-um of one-dimensional Schrodinger ope rators, Invent. Math. 135 (1999), 329-367. Y. Last and B. Simon, The essential spectrum of Schrodinger, Jacobi, and CMV operators, J. Anal. Math. 98 (2006), 183-220. Y. Last and B. Simon, Fine structure of the zeros of orthogonal polynomials, IV. A priori bounds and clock behavior, Comm. Pure Appl. Math. 61 (2008) 486-538. E. Levin and D. S. Lubinsky, Universality limits involving orthogonal polynomials on the unit circle, Computational Methods and Function Theory 7 (2007), 543-56 1 . E . Levin and D . S. Lubinsky, Applications of universality limits t o zeros and reproducing kernels of orthogonal polynom ials, J. Approx. Theory 150 (2008) , 69-95.

334

B.

[60] D.

SIMON

to Ann. of Math. [61] D. S. Lubinsky, Unive1·sality limits in the bulk for arbitrary measures on compact sets, to appear in J. Anal. Math. [62] D. S. Lubinsky, A new approach to universality limits at the edge of the spectrum, to appear in Contemp. Math. [63] A. A. Markov, Dbnonstration de certaines inegalites de M. Tchebychef, Math. Ann. 24 S. Lubinksy, A new approach to universality limits involving orthogonal polynomials,

appear in

and P. Nevai,

Bernstein's inequality in LP for

(1884), 172-180. [64] A. Mate

orthogonal polynomials, A nn . of Math .

[65]

A. Mate ,

0

1 and

(C, 1)

bounds for

V. Totik, Extension.9 of Szego's theory of orthogonal polynomials. (1987), 73-96. /\.. Mate, P. Nevai, and V. Totik, Szego's extremum problem on the unit circle, Ann. of Math. 134 ( 1 991), 433-453. P. N evai, Orthogonal polynomials, Mem . Amer. Math. Soc. 18 ( 1 979) , no. 2 1 3 , 185 pp. P. Nevai, Geza Freud, orthogonal polynomials and Christoffel functions. A case study, J. Approx. Theory 48 (1986), 167 pp. P. Nevai, V. Totik, and J. Zhang, Orthogo nal polynomials: their growth relative to their sums, J. Approx. T heory 67 (1991), 2 1 5-234 . F. Peherstorfer, Orthogonal and extremal polynomials on several intervals, in "Proc. Seventh Spanish Symposium on Orthogonal Polynomials and Applicatio ns ( VII SPOA ) " ( G ranada, 1 99 1 ) , J. Cornput. Appl. Math. 48 (1993), 187-205. F. Peherstorfer, Deformation of minimal polynomials and approximation of several intervals by an inveTse polynomial mapping, J. A pprox . Theory I l l (2001), 180-195. F. Peherstorfer, InveTse im.age,q of polynomial mappings and polynomials ur·thogonal on them, in "Proc. Sixth International Symposium on Orthogonal Polynomials, Special Func tions and their Applicati ons" (Rome , 200 1 ) , J. Comput. A ppl . Math. 153 (2003), 371-385. T. Ransford, Potential Theory in the Complex Plane, Press Syndicate of the Universi ty of P.

Nevai , and

III, Constr. Approx . 3

[66] [67] [68] [69]

[70]

[71] [72]

[73]

Cambridge, New York,

[74]

1995.

Self-Adjointness, Academ i c Press, New York ,

M.

Reed

and B.

Simon,

Methods of Modem Mathematical Physics, II. Fourier Analysis,

[75] C. Remling , The absolutely continuous

[76 ] v\'. Rudin, [7 7] B. Simon,

matrices, preprint.

1987.

The classical moment pmblem as a self-adjoint finite differen ce operator, Adv.

(1998), 82-203.

Simon, Ratio asymptotics and weak asymptotic measu1·es for orthogonal polynomials on

the real line,

[79] B. [80] B.

1975.

Real and Complex Analysis, 3rd edition, McGraw-Hill, New York,

in Math. 137

[78] B.

spectrum of Jacobi

J.

A pprox. Theory 126

(2004), 198-2 17.

Simon, Orthogonal Polynomials on the Unit Circle, PaTt 1:

Simon, OPUC on one foot, Bull. Amer. Math. Soc. 42

Classical Theory, AMS

Orthogonal Polynomials on the Unit CiTcle, Part 2: Spectral Theory, AMS Col

Colloqui um

[8 1] B. Simon,

(2005), 431-460.

Series, 54.1,

A merican .Mathematical Society, Providence, RI, 2005.

2005. [82] B. S imon , Pine structure of the zeros of orthogonal polynomials, I. A tale of two pictures, Electronic Transac tions on N umer ic al Analysis 25 (2006), 328-268. [83] B. Simon, Fine structure of the zeros of orthogonal polynomials, II. OPUC with competing exponential decay, J. Approx. Theory 135 (2005), 125-139. [84] B. Si mon , Fine structure of the zeros of oTthogonal polynomials, III. Periodic recursion coefficients, Comm. Pure Appl. Math. 59 (2006) 1042-1062. [85] B. Simon, Equilibrium measures and capacities in spectral theory, Inverse Problems and Imaging 1 (2007) , 71 3-772. [86] B. Simon, Rank one pelturbations and the zeros of paraorthogonaJ polynomials on the unit circle, J. Math. Anal. Appl. 329 (2007), .'376-382. [87] B. Simon, Two exten.9ions of Lubinsky's universality theorem, to appear in J. Anal . M at h. [88] B. Simon, Weak convergence of CD kernels and applications, to appear in Duke Math. J. [89) B. Simon, Szego 's Theorem and Its Descendants: Spectral Theory for L2 Perturbations of Orthogonal Polynomials, in preparation; to be publ is hed by Princeton University Press. loquium Series, 54.2, American Mathematical Society, Prov idence , Rl,

3 35

THE CHRISTOFFEL-DARBOCX KERNEL

[90] M.

SodiJ1

and P. Yuditskii,

Almost periodic Jacobi matrices with homogeneous spectrum,

Anal. 7 (1997), 387-435.

infinite- dimensional Jacobi inversion, tions,

J.

Geom.

and Hardy spaces of character-automorphic func

[91] H. Stahl and V. Totik, General Orthogonal Polynomials, in "Encyclopedia of Mathematics and its Applications," 43, Cambridge University Press, Cambridge, 1992. [92] 1'. StiP.ltjP.I'I, Q1AP.lqu.P..9 rechP.rch es Mt.r la thf.n rie des qnad.rn.tnres d.itr.s mf.r.aniqu.r.s, A nn. Sci. Ecole Norm. S up . (3) 1 (1884), 409-426. [93] G. Szego, Orthogonal Polynomials, Amer. Math . Soc. Colloq. Pub!., 23, American Mathe matical Society, Providence, RI, 1939; 3rd edition, 1967. [94] R. S zwarc , A counterexample to subexponential growth of orthogonal polynomials, Constr. Approx . 11 ( 1995), 381-389. [95] M. Toda, Theory of Nonlinear Lattices, 2nd edition, Springer Series in Solid-State Sciences, 20, Springer, Berlin, 1989. [96] V. Totik , A8ymptotics for Christoffel functions for· general measures on the real line, J. Anal. Math. 8 1 (2000), 283-303. [97] V. Totik, Polynomial inverse images and polynomial inequalities, Act a Math. 187 (2001), 139-160. [98] V. Totik, Christoffel functions on curves and domains, in preparation. [99] V. Toti k , Universality and fine zero spacing on general .� ets, in preparation. [1 00] W. F. Trench, An algorithm for the inversion of finite Toeplitz matrices, J. Soc. Industr. Appl. Math. 1 2 (1964), 515-522. [101] W. F. Trench, An algorithm for the inversion of finite Hankel m.a.tri ces , .J. Soc. Industr. Appl. Math 13 ( 1965), 1 102-1107. [102] M. Tsuji, Potential Theory in Modern Function Theory, reprinting of the 1959 original, Chelsea , New York, 1975. [103] J. L. Cllman, On the regular behaviour of orthogonal polynomials, Pro c . London Math. Soc. (3) 24 ( 1972), 1 1 9-148. [1 04] P. van Moerbeke, The spectrttm of Jacobi matrices, Tnvent . M ath. 37 (1976), 45 81. [105] M .-W . L. Wong , First and second kind paraorthogonal polynomials and their zeros, J. Ap prox. Theory 146 (2007), 282-293. [106] M.-W. L. Wong, A formula fo r inserting point masses, to appear in Proc. OPSFA (Marseille, 2007). [107] M .-W . L. Wong , in preparation. [108] J. Zhang, Relative growth of linear iterations and orthogonal po lyno mia ls on several inter vals, Linear Algebra Appl. 186 (1993), 97-115 . MATHEMATICS .

253-37,

CALIFORNIA INSTITUTE OF TECHNOLOGY,

U . S.A. FJ-mail

address:

bsimon
URL: http : //wwr.� . math . cal t e c h . edu/people/s imon . html

PASADENA,

CA 91125,

Proceedings of Symposia in Pure Mathematics Volume 79, �008

A Saint-Venant Principle for Lipschitz Cylinders Michael E. Taylor ABSTRACT. We show how a general program exposed by P. Lax appl ie s to t he study of the asymptotic be havior at infinity of a biharmonic fun<.:tion u on a half-infinite cylinder whose cross section 0 is a Lipschit?. domain, given a mild bound on u and vanishing Dirichlet data on the lateral boundary. It is shown that such a solution u exists, given D irichlet data on the base of the cylinder, (!,g) E HJ (O) $ L2 (0), and that u has an asymptot ic expansion in a series of progressively more rapidly exponentially dtJcreasing terms. To carry out this analysis, we make use of results of B. Dahlberg, C. Kenig, G. Verchota, J. Pipher, and V. Adolfsson.

Dedicated to V. G. Maz'ya on the Occasion of his 70th Birthday

1. Introduction L€t 0 c ]Rn-l be a bounded, strongly Lipschitz domain, and set n JR+ X 0 c IR" . Then n is a Lipschitz cylinder, with boundary an = ({ 0} X 0) u (I� + X ao). We want to study the behavior of solutions to the following Dirichlet problem for the hi-Laplacian �2 , where � = a; + �X = a; + L�-l a; . With y E R +, X E 0, j we look for u (y, x ) , solving =

�2u = O on n ,

(1 . 1)

u(O, x) = f (x) , a11 u (O , x) = g (x) , x E 0 , u ( y, x) = O, 8,/ u(y , x) = O, x E 80 , y 2 0 ,

where Ov is the unit normal to 80.

We

take

(1 .2)

f E HJ (O), g E £2(0) . This D irichlet problem for �2 has a unique solution u(y, x) in a class of functions we will specify later on, having some decay as y -) oo. We will show that u(y, x ) has an asymptotic expansion as a series of progressively more rapidly exponentially decreasing terms; see (4. 19)-( 4.20) for a precise statement. P. Lax [3] (cf. also [4] , Chapter 26, and [5]) provided an abstract context in which to prove the existence of such an asymptotic expansion, for rather general families of y-independent elliptic PDE on infinite cylinders. An interesting aspect of the analysis in [3] is that no existence theorem was needed. On the other hand, 2000 Mathema.tics Subject

Classification.

This work was supported by

35J40, 35840.

NSF grant DMS-045686 1 .

@2008

337

American

Ma.thcma.tical Society

338

MICHAEL E. TAYLOR

problems like the one mentioned above, one is just as interested in the existence of non-exploding solutions as one is in their asymptotic behavior. We will build on several works on Lipschitz domains done in the 1980s and 1990s to obtain such a solution to (1.1)-( 1.2). Then we derive the asymptotic behavior, using a variant of the methods of [3] . Actually, the existence result provides more structure, and this allows for a simpler endgame argument involving functional analy�;is and semigroups. The plan of the rest of this paper is as follows. In §2 we replace !1 by f!n = [0, R] x 0, and study the solution to �_2u = 0 on f2R, u(O, x) = f(.r,), 8y7�(0, x) = g( x), x E 0, (1.3) u (y, x) = 0, 8v u(y , x) = 0, x E 80 , y E [O, R] ,

for specific

u(R, x) = O , 8yu(R, x) = O, x E O, which is a Dirichlet problem for L�.2 u = 0 on the Lipschitz domain [!R · Results of [2] yield a unique solution to (1.3), with non-tangential maximal function bounds on n and Vu in L2 (Df!R)· \Ve establish further properties of the solution u, making also use of results of [6] , [7], and [1] . In §3 we use the results of §2 to find a solution to (1.1), also with non-tangential maximal function estimates, and satisfying

�o= j iu(y, x) 1 2 dx dy < oo,

(1.4)

0

and other regularity properties and bounds, which we delineate there. In §4 we study the asymptotie behavior of u(y, x) as y --> oo. Following the Lax program, we study a semigroup of operators. In this case, we study the semigroup SY on HJ (O) EB L 2 (0), defined by

(g) (8yu(u(y)y)) ,

( 1 . 5)

SY !

=

where u is the solution to (1.1) constructed in §3. In §5 we look at another semigroup, T5 : Y -> Y, where Y {u E H2(!1) : �2u 0, u and fhvu 0 on �+ (1.6) =

=

=

T8u( y, x) = u( y + s, x).

x

80},

This set-up is more closely parallel to that of [3] than the consideration of SY . We note parallel results for T8 , also making use of the results of §§2-3.

Remark. It is interesting to compare (1.1) with the Dirichlet problem for D.u = 0 on D, (1.7)

D.:

u(O, x) = f(x) , x E 0, u( y , x) = O, x E 80, y 2: 0.

The method of separation of variables represents the non-exploding solution to (1. 7) as (1 .8)

00

u(y , x) = 'I:, ](k )e - >.kYcpk (x) , k=l

A S AINT- VENANT PRINCIPLE FOR LIPSCHITZ CYLINDERS

33 9

where { IPk

: k � 1} is an orthonormal basis of L� ( 0) consisting of Dirichlet eigen functions of � x , ipk E HJ (O) , �x iPk = ->.}pk, 0 < At < .\2 S:: >.3 / oo, and ](k) = (!, 'Pk)£2 (0) · In such a case, >.k Ckl/(n-l ) , L; k >.�i](k ) 1 2 < oo, and (1.8) i� both asymptotic and convergent, in HJ (O) . Such a separation of variable� approach would work for solutionH to �2 u 0 if 0 were replaced by a compact manifold without boundary, or if the lateral boundary conditions in ( 1 . 1) were replaced by "'

=

u (y, x) = 0 ,

{1 .9)

but for ( 1 . 1 ) this method fails.

�xu(y , x ) = 0,

x E 80,

y

2: 0,

2. The Dirichlet problem on OR

as

Here we fix R E (0, oc ) and discuss solutions to (1.3), which can be rephrased

(2. 1 )

where aN is the unit normal to anR . Here !R = f on {0} X 0, !R = 0 on the rest of anR, while 9R = g on {0} X 0, 9R = 0 on the rest of anR . The hypothesis (1.2) yields

(2.2 ) Work of [2] yields a unique solution to (2. 1), smooth in the interior of OR, and satisfying

(2 .3 )

l l u * II L2(8flR) + II ( V'u) * II L2(&nn) ::::: C I!/R I I H1 (&flR ) + GII 9R I I L2(8nn) · Here, given a function v, continuous on the interior of On, we denote by v* the nontangential maximal function of v, v• (x) (2.4)

r (x )

= =

sup lv(z) l ,

zEI' (x)

x E 80n,

{z E flR : dist (z, x) ::::: K dist(z, 80 R ) },

for some fixed (large) positive K. The following additional information will be useful. PROPOSITION

2 . 1 . The solution to (2. 1) satisfying {2.2)-{2.3) also satisfies

(2.5) PROOF.

u

E H312 (0R)·

It is shown in Theorem 2.4 of [7] that, for such u ,

(2 .6) where (2.7)

A(V' u )(x)

(

One has

(2.8)

J dist (x, z)2-niV'V'u(z) l2 dz

=

(x)

J Xr(x) (z) dist(x, z)2-n dS(x)

ann

)1�

�

dist(z, 80R ) ,

340

MICHAEL E. TAYLOR

which, by Fubini'::; theorem, implie::;

(2.9)

j dist (z, &nR) j\7\i'u(zW dz :::; C / IA(V'u) (x) l2 dS(x).

OOR

OOR

From here, Proposition S of [1] yields (2.5).

D

We investigate higher regularity of n away from the top and bottom pieces of the boundary. We parametrize 0.R by z = (y, x ) , y E [0, R] , x E 0. Pick tp = r.p(y) E 00 ((0, R)). (2.10) PROPOSITION

2.2.

For u as in Proposition 2. 1, tp as in (2. 10), tpu E H5/ 2 -" (0.R),

(2.11) PROOF.

(2. 12)

'II t: > 0.

First note that �2 (r.pu) = (a� + � x )2 (r.pu) = (a� + 2a;�x + �;) (tpu)

where

Q = [a: , Mcp ] + 2�x [a; , Mcp] = Q 3 (y , ay) + �xQl (y , av ) , where Qj (y, ay) have compact y-support in y E (0, R) and order j. We hence have �2 (tpu) = Qu E H-312 (rtR) , tpu l anR = aN ( tpu) l anR = 0 (2.1 4)

(2. 13)

,

the degree of regularity of Q u following from (2 . 5) . Now Theorem 2.1 of [1J applies to (2.14), to give (2.11), with (2. 15) II'Puii Hs/2-•(r!R ) ::; Cs IIQull H-3/2 (f!R) . D

' In fact, Theorem 2.1 of [1] gives 5 2

3 2

-- < s < --.

(2. 16) We next establish the following result. PROPOSITION

2.3.

r.pa;;'u E H5 / 2-< (OR) ,

(2 . 1 7) PROOF.

have (2. 18)

Given u as in Proposition 2.1, tp as in {2. 10), m E z+ ,

Define ay,h by ay,hv(y, x )

and, since �2 (ay,hu)

=

=

'\/ t: > 0.

h- 1 [v(y +

h, x) - v(y, x)] . For small h

0,

(2 . 19) 6.2 (r.p&y,h u) = Q(&y,hu), bounded in H-3/2 -e(nR) · Since tp(y)8y,hu has vanishing Dirichlet data on &nR, (2.16) yields (2.20) li 'POy,huiiH5/2-c(f!R) :::; Ce ,

A SAINT-VENANT PRJNCIPLE FOR LIPSCHIT:t; CYLINDERS

34 1

with CE: independent of h . Taking h � 0, we have cpay ,h u � cpay u , and the bounds in (2.20) imply this convergence hold!:l weak* in H512 -= (DR) and This gives (2.17) for m

m.

llcpay uii H�/2 -•( nR ) :::;

c"'.

= 1 , and iterating this argument

gives

the

result for larger

D

The following corollary will be useful in §:�. COROLLARY 2.4. 000 ( [0, R])

such that

(2.21)

Then

tf;(y) = 1

Take u as in Proposition 2. 1. Also choose t/J 2R R for 0 :::; y :::; 3 ' 'lj;(y ) 0 for 3 y R. :::;

=

tf;(y) E

<:._

( 2.22) PROOF. A calculation parallel to (2.12)-(2.13) gives (2.23)

!::.2 (1/Ju) = Q3 (y , ay )u + !::.x Ql (y , Oy )u ,

where Qj (y, ay) have compact y support in (0, R) and order j . Then (2.22) follows from (2.17). D

3. The Dirichlet problem on D To solve (1.1), with f, g satisfying (1.2), we do the following. Pick R E (O, oo) and solve (1 .3) on DR. Call this solution v. Pick t/J = tf;(y) E 000 ( [0, R)) such that (2.21) holds, so, by (2.22), t::.2 (tf;v) = F E H11 2 -E:(DR),

(3. 1 )

with F vanishing for y � 2R/3. Extend F to D as 0 for y � R. We claim there is a unique w satisfying (3.2 ) Then the solution to ( 1 . 1 ) is given by

�1 = 'ljJV - W.

(3.3) To get (3.2), consider (3.4) so

L : HJ (O)

(3.5)

(Lw , w ) = ll t::. w ll�,2 (fl) •

Now (3.6)

w E H6 (D)

�

____.

H5 (D) * ,

Lw

= t::.2w,

V w E H5 (D) .

( -Llw, w) = (\?w, \?w) �

100 ll\7xw lli2 (0) dy

� All w ii �2(H) •

where A > 0 is the smallest eigenvalue of -t::.x on £2 (0) , with the Dirichlet bound ary condition. Since ( -t::.w , w) � l l t::.w ii P II wii P , we deduce that (3.7)

ll t::.w tli2 (0) � A2 ll wllh(n) '

V w E H6 (D).

342

MICHAEL E. TAYLO R

Now the Friedrichs method produces a positive, self-adjoint operator £, with V ( £ 112) = HJ (D.) , (3.8) and C is the self-adjoint extem;ion of �2 defined by the Dirichlet boundary condi tion. From (3.5) and (3. 7) we have C - A 2 I � 0. (3.9) In particular, C is invertible on L2 (f2), so we have (3.2) , with w = £-1 F E D(C) c D(£11 2 ) = H�(D.). (3.10) Once we have u in (3.3), the local regularity results of Propositions 2.2-2.3 readily extend. We get (3. 11) cp8�nu E H5/2 - " (f2) , V c. > 0, m E z+ , whenever cp = cp(y) E C0 ( (0, oo)). 4. The semigroup SY and asymptotic behavior

We define the semigroup SY : X ---7 X, (4.1) by (4.2)

SY

X = HJ (O) EB £2 (0),

(gf) = (8yu(yu(y)) ) ,

where u = u(y, x) is the solution to (1.1) constructed in §3. Given that a;;u E H5/2-e([a, b] x 0), Vc. > 0, m E z+ , (4.3) whenever 0 < a < b < oo, we see that SY is strongly continuous in y E (0, oo). Whether SY is strongly continuous at y = 0 requires further investigation , but since our focus is on the behavior as y __, oo, we leave this point. From (4.3) we obtain SY : X ---7 H512 - < ( 0) EB H512 -< ( 0), ( 4.4) for all c. > 0, whenever y > 0. In particular, y > 0 ==} SY is compact on X. ( 4.5) In particular, the spectrum of 81 has only 0 as an accumulation point . 4. 1 . Jf ( E Spec 81, then 1 (1 < To begin, since 81 is compact,

PROPOSITTON PROOF.

0 # ( E spec 81

� �

(4.6)

�

3

nonzero

sk (�)

=

G)

(k

1.

E X such that 81

(;)

(;) = ( (;)

u(y + k, x) = ( k u(y , x),

where n i s the solution to (1.1) constructed in §3. In such a ca.'>e, (4.7)

1= j ju(y, xW dx dy = l( l2k 100 j i u(y , xW dx dy. 0

0

A SAINT-VENANT PRINCIPLE FOR LIPSCHITZ CYLINDERS

But by (3.2)-(3.3) we have u E L2 (JR+ k -+ oo. This forces 1(1 < 1.

x

343

0), so the left side of (4.7) tends to 0 as 0

To proceed, fix c = e -K E (0, 1), and let Pe be the projection of X onto the of Sp ec S1 lyin.u; inside De; = {( E c : I C I < c}, i.e.,

part

(4.8)

Prc: =

2�i /((I - si )- 1 d(, ''I<

where "fe = oDe or, if a point in Spec S 1 lies on oDe , take 'Ye = oDf: - ry for appropriate 7] < < c. Then the range of Q10 = I - P10 is a finite dimensional subspace XE of X , on which SY acts as a semigroup. One has

( 4.9) where A.= is a linear operator on the finite dimensional space X.= , with

(4.10)

Spec Ae =

{.Xl, . . . , .XN(el } ,

where B > 0 is taken so that

Re .X.i � B > 0 ,

( 4.11)

One can decrease c and increase the list {>.1 , . . . , ,\v(.=) } ; note that

( 4 .12)

Re Aj -+ +oo,

as j -+ oo.

Now for each Aj t here arc associated eigenvectors of A " , and also possibly general

ized eigenvectors, so we have (possibly with repetitions of some of the eigenvalues

Aj)

(4.13) for

()

SY Qe 1 g

=

N(e:) M (j)

L L aje ye e- >.j y

j=l

P.= l

cert ain functions 'Pjt(x) and '1/Jje(x). It remains to analyze, as y -+ oo,

(�'liJJJRe ((�))) ,

(4. 14)

a sernigToup on the range of Fe . Note that

(4.15)

Spec z;

c

{( E C : I C I < c}.

The spectral radius formula implies

(4.16) hence

( 4.17)

�

lim sup IIZ�II �) � n -HXl

< e - n (K- 1 ) ' l l zen ll L:(X) -

for n � v (c ) , sufficiently large.

Now if n+ 1 � y � n + 2 , using the fact that {SY : y operator norm on X ,

( 4.18)

c,

E

[1 , 2] } has uniformly bounded

) ) ) I I ZJ' I L c(X ) ::; I I Z� IIc (x ) II ZY- n iL c (X ) ::; ce- n ( K- l ::; ce - (Y -2 (K - l

We have the following conclusion (after some minor relabeling) .

344

MICHAEL E. TAYLOR

THEOREM 4.2. Given c = e - K E (0, 1), the solution u( y , x ) to in § 3 has the behavior N(e) M(j) (4.19) u (y, x ) = aj e yle->.jylpjt(x) + llle (y, x ),

(1. 1) constructed

LL

where

(4.12) holds and

(4.2 0)

j=l e=I

II W"e (Y, · ) llnJ (O) = O (e -KY),

We have

y ---+ oo.

(4.21) Remark. Making one more use of (4.4), we can improve (4.20) to (4.22) l l llle ( Y, · ) II Ho/2 -> (o) = O ( e - KY ) , y ---+ oo ,

for each o > 0. Other bounds, involving £P-Sobolev spaces and Besov spaces, can be deduced from regularity results of [1] and [8] . Of course, if 80 has additional regularity, one has further estimates, in stronger norms. 5. Another semigroup We define T8 for s � 0, acting on functions on 0 by (5.1) T8u (y , x) = u(y + s, x). We have T8 acting on various function spaces, such as £ 2 (0) . Here we investigate the action on Y = {'u E H2 (0) : !::!.. 2 u = 0, u and ON U = 0 on �+ x 80}. (5.2 ) We denote the restriction of rs to Y by ys . The study of T8 : Y ---+ Y is more closely parallel to the general set-up of [3] than the study of SY : X ---+ X . One advantage of T8 is that it is obviously a contraction semigroup on Y, strongly continuous in s E [0, oo) . On the other hand, an advantage of SY is that it incorporates an existence result for the Dirichlet problem. The next result, parallel to (4.5), makes use of result�; of §§2-3 as much as (4.5) does. PROPOSITION 5.1. For s > 0, T" : Y ---+ Y is compact. PROOF. Consider

(5.3)

pu( x) =

( u(O, x) ) 8y u(O, x)

,

p : Y ---+ H312 (0)

n

HJ (O) EB H112 (0) .

Since X = HJ (O) $ L2(0 ) , we see that (5.4)

p

:

Y

---+

X is compact.

Now, given u E Y, s > 0, we have T8U = T8Y:.pu, (5.5) where Y:. is the solution operator to (1.1) constructed in §3. Results of §3 imply T'Y:. : X ____. Y, V s > 0, (5.6)

A SAINT-VENANT PRINCIPLE FOR LIPSCHITZ CYLINDERS

345

so (5.5) represents 78 : Y -+ Y as a composition of a compact operator and a 0 continuous operator, for each s > 0. The next result paiallels Proposition 4.1. LEMMA 5.2. If ( E

Spec T1 , then 1(1 < 1 .

T1 is compact on Y, 0 -1 ( E Spec 71 ::::} 3 nonzero u E Y such that 'T1 u = (u ::::} u(y + k, x) (ku(y, x).

PROOF. Since

( 5 . 7)

Again (4.7) holds, and implies 1(1

<

1

=

0

The path from here to the asymptotic expansion (4.19) is quite parallel to that taken in §4.

References [1] V. Adolfsson and J . Pipher, The inhomogeneous Dirichlet problem for l:l2 in Lipschitz domains , J. Funct. Anal. 199 ( 1998), 137-190. [2] B. Dahlberg, C. Kenig and G. Verchota, The Dirichlet problem for the biharmonic equation in a Lipschitz domain, Ann. lnst. Fourier (Grenoble) 36 (1986) , 109-135. [3] P. Lax, A Phragmim-Lindelof theorem in harmonic analysis and its applica!.ions to some questions in the theory of elli ptic equations, Comm. Pure Appl. Math . 10 (1957), 361-389. [4] P. Lax, Functional Analysis , J. Wiley, New York, 2002. [5] P. Lax, Abstract Phragmen-Lindelof theorem and Saint-Venant's principle , Abel Lecture, University of Oslo, May 2005. [6] J. Pipher and G. Verchota, The Dirichlet problem in LP for the biharmonic equation on Lipschitz domains, Amer. J. Math. 114 (1992), 923-972. [7) J. Pip her and G. Verchota, Area integral estimates for the biharmonic operator in Lipschitz domains, Trans. AMS 327 (1.991), 903-917. (8] z. Sben, The LP D irichlet problem for elliptic systems on Lipschitz domains, Math. Research Letters 13 (2006), 143-159.

:MATHEMATICS DEPARTMENT, UNIVERSITY OF NORTH CAROLINA , CHAP8L HILL, NORTH CAR OLINA

27599

E-mail address:

metCmath . unc . edu

PToceedings of Symposia in Purl::! Mathematics

Volume 79, 2008

Wavelets in function spaces Hans Triebel Dedicated to Professor Vladimir Maz'ya on the occasion of his 70th birthday. ABSTRACT. The paper deals with wavelet bases in function spaces clidean n-space, on the n-torus, and on diverse types of domains.

on

Eu

1. Introduction This survey deals with the symbiotic relationship between (compactly sup ported) wavelets on JR"" on the one hand and the recent theory of function spaces on Rn , on (smooth and rough) domains and on manifolds in IR.n , governed by build ing blocks, on the other hand. We concentrate on detailed descriptions and outline preferably only those ideas which illuminate how closely some basic constructions of wavelet theory are interwoven with function spaces. The latter will be done in diRtinguishcd remarks called Discussions. We give references if the corresponding assertions are available in the literature. But most of the results are presented here for the first time. Then we must refer for (technical) details to forthcoming publications, especially to 135]. Section 2 deals with the spaces A;q where A = B or A = F on IR.n and (briefiy) on the n-torus ']['n and their wavelet expansions under natural restrictions for the parameters involved. In Section 3 we describe wavelet systems and intrinsic wavelet bases for Lp-spaces on arbitrary domains. For a natural class of domains, culled E-thick domains (covering in particular Lipschitz domains), we get in Section 4 common intrinsic wavelet bases for related scales of A�q-spaces. This will be complemented in Section 5 by more specific assertions for corresponding A�q-spaces on coo manifoldR and on ceo domains. In Section 6 we give some additional references and add a few comments.

2. Spaces on IR.n and ']['n

2.1. Definitions. We use standard notation. Let N be the collection of all natural numbers and No = N U {0}. Let IR.n be Euclidean n-space where n E N. Put JR. = IR.1 , wherea.s C is the complex plane. Let S(IR.n ) be the usual Schwartz 1 991 Mathematic.s Subject Classification. 46E35, 42C40. Key words and phrases. function spaces, wavelet bases.

347

@2008 American Mathematical Society

HANS TRIEBEL

348

space and S' (IW') he the space of all tempered distributions on :!Rn . Furthermore, Lp(JRn) with 0 < p ::::; oo is the standard quasi-Banach space with respect to the Lebesgue measure in :!Rn , quasi-normed by II/ IL (R.n ) ll = p

(in

1p l f(x) I P dx) 1

with the usual modification if p = oo. As usual, Z is the collection of all integers and zn where n E N, denotes the lattice of all points m = (m 1 , . . . , mn) E lRn with mj E Z. Let N0 where n E N, be the set of all multi-indices, with

Ctj

E No

n

and lcxl = L Ctj j=J

If X = (x , . . . , Xn ) E R.n and fJ = ({31 > . . . , f3n ) E N0 then we put 1 (monomials) .

If cp E S(JR") then (2.1)

iP ( O

=

(F cp) ( O

=

(211' ) - n/2

[ e- ix� cp (x) dx , }[f.n

� E :IR",

denotes the Fourier transform of cp. As usual, p- 1 r.p and cpv stand for the inverse Fourier transform, given by the right-hand side of (2.1) with i in place of -i. Here x� denotes the scalar product in IR". Both F and p - 1 are extended to S' (!Rn) in the standard way. Let cp0 E S(!Rn) with (2.2 )

cp0 (x) = 1 if lx l ::::; 1

and cp0 (y) = 0 if IYI 2: 3/ 2 ,

and let (2.3)

X

E :!Rn ,

k E N.

Then ��0 cpj (x) = 1 in !Rn is a dyadic resolution of unity. The entire analytic functions (cpj j) v (x) make sense pointwise for any f E S'(:!Rn ) .

DEFINITION 2 . 1 . Let cp = {cpj }�0 be the above dyadic resolution of unity. (i) Let 0 < p ::::; oo, 0 < q ::::; oo, s E R Then B �q (!Rn ) is the collection of all f E S' (!Rn) such that

(2.4)

II! I B;_, (R" l ll.

�

(� =

2; " 1 1 ( �Jl v I L,(R") I I '

)

' ''

< 00

(with the usual modification if q oo) . (ii) Let 0 < p < oo, 0 < q ::; oo, s E R Then F;q (!Rn ) is the collection of all f E S' (JRn) such that (2.5) (with the usual modification if q = oo) .

< oo

WAVELETS IN FUNCTION SPACES

349

REMARK 2.2. The theory of these spaces may be found in [27, 29, 32]. In particular these spaces are independent of admitted resolutions of unity

with A E: {B, F} if the assertion considered applies equally to s;q (!Rn) and F;q (!Rn) {always with A;q (!Rn )

( 2. 6)

p

< oo for the F-spaces) . We remind of a few special eHScs and properties referring for details to the above books, especially to [32, Section 1 . 2 ].

p < oo and k N0 . Then w; (IR") = F;, (1Rn ) (2.7) are the classical Sobolev spaces, usually equivalently normed by {i) Let 1 <

E

2

IIJ IW;(JRn ) ll =

(2.8) and based on the

spaces Lp(lRn).

(

L li D "' J ILp (IRn) II P

i al:<=;k

)

l/p

Paley-Littlewood assertion Lp(IRn) = F�,2 (1Rn) for the Lebesgue

(ii) For all admitted

s , p, q

and

o

E IR,

I.,. A�q (IRn) = A��.,.(IRn) with Iuf

=

((1 +

l� l2t12f)

v

is an isomorphic map (equivalent quasi-norms). Nowadays one calls

H;(IRn ) = L .• Lp (IRn ) = F;,2 (1Rn ) ,

s E IR ,

Sobolev spaces with the classical Sobolev spaces

1
oo ,

1<

k

p

<

oo,

E N0,

as special cases. (iii) Let

(�1f)(x) = f(x + h) - f(x), (��+1 f) (x ) = �k (�� !) (x ), where x E !Rn, h E IRn, l E N , be the iterated differences in !Rn. Then the

Zygmund spaces

C8 (1Rn )

=

Holder

s > 0,

B�= (IR"') ,

can be equivalently normed by

llf IC·' (!Rn)llm = sup lf(x)l + sup lhl-s �� f(x) l , xEJRn

0 < s < m E N,

where the second supremum is taken over all x E IRn and h E IRn with 0 < l h l S: 1. (iv) The spaces B;q (IRn) according to Definition 2 . 1 are called Let 0 < p, q S:

(2.9) where by

a+

=

max(a , 0) if

(2. 1 0) II/ I B:, (R " ) IIm

�

a

oo ,

n

E R Then

(� 1) -

Besov spaces.

+

<s<m

E

N,

s;q (IRn) can be equivalently quasi-normed

II/ I L,(IR" ) II +

uhj
IW" Ill>;:' f I L, (R") II'

��. )

l /o

350

HANS TRIEBEL

and

(lo

II/ I B;q {Rn) ll:r. = II/ I Lp (Rn ) ll +

(2. 1 1 )

(with the usual modification if q

B�q(Rn) are the

=

dt { 1 rsq sup ll�h' f I Lp( R" ) II t

oo). If 1

classical Besov spaces.

lhl5t

 0, then

Otherwise we refer to [29, Chapter 1] and [32, Chapter 1] where one finds the history of these spaces, further special cases and classical assertions.

2.2. Wavelets on Rn . \Ve suppose that the reader is familiar with wavelets on R" of Daubechies type and the related multiresolution analysis. The standard references are [11, 21, 23, 37]. A short summary of some relevant aspects may also be found in [32, Section 1 .7] . We give a brief de!:>cription of some basic notation. As usual cu(IR) with u E N collects all {complex-valued) continuous functions on R having continuous bounded derivatives up to order u (inclusively) . Let (2.12) ·u E N, ¢F E Cu (R) , '1/JM E Cu(R) , be real compactly supported Daubechies wavelets with

1 '1/JM (x) xv dx

(2.13)

=

for all v E No with v < u.

0

Recall that '1/JF is called the scaling function (father wavelet) and '1/JM the associated wavelet (mother wavelet). \Ve extend these wavelets from R to Rn by the usual tensor procedure. Let u E N and (G1 , . . . , Gn) E G0 { F, M } n G which means that Gr is either F or lvl. Let =

=

j E N,

which means that G,. is either F or M where * indicates that at least one of the components of G must be an M . Hence G0 has 2n elements, whereas GJ with j E N has 2n l elements. Let L E N0 and (2. 14)

-

where j E N0.

n

\Vc

PROPOSITION

(2. 15)

r=l

always assume that '1/JF and '1/JM in (2.12) have L2 -norm 1. 2.3.

Let L E N u and u E N. Then

{ w�.m :

is an orthonormal basis in L

2

jE

(Rn )

.

No,

G E GJ , m E

zn }

REMARK 2.4. This is a cornerstone of (inhomogeneous) wavelet theory. We refer to the above-mentioned books. In JRn one may choose L 0. But later on when it comes to domains then it will be import�tut that one can choose L E No large. The L 2 -normali!:iation of wb,., is natural. But for our later purposes it is convenient to switch to an L00-normalisation. To prepare what follows we remark that any f E L2 (Rn ) can be represented as =

(2. 1 6)

00

f = L L L >.t;,G T j n/ 2 wb,m j =O GEGi mEZ"

35 1

WAVELETS IN FUNCTION SPACES

with (2.1 7)

and (2. 18)

(Recall that the wb ,m 's are real) . 2.3. Wavelet bases in A�q(!Rn). \Ve extend the wavelet representation ac cording to Proposition 2.3 and (2. 16)-(2.18) from L2 (1Rn) to A�q(!Rn) where A = B or A = F and E JR, 0 < p, q ::::; oo (with p < oo for the F-spaces). First we need a substitute of the sequence spaces £2 in (2.18). Let Xi m be the characteristic function of a cube Qjm in !Rn with sides parallel to the axes of coordinates, centred at 2-j m and with side-length 2-j+l where m E zn and j E N0 .

s

DEFINITION

all sequences ( 2.1 9) such that of

2.5. Let s E JR, 0 <

p

:S

oo, 0 <

q

:S

oo.

Then b�q is the collection

and f;q is the collection of all sequences (2.19) such that li). l f;q ll =

( _L

21sq j).�a

J ,G,m

with the usual modifications if

p

REMARK 2.G. One ha.<> b;P right-hand side of (2. 18) .

= oo =

Xim CW)

and/or q

f;v . With

11 q

I Lv(lRn ) < oo

= oo. s

= 0,

p

=

q

=

2, one gets the

We use standard notation naturally extended from Banach spaces to quasi Banach spaces. In particular {bj }� 1 c B in a <.:omplex quasi-Banach space is called a if any b E B can be uniquely represented as

basis

b = L Ai bi , j= 1

(2.20) A basis

00

unconditional basis if for any rearrangement

{bj }_f=: 1

is called an (one-to-one map of N onto itself)

(2.21)

( convergence in B).

Aj E C

00

{bcr(j)}f=1

b = L Acr(j) bcr(j) j =l

i::; again a basis and

0'

of N

(convergence in B)

for any b E B with (2.20). Standard bases of separable sequence spaces as con sidered in this paper are always unconditional. We refer to [lj for details about bases in Banach (sequence) spaces. Similarly as in (2.20), (2.21) we speak about

HANS TRIEBEL

352

means convergence in A;q (K ) for any ball K in �n. Let a�q be either b�q

unconditional convergence in S'(JRn) and in A�q(�n ) . Local convergence in A�q(�n)

In any case ·u E N in (independently of L)

(2.12) and hence in (2.14) will be chosen in such a way that or

J;q .

(2.22)

converges unconditionally in S'(lR.n ) and locally in any A;q(�") with rr < s. This justifies to abbreviate (2.22) by

(2.23) We use the nowadays standard abbreviations and

(2.24) where 0 < p, q �

oo

and b +

=

max(b, 0)

if

rr

(

1

pq = n min(p, q)

b E JR.

2 . 7. (i) Let 0 < � 0 < q � oo, s JR, wwuelets in (2.14) with L E N0, based on (2.12), (2.13), with THEOREM

p

oo,

E

-1

)

+

and let

W�,m

be the

u > max(s , CJp - s).

Let f E S' (�"). Then f E B�q (JR.") if, and only if, it can be represented as

(2.25)

j ,G ,m

unconditional convergence being in S' (JRn) and locally in any space B;q (lR.n ) with s. The representation (2.25) is unique,

a<

)..�G = 2j nf2

(2.26)

(!, wb ,m)

and (2.27)

n

is an isomorphic map of B;q(JR. ) onto b;q . If, in addition, p < oo, q < oo, then { wb,m} is an unconditional basis in B�q (R.n ) . (ii) Let 0 max (s , rYpq - s). Let f E S' (JR.n). Then f E F:q (lR.n) if, and only if, it can be represented as (2 . 28)

j,G,m.

unconditional convergence being in S'(JRn) and locally in any space Fh (lRn ) with (2.28) is unique with (2.26). Furthermore, I in (2.27) is an isomorphic map of F;q (JR"') onto f;q . If, in addition, q < oo , then { w b, m } is an unconditional basis in F;q (lR11') .

a < s . The representation

WAVELETS IN FUNCTION SPACES

353

Discussion 2.8. As said in the Introduction there is a symbiotic relationship between some aspects of wavelets and the recent theory of function spaces based on building blocks. In particular the above wavelets wb,m may serve simultaneously as atoms and as kernels of local means. Atomic representations in function spaces us used nowadays go essentially back to [14, 15] . But more details about the somewhat involved history of atoms may be found in (29, Section 1.9 ] . By the sharp version of atomic representations according to [32, Section 1 . 5 . 1 ] it follows that (2. 25) is an atomic expansion based on the normalised atoms G E GJ .

(2.29)

As far as the required cancellations for the atoms are concerned we remind of

(2.30)

{

}ri{n

xf� w{;,m(x) dx = O

if j E N and 1 .8 1 <

u,

as a consequence of (2. 13) and (2.14) . Then it follows from the atomic representa tion t.heorem that f E S'(!Rn ), given by (2.25), belongs to B;IJ(!Rn) and

(2.31) On the other hand, (2.26) may be considered as local means (appropriately inter preted)

(2.32) Again cancellations for the kernels kfm of type {2.30) arc indispensable. Equiv alent quasi-norms in function spaces A;q (Rn) in terms of local means have some history going back to [28] . This has been presented in a more elaborated version in [29] . But a sharp assertion which can be used not only for function spaces of the above type on !Rn but also also on smooth and rough domains has been obtained only recently in [34] (and [35, Section 1.1.3]) with (31] and [32, Section 3.1] as forerunners. Based on these observations one gets by (2.32) for f E B;q(IRn) that

I IA l b;q ll

::=; C ll.f IB;q (JR n) jj .

By (2.31) both quasi-norms arc equivalent. Similarly for the spaces F;q (!Rn). Of course these sketchy arguments are far from being a rigorous proof covering all technicalities. This may be found in [32, Section 3.1], [34] and will be the subject of [35] . We mainly wanted to illuminate the double role played by the wavelets ..:Pb,m as atoms and as kernels of local means.

This is the crucial observation, the rest are technicalities. REMARK 2.9. Proposition 2.3 is the starting point of the wavelet theory in function spaces. It is natural to ask whether (2. 15) remains to be an (unconditional) basis in other spaces on IRn . First candidates are L p(IRn) with 1 < p < oo but also related Sobolev spaces and classical Besov spaces. Something may be found in the above-mentioned books [11, 21, 23, 37] . One may also consult [32, Remarks 1. 63 , 1 . 65, 1 . 66, pp. 32-35] for more details and further references. An extension of this theory to all spaces A�q(IRn) goes back to [20, 31, 34] .

HANS TRIEBEL

354

2.4. Wavelets on 'II'n . By rule of thumb wavelets and function spaces on !Rn have natural periodic counterparts on the n-torus 'II'n , (2.33) (opposite points are identified in the usual way). We give a brief description for sev eral reasons. There is a huge literature about periodic wavelets but only very little as far as wavelet bases in periodic function spaces is concerned, mostly restricted to L2 ( i!') , where 'II' = 'li' 1 is the 1-torus. We refer in this context to [1 1 , pp. 304/305] , [2 1, Section 7.5.1], [37, Section 2.5]. So it seems to be desirable to give precise definitions and assertions for the full scale of the periodic counterparts A;q ('JI'n) of the above spaces A;q (IR.n). But there is also some use later on in Section 5 in connection with wavelet bases for some spaces A;q (O) in (bounded C00) domains n in Rn , where the boundary is diffeomorphic to an (n - 1)-torus (with n = 2 as the case of preference) . Let D '('JI'n) be the space of (periodic) distributions on 'JI'n . We assume that the reader is familiar with basic assertions about these distributions. Recall that f E D' ('JI'n) can be represented as f = L am ei21!' mX) (2.34) m.Ezn

where the Fourier coefficients am E C are of at rnm;t polynomial growth, lam l ::; c(1 + l m l)"' for some c > 0, x > 0, and all m E zn . The theory of periodic distributions and related periodic spaces B�q ('II'n ) and F;q ('JI'n) has some history which is not the subject of this survey. We rely on [24, Chapter 3] and [27, Chapter 9] where one finds also further references. Let { 'Pj }� 0 be the same dyadic resolution of unity in Rn as in (2.2), (2.3) and in Defi nition 2.1. Let f E D'('JI'n), given by (2.34) , be extended periodically to JRn . Then f E S' (lRn) (using the same letter f) and m ( j f)v (x) = '2: am 'Pj (2rrm) ei2n x

'P

m.Ezn

trigonometrical polynomials. This justifies the following periodic counterpart of Definition 2.1. arc

2.10. Let cp = { cp1}�0 be the above resolution of unity in JRn (i) Let 0 < p ::; oo, 0 < q ::; oo, s E JR. Then B�q ('JI'n) is the collection of all f E D'('JI'n), given by (2.34), such that DEFINITION

(with the usual modification if q = oo ) . (ii) Let 0 < p < oo, 0 < q ::; oo, s E JR. Then F;q ('JI'n) is the collection of all f E D'('JI'n), given by (2.34), such that 'l l/ q am cpj ( 2rrm) e i21rmx 2jsq l l f I F;q ('li'n ) II'P = I Lp ('II'n ) < 00

(f: 1 2:: J =O

mEZ"

l)

WAVELETS IN FUNCTION SPACES

( with the usual modification if q

355

oo ) .

=

REMARK 2.1 1 . One has a periodic version of Remark 2.2, including the special cases mentioned there.

There are natural periodic counterparts of t he related sequence spaces and wavelet expansions. But the rigorous justification requires some care which may be found in [34] and to [35, Section 1]. We restrict ourselves to a description. Let wb ,m he the same wavelets as in (2.14) where we choose (and fix) L E N0 such that supp IJ!�, o

Let

C

{x E lRn : l x l < 1/2} ,

IP'j = { m E zn

G E G0 = {F, M}n.

0 $ m r < 2i+L } , be the 2 (i + L)n lattice points in 2i +LTn. Let (2.35)

w{J�:<x ) =

:

j E No,

2:: w�,m < x - z) = z:.= wb,m+2J+q (x),

with j E N0 and m E IP'j be the periodic extension of the distinguished wavelets wb m with off-points 2 -j - L m E Tn, restricted afterwards to Tn . Then one has the foll;wing counterpart of Proposition 2.3. PROPOSITION

2.12. Let u E N in (2. 12), (2.13) and (2.14), (2.35) . Then

{'I!{]�::

is an oTthonormal basis in REMARK

2.13. Let

:

j

E No ,

£2 (Tn).

G E GJ , m E

IP'j }

(J, g)'lr = r f(x) g (x) dx

}Tn

be

the dual pairing in ( D(Tn) , D' (Tn ) ) , appropriately interpreted. Similar as in (2.22), (2.23) (and with the same justification as there) we abbreviate

( 2.36)

,per = ""' ).._J ,G 2 -jn/2 \I!j,per ""' ""' )..j,G 2 -jn /2 \I!j�m ""' G� � � � � m j=O GEG.i mEl'j j ,G.m 00

m

in what follows. First we remark the obvious counterpart of (2. 16)-{2. 18): Any f E L2 (T") can be represented by f = ""' )...j ,G 2 -jn/2 \I!j ,per � m G,m

j,G,m

with and

(Recall that w{J�::; is real). The periodic counterpart of the sequence spaces in Definition 2.5 can be de scribed as follows.

356

HANS TRIEBEL DEFINITION

2.14. Let s E !R, 0 < p ::; oo, 0 < q ::; oo. Then b;;rr is the

collection of all sequences

(2.37)

,\

such that

=

{,\{_;.0 E C : j E N0,

G E Gi, m E lP'j }

and f;;r is the collection of all sequences (2.37) such that r

< oo with the usual modifications if p = oo and/or q = oo, where Xim is the characteristic function of a cube with the left comer 2-J-Lm and of side-length 2-j -L (a subcube of 'fn). After these preparations one gets now the following counterpart of Theorem 2. 7. Recall the abbreviation (2.:35) . Furthermore, O'p and O'pq have the Harne meaning aH in (2.24) . Let u E N be as in (2.12), (2.13) and (2.14), (2.35). THEOREM

2.15. Let { wb;:;} be the orthonormal basis in L2 ('fn) according to

Proposition 2. 12. (i) Let 0 

rnax(s, O'p - s) .

Let f E D' ('fn). Then f E B�q ('Jl'n) if, and only if, it can be represented as i,G j /2 j,per f= � (2 . 38) � ).. m 2- n w G,.,n. '

j,G,m

unconditional convergence being in D' ('IT'n) and in any space B;q ('IT'n ) with < s. The representation (2 38 ) is unique, 0'

.

(2 . 39 ) and

(2 40) .

is an isomorphic map of B;q('fn) onto b;;J'er . IJ, in addition, p < oo, q < oo, then { wb;;�} is an unconditional basis in B;q ('fn). (ii ) Let 0 max ( s, O'pq - s). Let f E D' ('fn). Then f E F;q ('JI'n) if, and only if, it can be represented as � ,\j,G ,per ' f � (2.41) m 2-i n/2 iJ!jG,m =

j,G,m

WAVELETS

IN

357

FUNCTION SPACES

unconditional convergence being in D'('JI'n) and in any space F;q ('JI'n) with u < s. The representation (2.41) is unique with (2.39) , and I in (2.40) is an isomorphic is an uncondi map of F;q (1I'11) onto f;:rr . If, in addition, q < oo, then tional basis in F;q ('JI'n ) .

{ llr·b�:·}

Discussion 2.16. This is the direct and to some extent expected periodic counterpart of Theorem 2. 7 . Basically one extends functions and distributions f E D'(1'") periodically to IRn. But these extended distributions do not belong to any space A�q(IRn) with p < oo (with exception of f = 0) . One has the same unpleasant effect if one periodises the 1Rn-wavelets as in (2.35) with x E !Rn in place of x E 1'" . Thi�:> obstacle can be circumvented if one deals first with suitable weighted spaces on !Rn. Let 'lL'a ( x

and

) = (1 + j xj 2)"12 ,

X E !Rn ,

A�q (IRn , 'Illa ) = {f E S' (IRn)

:

a E IR,

waf E A�q (IR" )} ,

naturally quasi-normed. There is a complete counterpart of Theorem 2.7 with the . same wavelets iJ!3a , ,m and the same restrictions for u and suitably modified sequence spaces. This goes back to [16] and may be found in [32, Section 6 . 2] . As for a refined version needed in the above context we refer to [34]. If 0 < p � oo and a < -njp then

A;;ier (IRn, wa ) = {f E A�q (IRn , wa ) : /( ·) = f( · - m) , m E zn } is the closed subspace of A� q (IRn, Wa) consisting of the indicated periodic distri butions on !Rn. It is isomorphic to A;q (1'" ) . First one proves a representation of

these distributions on !Rn in terms of the wavelets iJ!&P�: according to (2.35) in IR'n . Reduction to 'JI'n gives the above theorem. We refer t� [34] and [35]. 3. Spaces on arbitrary domains

3.1. Definitions. The remaining sections of thi�:> survey deal with wavelet bm:;es for function spaces on domains. First we fix some notation. Let n be an arbitrary domain in !Rn. Domain meam open set vvithout any further restrictions. Then Lp(!l) with 0 < p s; oo is the standard quasi-Banach space of all complex valued Lebesgue rnem:;urable functions in n such that

I (ln lf(x)IP dx)

II! I L ( D ) = p

l/ p

(with the natural modification if p oo) is finite. As mmal , D(D) = C0 (n) stands for the collection of all complex-valued infinitely differentiable functions in JR!n with compact support in n . Let D' (!1) be the dual space of all distributions in D . Let g E S' (!Rn ) . Then we denote by g j !l its restriction to n, =

g (cp) for

(g j D)(cp)

=

in Definition 2.1.

as

introduced

DEFINITION 3.1. Let !1 be an arbitrary domain in JR. " with D =/= IR" and let 0
oo ,

0 < q s; oo,

s E IR,

HANS TRIEBEL

with p < oo for the F-spaces. ( i) Then A;q (O)

=

{f E D'(O) : f

=

g i O for some g E A�q (IR11 ) } ,

IIJ IA �q (O) II = inf II Y I A�q (lRn ) l i , wheTe the infimum is taken over all g E A�q(IR11) with gjO

(ii) Let Then

{

A�q (f!) = f E D' ( O)

:

=

f = g j O for- some g E

II! I A;'l ( n ) l l = in£ 119 I A;q ( !Rn) ll

where the infimum is taken over all g E A�q(fi) with gjO

=

f

.

.4;'1(11) } , f.

3.2. Part (i) is the usual definition of A�q (O ) by restriction. The spaces .4;q(�1) are closed subspaces of A�q (JRn). They can be identified with .4;q (O) if {h E A�(IR11) : supp h c 80 } = {0} what in general is not the case (especially not if JDO I > 0) . If n is a bounded Lipschitz domain according to Definition 4.1 below, then one has in a few cases in trinsic quasi-norms and characterisations. This applies in particular to the classical Sobolev spaces k E N0, 1 < p < oo, with REMARK

1 11 w;,, ( nl ll

�

111 1 w; ( nl 1 1

�

( �, ,

li D" 1 IL,(!!J II' 'I'

)

in analogy to (2.7), (2.8), but also to the Besov spaces B;q (O) with (2.9) and an 0-version of (2. 11). Details, references and also intrinsic characterisations of some spaces F;q ( n) may be found in [32, Theorems 1 . 1 18, 1.122, pp. 74/77J . 3.2. Wavelet systems. It is the main aim of this survey to describe wavelet bases for function spaces on domains. For this purpose one asks for wavelet systems preserving as much as possible of the distinguished wavelet bases in lR11 and 1rn as used in the Propositions 2.3, 2.12 and in the function spaces A�(IR11) and A;q (1I'11 ) . Let n be an arbitrary domain in !Rn with n =f. IR11• Balls in IR11 centred at X E rn; n and of radius (2 > 0 are denoted by B( x , Q ) . If r 1 and r 2 are two sets in IR11 then (3. 1) Let for some positive numbers c1 , c2 , c3, (3.2) Zo {xt E n : j E No; r = 1 , . . . , Nj }

where Nj (3.3)

N

E=

=

N U {oo} such that j E No,

r =f. r',

359

WAVELETS I'l" FUNCTION SPACES

and

(3.4)

j E No,

where r = aD stands for the (non-empty) boundary. It is always assumed that the positive nnmberH r-1 , r-2 , c3 are sufficiently small in dependence on n such that for any j E No there are points with Based on we extend now to

(3.2)-(3.4).

(2.14) (3.5)

\]ijG,m ( :z: ) •

=

2 (j +L) n/2

n IT oi•c "t'

r=l

r

(2j+Lx

(2.12), (2.13)

r

- mr

)

'

for all j E N0, G E {F, M}n. Although not discussed in detail it is always assumed that L E No in (:3.5) is fixed such that for given (small)

E. > 0,

(3.6)

Let F

j E No , =

m E zn .

{F, . . . , F } E {F, M}n .

3.3. Let 0 be an arbitrary domain in JRn with 0 =/: JRn and let Zo be as in (3.2)-(3.4). Let L E N and u E N be as in (2.12), (2. 1 3) and (3.5), (3.6). Let K E N, D > 0 and > 0. Then (3.7) where Nj E N, (3.8) j E No , r = l., . . , Nj, with B (xt, c22-j) as in (3.1), is called a u-wavelet system (with respect to D) if it consists of the following three types offunctions. (i) Basic wavelets (3.9) for some G E {F, .l\J}n , m E zn . (ii) Interior wavelets n..j - '*' G,m' j E N, dist (x�, r) :2 rj ' (3. 10) for some G E {F, M}n• , m E (iii) Boundary wavelets (3.11) �j "' j E N, dist (x�, f) < 2 -j, L....J d:tn ,m' ,m , DEFINITION

r:4

.

�r -

r

for some m

=

=

lm-rn' I :SK

rn(j,

L:

,T,J

C4

71/' .

. \l!L F

,

C4

r) and r1!.,. ,m' E lR with j i d!r,, m ' l :S D and supp w�, m-' B (x�, c2 T ) .

lm-m'I:SK

c

3.4. By construction all wavelets �{ are reaL Only E N is of interest. All other fixed numbers L, K, D and the constants c1, . . . , c4 are technical ingredients. By what follows they depend on u, but not on D. This may justify the REMARK

above notation.

u

360

HANS TRIEBEL

3.3. Wavelet bases in Lp(O). First we ask for counterparts of Propositions 2.3 and 2.12 in arbitrary domains 0 based on u-wavelet systems according to Def inition 3.3. D EFINITION 3.5. Let 0 be an arbitrary domain in ]R n wdh 0 -:f. lR" and let u E N . Then

with NJ E N is called an orthonormal u-wavelet basis in L ( O) if it is both a u-wavdet system according to Definition 3. 3 and an orthonormal basis in £ (0 ) . REMARK 3.6. In other words, for given u E N one asks for constants z

2

L, K, D, c1 , . . . , c4, d�,m' such that one finds a related u-wavelet system as intro duced in Definition 3.3 which is also an orthonormal basis in £2 (0) . PROPOSITION 3. 7. Let 0 be an arbitrary domain in IR.n with 0 -:f. ]Rn. For any u E N there are orthonormal u-wavelet bases in £ 2 ( n ) according to Definition 3. 5. REMARK 3.8. This proposition plays the same role for function spaces on do mains as the Propositions 2.3, 2.12 for spaces on lR" and 1'". The close connec tion between wavelets and function spaces resulting in the above assertions will be outlinend below in Discussion 3.12. There is a more or leHs obvious counter part of (2. 16)- (2.18). If { 4>t } is an orthonormal u-wavelet basis in L2 (0) then any f E L 2 (0) can be represented as oo

J = L: L: >.t Tjnf2 4>t

(3. 12) with

NJ

j=O r=l

).,� = >.� ( f) = 2jnf2 (.f, 4>t) = 2jnj 2

and II/ 1£, (!1) I

�

(t, �

k f(x) 4>�(x) dx

2 -'nl!,i l'

-

)

,

, ,

By (3.9)-(3.11) based on (3.5) the functions 2 jn/2 4>� are L00-normalised what is convenient for our

later considerations.

The extension of Proposition 3. 7 and of the representation (3. 12) to other func tion �>paces on domains requires appropriate counterparts of the sequence spaces b�q and f;q in Definition 2.5. Let Xir be the characteristic functions of the balls B(x�, c2 2 -j) in (3.8) . DEFINITION 3.9. Let n be an arbitrary domain in ]Rn with n "1- IPI.n and let Z n be as in (3.2)-(3.4) . Let s E IR, 0 < p :S oo, 0 < q :S oo. Then b�q (ll.n ) is the

collection of all sequences

(3.13)

such that

361

WAVELETS IN FUNCTION SPACES

and f;q (Zn) is the collect·ion of all sequences (3.13) such that ; 2; " I At x,. ( l l ' I L. ( !l ) < oo IIA IJ;, (zo) I

)

(t, �

�

' '

with the usual modification if p = oo and/or q = oo. REMARK 3.10 . The structure of the sequence spaces f;q (Zn} is somewhat com plicated. A relevant discussion may be found in [32, Section 1.5.3] . One has =

b�P (Zn)

As usual

n such that

s

f;p (Zn),

E

IR, 0 < p �

oo.

Li0c(n) collects all complex-valued Lebesgue-measurable functions in

L lf(x) l dx <

for any bounded domain w with

00

w c

n.

THEOREM 3.11. Let 0 be an arbitrary domain in !Rn with 0 -=/= !Rn . Let 1 < p < oo and u E N. Let (3.14) with Ni E N, { if.lt j E No ; r = 1 , . . . , Ni } be an orthonormal u-wavelet basis in £2(0 ) according to Proposition 3. 7 and Defi nition 3. 5. Then Lp(O) is the collection of all f E L�oc(n) which can be represented as :

oo

Ni

f = L L )..� Tj n/2 if.Jt,

(3. 15)

j=U r=l

Furthermore, { if.Jt } is an unconditional basis in Lp ( 0 ) . If f E Lp (n) then the representation (3.15) is unique with ).. = >.(!), >.t (f) = 2jn/2 ( !, ibt.) = 2j nf2

(3.16) and

J

:

f � >.(f)

=

1n f(x) � (x) dx

{ 2jn/2 (!, ibt.)}

is an isomorphic map of Lp(O) onto fg,2 (Zn) (equivalent norms) . Discussion 3.12. Again there is a striking interplay between wavelets and building blocks in function spaces. We give an idea bow to prove Proposition 3. 7 and the above theorem. First one decomposes n in Whitney cubes Qln centred at some points 2-1m E 0 (m E zn, l E No) and of side-length 2-1 , 00

(3.17)

(modification for l = 0). Then

(3. 18) where

(3.19 )

fz,. Xtr

l,1"

is the characteristic function of Qlr and

= Xtr f,

HANS

::Jti2

TRIEBEL

9lr one has a canonical situation which admits a uniform 2. 7. Pulling back one gets flr in a (small) neighbomhood of Qlr · Clipping together

For the dilated functions

application of the �n-expansion according to Theorem wavelet expansions for

ii!-{-;.,m in

these expa,nsions (with some tm:hnical care) and using that the wavelets

(2. 14) are well-ada,pted to dyadic dilations and translations one gets expansions for f which are first steps towards ( 3 . 1 5 ) . But there are some obstacles. In contrast to

wb m in(2.25) with j E N , Lhe dilated starting ii!� , m do not fulfil the moment conditions needed for the

the dilated terms originating from i nterpretation as atoms in

terms originating from

Lp(JRn).

But this applies only to the starting terms

It

and this difficulty can be removed by direct Lp-arguments.

is more serious that

other dilated wavelets) . This is the point where the multiresolution structure of the

just these dilated starting terms spoil the desired orthogonality (in contrast to the wavelets is of great service. Let the origin in the one-dimensional case

'lj;c where

breaking point with the dilation With

GE

at the left and the dilation

(21 +Lx - m)

=

n

= 1 be a

at the right.

L cfm ,t 1/JF (2l+l+L x - t)

consisting of finitely many elements with supp

1/Ja

2-l-l

{F, M} as in ( 2 . 1 2 ) one has ncar the origin the multiresolution

1/Ja

property

2-1

¢F (21 + 1 +t

tEZ

-t )

·

c supp

'lj;p

,Pn

(21+L · -m ) .

VVith some local orthogenalisation at the origin one can remove the disturbing terms

(21+£ -m) ·

a,t the expense of

(21+1+L -t) .

If n :2':

·

In case of

n

=

1

all

breaking points are isolated endpoints of intervals and the above orthogonalisation can be done at each such point separately. structure of to

n

2 then one can rely on the product

wb, m in (2 . 14) which transfers the orthogonalisation from one direction

directions (applying Fubini's theorem) . Some care is necessary, especially in

corner points and along edges. But all this can be done and results finally in the boundary wavelets

if?�

according to (3. 1 1 ) .

In this way one

proves first Proposition

3 . 7 and afterwards (with the help of the indicated atomic arguments) Theorem 3.11.

The first step of this procedure (without the final orthogonalisation) was

done in [33] , where we denoted the outcome as a Proposition 3.7 and Theorem proofs will be given in

3.1 1

para-basis.

The above versions of

arc published here for the first time. Detailed

[35].

Discussion 3 . 1 3 . The above arguments for the spaces Lp (O) in arbitrary do n rely mainly on the localisation (3. 18) , the homogeneity ( 3 . 1 9 ) and the 1Rn-wavelet theory. There is little hope that other spaces A�q (O) and A�q (n) ac mains

cording to Definition 3 . 1 in arbitrary domains fit in this scheme.

But there is a

remarkable modification which even lays the foundation of all what follows. sketch the basic ideas. Let

( 3.2 0 ) Let

CTpq

0
be as

in (2.24)

oo,

0
We

and let

oo,

> Clpq·

Qlr be the (roughly) indicated Whitney cubes in (3.17) and let 0 -::; (!lr E D(O) s

be a related resolution of unity, say, (3 .21)

supp

(Jlr c 2Qlr c n,

L f?lr (x ) = 1 lr

if

X

E n.

363

WAVELETS IN FU:'-ICTION SPACES

Then the refined localisation space F;qrloc(n) consists of all f E Lioc(n) such that

(3.22) This quasi-Banach space is independent of admitted resolutions of unity 12 = { 121r} (equivalent qua..'i i-norms) . Furthermore for p, q, s in (3.20) one has the homogeneity

0 < .A ::; 1,

(3.23) for all

f E F;q (�Rn )

with supp f C {y E !Rn : IYI < .A } . Then (3.22) (by definition) and (3.23) are the counterparts of (3.18) , (3. 19). One gets by the same outlined arguments as for Lp(O) that { apq in (3.14) is a wavelet basis in F;{loc ( n ) with the cxpansionl:l (3.15), (3. 16) now with .A E f;q (Zn) in place of f� 2 (Zn). This theory started in [33 ] (where we denoted , F;qrloc(n) as F�q (O)). There we got para-bases. As for final results in terms of the above orthogonal basis with the related sequence spaces J;q(Zn) according to Definition 3.9 we refer to [35]. 4. Spaces on E-thick domains

4.1. Classes of domains. So far we described wavelet bases for function spaces A;q on lEt" and 'JI'n . ln case of arbitrary domains n we have a satisfactory wavelet representation for Lp (O), 1 < p < oo, according to Theorem 3.1 1 and we indicated in Discussion 3.13 an extension of these assertions to refined localisation spaces F;{loc (n) . But in general these spaces do not coincide with corresponding spaces F;q ( n ) or .F;q(n) according to Definition 3.1 . All spaces A;q(n) , Ji;q (n) introduced there originate from the related spaces A;q (JR.n) which are governed by atoms and kernels of local means. But these elementary building blocks require not only some minimal smoothness but also some cancellations (moment conditions). This is well reflected by the wavelets q,J0.,rn in IR.n (and also in 'JI'n ) playing a double role as atoms and kernels of local means as outlined in Discussion 2.8. In case of domains the basic wavelets
:

i E I} means

for all i E I.

As usual, for 2 :::; (4. 1 )

n

E N,

!Rn-l 3 x'

f--->

h(x ' ) E lR

that

there are

364

HANS

TRIEBEL

is called a L-ipschitz function (on Rn- 1 ) if there is a number c > 0 such that (4.2) ih(x') - h(y')l :$ c lx' - y' l for all x' E Rn- 1 , y' E Rn - l . If h is infinitely differentiable and all derivatives are bounded then h is called a coo function. The distance dist (r1 , r 2 ) between two sets in !Rn has the same meaning as in (3.1). DEFINITION 4.1. (i) Let n E .N. A domain ( open set) in JR n with {1 f. JRn

and r = an is said to be E-thick (exterior thick) if one finds for· any inteTior cube Qi c n with =

(4. 3 )

j 2: Jo E N ,

complementing cube Q e c nc lRn\n with l( Qe) 2 -j ' dist ( Qe, r) dist (Qi Qe) 2- j j 2: )o E N. ( 4.4) (ii) Let 2 :$ n E N. A Lipschitz graph domain (C00 graph domain) in IRn is the collection of all points ( Xn ) with X1 E JRn - l and h(x' ) < Xn < 00 1 wher·e h(x') is a Lipschitz funct·ion according to (4.1), (4.2) (a coo function) . (iii) Let 2 :$ n E N. A bounded Lipschitz domain (bounded c= domain) in Rn is a bounded domain n in IRn where the boundary r can be covered by finitely many open balls Bj in Rn with j 1 , . . . , J, centred at r such that a

=

,..._

,..._

1

�

1

' x ,

Bj

n

=

n

=

Bj n ni

for j

=

1,

. . . , J,

where nj are rotations of suitable Lipschitz graph domains (Coo graph domains) in

JRn .

REMARK 4 . 2 . The equivalence constants in (4 . 3 ) , (4.4) are independent of j. In other words, a domain n is called E-thick if for any choice of positive numbers c1 1 c2 , c3 , c4 and jo E N there are positive numbers c5 , c6 , c7, c8 such that one finds for each interior cube Qi c n with c1 Tj :$ l(Qi) :$ c2 2 - j , c3 Tj :$ dist ( Q'i , r) :$ c4 Tj , j 2: Jo , an exterior cube Q e c nc with ( 4.5) C,5 2- j :$ l ( Qe) :$ C6 2 -j , C7 2-j :$ dist ( Qe , f) :$ dist ( Qi, Qe) :$ cs Tj , j 2: j0 . One checks easily that n is E-thick if, and only if, one has (4.5) for some cs, . . . , cs for the standard Whitney cubes as in, say, [25, Theorem 3, p. 16, Theorem 1, p. 167]. By a Lipschitz domain we mean either a Lipschitz graph domain or a bounded Lipschitz domain. Quite obviously, any Lipschitz domain in Rn is E-thick. On the other hand, E-thick domains may be rather irregular. It may happen that 1r1 > 0. There are E-thick domains with fractal boundaries, for example, the snowflake curve. A discussion may be found in [33, 35] . In addition to E-thick domains we introduced in [35] also !-thick (interior thick) domains, where, roughly speaking, the above cubes Qi and Qe change their roles. Conditions of similar types (E-thick and !-thick) have been used several times in literature in connec tion with function spaces and POE's . First we refer to the corkscrew property of non-tangentially assessible domains according to (18]. Details and generalisations (domains of class S) may be found in [19, pp. 4,8] . In connection with Sobolev

WAVELETS

365

IN FUNCTION SPACES

and Poincare inequalities in function spaces (preferably Sobolev spaces) conditions of the above type play a role resulting in John domains and plump domains. De tails, references, examples and discussions may be found in t12, Section 4.31. In connection with atomic representations of function spaces in rough domains we in troduced in [36] (exterior and interior) regular domains. They are similar (but not identical) with the above (exterior and interior) domains and the other types of domains mentioned above. This may also be found in [13, Section 2.5). In (36, 13] we referred also to other classes of domains in the literature. 4.2. Wavelet bases in .F;q (n). As roughly indicated in Disussion 3.13 for the refined localisation spaces F;qrloc (n) according to (3.20)- (3.22) in arbitrary domains 0 one has a counterpart of Theorem 3.11. If 0 is E-thick then these refined localisation spaces coincide with the spaces .F;q (O) according to Definition 3.1. \Ve formulate the outcome and discuss afterwards the key ideas. We incorporate now F�00(0) = .8�00(0). Let /;q (Zn) be the sequence spaces as introduced in Definition 3.9 and let O"pq be as in (2.24).

4.3. Let n be an E-thick domain in ]Rn according to Definition .F 4 . 1 (i) and let ;q (n) with s > o-pq • O < q � oo, (4.6) O s,

=

oo

) be the spaces as introduced in Definition 3. 1 . Let for u E N with

with Nj E N { � : j E No ; r = 1, . . . , Nj } be an orthonormal u-wavelet basis in L2(0) according to Propo:;ition 3. 'l and Defi nition 3. 5. Then n n max(l , p) < v .::; oo , s - - > - - , (4.7) v p

(what means v = oo if p oo). Furthermore, f E Lv (fl) is an element of .F;q (O) if, and only if, it can be represented as =

(4.8)

j

=

NJ

oo

L L A� Tjn/2 ClJ�. ,

j=O r=l

absolute ( and hence uncondil'iona� convergence being in L, (n) . If f E the representation (4.8) is unique with A = A(/),

(4.9)

I

:

f�

then

2jn/2 l f(x) i.(x) dx A(j) = { 2j n 2 (!, t )}

A� (f) = 2jnf2 (!, �)

and

F';q(O)

=

/

is an isomorphic map of .F;q (O) onto f;q (Zo.). lf p < oo, unconditional basis in F';q(Q).

q

< oo then { f. } is an

Discussion 4.4. First we remark that (4. 7) is a continuous Sobolev embedding. It ensures that f E .F;q (O) admits an expansion of type (3.15) with /� 2 (Zo. ) in place , of f�. 2 (Zn) (locally if p oo ) . By Definition 3.1 one gets f E i';q (O) from =

(4. 10)

f = giO

with g E F:q(fi)

C

F;q(lR11).

366

HA�S TRIEBEL

This reduces (4.8 ) to corresponding expansions in �n as considered in Theorem 2.7 and Discussion 2.8. One does not need moment conditions for the atoms in (2.29) with
(4 1 1 )

J.t (f)

.

= in

kjr

kt, (y) f (y ) dy,

=

2jn/2 �j '*' 1· '

one needs moment conditions of type (2. 30 ) with s. The kernels generated by the basic wavelets (3.9) and the interior wavelets (3.10) fit in this scheme, but not the kernles k? with the boundary wavelets
=

is an admitted �"-kernel having in particular the required moment conditions. Furthermore with 9 as in ( 4. 10) (in particular supp 9 c n) one gets

{

(4 1 3 )

locn

.

kt (x) g(x) dx =

n

1 kt (y) j( y) dy = :\t (f).

l\ ow it might be at least plausible that the above proposition is closely connected with Theorem 2.7.

REMARK 4.5. There is a counterpart both of Prop osi ti on 4.3 and of Discussion 4.4 for the spaces i.i;q (O) with 0 < p, q :::; oo and s > up . But all this will be covered by Theorem 4.8 below. The preference given to i';q (n) comes from the little history of the so-called refined localisation property which can be written as

F;;/loc (O)

=

.F;q (fl)

for E-thick domains

n,

with s,p, q as in ( 4.6) where we outlined in Discussion 3.13 what is meant by F;;/loc (n ). We proved this property first for bounded coc domains in 130, Theorem 5 . 1 4 pp. 60/6 1J and then for bounded LipHehitz dornaius in 132, Proposition 1.20, pp. 208/209] . ,

4.3. Wavelet bases in A�q (fl) . We are now ready for the main result of Section 4. Let A;q (fl) and .A;q (fl) be the spaces introduced in Definition 3.1 for arbitrary domains 0 in �n with n =/= �n , considered as subspaces of D' (n) . Let as before

up = n

where b+ =

1

rna.x(b, 0) if b E �

DEFI�ITION

4 . 1 (i ) . Then (4. 14)

(� - )

4.6. Let

n

+

and upq = n

(

.t

mm p, q)

-1

)

+

be an E-thick domain in �n according to Definition if 0 upq ) s 0, s < 0,

=

367

WAVELETS IN FUNCTION SPACES

and

·if 0 ap, if 1 < p < 00' 0 < q s 00' s = 0' if 0 < p s oo, 0 < q s oc , s < 0 .

(4.15)

REMARK 4. 7. We wish to extend the wavelet expansions according to Proposi tion 4.3 from ff;q (O) to the above spaces A�q(O). At first glance this looks a little bit suspicious. All wavelets <��t according to Definition 3.3 have compact supports in n. Then one can hardly expect wavelet expansions of type (4.8) in spaces A�q(O) having boundary values. This will be the subject of Section 5. However this possi ble counter-argument does not apply to the above spaces. This is quite obvious for the spaces A�q (O). Let s < 0 and, say, 1 < p < oo, 0 < q < oo. Then the restriction D(.IRn) I O of D(.IRn) to n is dense in A�q (n). But any function cp E D(.!Rn) I O can be approximated in Lp(O) by functions belonging to D(O) . Since s < 0 this is al�:>o an approximation in A�q(O). Hence D(O) is dense in this space. These arguments can be extended to other cases covered by the above definition.

In what follows the technicalities (unconditional and loeal convergence ) arc the same as in connection with Theorem 2 . 7. Let /;q(Zo) and b;q(Zo) be the sequence spaces according Lo Definition 3.9. THEOREM

Let for u E N,

4.8. Let 0 be an E-thick domain in IR." according to Definition 4 . 1(i ) . {
with

Nj E N,

be an orthonormal u -wavelet basis in L2 (f2) according to Proposition 3. 1 and Defi nition 3.5. (i) Let P;q (O) be the spaces in (4. 14) and let u > max(s, apq - s),

Then f E D'(O) is an element of F�q (O) if, and only if, it can be represented as

(4.16)

f=

oc

Ni

� � >.t rin/2
j =O r=l

unconditional convergence being in D' ( 0) and locally in any space F�q ( 0) with a < s. Furthermore, if f E F�q (n) then the representation (4. 16) is unique with ).. =

>. (!),

(4 . 1 7 ) and ( 4. 18)

J

:

I

f-t

>.(f) =

{ 2jn/2 (!, <��n }

is an isomorphic map of F�q (O) onto J;q(Zn). If, in addition, q < oo, then { max(s, aP - s),

s =I= 0.

HANS TRIEBEL

368

Then f

E

D'(f!.)

(4.19)

is

an

element of B�q(f!.) if, and only if, it can be represented as oo

Nj

J = I: I: ;:..t Tjn/2
unconditional convergence being in D'(f!.) and locally in any space B�q (D) with < s . HrdhemwTe, if .f E B�q (f!.) then the representation (4.19) is unique with ), = ).. ( j) as in (4. 17), and I in (4. 18) is an isomorphic map of B�q (D) onto b;q(Zn) . If, in addition, p < oo, q < oo, then { .Pn is an 7lnconditional basis in a

R�q(n) .

Discussion 4.9. As indicated above, (4. 19) wilh s >

ap and (4. 1 6) with s > are atomic decompositions, no moment conditions are needed. But for the coefficient.;; >..{ , int erpr eted as local means one needs moment conditions for the kernels k? in ( 4. 1 1 ) . Since fl. is E-thick one can circumvent this diffieulty by ( 4. 1 1 ) (4. 13). If s < 0 then one does not need moment conditions for the kernels k? , but for the atom;;. However thP- necessary repair for the boundary wavelets ? in (3. 1 1 ) can be done i n the same way as in ( 4. 12). Otherwise the above Lheorern is published here for the first time. A detailed proof will be given in [35] . So far we excluded s = 0. Then the assumptions for fl. rum;t be strengthened and some arguments from fractal analysis enter on the scene. We refer again to [35] for details. This covers in particular Lipschitz domains. Here is a corresponding formulation for this special case. apq

4.10. Let n be a Lipschitz domain in !Rn with n ::: 2 according to Definition 4 . 1 or an interval ·if n = 1 . Then Theorem 4 . 8 remains valid for all spaces F�q (f!.) in (4.14) and all spaces B�q (f!.) in (4. 15) (including s = 0) . COROLLARY

REMARK 1. 1 1 . As mentioned in Remark 4.2 Lipi:ichitz domains are E-thick.

5. Spaces on c= domains 5.1. Preliminaries, spaces on manifolds. All constructions of wavelet bases so far in !Rn , 'll'n , arbitrary dornaim; and E-thick domains rely on dual pairings, say, (D(f!.), D'(D)), and the related interplay between atoms on the one hand and local means on the other hand. This is well reflected by the corresponding wavelets, say iP� in Definition 3.3, having compact supports in the underlying d omain fl.. This type of arguments cannot be extended to spaces A;q (f!.) having boundary values at. r = DD. We discussed this point briefly in Remark 4.7. In particular, there is no duality for these spaces within (D(D) , D' (D)). One can try t.o circumvent these difficulties by decomposing A�q (O) into an interior part, say, A� q (D) , to which the above considerations can be applied, and trace spaces on r. In rough domains or for spaces A �q ( f!. ) with p < 1 such an approach does not work. For this reason we restrict ourselves Lo bounded c= domains n according to Definition 4 . 1 and to spaces A�q (D) with p ::: 1, q ::: 1. Furthermore we give only an outline of a few basic ideas and assertions and shift a more elaborated presentation to a later occasion. We refer in particular to [35] . First we introduce function spaces on compact d-dimensional coo manifolds r (without houndary), where d E N. Let {Vj, 1j11 }f= 1 with J E N be an atlas of r

369

WAVELETS IN FUNCTION SPACES

consisting of a covering r =

u�=l vj and homeomorphic maps

'¢j : vj -¢=:} uj = �j (VJ) c R.d , J = 1 , . , J, onto open connected bounded sets in R.d , say in the unit ball U in JRd , with the u:;ual compatibility conditions converting r inLo a compact coo manifold. Then D(f) = c= (f) and D' (r) have the usual meaning. Let {Xi }j= 1 c D(r) be a resolution of unity with (5.1)

_

HUpp

Xj c

and

Vj

If f E D' (f) then

J

L: x.i h) = 1

j= l

if

_

,

E r.

j = 1, . . . , J.

5.1. Let d E N and let r be a compact d-dimensional c= manifold. { B, F} and s E R., 0 < p, q ::; oo (p < oo for the F -spaces) . Then

DEFINITION Let

A

E

A;q (r ) = { f

E D' (r) :

and

(xjf) o 'ljlj 1 E A�q (R.d) , j = 1, . . . , .!} J

I I ! I A;q (r ) ll = I: I! Cxif) o '¢j 1 IA�q (IRd ) ll -

j=l

REMARK 5.2. It follows by standard arguments (diffeomorphic maps, point w is e multipliers) that these spaces are independent of compatible atlasses and related resolutions of unity. Blli:iically one can transfer the wavelet expansions for the spaces A�q(JRd) according to Theorem 2.7 via (5.1) to A;q(r) using that ii!i:;,m in (2.25), (2.26) has compact support. The outcome is in general not a wavelet basis but a wavelet frame which is sufficient for many purposes. But in some cases one gets even bases. For details we refer again to [35). We describe an example on which we rely afterwards.

REMARI< 5.3. (Wavelet bases on curves). The boundary r of a bounded coo domain 0 in the plane IR2 (planar domain) consists of finitely many dosed connected c= curves, which are diffeomorphic to the 1-torus 1!' = 'JI' 1 in {2.33). The wavelet bases from Proposition 2.12 and also the wavelet expansions according to Theorem 2.15 can be transferred to r , where the coo distortion factors originating from the diffeomorphic maps can be incorporated in the wavelets. Then one gets orthonormal wavelet bases { ci>V } on r similar as in {3.7), (3.8) based on the counterpart Zr of (3.2), (3.3). Afterwards one obtains wavelet bases for all spaces A;q(r) of the same type as in Theorem 2 . 15 with n 1 . =

5 . 2 . Decompositions. Let 0 be a bounded coo domain in IR " and let r = an be its boundary. Let A�q(n) and A�q (O) be as in Definition 3.1 and let A�q (r) be their counterparts on r as introduced in Definition 5. 1 . First we recall some more or less known trace assertions and decompositions for the spaces

with The linear bounded

trace

1 ::; p <

operator trr,

oo,

1 ::; q < oo , s > ljp .

370

HAKS TRIEBEL

has the usual meaning. Let K = [s - ;1- be the largest integer K K < s - l. Let v be the (outer) normal on r. Then () g 1 K = [s - - ] - , tr�'P : g ,...... trr ovk : 0 � k � K , (5.2) p k maps

p

{

E

N0 with

}

K

II s;; r;-k(r)

s; ( O ) onto q

1

k=O

K

and

II s;; "-k(r).

F;q (O) onto

1

k=O Futhermore there are linear and bounded extension operators extL� p ·. {go , . . . , gK } I-> g (5.3) mapping K

IT s;; �- k (r)

(5.4)

into s;q ( n)

k=O

and

K

II s;; i -k(r)

(5.5)

into F�(n)

k=O

such that

trr's p o extr's p

=

id ,

identlty m •

•

K

Il Bpsq- lP -k ( f ) . k=O

We refer to [27, p. 200] with corresponding assertions in [26] as a forerunner. But the extension operators constructed there are not good enough for our purpose. One needs wavelet-friendly extension operators extending functions 9 on r with supports in an c-neighbourhood in f of some point "'/ E f into functions with supports in a corresponding c-neighbourhood of "'! in n. This can be done but will be shifted to !35] . Its paves the way to clip together wavelet bases of the spaces A�q (O) according to Theorem 4.8 with wavelet base.s for s;q (r) (if exist ). This procedure relies on the following assertion. Recall that A.;q (O) is the completion of D(O) in A;q (O), whereas l;q (O) has the same meaning as in Definition 3.1. PROPOSITION 5.4. Let n be a bounded c= domain 'tn JR.n accm·ding to Defini tion 4, 1(iii) and let r = 80, Let 1 � p < oo ,

1�

q

1 0 < s - - It No ,

< oo,

p

Then

(5.6) with tr�'P as

( 5 . 7)

in

(5.2 ) .

Furthermore, s;q (n) = i3;q (n)

x

K

II s;; r;-k(r)

k =O

1

3 71

WAV ELETS IN FUNCTION SPACES

and F%9 (0)

(5.8)

REMARK 5.5. If 1 :::; p, q

=

i';q(n)

< oo

x

K

k II B;; r; - (r) . k=O 1

then (5.6) can be complemented by

-1 <

X;q(n) = fi;q(n) = A�q(n),

(5.9)

s - p� < o.

Both (5.6), (5.9) are covered by [27, p. 210] (and the related proof) with [26, pp. 317 /318] as forerunners. The decomposition (5 .7) , (5.8) must be understood as follows. Let ext�·P be as in (5.3)-(5 .5). Then

P

= ext�·P

is a projection and extf P : ext�'P :

o

tr�·P

K

:

A� (D)

k II B;; r; - (r) k=O k II B;; " - (r) k=O K

1

1

<---t

A;q (O)

{==:>

PBZq(D)

{==:>

PF;q (n)

are isomorphic maps. Furthermore, id - P is a projection of A�(O) onto X;q (O) . It is just this decomposition of A;q (D) into two complemented subspaces which paves the way to clip together wavelet expansions in .A;q (n) and in B;q(r) .

5.3. Wavelet bases in A�q (D). Let n be a bounded coo domain in JR.n. We are looking for wavelet bases for the spaces covered by Proposition 5.4. In contrast to the wavelet bases considered so far, for example in Theorem 4.8, the boundary r an must come in now. First we modify Zn in (3.2)-(3.4) and the sequence spaces according to Defi nition 3.9. We use the same notation as there. In particular, B(x, Q) stands for a ball centred at x E ffi.n and of radius Q > 0.

=

DEFINITION 5.6.

Let D be a bounded domain in ffi.n and let Z11 {xt E D : j E No; r = 1 , . . . , Nj } , typically with Nj ,...., 2Jn, such that for some > 0, 'xi - d i > Ti j No, r =f. r'. Let be the characteristic function of B(xt , 2-j ) for some s IR, 0 < q Then b;q (Z11) is the collection of all sequences (5.10 ) .A = { .At E C : j E No; r = 1, . . . , Nj } such that =

c1

E

X.i r

r

p,

:::;

oo .

r' - c1

'

E

C2

n

n

C2

> 0.

Let

372

HANS TRIEBEL

and f;q ('J.P.) is the collection of all sequences (5.10) such that II A l f;,
( modification if p

=

(t, �

�

2'"' 1 :.: )(j.( W ·

oo and/or q = oo).

)

' ''

ILp(fl) < oo

REMARK 5.7. Thi
zn

DEFINITION 5.8. Let fl be a bounded { x{} be ns in Definition 5. 6.

C00 domain in

R.n .

Let u E N and let

=

(i) Then {
:

j E Nn ; r = 1 , . . . , Nj }

c

c u (n)

is called a u-wavelel system ('with r-espect to 0) if for some c3 > 0 and c4 > 0, j E N0; r = 1 , . . . , NJ ,

and

j E No; r = 1 , . . , N1 , I D0
,

.

REMARK 5.9. For the spaces in Proposition 5.4 one cannot expect to find common u-wavelet bases originating from an orthonormal wavelet basis in L2 (D) a. '> in Theorem 4.8. This may explain the difference of the above part (ii) and Definition 3.5. After these preparations we can now formulate the main result of Section 5. As before we write A;q ( D ) with A E {B, F} and similarly a;q(z n ) with a E {b, f} if the assertion applies equally to the B-spaces and F-spaces. THEOREM 5.10. Let m E No . Let ( 5. 1 1 )

s = m+u

1 1 with 1 � p < oo and - - 1 < (]' < -. p

p

(i) Let D = 1 = (a, b) with -oo < a m there is a common u-wavelet basis accord·ing /,o Definition 5.8(ii) for all spaces A�q(I) with 1 � p, q < 00 and s as in (5. 1 1 ) . Furthermore,

N1 1 = 2::: :�:::t:-� u) 2-j 12 t oc

j=O T=l

and f � >-(!) is an isomorphic map of A�q(I) onto a�q (:l/ ) . (ii) Let n be a bounded coo domain in the plane R.2 . Then for any u E N with

WAVELETS IN FUNCTTO'< SPACES

373

u > m there 1:s a common u-wavelet basis according to Definition 5.8(ii) for all spaces A�q(n) with 1 :S: p, q < oo and s as in (5.11) . HtTthe'Tmorc, oo

Ni

f = L L .Xt( f) Ti
and f r--. .X(f) is an isomorph·ic map of A;q (f2) onto

a;q(zn) .

Discussion 5 . 1 1 . We give an idea how to prove part (ii). First one decomposes

according to Proposition 5.4 ( modified by (5.9) if m = 0) and Remark 5 . 5. Theu one clips together the wavelet bases from Theorem 4.8 and Remark 5.3 via wavelet friendly extension operators as discussed above. Details are shifted to a. later occa sion. We refer in particular to [35] .

6. Comments We add a few references and comments. As far as new results are concerned we refer for details and proofs to [3 5] . Some references for wavelet bases in function spaces on JRn and 2.9 and at the beginning of Section 2 .4. Quite ob viously wavelet bases for function spaces on intervals, cubes and Lipschit;� domains attracted a lot of attention. We refer to [21, 6, 7, 8, 9) and in particular to [5, 17] and the literature given there. But the methods are different from what we outlined here. COMMENT 6 . 1 .

1I'n have been given in Remark

CoMMENT 6.2. In [3, 4, 2] Ciesielski and Figiel constructed common spline bases for Sobolev spaces and Bcsov spaces on compact d-dimensional coo manifolds r (including the closure n of bounded coc domains f2). The method is based on a rather sophisticated decomposition of r (or n) into finitely many domains which are diffeomorphic images of cubes in !Rd . Bases on cubes are shifted in this way to r and glued together. Similar procedures have been used to construct wavelet bases on (smooth) manifolds and on (smooth) domains admitting the required dom ain decomposition. Descriptions may be found in [8, Section 10], [9, Section 9] , [5, p. 1 30] with a. reference to the original paper [10) . It remains to be been whether one can use these results in connection with the above considerations especially in Section 5. CoMMENT 6.3 . One may a..-;k whether one can extend Theorem 5.10 from one and two dimensions to higher dimensions based, for example, on the decompositions according to Proposition 5.4. But the topology of t.he connected components of bounded C00 domains in !Rn with n ;::: ;) is much more complicated than in case of bounded coo domains in the plane !R2 . If one has wavelet bases in suitable spaces B;q (r) with r = 8f2 then one can argue as indicated in Discussion 5.11. If r (or one of its components) is diffeomorphic to an d-torus (d = n - 1), then one can apply Theorem 2 . 1. 5 . If

r = §d = { x E

JRd+ l

:

lxl

=

1} ,

2 :S: d E N,

HANS

374

TRIRBEL

is a d-sphere, d = n 1 , or a diffeomorphic image of §d , then one can construct wavelet bases of the desired type in l 1 a > P - 1 , a - p- � No ,

-

l=

1 , . . . , d - 1 . This can be done by induction with respect to dimensions start for ing from Theorem 5.10(ii) which causes the curious (and presumably unnatural) restrictions for a . COMMENT 6.4. B y Definition 3 . 1 all spaces A�q (n) on arbitrary domains n are defined by restriction of A�q(JR.n) to n. As mentioned in Remark 3.2 one recovers in case of bounded Lipschitz domains n the classical Sobolev spaces w; (n) = { ! E Lp(r2) : Da f E Lp(r2) , l n l :=::; k} , with lfp II! IW; (n ) ll = L liDo f I Lv ( n) IIP JaJC:: k where 1 < p < oo and k E N. However for rough domains n having, for example, peaks and cusps these spaces do not fit in the scales A�" (n ) . The standard reference for Sobolev spaces in rough domains is [22] . But one can say something. Let { �t.} be the same orthonormal u-wavelet basis for Lp(r2) as in Theorem 3.11 where n is an arbitrary domain in lR". One can apply Theorem 3.11 to each D0 f E Lp(n) with l a l :=::; k. Then it follows that W;'(n) is the collection of all f E Lp(f!) with its Lp-representation (3.15), (3.16) such that >.(f) k E !2,2 (Zn ) where >.{ (f) k = 2jn/2 L I (f, D" �{) I

(

)

JC>!J9

(equivalent norms). Sometimes �� are called vaguelettes. It remains to be seen if this observation is of any use. COMMENT 6.5. We excluded in Proposition 5.4 and Theorem 5.10 the case s - � E N0 • But there are some negative results. Let again n be a bounded c= domain in JRn and 1 < p, q < oo. Then D(rl) is dense in A��P (f!) . On the other hand there is no orthonormal u-wavelet basis according to Proposition 3.7 which is also a basis in A��P (n) (in contraHt to A"!�P(n) according to Theorem 4.8). COMMENT 6.6. We mention a second negative result. By Remark 2.2(iv) the spaces B�(l�n ) with p, q, s as in (2.9) can be equivalently quasi-normed by (2.10), (2. 11). This is no longer valid if s < n 1 + . But one can define corresponding spaces as subspaces of Lp (lRn), B�q (JR.n ) {J E Lp(JR.n ) : IIJ IB�q (JRn) ll m < 00 } where 0 < s < m E N, 0 < p, q oo, (6. 1) and

=

(� - )

<

375

WAVELETS IN FUNCTION SPACES

These are quasi-Banach spaces which are independent of m. References to the might be found in [32, Section 9J . One may ask whether these spaces have (wavelet) bases or frames for the full range of p, q , s in (6.1). But this is impossible since the coefficients in such expansions must be linear and continuous fnndionals in B;q (�11). However one can prove that

literature

B�q (IRn )' = {0} if 0 < p < 1, 0 <

q

< oo, 0 < s <

n

(� - 1 )

References [1] F. Albiac, N.J. Kalton. Topics in Banach space theory. Springer, New York, 2006 [2] z. Ciesielski. Spline bases in classical function spaces Oll compact c= manifolds, III. In: Constructive Theory of Functions, Proc. Intern. Conf. Varna. Publishing House Bulg. Acad. Sciences, Sofia, 1984, 214-223 [3] Z. Ciesielski, T. Figiel. Spline bases in classical function spaces on compact coo manifolds, I. Studia Math. 76 (1983), 1-58 [4] Z. Ciesielski, T. Figiel. Spline bases in classical function spaces on compact C00 manifolds, II. Studia Math. 76 (1983), 95-136 [5] A. Cohen. Numerical analysis of wavelet methods. North-Holland, Elsevier, Amsterdam, 2003 [6] A . Cohen, W. Dahmen, R. DeVon�. Multi scale decompositions on bounded domains. Trans. Arner. Math. Soc. 352 (2000), 3651-3685 [7] A. Cohen, W. Dahmen, R. DeVore. Adaptive wavelet techniques in numerical simulation. In: Encyclopedia Computational Mechanics. Wiley, Chichester, 2004, 1-64 [8] W. Dahmen. Wavelet and multiscale methods for operator equations. Acta Numerica 6 (1997), 55-228 [9] W. Dahmen. Wavelet methods for PDEs - some recent developments. Journ. Computational Appl. Math. 128 (200 1 ) , 133-185 [10) W. Dahmen, R. Schneider. Wavelets on manifolds 1: construction and domain decomposition. SIAM Journ. Math. Anal. 31 (1999), 184-230 [11) I. Daubechies. Ten lectures on wavelets. CBMS-NSF Regional Conf. Series Appl. Math., SIAM, Philadelphia, 1992 [12) D.E. Edmunds, W.D. Evans. Hardy operators, function spaces and embeddings. Springer, Berlin, 2004 [13) D.E. Edmunds, H. Triebel. Function spaces, entropy numbers, differential operators. Cam bridge Univ. Press, Cambridge, 1996 (sec. ed., 2008) [14] M. Frazier, B. Jawerth. Decomposition of Besov spaces. Indiana Univ. Math. Journ. 34 (1985 ) , 777-799 [15) M . Frazier, B. Jawerth. A discrete transform and decompositions of distribution spaces. Journ. Funct. Anal. 93 (1990) , 34- 170 [16) D . D . Haroske, H. Triebel. Wavelet bases and entropy numbers in weighted function spaces. Math. �achr. 278 (2005), 108-132 [17] J.A. Hogan, J.D. Lakey. Time-frequency and time-scale methods. Birkhauser, Boston, 2005 [18] D. Jerison, C. Kenig. Boundary behavior of harmonic functions in non-tangentially accessible domains. Advances Math. 146 (1982), 80-147 [19] C.E. Kenig. Harmonic analysis techniques for second order elliptic boundary value problems. CBMS, Regional Conf. Series Math. 83. Arner. Math. Soc., Providence, 1994 [20] G . Kyria.zis. Decomposition systems for function spaces. Studia Math. 157 (2003), 133-169 [21) S. Mallat. A wavelet tour of signal processing. Academic Press, San Diego, 1999 (sec. ed. ) [22) V.G. Maz'ya. Sobolev spaces. Springer, Berlin, 1985 [23) Y. Meyer. Wavelets and operators. Cambridge Univ. Press, Cambridge, 1992 [24] H.-.J. Schmeisser, H. 1Hebel. Topics in Fourier anal ysis and function spaces. Wiley, Chichester, 1987 [25) E.M. Stein. Singular integrals and differentiability properties of functions. Princeton Univ. Press, Princeton, 1970 [26] H. Triebel. Interpolation theory, function spaces, differential operators. North-Holland, Am sterdam, 1978. ( Sec. ed. Barth, Heidelberg, 1995) [27) H. Triebel. Theory of function spaces. Birkhauser, Basel, 1983

HANS TRIEBEL

376 [28] H.

Triebel. Characterizations of Besov-Hardy-Sobolev spaces:

Approximation Theory 52

[29] [30] [31] [32] [33]

(1988), 162-203

a un i ed approach.

fi

Journ.

1992 2001

H. Triebel. Theory of funct.ion spa<;es II. Birkhauser, Basel, H. Triebel. The structure of functions. Birkhauser, Basel,

H. Triebel. A note on wavelet

function spaces. In:

Center Pub!. 64, Warszawa, Polish Acad.

bases in

H. 'l'riebel. Theory of function spaces III.

Orlicz Centenary Vol., Banach

Sci., 2004, 193-206 Bi rkhauser , Basel, 2006

H. Triebel. \Vavelet par�bases and sampling numbers in function spaces on domains. Journ.

Complexity 23 (2007), 468-497 [34] H. Triebel. Local means and wavelets in function spaces. Proc. Conf. Function 8po.ces 8, B�dlewo, 2006. Banach Center Pub!. 79 (2008), 21!>-234 [35] H. T'riebel. F\mction spaces and wavelets on domains. E ur ope an Math. Soc. Publishing House, Ziirich, 2008 [36] H. Triebel, H. WinkelvoE. Intrinsic atomic characterizations of function spaces on domains. Math. Zeitschr. 221 (1996), 647-673 [37J P. Wojtaszczyk. A mathematical introduction to wavelets. Cambridge Univ. Press, Cam bridge 1997 ,

MATHEMATISCHES

GERMANY

INSTITUT,

FRIEDRICH-SCHILLER-UNIVERSITAT

E-mail address: triebel!Qminet . uni -j ena. de

JENA,

0-07737

JENA,

Proceedings of Symposia in Pure Mathematics Volume 79, 2008

Weighted norm inequalities with positive and indefinite weights I. E. Verbitsky This paper is dedicated to Vladimir Maz'ya on the occasion of his 70-th birthday.

ABSTRACT. Integral inequalities of the type

[ [Tu[ [ Lq (dw) $ C [ [u[[LP(do-)•

and their variations are considered. He1·e

w,

u a

E

are

LP(da),

p i ive os t

dx is Lebesgue

measures on

Rn ,

A survey of some

and T is an i ntegral operator with nonnegative kernel .

recent results is given with an emphasis on the so-called trace inequalities where da

=

pioneering work

on

measure.

lt

reflects the impacL of V. Ma:.:'ya's to spectral theory, linear

this problem and its applications

im c i fi i or i�2( Rn) b [ [u[ [�2(R")' I JR" [u[2 wl [ [vui f g ly r ll m h m tc iff t a c

and nonlinear PDE.

An port ant special case where q = p = 2 and T = associated with the S hrodinger operator H = -b. + w and

W1•2(R") c L2(w}

(1 b. ) - 112 is the imbedding -

if w � 0. For nde n te weights" w , the following quadratic form inequality is equivalent to the fundamental notion of the relative form boundedness of "

the potential energy

Ho =

-

b.:

[ (w u , ull

pe ator w with respect to the kinetic energy op erator

::;

=

for some a , b > 0. Here

sign,

or more

enera l

w

,

a

+

a

is a o cally integrable

below for

[MV2] .

at e

ics, was recently solved by Maz'ya and Vcrhitsky ficients

unction which may change distribution. This form

ea - 01' complex-valued

boundedness problem, which is important to

u E C0 (Rn ) ,

a i al quantum mechan It will be discussed

ge neral second order d eren i l operators with distributional (not ne essarily elliptic ) in place of the Laplacian [MV6].

coef

CONTENTS

1. 2. 3.

Introduction Basic trace inequalities

Dyadic and radial trace

inequalities and nonlinear potentials

378 382 387

1991 Mathematics Subject Classification. 3 1 8 1 5 , 35J10, 35J60, 42B25. norm inequalities, indefinite weights, nonlinear potentials,

Key word.� a.nd phrases. \'Veighted

ene al

furm boundedness, Schrodinger operators, g The author was support ed in part

by

r

second-order differential operator�.

NSF Grant DMS-0556309.

377

(92008 A m P r i c:a.n lvtat. h�matic:a.J Soc:it"t.y

378

I. E. VER.BITSKY

4. Indefinite weights and Schri:idingcr operators 5. Infinitesimal form boundedness, Trudinger's and Nash's inequalities 6. Form boundedness of general second order differential operators References

391 395 399 404

1 . Introduction We will consider weighted norm inequalities of the type ( 1.1)

l lT f l i L•( dw)

(1.2)

J JT/J J M (dw)

S::

C 11/I ILP(da) ,

f E £P(da'),

C J J /J I LP( dx) >

J E £P( Rn) ,

where w and a are positive Borel measures, and 0 < p, q S:: oo , for a number of "positive" operators T. We are particularly interested in integral inequalities for such operators T as Riesz potentials lex = ( -.6.) - "i" (0 < a < n ) , Bt>.ssel potentials Ja = (1 - .6.) - � (a > 0), and more general convolution operators on the Euclidean space Rn with radially decreasing kernels, as well as their dyadic models. Other important operators, e.g., various maximal operators, Green 's potentials, Poi.<>son integrals, etc. can be studied using similar approaches. (See, e.g., [KSJ, [KV] , [VW2] , [Vl] , [V2] .) We will treat in detail t.he less studied "upper triangle" case where q < p [COV1]-[COV3] , [MN] which is intimately connected with modern nonlinear potential theory. A special consideration will be given to the �o-called trace inequalities of the type S::

with an arbitrary measure dw on the left-hand side, and with Lebesgue measure dx on Rn in place of da on the right. This important special case covers many classical inequalities, as well as a great number of applications. A deep study of trace inequalities with an emphasis on necessary and sufficient conditions was pioneered by Vladimir Maz'ya starting in the early 1960s [Ml], [M2] . Let n c Rn be an arbitrary open set, and let e be a compact subset of n. \Ve will need the Wiener capacity relative to the domain n defined by: ( 1 . 3)

cap (e, n)

=

inf

{In J V'u l2 dx :

u E C0 (n) , u > 1 on e

For a nonnegative Borel measure w on n, we set (1 .4) and (1. 5)

c1 (w, n) = sup

{ l_.,. Ju(x)l 2 {

c2 (w , n) = sup

dw

:

}

u E C0( il ) , IIV'u i i P CO)

w(e ) : cap (e, n )

.

S:: 1

}

where the supremum above is over compad. sets e c n of positive capacity. As was shown in [Ml] (see also [M4] , Sec. 2.5) , (1.6)

WEIGHTED NORM INEQUALITIES

3 79

where the constants 1 and 4 are sharp. Measures w obeying conditions of the type c2 (w, D) < oo which characterizes the imbedding L1•2(0 ) c L2(D, ch;.;) will be called Maz'ya measures (or sometimes admissible measures) . The class of Maz'ya measures associated with (1 .2) in the case of Riesz poten t i als T = Iu for 0 < a < n on n Rn was characterized completely in terms of Riesz capacities by Adams, Dahlberg and Maz'ya (see [AH] , [M4] ) in the 1970s. Alternative characterizations were given later by Kerman and Sawyer [KS] who used more precise local energy conditions, and Maz'ya and Verbitsky [MVl] in terms of yet more localized pointwise inequalities. The latter turned out to be especially well adapted for applications to nonlinear PDE (HMV] , [KV] , [PV1] [PV3], [VWI]. These results are discussed in Sec. 2. In Sec. 3, we consider two weight inequalities and nonlinear potentials for the so-called dyadic model [COV2], [ C OV3] , [NTV] where T is the integral operator =

Tf(x) =

(1. 7)

{

Kv (x, y) f(y) da(y)

{

k ( i x - y i ) f(y) da(y) ,

fw,

with the kernel Kv (x, y) = L Q E'D K(Q)xQ (x) xQ (y) . Here v = {Q} is the family of all dyadic cubes on Rn, K(Q) are arbitrary nonnegative constants, and XQ are characteristic functions of Q. Weighted norm inequalities for general convolution operators

(1.8 )

Tf(x)

=

fw,

where k = k(r ) is an arbitrary positive nonincreasing function of r > 0, can be deduced from the dyadic model. This makes it possible to answer a question raised in (AH] , Sec. 7.7, of how to define an analogue of Wolff's potential for general radially decreasing kernels. In Sec. 3 we give an appropriate definition of Wolff 's potential, which is far from being obvious, and prove Wolff's inequality for radially decreasing kernels [COV2], [COV3] . We observe that Woltr's potentials and their modifications have become an important tool in modern theory of quasilinear and fully nonlinear PDE [KiMa] , [L] , [MaZi] , (PVI]--[PV3] , [TW] . In Sec. 4 we will present recent developments [MV2] concerning trace inequal ities with "indefinite" weights of the type:

where (1. 10)

I Ln w l � I L, wl � a l u l2

(1.9) n

C l l \7 u l li2(R" ) '

u E C8" (Rn),

:2: 3, and their inhomogeneous analogues: lu l

2

l l \7 u l l i2 (Rn) + b l l u l l i2(Rn ) •

for some positive constant:-; a , b, where w E D'(Rn), n 2 1 . Here D(Rn) = C0 (Rn ) , and the left-hand sides are understood in the sense of distributions. Both inequalities (1 .9) and (1. 10) are important to the Schrodinger operator theory. The first inequality expresses the domination (in absolute value) of the potential energy associated with w by the kinetic energy, while the second one is equivalent to the classical notion of the relative form boundedness of w with respect to the kinetic energy operator Ho = -�. This concept is used in the so-called KLMN theorem which makes it possible to define a self-adjoint operator H = Ho+w so that the quadratic form domain Q(H) coineides with Q(H0) provided w is real-valued. See, e.g., [EE] , [RS] , [Sch] . It is worth mentioning that the

I. E. VERBITSKY

380

quadratic form inequality ( 1 .10) is equivalent to the boundedness of the operator H : W l,:t (Rn )

__..

w - 1 ,2 (Rn ) .

When the form bound a > 0 i n ( 1 . 10) can be arbitrarily small (with b depending on a) w is said to be infinitesimally form bounded relative to C. This notion is used extensively in mathematical quantum mechanics (see [RS] , [Sch] ) . Another important version of these form boundedness properties occurs when one replaces the nonrelativistic kinetic energy, namely I I V'u l l i2 (Rn ) ' with its relativi:;tic counterpart, I I ( -� + 1 ) 114u l l i2 (Rn ) associated with the Sobolev space W 1f 2 ,2 (Rn ) of order 1 /2 . The form boundedness problem for H = -� + w where w E D'(Rn), along with its infinitesimal version, was recently solved by Maz'ya and Verbitsky [MV2] [MV6] . The main idem; discussed below can be expressed as follows: (1 .9) holds for a distributional potential w if and only if it holds for I V' � - 1 w l 2 in place of w. This nonlinear transformation w --> IV'� -1wl 2 reduces the form boundedness problem for "indefinite weights" to the well-studied case of nonnegative weights. In Sec. 4 we will outline a proof of this form boundedness criterion for H = -� + w based on sharp estimates of equilibrium potentials and related weighted norm inequalities [MV2] .

Similarly, the class of w E D' (Rn) associated with the relativistic form bound edness of 1i = ( -� + 1 ) 112 + w is invariant under the transformation w --> 1 ( 1 �)- 112wl 2 . The relativistic problem can be reduced to a similar nonrelativistic one for the operator H = -� + w with a distributional potential w on a higher dimensional Euclidean space as was shown in [MV4] . In [MV5] necessary and sufficient conditions were obtained for the infinitesimal form boundedness of the potential energy operator associated with w with respect to the kinetic energy operator H0 = -6 on L2 (Rn) . Here w is an arbitrary real or complex-valued potential (possibly a distribution). Furthermore, a related form subordination property of Trudinger type (see [Tru] , and also [Sim] , [RSS] , and the literature cited there) is characterized explicitly in [MV5] . More precisely, we will present in Sec. 5 a characterization of the class of po tentials w E D'(Rn) which are -�-form bounded with relative bound zero, i.e., for every € > 0, there exists C(t: ) > 0 such that (1.11)

For complex-valued w,

D(H)

c

it

follows that II is an m-sectorial operator on L2 (Rn) with

W1 , 2 (Rn) ( [EE], Sec. IV.4).

The characterization of (1.11) obtained in [MV5] uses only the functions I \7 ( 1 -

A ) - 1 w l and 1 ( 1 - A)-1 w l , and is based on the representation :

(1. 12)

w = div f + 1,

In particular, f E Lf0c (Rn)",

f(x) = -\7(1 - �) - 1 w, 1 = (1 - 6)-1 w.

1 E Lfoc (Rn) , and, when

n ;::::

3,

(1.13)

once ( 1 . 1 1 ) holds. Here BJ (x0) is a Euclidean ball of radius b centered at xo.

WEIGHTED

NORM

38 1

INEQUALITIES

In the opposite direction, it follows from the results of [MV5] that ( 1 . 1 1) holds whenever (1 . 14)

sup t52r-n

lim

XoER"

o-+0

(

)

r f !f(xW + h(x) l dx = o, .IBa(xo)

for some r > 1. Such potentials form a natural analogue of the Fefferman-Phong class [F] for the infinitesimal form boundedness problem, where cancellations be tween the positive and negative parts of w come into play. It includes functions with highly oscillatory behavior as well as singular mea.•mres, and contains properly the class of potentials based on the original Fefferman-Phong condition where lwl is used in {1. 14) in place of lfl 2 + h'l- Moreover, one can expand thiH cla...:;s fur ther using a sharp condition due to Chang, Wilson, and Wolff [ChWW] applied to lfl2 + hi· In the proofs given in [MV5] considerable technical difficulties have been overcome using sharp estimates for powers of equilibrium potentials, factorization of functions in Sobolev spaces, and theory of Av-weights, along with appropriate lo calization arguments, and good understanding of trace inequalities for nonnegative potentials w. In [MV5] , we also study quadratic form inequalities of Trudinger type where C(t:) in ( 1 . 1 1 ) has power growth, i.e., there exists Eo > 0 such that ( 1 . 1 5)

for every E E (0, co), where f3 > 0. Such inequalities appear in studies of elliptic PDE with measurable coefficients, and have been used extensively in :spectral theory of the Schrodinger operator [AS] . As it turn�> out, it is still possible to characterize (1. 15) using only lfl and I 'Y I defined by ( 1 . 12 ) , provided /3 > 1 . In this case (1. 15) holds if and only if both of the following conditions hold: sup o2�:;� - n

(1.16)

xoER" 0<6<6o

sup offfi -n

( 1 . 1 7)

XoERn

O
!fcxW dx < + oo, f }Bo(xo)

r

}B.; ( x o )

!'Y(x)! dx < +oo,

for some oo > 0. However, in the case /3 :::; 1 this is no longer true. For {3 = 1 , ( 1 . 16) has to be replaced with the condition that f is in the local BMO space, or respectively is Holder-continuous of order i�g if 0 < /3 < 1 . In the homogeneous case Eo = +oo, (1. 15) is equivalent to the

multiplicative

inequality: ( 1 . 1 8)

where p 1!13 E (0, 1 ) . In spectral theory, (1. 18) is referred to as the form p-subordination property (see, e.g., [RSSJ, Sec. 20. 4 ) . For w, where w iH a locally finite measure on R" , inequality (1.18) is known to hold if and only if

nonnegative

{ 1 . 1 9) where the constant

c

does not depend on 8 > 0 and

x0 E

R" ( [1\13] , Sec. 1.4. 7) .

382

I. E. VERBITSKY

For general w, we obtain the following result : If p > !, then (1.18) holds if and only if V' � - l w lies in the Morrey space C.2 • " (Rn), where >. n + 2 4p. For p �' it holds if and only if V'� -lw E BMO(Rn), and for 0 < p < �' whenever V' � -lw E Lip 1 _2p (Rn ) These different characterizations are equivalent to (1.19) if w is a nonnegative measure. As a consequence, we are able to characterize those w which obey an analogous inequality of Nash's type: =

-

=

.

(1.20) where p E (0, 1). In fact , the preceding inequality has two critical exponents, p. * � · For 0 p. , the validity of ( 1 . 20) is equivalent respectively to: V' � - 1 w E Lipn+ l -p( +2 ) (Rn) n if p. . = 3n + 2 - 2p(n + 2) if p * < p < 1. The form boundedness problem for the general second order differential operator

=

=

( 1 .21)

C. =

n

L

i,

j=l

aiJ f)J}J + L j=l

bJ Oj + c,

where aij , bi , and c are real- or complex-valued distributions was solved in [MV6] . Here C. is not even assumed to be elliptic. \Ve will discuss in Sec. 6 quite complicated necessary and sufficient conditions for the quadratic form inequality

(1.22) to hold for some a , b > 0. It is easy to see that the symmetric part of the matrix (aij ) must be uniformly bounded, and the skew-symmetric part reduces to the first order terms. The main problem here is to inve::;tigat.e the interaction between the first-order and zero-order terms. The proofs make use of compensated compactness arguments (a vector-valued version of the div curl lemma) , along with the gauge transform involving powers of equilibrium potentials. Applieations to multidimensional Riccati's equations, quasilinear and fully non linear PDE, global estimates of Green's functions, etc., can be found in [FV] , [HMV], [KV] , [ Ma Z i] , [PV1] -{PV3] . -

2. Basic trace inequalities

We start with the following important theorem [M3] .

THEOREM 2 . 1 . (Ma:.-:'ya) Let 1 < p < 00. Let n be an arbitrary open set in Rn ' and let w be a nonnegative locally finite Borel measure on n. Then the inequality

(2 . 1 )

holds if and only if, for any compact set E c n, (2 . 2)

w ( E)

:::;

C cap1,p (E, n),

383

WEIGHTED NORM INEQL"ALITIES

where C is a constant which is 'independent of E. Here cap1 ,p (E, n) is the capacity defined by (2.3)

cap 1 ,p (E, n) = inf

{ fo l'\7u iP

dx :

u ( x ) � 1 on E, u E Cgo(n) .

}

Theorem 2.1 has numerous applications in harmonic analysis, operator theory, function spaces, linear and nonlinear PDE's, etc. (see, e.g., [AH] , (FV] , [M4] ,

(MSh] , [MaZi], [PV2]). For simplicity, we will only consider some analogues of Theorem 2 . 1

n = R"' for Riesz potentials defined by Ia f

=

( - A) - � f = c( n , a: ) ( l l a - n * !),

·

in

the case

0 < a: < n,

where c ( n , a:) is a normalization constant. We also set la (f duJ) ( x) = c( n , a)

1

f(y )

· -a dw ( y ) , R" I x - y I n for potentials with a Borel measure duJ in place of dx, and law = Ia ( ldw) if f = 1 on Rn . The Riesz t:apacity of a measurable set E c Rn is defined by ( 2.4)

(2.5 )

Capa ,p ( E ) = inf

{ fo l giP dx : Iag(x) � 1

on

E,

g E

}

L�{Rn) .

In the case a: = 1 it is known that C ap1 ,p (E) :::::: cap1 p (E, Rn) for compact sets E, , where cap 1 p (-, Rn) is defined by (2.3), and constants of equivalence depend only , on p (see (AH]) . The following theorem for a: = 1 and q = p is equivalent to Theorem 2.1 when n = Rn . THEOREM 2.2. (D. Adams-Dahlberg-Maz 'ya) Let 1 < p < and let 0 < a: < Let w be a nonnegative Borel measure on R"' . Then the following statements are equivalent. (i) The inequality oo

n.

(2.6)

holds where C is a constant which is independent of f . (ii) For every compact set E C Rn , (2 . 7)

w (E) � C Capa,p (E),

where C is a constant which is independent of E. The statement {i)=}(ii) in Theorem 2.3 is obviou:o; the converse follows from the so-called strong capacitary inequality

100 Capa,p({x : llaf(x) l > t}) tP -1 dt

�

C l lfl li,P , f

E

LP(Rn ) ,

discovered by V . Maz'ya i n 1972 for a: = 1. In a series of papers by D . Adams, B. Dahlberg and V. Maz'ya in the late 70-s, the preceding inequality was established for aU a: > 0 and p > 1 . Another proof valid for more general convolution operator::; is due to K. Hansson. (See [AH] , [M4] , [H].)

384

I.

E.

VERBITSKY

REMARK 2.3. Condition (2.7) combined with a standard estimate of the ca pacity from below, C ar ,p E ) ?:: C JEJ1- � , immediately yields the following well known sufficient condition: holds if (E) � C JEI1 - � , for every compact set E. This can be restated in terms of weak spaces: cU.u = p(x) dx, where p E L"•00 (Rn), for r = ; P.

a (

(2.6)

w

Lr

A substantial improvement of the sufficient condition mentioned in Remark 2.3 was found by C. Fefferman and Phong [F]. THEOREM

2.4. If dJ.u = p(x) dx where

l pl+< dx

(2.8)

� C JB J 1 - "P\';+-�)

,

E

> 0,

A sharp version of the Fefferman-Phong condition is clue t.o Chang, Wilson, and Wolff [ChWW] .

for every ball B

=

Br (x), then (2. 6) holds.

2.5.

REMARK An improved version of both the Fefferman-Phong and Chang \Vilson-Wolff conditions where w is not necessarily absolutely continuous with re spect to Lebesgue measure is given in (MVl] (see Corollary below).

2.10

It is easy to see that the capacitary condition (2. 7) is equivalent via duality to the inequality (2. 9)

for every compact set E C Rn, where � + -/;; = It was noticed by Kerman and Sawyer [KS] that in this dual form it is enough to restrict oneself to E = where = is a hall (or cube) in Rn.

1.

B

B Br(r)

THEOREM (Kerman-Sawyer) Let 1 < p < oo and let 0 < a: < n. Let be a nonnegative Borel measure on R" . Then the trace inequality (2. 6) holds if and only if

2.6.

w

(2. 10) for every ball B

=

r x) .

B (

The following theorem [MVl] shows that, in a sense, balls may be replaced with single points x.

B Br(x) in (2. 10) =

THEOREM 2 . 7. (Maz'ya-Verbitsky) Let < p < oo and let 0 < a: < n. Let w be a nonnegative Borel measure on Rn . Then the following statements are equivalent. (i) The trace inequal-ity {2. 6) holds. (ii) For every compact set E C Rn , the capacitary inequality (2. 7) holds. < oo dx-a. e. and {iii)

1

law

(2.11)

Ia[(I,w)P'](x) � C Iaw(x)

dx-a. e.

(iv) The trace inequality (2. 6) holds with w replaced with the absolutely contin uous measure dv = dx, or, equivalently,

(Iaw)P'

(2. 12)

v (E

)

=

l (Iaw )P' dx � C Cap0,p (E) ,

where C is a constant which is independent of a compact set E.

WEIGHTED

NORM

385

INEQUALITIES

REMARK 2.8. A simple direct proof of Theorem 2.7 from which Theorem i::; deduced as a corollary can be found in [V4] (see also [VW2]) .

2.6

Another useful characterization of Maz'ya measures in terms of discrete Car leson measures was given in (V4]. COROLLARY 2.9. (Verbitsky) Let 1 < p < oo and let 0 < a < n . Let w be a nonnegative Borel measure on Rn . Then any one of the conditions {i)-(iv) of Theorem 2. 7 is equivalent to the following pmperty: (v) For every dyadic cube P,

)p' IQI :s; Cw(P), � ( I Qw(Q) il-�

(2. 13)

where the sum is taken over all dyadic cubes Q contained in P, and C does not depend on P. w

The following sufficient conditions which are applicable to broader cla.o.;::;es of than those considered in Remark 2.3 and Theorem 2.4 follow from Theorem 2.7

(iv) .

COROLLARY

2 . 10 . (i) If [Otw

E

L8•00 (Rn ) , where s

a(pn- 1 )

I

then the trace

inequality (2. 6) holds. ' {ii) lf t > 0 and J�(!Otw) Cl+<)P dx :s; C I B I 1 _ "'PC�+•l , for eveT'IJ ball B, then (2. 6) holds. =

We now consider the trace inequality (2. 14)

for q # p. In the classical "lower triangle case" q > p there is a definitive charac terization of (2.14) due to D. Adams (see [AH] , [M4] ) . THEOREM

trace inequality

2.1 1 . (D. Adams) Let 1 0 and x REMARK

E

(n- e>p)q P

Rn .

< 1p - Q. then (2. 14) holds only for w = 0. n \'\Then 1q = p1 - Q. and w is Lehesgue measure Theorem 2.11 gives the celebrated n Hardy Littlewood-Sobolev inequality for fractional integrals.

2. 12. If a � � or P

1 q

We next define the following nonlinear potentials which play an important role in modern potential theory and applicatiom; to linear, quasilinear and fully nonlinear partial differential equations (see [AH], [L], [KiMa], [MaZi] , [TW]) . Let

(2.16) and

(2.17)

Ww(x)

=

WOt ..P w (x) =

['XJ Jo

[

]

w (Br (x ) ) p' - 1 dr r n -Otp

r

.

386

I. E. VERBITSKY

We shall use the notion of the energy £(w) defined by

=

Ea: p (w ) of a measure ,

w

on Rn

(2.18)

Note that in the case p potential:

=

2 both Ww and Vw coincide with a classical linear

(2.1 9)

The nonlinear potential Va:,p (p =/= 2) was introduced and studied in detail by Havin Maz'ya, Reshetnyak, and Meyers, and Wa:,p by Adams-Meyers and Hedberg-Wulff (see [AH], [HWJ , [Ml]). The latter is usually called Wolff's potential because of its prominent role in the following Wolff's inequality that appeared in [HW] : (2.20)

Ea:,p (w ) =

( Va:,p w dw � C ( Wa:,p w dw, Jan Jan

where C is a constant which depends only on p, a, and n . The converse inequality is obvious since (2.21) where C is a constant which is independent of x and w (see [Ml], Sec. 7.2.2) . There are alternative proofs of Wolff's inequality (see [AH] . Sec. 4.5) , but the original proof presented in [HW) , which can be easily simplified nsing an elementary "integration by parts" lemma [V4] , remains most direct and leads to important applications to nonlinear PDE [PV2] , [PV3] . REMARK 2.13. It follows from Theorem 2.2 (in the dual form (2.9)) and Wolff's inequality that the trace inequality (2 . 14) for q = p holds if Wa: p w is uniformly bounded. The converse is clearly not true. ,

We next consider the difficult case q < p. A capacitary characterization of (2.14) in this case was obtained by Maz'ya and Netrusov [MN].

w

THEOREM 2. 14. (Maz'ya-Netrusov) Let 1 < p < oo and 0 < q < p < oo. Let be a positive Borel measure on R" and let ¢(t) inf {Capo ,p (F) : w(F) 2: t } .

Then (2. 14) holds if and only if

100 [ ¢�;)q]

=

p�q

�t <

00.

An alternative noncapacitary characterization of (2 . 14) for 0 < of Wolff's potentials was given in [COVI] , [V4] .

q

 1 ; Verbitsky q > 0) Let 1 < q < p < oo. Let w be a po8'itive Borel measure o n Rn . Then (2. 14) holds if and only if

p < oo and 0 < (2 . 2 2)

Wa,p w E L

q(p·-l) v-•

(dw).

All theorems stated above can be carried over to general convolution operators with nonnegative radial kernels [COV2] , [COV3] (see Sec. 3), as well as to integral operators with "quasi-metric" kernels [KV] .

387

WEIGHTED NORM INEQUALITIES

3. Dyadic and radial trace inequalities and nonlinear potentials In this section we concentrate on integral operators with general dyadic and nonincreasing kernels, as well as the corresponding nonlinear potentialt;, and two weight inequalities. Continuous versions for convolution operators with radial kerneb are deduced from the dyadic models using averaging over shifts of the dyadic lattice (see [COV3] ). Let D = { Q } be the family of all dyadic cubes Q in Rn , and K : D --t R+ . The kernel Kv (x, y) on Rn x Rn is defined by radially

KD (x, y)

(3.1 )

=

L K (Q) Xq(x) Xq (y) , QED

where XQ is the charaeteristic function of Q E D. Let a be a locally finite positive Borel measure on Rn, and let f E Lfoc ( da) . We define the dyadic integral operator: Ti
(� .2 )

In case f

=

=

r KD (x, y)f(y) da (y) = L K(Q)xq(x) }Rn Q ED

1, we set TKn [a](x)

=

1Q f da.

L K(Q) u(Q) Xq (x) . QED

If 0 < q, p < +ex:, and a and w are locally finite Borel measures on Rn , the corresponding dyadic trace inequality is given by: f E £P(da) .

(3.3) Assume for a moment that to the inequality:

q, p

> 1 . Duality then gives that (3.3) is equivalent

(3.4)

g

E

u' (dw ).

The quantity on the left-hand side of (3.4) is a generalized version of the discrete energy of dv = gdw. For positive locally finite Borel measures v and u on Rn, the discrete energy associated with v and a is defined by (cf. [HW] ) :

(3.5)

&�. a [v]

=

�

R"'

' (TKn [v] )p da =

1

Rn

(

L K(Q) v(Q) Xq(x) QED

)

p

'

da( x) .

Fubini's theorem gives an alternative expression for &K o- = E�, o- [v]

=

r TK'D [(TK'D [v])P' -1da] dv, }Rn

where TK'D [(TK'D [v])v' -1 da] is a dyadic analogue of the nonlinear potential of Havin Maz'ya originally defined for da = dx and with Riesz kernels in place of KD (see [AH), [M4] ). In the special case where da is Lebesgue measure on Rn , K (Q) = IQi l \a/n), 0 < a < n and I Q I i s the Lebesgue measure of Q , i.e. , when Kv is a discrete Riesz

E. VERBITSKY

I.

388

kernel on Rn, Hedberg and Wolff introduced a dyadic nonlinear potential defined by: W� dx [v] (x) =

(3. 6)

2:.::

(;��)p'-l XQ(x) .

QEV Q of Q . ) A

(Here eQ denotes the side length dyadic version of Wolff's inequality established in [HW] shows that, for 1 < p < +oo, (3 . 7)

Con�equeutly, the trace inequality (3.3) holds for q = 1 , 1 < p < +oo, and da = dx if and only if W� dx [w] is in L 1 (dw) . For 1 < q < p < +oo, as was shown in q(p-1)

[COVl] , (3.3) holds if and only if W� dx [w] E L p - q (dw ) . We next define a suitable nonlinear potential associated with a pair of measures , a v and the kernel Kv so that it is applicable to characterization of the trace inequality (3.3) . Let v and a be positive locally finite Borel measures on Rn. We denote by K(Q) the function " K ( Q')a ( Q')XQ' (x ). K (Q) (x) = 1 � (3. 8)

cr( Q ) Q CQ '

For x E Rn , we set

(3.9)

W�, a [v] ( x)

=

)

t 2:.:: K ( Q) a ( Q) (j{Q K (Q )(y) dv(y) p' - XQ(x).

QEV

It is worthwhile to observe that several other natural alternatives to Wf, 17 [w] discussed in [COV2] fail to satisfy the desired analogue of Wolff's inequality. In [COV2] , the following Wolff-type inequality was established for an arbitrary positive measure v on Rn, and dx in place of dcr: C1 £r, dx[v] ::;

{

}Rn

wr,dx [v] dv ::; C2 £r,dx[v],

where Ct , C2 are constants which do not depend on v . Its generalization to a two-weight setting is also true [COV3] .

THEOREM 3 . 1 . Let K : 1J -t R+ . Let 1 < p < + oo , and let v and a be locally finite positive Borel measures on Rn . If £'f 17 [v] and W'f u [v] are defined respectively by (3.5) and (3.9), then

,

C1 £f, a [v] ::;

{

}Rn

'

Wf, ,. [v](x) dv(x) ::; C2 E'f, ,. [v] ,

where c1 ' c2 are constants which do not depend on v and

We now state a two weight inequality proved i n [COV2] for 1 < q 1 .

w

cr .

THEOREM 3 . 2 . Let K : 1J -t R+ . Let 0 < q 0 such that the trace inequality

hold8.

r I TKn [fda] i q (x) dw (x ) ::; c llflll,(d�)' }Rn Then WTJ (dw) . K. � [w] E L

f E £P(da),

q(p-1) p-o

(ii) Conver·sely, if W�. 17 [w] E L ·��•1 > (dw) then the preceding trace in�quality holds provided the pair ( K, a ) satisfies the dyadic logarithmic bounded oscillation condition (DLBO) : sup K( Q) (x) :::; A inf K(Q) ( x ) ,

xEQ

xEQ

where A does not depend on Q E D. If q = 1 then statement (ii) holds without the restriction (K, a) E DLBO. (In this Theorem 3.2 is, by duality, an immediate consequence of Theorem 3. 1 . ) The proofs of Theorems 3.1 and 3.2 are based on the following important lemma [COV2]. Let 1 < s < + oo , A = ( >.Q ) Q e TJ , AQ E R+ , and let a be a positive locally case

finite Borel measure. We assume that >.Q that 0 · oo = 0. We define A, (A)

�

A2 (A) =

A3( A) = LF;MMA

=

0 if a( Q) = 0, and follow the convention

L" (� a��) Xq (x) ) L >.Q

[ sup la xEQ n

da(x ) , s

(�L ) ( L ) a( )

QED

'

a

Q'CQ

(lQ)

AQ ·

Q ' CQ

>.Q'

-1 '

s

da(x) .

3.3. Let a be a positive locally finite Borel measure on Rn . Let 1 < s < oo. Then there exist constants Ci > 0, i = 1, 2, 3, which depend only on s, such that, for any A = (>.Q ) Q e TJ, >.Q E R+ , A1 (A) :::; C1 Az(A) :::; C2 A3(A) ::; C3 A1 (A) .

Theorem 3.1 is deduced from Lemma 3.3 in the case

s = p'.

>.Q

=

K(Q)w ( Q) a ( Q ) and

We next treat continuous versions of the above theorems for integral operators with radial kernels, Tk [f da] ( x ) =

{

Jan

k ( ix - yl) f(y) da(y).

Here k = k ( r) , r > 0, is an arbitrary lower semicoutinuous nonincreasing positive function. The corresponding nonlinear potential is defined by Wk, u [w](x) =

where

-

{

+oo

Jo

k (r) a (Br (x))

(1

Br(x)

k(r) (y) dw (y )

ds k (·r) (x ) = a 1r ( ) Jor k ( s) a (B s (x) ) � , ) (B x

)p'-l

d

_!_ , r

I. E. VERBITSKY

390

for x E Rn, T > 0.

THEOREM 3.4. Let 0 < q < p < +oo, 1 < p < oo, and lei. w and a be nonnegative Borel measures on Rn . A ssume that a satisfies a doubling condition, and the pair- ( k, a ) has the following logarithmic bounded oscillation property (LBO) : yEBr (x)

sup k(r) (y) � A

(3. 10)

.Ln

holds if and only

inf

k(r)(y) ,

E Rn, r > 0. Then the trace inequality f E LP(da ) , I Tk [fda] iq dw � C ll fii1P(da}'

where A does not depend on x (3. 1 1)

yEBr(x)

if Wk,,- [w] E L q(p-1) v-·� (dw ) .

The (LBO) property is satisfied by all radially nonincreasing kernels in the case a satisfies a reverse doubling condition (see [COV2] ) . In particular, the following theorem holds for convolution operators Tk [/] k*f and da = dx. We define the corresponding \Volff's potential by

da = dx, and also by Riesz kernels k(x) = lxla-n, 0 < a < n, if

=

(3.12)

Wk [w] (x)

=

roo o k(r) k(r·) ,:1 w (Br(x)) v!:l rn-l dr, J

1 k(r)(x) = n

where

_

(3.13 )

r

for x E Rn, r > 0.

ir k( s) sn-l ds , o

THEOREM 3.5. Let 0 < q < p < +oo and 1 < p < oo. Let w be a nonnegative Borel measure on Rn. Suppose k = k( I x - y I), where k( r) is a lower semicontinuous nonincreasing positive function on R+, and Wk [w] is defined by (3. 12). Then the trace inequality (3.14 ) holds

if and only if

(3. 15)

Wk[w] E L � (w). p-o

REMARK 3.6. For Riesz kernels, a proof of Theorem 3.5 was given in (V4] . Some technical details related to passing from a discrete to continuous version using shifts of the dyadic lattice, as well as generalizations, can be found in [COV3] . REMARK 3.7. A characterization of (3. 14) for Riesz or Bessel kernels in terms of capacities was given in (MN] (see Sec. 2). The special case q = 1 of Theorem 3.5 leads by duality to Wolff's inequality for radially nonincreasing kernels [COV2] , [COV3) . THEOREM 3.8. Let 1 < p < oo. Let w be a nonnegative Borel measure on Suppose k = k(lx - y l), where k(r) is a lower semicontinuous nonincreasing positive function on R+ , and Wk[w] is defined by (3. 1 2). Then there exist positive constants cl ' c2 which depend only on k / p and n such that

Rn.

(3 . 16)

Ct l lk * wl l�v' (Rn)

�

ln Wk[w) dw � C2 ll k * wi i�P' (R")'

WEIGHTED NORM TNRQUALITIES

39 1

Theorem 3.8 demonstrates that (3. 12) is an appropriate definition of Wolff's potential for radially nonincreasing kernels. This solves a problem posed in [AH] , p. 214. 4. Indefinite weights and Schrodinger operators

We start with some prerequisites for our main results. Let D(Rn) = C
[l_, (lxl -2 lu(x) l 2 + IV'u(x) l 2 ) dx]

1

2

In this section, we assume that n � 3, since for the homogeneous space L 1 • 2 (Rn ) our rt>Bults become vacumL<> if n = 1 and n = 2. Analogous results for inhomoge neous Sobolev spaces WL2 (Rn) hold for all n � 1. For V E D'(Rn), consider the multiplier operator on D(Rn) defined by < V u. v > := < V, u v > ,

u, v E D(R") , where < · , · > represents the usual pairing between D(Rn) and D' (Rn). Let us denote by £-L2(Rn) = £1•2(Rn)* the dual Sobolev space. If the scsquilinear form < V · > is bounded on £ 1 • 2 (Rn ) £1•2 (Rn): (4. 1 )

(4.2)

x

·,

where the constant c is independent of u, v E D(Rn), then V u E L - 1 •2(Rn), and the multiplier operator can be extended by continuity to all of the energy space £ 1•2 (Rn ) . (As usual, this extension is also denoted by V . ) We denote the class of multipliers V : £1•2(Rn) --. £ - 1 •2(Rn) by

M ( L t ,2 (Rn ) -t £ -t,2 (Rn)). Note that the least constant in (4.2) is equal to the norm of the multiplier operator: I I V I I M(L' ·,(R")-.L-12(R")) = sup { 11 Vul i£-1.2(R") : llullv.2(R") :::; 1 } . For V E M(£1•2(Rn) --. L -1•2(Rn)), will need to extend the form < V, u v > defined by the right-hand side of (4.1) to the case where both u and v are in c

£ 1•2 (Rn ) . This can be done by letting

we

< V u, v >= Nlim < V uN, VN > , -+oo

where u = limN-oo UN , and v = limN-.oo VN , with uN , VN E D(Rn) . It is known that this extension is independent of the choice of UN and 1'N · We now define the Schrooinger operator H = Ho + V, where Ho = -Ll, on the energy space £1•2 (Rn). Since H0 : £1•2 ( Rn) --. £-1•2 (Rn) is bounded, it follows that H is a bounded operator acting from £1• 2 (Rn) to L - 1 • 2 (Rn ) if and only if V E .M (£ 1 , 2(Rn) --. £ -1 •2 (Rn )) . Clearly, (4.2) is equivalent to the boundedness of the corresponding quadratic form: I < V u , u > I = I < V,

lu l2 > I :::; c i i'Vull i2(R" ) •

I.

392

E. VERBITSKY

where the constant c is independent of u E D(Rn). If V is a (complex-valued) measure on Rn, then this inequality can be recast in the form: (4.3) For positive distributions (measures) V, this inequality is well studied (see Sections 1 and 2). We now state the main result for arbitrary (complex-valued) distributions V. By Lfoc(Rn)n = Lfoc (Rn) ® C" we denote the space of vector-functions :f = (ft . . . . r ) such that ri E L�c (Rn), i = 1, . . . , n. THEOREM 4.1. Let V E D' (Rn ) . Then V E M(Ll.2(Rn) ,

n

the inequality

__.

L-1•2(Rn)), i. e.,

I < V u, v > I S:: c l lul l u . 2 (R"J l lvi1Ll, 2 (R" l

(4.4)

holds for all u, v E L1•2{Rn), if and only if there is a vector-field :f E qoc (Rn) such that v div r, and =

(4.5) for all

f 2 1f(x ) l 2 dx s:: c f I Vu(xW dx, jR" 1 '1L(x) l }Rn

u E D(Rn). The vector-field :f can be chosen in the form :f

=

\7 .6. -l 17.

REMARK 4.2. For :f = \7 .6.-1 V , the least constant C in the inequality (4.5) is equivalent to I IV I I 2M(Ll,2 (R" )-L-1.2 (R")) .

The proof of the "if" part of Theorem 4.1 is easy as long as V is represented in the divergence form. This idea was discovered by mathematical physicists in the 1970s (sec [MV2] , p. 265) . Indeed, suppose that V div f , where f satisfies (4.5). Then using integration by parts and the Cauchy-Schwartz inequality we obtain: =

I < v u , v > I = I < v, u v > I

f, v vu > + < r, ·u. vv > I S:: l l fv i i L2(R")" l l \i' u l l u (R"' ) + l l fu i iL2(R" ) l l\i'vi i P(R") S:: 2 VC [IV'u[ [P(R") [ [V'vl lucR") : =

I <

where C is the constant in (4.5) . The proof of the "only if" part of Theorem 4.1 given in [MV2] is much more complicated. It is based on a combination of methods of harmonic analysis and potential theory. We outline the main ideas in a series of lemmas and propositions stated below. In the first lemma, which is only a preliminary estimate, it is shown that f = \7 .6,- l y E Lf0c (R1' )n , and a crude estimate of the average decay of f at infinity is given. We observe that expressions like \7 .6. - l V should be understood in a specific sense, e.g. , in the sense of weak BMO convergence. The latter is discussed in detail in [MV6J . LEMMA 4.3. Suppose that

( 4.6)

393

WEIGHTED NORM INEQuALITIES

Then :f = \7 � - l y E Ltoc(Rn )n, and V = div :f in D'. Moreover, for any ball BR (xo) (R > 0) and f > 0, (4.7)

}{

BR(xo )

j :f(xW dx :::; C (n, f) Rn-2+ of equilibrium potentials ast>ociated with a set of positive capacity.

PROPOSITION 4.4. Let (j > ! and let P = Pe be the equilibrium Newtonian potential of a compact set e C Rn of positive capac·ity. Then

(4.8)

i ! V'P i !L2( R")

REMARI< 4.5. For 8 :S: PROPOSITION 4.6.

v E L 1•2(Rn ) . Then

(4.9)

I ! 'Vv li£ 2 (R" ) :S:

2

=

� Jcap (e) . 26 - 1

�' it is easy to see that \7p& ¢ L2 (Rn ) .

Let 8 > 0, and let v be a real-valued function such that

r

}R

"

' \i' (v P� ) (x) l 2 P dx 20 :S: (x)

''2

( 8 + 1 ) {48 + 1 ) l i'Vv £2(R")·

The proof of the preceding inequalities is based on multiple integration by parts, along with the following estimates for Newtonian potentiaL'> of positive (not necessarily equilibrium) measures w.

PROPOSITION 4.7. Let w be a positive Borel measure on Rn such that P(x) = l2w (x) ¢. oo . Then the following inequalities hold:

(4.10 ) and

v E D ( R" ) ,

2 j\7 P(xW r 2 Jn n v ( x) P (x) 2 dx :::; 4 I I 'Vv l i £2 (R" ) '

(4. 11) REMARK 4.8. The constants 4 and 1 respectively in (4. 10) and (4. 11) are sharp (see [MV2]) . REMARK 4.9. An inequality more general than (4.11), for Riesz potentials of order a E (O, n) and Lp norms (with nonlinear Wolff's potential in place of P (x)) , but with a different constant was proved in [V4].

Vle will also need the following proposition which is deduced from the facts that P(x)28 is an A2 weight (this was proved earlier in [MVl]) , and that the Riesz transforms are bounded in weighted L2 spaces with such weights [St] . (See details in [MV2] .) PROPOSITION 4.10. Let w = � - 1div ¢' where 1 < 28 < n�2 • Then

(4.1 2)

¢'

E

D

0

C".

2 dx r r - z dx Jn" j\i'w (x) j P (x) 21i :S: C( n, 8) }R " j¢(x) j P ( x ) 2� ·

Suppose that

I . E. VERBITSKY

394

\Ve now sketch the proof of the "only if" part of Theorem 4. 1. S upp ose that I < V, u v > I :S: I IV I I M(J,' ·2(R"J-D;-' C R")) l l"vu i i P CR"") I I Y'vi i P C R"" J ·

($ = ( ¢ 1 , . . . , ¢ n) b e a n arbitrary vector-field in V ® en, and let w = �-l div ($ = - J2 div {$, (4.13)

Let

so that

¢ = V'w + s, div s = 0. Note that w E C00 (Rn ) n L1,2(Rn ) , since w(x) = O( l x l 1 -n) and I Y' w(x) l = 0 ( lx l -n)

Hence,

I < V, W > I = I < f, ¢ > I =

l ln

as

lxl --> oo .

f(x) ;f(x) dx ·

l

where by Lemma 4.3, f E LfocCRn) n . We pick 8 so that 1 < 28 < n:::_2 , and factorize w (x) = u (x) 'u(x), where w (x) v = P(x 6 " ) Consequently, -

-

0 I < r , ¢ > I :::; I I V I I M(Ll·Z (R")-- > L21(R" )) I I V'P I I £2( R" ) I I V'vi iL2{ R" ) · B y Proposition 4.6 { 2 dx 2 dx 2 r 0 I I V'vi i£2 (R") :S: J j V'( vP ) (x) l P(x) 2 5 = J " IY' w(x) J P(x) 26 < oo . R R" From this, applying Proposition 4.4, we obtain:

I .Ln

f(x) · ¢(x) dx :S: 0 (1 - 28) - � I IV I I M(£1·2(R" )--.L2" l (R" ))

l

x

cap (e

Notice that by Proposition

Hence,

1 )2

4.10,

(

{

2 dx JR" I V'w(x) l P(x) 28

.

{ 2 dx 2 dx r }Rn IV''w (x) l P (x) 26 s C( n , o) }Rn l ¢(x) l P(x) 25 .

x

cap (e) �

(J�..

2 l ¢(x) j

From the preceding inequality we deduce

(L,

)�

2 lf(xW P(x ) 5 (x) dx

)

1

2

:s: C(n,

� 5) p 2

1 2 •

o) I I V I I cM(£L2 (R")--> L-1,2 (R" ) ) cap (e) � .

Note that P is the equilibrium potential of e , and hence P(x) � 1 dx-a.e. on e. Thus,

WEIGHTED NORM INEQUALITIES

395

for every compact set e c R11, and by Theorem 2.2 this gives ( 4.5), which completes the proof of Theorem 4 . 1. There is an analogue of Theorem 4.1 in terms of ( -D.)- 1 1 2 V .

THEOREM

4. 1 1 . Under t.h.e a.ss?J.m.ptirYn.<; of Theorem. 4 . 1, V E M (L 1 2 ( Rn ) ---+ L - 1•2 ( Rn ) )

if and only if (-D. ) -112 V

•

i t follows that

M(L1•2 ( Rn ) ---+ £2(R11 )) .

E

5. Infinitesimal form boundedness, Trudinger's and Nash's inequalities Throughout this section we will be using the following notation and conventions. We denote by W1•2 (R11) the Sobolev space of weakly differentiable functions on Rn (n 2: 1 ) such that

JJ u l lw1 2(an)

=

J JuJJP( an) + JJ'Vu JJP (an)

< +oo ,

and by w- 1 • 2 (R11) = W 1 • 2 (R11)" the dual Sobolev space. For a compact set e C R11 , the capacity associated with W 1 •2 (R11) is defined by cap (e) = in£

{ l lu l l�rt- 2 (an)

:

u

E

Cg' (Rn ),

u(x) >

f E L]0c(R11) such that I IXBl(xo) f iiLr(an ) < 00.

For 0 < r < oo, we denote by L�nif(Rn) all

IIJI IL� Dif

sup

=

xoERn

}

1 on e .

By U(R11)11 = U(R11) :29 en we denote the class of vector fields :f = {ri }j= 1 R11 ---+ en, such that rj E Lr(R"), j = 1 , 2, . . . , n, and use similar notation for other vector-valued function spaces. By M + (R" ) we denote the class of nonnegative locally finite Borel measures on R11 • If V E V'(R11) is nonnegative, i.e., coincides with w E M + (R11 ) , we write fan J u (x ) l 2 dJ..V in place of (V, J u J 2) = (Vu , u) for the quadratic form associated with the distribution V, if u E C0 (Rn). Sometimes we will use fan J u(x) l2 V(x) dx in place of (Vu, u) even if V is not in L foc (Rn) . We set 1 r f (x) dx mB ( f ) = for a ball B

C

TBl ./B

Rn, and denote by BMO(R") the class of f E Lioc (R11 ) for which sup

xuER" , ti>O

�

I Bli

Xo ) I

r

.JB6(xo)

lf( x ) - mn6(xo) U)I" dx < +oo,

for any 1 .::; r < +oo. It follows from the John-Nirenberg inequality that this definition does not depend on the choice of r E (1 , +oo) . We will also need an inhomogeneous version of BMO(R11) (the so-called local BMO) which we denote by bmo(R11) . It can be defined in a similar way as the set of f E L�nif(Rn ) such that the preceding condition holds for 0 < o .::; 1 (see (St] , p. 264) . The Morrey space _Lr. .>. (Rn) ( r' > 0, A > 0) consists of f E L!oc (Rn) such that sup

xoERn , 6 >0

1

r lflr dx < +oc . J B.s (xo)J-n JB8 ( x o) >.

396

I. E. VERBITSKY

In the corresponding iuhomogeneom; analogue, we set 0 < /j � 1 in the preceding inequality. It will be clear from the context which version of the Morrey space is used. We now state the main results of [MV5] .

.

5 1 Let V E V'(Rn), n ? 2. Then the following statements hold. (i) Suppose that V is represented in the form:

THEOREM

.

(5.1)

where f E Lfoc (Rn)n and 'Y E Lfoc(Rn) satisfy respectively the conditions: (5.2)

lim

o_.+D xoE R" . 1 1m

(5.3)

f8

sup sup u

sup sup

o�+DxoE R"

6

(xo ) Jf(xW Ju(x)i2 dx

2 J J V'uJ J£2 (B�(x0) )

f86 (xo) lr(x) J J u (x) i 2 dx

2 I J Y'uJI P (Bo�(xo) )

=

=

0,

0,

where u E CQ'" (B0 ( x0)), u =:/= 0. Then V is infinitesimally form bounded with respect to -6.. , i. e., for every E > 0 there ex·ists C(E) > 0 such that (1.11) holds. (ii) Conversely, suppose V is infinitesimally form bounded with respect to 6.. Then V can be represented in the form (5.1) so that both (5.2) and (5.3) hold. Moreover, one can set f = -Y' ( 1 - 6.. ) -1 V and 1 = ( 1 - 6.. ) -1 V in the representation u

-

.

(5. 1 ) .

REMARK 5.2. I n the statement of Theorem 5.1 one can replace conditions (5.2) and (5.3) with the equivalent condition where J(1 - 6.. ) - � V J 2 is used in place of J f J 2 in (5.2). The importance of Theorem 5.1 is in the means it provides for deducing explicit criteria of the infinitesimal form boundedness in terms of the nonnegative locally integrable functions J fl2 and l r l · THEOREM

5.3. Let V E V' (Rn ) , n ? 2. The following statements are equiva

(i) V is infinitesimally form bounded with r-espect to (ii) V has the form (5. 1) where f = -\7(1 - 6.. ) - 1 V, measure w E Af+ (Rn) defined by lent:

(

-6.. . 1 =

(1 - 6._)-1 V, and the

)

dw = l f (x W + !r( x) J dx

(5.4)

has the property that, for ever-y E

>

0, there exists C( f) > 0 such that

J u (x W dw � € J J V'uJ Ji2( R" ) + C(t: ) J J V'uJJl,( R") '

{ Jnn (iii) For w defined by (5.4),

(5.5)

(5.6)

. hm

J-.+U P.o: diam Ra _< {) sup

1 -(R0 ) W

w ( P)2 -0 � I P J 1 - l. Pt:;.Po

Vu E C0 (Rn) .

"'

n

'

where P, Po are dyadic cubes in Rn, i.e., sets of the form 2i(k + [0, l) n ), where i E Z, k E Z".

397

WEIGHTED NORM INEQUALITIES

{iv} For w defined by (5.4), o-++O " ' di am e�o

(5.7)

lim

sup

w(e) -- =

cap (e)

0,

where e denotes a compact set of positive capacity in Rn . {v) For w defined by (5.4),

ii wB6 (xo ) ��- 1 2 (Rn) = 0, w(Bt� (xo)) o-++ 0 xo ERn restriction of w to the ball B,s(xo). .

(5.8)

hm

sup

where wB6 (xo) is the (vi} For w defined by (5.4) , (5.9)

where G1 * w = (1 - t!,) - 1 w is the Bessel potential of order 1.

It is worth noting that although Theorem 5.3 holds in the two-dimensional case, its proof requires certain modifications in comparison to n ;::: 3. In the one dimensional case, the infinitesimal form boundedness of the Sturm-Liouville oper + V on L2 ( R1 ) is actually a consequence of the form boundedness. ator H =

-l;.

THEOREM 5.4. Let V E 1)' ( R1 ) . Then the following statements are equivalent. (i) V is infinitesimally form bounded with respect to i. e., (ii) V is form bounded with 1espect to I( V u, u ) [ :::; canst l lull�fl.2( ' ' 'r/u E cgo (R 1 ) .

�.

d�2 ,

R)

{iii} V can be represented in the form V

(5. 10)

sup

xE R 1

=

�;; + "'f , whe1e

1x+l (jr(x) 2 + i"'f(x) ) dx < l

l

x

+ oo .

(iv} Condition (5. 10) holds where

r(x)

=

{

jR '

sign (x - t) e - l x-tl V (t) dt,

'Y(x)

=

{

jR'

e - ! x - t l V(t) dt

are understood in the distributional sense. The statement (iii)=> (i) in Theorem 5.4 is known ( [Sch], Theorem 1 1 .2.1), whereas (ii)=>(iv) follows from [MV2] . We now state a characterization of the form subordination property (1 .15). It was formulated originally in [Tru] , in the form of the inequaliLy:

(5. 1 1)

i ( Vu, u) l

:S

€

i [ Vu[ [i2(Rn) + C€- v l lu l l i•(Rn)'

'r/u

E Cgo (Rn) ,

for V ;::: 0 . Such V are called €v -compactly bounded in [Tru] . It follows from Nash's inequality that (1. 15) yields (5.11) with '{) = n!2 {3 + � ; the converse is also true, provided v > �, and is deduced using a localization argument. In the critical case '{) = � , (5.11) holds if and only if V E L00 (Rn ), while for 0 < v < � ' it holds only if v = 0 . Necessary and sufficient conditions for ( 1 . 1 5 ) , or equivalently (5.11) with v = �{3 + � (see [MV5] ) , can be formulated in terms of Morrey-Campanato spaces

398

L

E. VERBITSKY

using mean oscillations of the functions f and rems 5.1�5.4.

1

which have appeared in Theo

5.5. Let V E V'(Rn), n 2: 2, and let 0 < f3 < +oo. (i) Suppose there exists t:o > 0 such that ( 1.15) holds for e11ery t: E (0, t: 0 ) . Then V can be represented in the form THEOREM

(5.12)

V

=

div f + ')',

where f = - V' ( l - �)-1 V E Lf0c (Rn) n and 1 = ( 1 - �)- 1 V E Lf0c (Rn) . Moreover, there exists So > 0 such that (5.13)

{

}Bo(xo)

lf(x) - ffi£� (xo) (f) l2 dx :S c Sn� 2 �+i ,

0 < S < Jo,

r b(x) l dx :::; c sn� ff!-r ' 0 < 8 < 6o , }Bo (xo )

(5. 14)

where c does not depend on xo E Rn and So . Furthermore, f E L�nif(Rn) n if (3 2: 1, and f E L= (Rn) n if 0 < f3 < 1. (ii) Conversely, if V is given by (5.12) where f E Lf0c(Rn)n, 1 E Lfoc (Rn) satisfy (5.13), (5. 14) for all 0 < S < So then there exists Eo > 0 s·uch that (1.15) holds for all 0 < t: < t:o . REMARK 5.6. (a) In the case (3 = 1, it follows that (5. 13) holds if and only if f E bmo(Rn) n . In other words, V E bmo_ 1 (Rn), where bmo_1 (Rn) can be defined as the space of distributions f that can be represented in the form f = div § where § E bmo(Rn)n. We ohf;erve that bmo�1 (Rn) = F.:.•{"' (Rn), where F!;,·q stands for the scale of inhomogeneous Triebel�Lizorkin spaces (see, e.g., [KT], [T] ) . (b) I n the case 0 < f3 < 1, (5.13) holds if and only if f is HOlder-continuous: lf(x) - f(x' ) l :S c l x - x'l i3Tl , -

-

1 - iJ

lx - x' l <

Oo .

For f3 > 1, (5. 13) holds if and only if f lies in the inhomogeneous Morrey space /3-1 C 2 , n� 2 13 + 1 ( Rn t , i.e., - )l �Bo (xo) lr(x

2

dx

� c bn- 2 .!L::l

ll+ l ,

0 < o < Oo .

•

These statements follow from the known characterizations of Morrey spaces. Note 213 that, according to (5. 14) , 1 E £1 , n � P+I (Rn). REMARK 5.7. (a) An immediate consequence of Theorem 5.5 is that, for all (3 > 0, (1.15) is equivalent to the following localized energy condition:

1 1 (1 - � ) - � ( 17b, x0 V ) l l l2(Bo ( :ro ) ) :S c on- 2 ��i , 0 < o < Oo, Xo E Rn, where 1/Q, x0 (x) = ry (o-1 (x - x0) ) ; here ry is a smooth cut-off function such that 17 E c= (B1 (0) ), 0 :S TJ :S 1, and ry = 1 on B (0) . i (b) A similar energy condition, is

1 1 ( 1 - � ) - � ( TJ&, xo V ) l ll2 (Rn) :::; c on - 2 �+� ' 0 < 0 < Oo, Xo E R", sufficient, but generally not necessary in the case n = 2.

WEIGHTED NORM

399

INEQUALITIES

We next state a criterion for the multiplicative inequality (1. 18) to hold, which is equivalent to a homogeneous version of (1. 15) with t:o = +oo and p 13�1 . =

E V'(Rn), n 2: 2, and let 0 < p < 1 . (i} Suppose that (1. 18) holds. Then V can be represented in the form

THEOREM 5.8 . Let V

(5. 15) where f (5. 16)

=

V1.6. - 1 V,

V

=

div f,

and one of the following conditions hold:

f E BMO(Rn) n

if

p

=

�;

f E Lip1 _2p (Rn)n

f if(xW dx -::: c <>n+ 2- 4v, jBs (xo) where does not depend on x0 E Rn , 6 > 0. (5. 17)

if 0 < p < �;

if � < p < 1;

(ii) Conversely, if V is given by (5.15) where f E L?oc ( Rn )'� and satisfies (5.16), (5.17), then (1. 18) holds. c

REMARK 5.9. In Theorem 5.8, the "antiderivative" f = V1.6. -1 V can be re placed with (-.6.) - � V . Furthermore, a corollary we deduce that ( 1 .18) holds if and only if V E BM0_ 1 (Rn) = P:·� (Rn) for p = � ' where BM0_ 1 (Rn) is defined as above in the caBe of its inhomogeneous counterpart bmo_I (Rn). (See, e.g., [KT) where this space is thoroughly studied in the context of Navier-Stokes equation�:�.) For 0 < p < �' ( 1 . 18) holds if and only if V E B�2;' (Rn) for 0 < p < �- Here .fr:;.,q and B�,q are homogeneous Triebel-Lizorkin and Besov spaces respectively (see aB

[T] ) .

In the case p ! , statement (ii) of Theorem 5.8 (sufficiency of the condition BMO) is equivalent via the 1-£1 - BMO duality to the inequality Vu E Ctf (Rn). (5. 18) l l u Vui iHl (R" ) 'S c l l u i i £2 (R") 1 1Vui i £2(Rn) , Here 1-£1 (Rn) is the real Hardy space on an ([St]). The preceding estimate yields the following vector-valued inequality which is used in studies of the Navier-Stokes equation, and is related to the compensated compactness phenomenon [CLMS) : (5. 19) div it = 0, l l (it · V ) itl l1il (R" ) <::: c l liti i £2 (R") 1 1 Viti i£2(R" ) • for all it E C0 (Rn)11• f

=

E

6. Form boundedness of general second order differential operators In this section we discuss analytic characterizations of form boundedness for the general second order differential operator n n (6.1)

£

=

L

i,j=l

aij 8i8i +

L b.i 81 + c,

j =l

where aij , hi , and c are real- or complex-valued distributions, on the Sobolev space W1• 2 (Rn), and its homogeneous counterpart L1• 2 (Rn). These results were ob

tained in [MV6) . One of the motivations is to give a criterion for the relative form boundedness of the operator b · V + V with distributional coefficients b and V with respect to the Laplacian .6. on L2(R"). This ensures, in view of the so-called KLMN Theorem

400

I.

E. VERBITSKY

(see [EE] , Theorem IV.4.2; [RS] , Theorem X.l7) , that £ = A + b · V + V can be defined, under appropriate smallness assumptions on b and V, as an m-sectorial operator on L2 (Rn) so that its quadratic form domain coincides with W 1 • 2(Rn) . I n particular, we can deduce a characterization of the relative form boundedness for the magnetic Schrooinger operator M

(6.2)

= (i v + a)2 + v,

with arbitrary vector potential a E Lfoc(Rn)n, and V E D' (Rn) on L2 (Rn) with respect to �This approach is based on factorization of functioru; in Sobolev spaces and integral estimates of potentials of equilibrium measures discussed above, combined with compensated compactness arguments, commutator estimates, and the idea of gauge invariance. vVe are able to treat general second order differential operators, and establish an explicit Hodge decomposition for form bounded vector fields. It is worth mentioning that in this decomposition, the irrotational part of the vector field is subject to a more stringent condition than itH divergence-free counterpart. We observe that no additional assumptions (like ellipticity) are imposed on the coefficients of £. In particular, withm1t loss of generality we can and ,.n.n assume that the principal part of £ is in the divergence form, i.e., (6.3)

£ u = div (A V'u) + b · Vu + V u ,

where A = (aij )�.i=l E D'(Rn)nxn, b = (bj )J=l E D'(Rn)n, and V E D' (Rn). We will present necessary and sufficient conditions on A, b, and V which guar antee the boundedness of the sesquilinear form associated with £: (6.4) where the constant C does not depend on u, v E Ccf (Rn). Here L1 2 (R"') is the completion of (complex-valued) Ccf (Rn) functions with respect to the norm llull u . 2(R") = I I Vu i i L2(R")· Equivalently, we will characterize all A, b, and V such that •

(o. 5 )

is a bounded operator, where L - 1•2(Rn) L1• 2(Rn)* is a dual Sobolev space. In the special case where A, b and V are locally integrable, the form bounded ness of £ may be expressed in the form of the integral inequality =

(6.6)

I L" (-(A Vu) · 'Vv

+ b · 'Vu v + Vuv)

dxl

:::;

C iiul l u. 2(R") I Ivllu. 2 (R") •

where the constant C does not depend on u , v E Ccf (Rn). Sometimes it will be convenient to write (6.4) in this form even for distributional coefficients aij , bj , and v.

To state our main results, it is convenient to introduce the clasH of Maz'ya rnea.
WEIGHTED NORM INEQUAL[TrES

401

where the conotant C does not depend on u. We will call measures w E wt� 2 (Rn.) admissible. For admissible V (x) dx with nonnegative density V E L�oc (Rn), we will write V E W1� 2 (Rn'). Inequalities of this type (with w possibly singular with respect to Lebesgue measure) have been thoroughly studied. A straightforward consequence of (6.7) is 1 2 that if w E Wl+ (Rn) then (6.8)

dw(y) � const r"- 2 ,

1

ix-yl 0, x E Rn, if n 2 3, and w = 0 if n = 1, 2 (see e.g. [M4], Sec. 2.4). A close sufficient condition on V E L�0c (Rn), V 2 0, which ensures that V E wt� 2 (Rn), is provided by the Fefferman-Phong cla..'lS (6. 9)

1

ix-yi
vt+• dy

� const rn- 2 (1+•) ,

where E > 0, and the constant does not depend on r > 0, x E Rn. A complete characterization of the class of admissible measures wt� 2 (Rn ) can be expressed in several equivalent forms discussed above. These criteria employ var ious degrees of localization of w, and each of them has its own advantages depending on the area of application. We now otate the main form boundedness criterion [MV6] . For A = (aij ) , let At = (aji) denote the transposed matrix, and let Div : D'(Rn)nx n --> D'(Rnt be the row divergence operator defined by

n

Div(aij ) = (L: Oj aij)'?=l ·

(6. 10)

j= l

6. 1 . Let L = dh· (A V' · ) + b V' + V, where A E D' (Rn)nx n , b E D'(Rn)n and V E D' ( R" ) . n 2 2. Then the following statements hold. (i) The sesquilinear form of L is bounded, i.e., (6.4) holds if and only if � (A + At ) E Ux:; (Rn )nxn , and b and V can be represented respectively in the form THEOREM

·

b = c + Div F,

(6. 1 1)

V = div h,

where F is a skew-symmetric matrix field such that

(6.12)

F - � (A - At ) E BMO( Rnt x n ,

whereas c and h belong to (6 . 13)

Lroc(Rn)n,

and obey the condition

lcf + lhl 2 E

wt� 2 ( Rn) .

(ii) If the sesquilinear form of L is bounded, then c, F, and h in decomposition (6. 11) can be determined explicitly by h = V' (D.. - 1 V), c = V'(A- 1 div b) , (6 .1 4)

(6. 15) where (6.16) and

(6.17)

402

I. E. VERBITSKY REMARK

placed with

Condition

6.2.

in statement (ii) of Theorem

(6.16)

may be re

6.1

(6. 18) which ensures that decomposition (

REMARK

� (A + At ) E

In the case n

6.3.

L00 (R2 ) 2 x 2 ,

REMARK

6.11) holds. = 2, we will show that (6.4)

b - � Div (A - At)

holds if and only if

E BM 0 _ 1 (R2 ) 2 , and

V

=

div b =

0.

Expressions like \7(�-1div b) , Div ( � - 1 curl b') , and \7(�-1 V)

6.4.

ru;ed above which involve nonlocal operators are defined in the sense of distributions . This is possible,

as

is shown in

[MV6] ,

since �- 1div b, �-1curl b, and �-1 v

can be understood as the limits in the sense of the weak BMO-convergence (see

[St] , p. 166) of, respectively, � -1 div (1/JN b) , � -1 curl ('1/JN b) , and � -1 ('1/JN V) as N --+ + oo . Here ¢N is a smooth cut-off function supported on {x : l xl < N}, and the limits above do not depend on the choice of It follows from Theorem

1/JN .

that £ is form bounded on

6.1

L1· 2 (Rn)

x

L 1• 2 (Rn)

if and only if the symmetric part of A is essentially bounded, i.e . , � (A + At) E

L "" (Rn) n x n ,

and b1 ·

\7 + V is form bounded, b1

(6.1 9) if

=

-

-

b-

In particular, the principal part

where

2 Dtv(A - A ) . ·

I

Pu

=

t

div (A \lu)

is form bounded if and only

t oo nxn � (A + A ) E L (Rn ) (6.20) , Div �(A - A1) E B:M0 1 (Rn )n. (6.21 ) A simpler condition with � (A - At) E BMO(Rn ) n xn in place of (6.21) but generally not necessary, unless n :::; 2.

is sufficient ,

Thus, the form boundedness problem for the general second order differential

operator iu the divergence form

(6.22)

£

=

b

·

\7 + V,

As a corollary o f Theorem

i.e., for all

u, v E CQ" (R") ,

l l,

(6.23)

(6.3)

b E D'(R")",

6.1,

b

holds, where � - 1 curl b E

div b

=

Jlfx-yi
> 0, V = 0.

r

x

=

combined with

E

b \7 + V ·

is form bounded,

l

\7 ( � -1div b) + Div (L:l-1curl

BMO(R")nxn,

and

[ I V' (� - 1 div EW + IV' (�

E Rn,

in the case

We observe that condition

by L:l - 1curl b

E D'(R").

(b · \lu v + Vu v) dx :::; C l l u l l u. 2 (R") I Iv i i£1. 2 (R") •

(6.24)

for all

V

deduce that, i f

we

then the Hodge decomposition

(6.25)

is reduced to the special case

L:l-1 V E

(6.25)

n

:;:::

b)

- l vW l dy :::; canst rn-2,

3;

in two dimensions, it follows that

is generally stronger than � - 1 div b E BMO

BMO, while the divergence-free part of

BMO, for all n :;:::

2.

b is

characterized

403

WEIGHTED NORM INEQUALITIES

A close sufficient condition of the Fefferman-Phong type can be stated in the following form:

1

(6.26)

lx-yl
[ I V7( .6. -1div bW + IV7(.6. - 1 V ) l 2 ]1+ ' dy s; const rn -2 (1 +<) ,

for some f. > 0 and all r > 0, x E Rn . This is a consequence of Theorem 6.1 coupled with (6.9), where 1 (.6. -1div b) i2+ 1V'(.6.-1 V) i2 i s used i n place of V. Sharper conditions of the Chang-Wilson-Wolff type are readily deduced from Theorem 6. 1 by combining it with the results of [ChWW] . It is worth mentioning that the class of potentials obeying (6.26) is substantially broader than its subclass

1

(6.2 7)

lx-yl
( l bl 2 + l V I) 1 +< dy '.S: canst rn -2 (1+<).

The sufficiency of the preceding condition for (6.23) is deduced by a direct appli cation of the original Fefferman-Phong condition and Schwarz's inequality. More generally, (6.23) clearly follows from a cruder estimate,

2

{ l u l 2 ( l bl 2 + l V I) dx '.S: const ll u l l i t , (R }Rn which is equivalent to lbl 2 + l V I E 9JL� 2 (Rn). (6.28)

"

l'

u E C3"' (Rn),

However, by replacing (6.23) with (6.28), one !>trongly reduces the class of admissible vector fields b and potentials V . Various examples of this phenomenon are given in [MVl], (MV6] . The main difficulty in the proof of Theorem 6.1 is the interaction between the quadratic forms associated with V - � div b and the divergence free part of b. To overcome this difficulty, one needs to distinguish the class of vector fields b such that the commutator inequality

I Lu,nv bE· (uC0\i'v(R- v V'u) dxl

(6.29)

s; const l lui i LI , > (R")

IJvJ I£L 2(Rn)

holds for all n). In the important special case of irrotational fields where b V7 f, the preceding inequality is equivalent to the boundedness of the commutator [f, .6.] acting from £ 1 • 2(Rn) to L - l , 2 ( Rn ) . A complete characterization of those b which obey (6 . 29) is obtained using the idea of the gauge transformation ((RS] , Sec. X.4) : V7 -j. e- i.>. V7 e+ i>. ' =

where the gauge .-\ is a real-valued function which lies in L��? (Rn). The problem of choosing an appropriate gauge is known to be highly nontrivial. In the present paper, ..\ is picked in a very specific form:

..\ = r log (Pw), 1 < 2T < n�2 , n ;::: 3, where is a constant, and Pw = ( -.6.)- 1 w is the Newtonian potential of the equi T

librium measure w associated with an arbitrary compact set e of posiLive capacity. We will verify that , with this choice of A, the energy space L1• 2(Rn) is gauge invariant, and the irrotational part c V (.� -1div b) of b obeys =

1 IC1 2 dx

s; const cap ( e) ,

404

I.

E. VERBITSKY

where the constant does not. depend on

JcF

E

9Jt� 2 (Rn ) .

to BMO, and

b = c + Div F.

e.

This is known to be equivalent to

Jn addition, a careful analysis shows that

F = �-I curl b belongs

These conditions combined turn out to be necessary

and sufficient for (6.29 ) .

[MV6]

In

applications are given t o the magnetic Schrodinger operator

M

de

V + JeW

and a V' are form bounded. Thus, the form boundedness criterion for M can be

fined by (6.2) .

It

is shown that

M

is form bounded if and only if both

·

deduced from Theorem 6 . 1 .

These results are extende d to the Sobolev space

sary and sufficient conditions are given

([MV6] ,

C : Wl, 2 (R " ) _,

lV1• 2 (Rn ) . In particular, neces

Theorem

5.1)

for the boundedness

of the general second order operator

This solves the

w -1 , 2 (Rn ) .

M, with respect to the Laplacian on L 2 (Rn). Th e

relative form boundedness problem for £,

magnetic Schrodinger operator

and consequently for the

proofs involve an inhomogeneous version of the div-curl lemma (see

[MV6] ,

Lemma

5. 2 ) . Other fundamental properties of quadratic forms associated with general dif ferential operators, e.g., relative compactness, infinitesimal form boundedness, in equalities of Trudinger's and Nash's type can be characterized using a similar ap

proach ( see

[MV6] ) . References

D.R. Adams and L.I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1996 M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrodinger [AS] operator�, Comm. Pure Appl. Math. 35 ( 1982) 209-273. [COV1] C. Cascante, J.M. Ortega, and I.E. Verbitsky, On imbedding potential spaces into Lq(dw), Proc. London Math. Soc. 80 (2000), 391-414. [COV2] C. Cascante, J.M. Ortega, and I.E. Verbitsky, Nonlinear potentials and two weight trace inequalities for general dyadic and radial kernels, Indiana Univ. Math . .J . 53 (2004) , 845-882. [COV3] C. Cascante, J.M. Ortega, and I.E. Verbitsky, On LP - Lq trace inequalities, J. London Math. Soc 74 (2006), 497- 5 1 1 . [ChWW] S.-Y.A. Chang, J.M. Wilson, and T.H. Wolff, Some weighted norm inequalities con ceming the Schrodinger operators, Comment. Math. Helv. 60 ( 1985), 217-246. [CLMS] R. Coifman, P.L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. , 72 (1993), 247-286. [EE] D.E. Edmunds and W.D. Evans, Spectr·al Themy and Differential Operators, Clarendon Press, Oxford, 1987. [F] C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc., 9 (1983), 1 29-206 M. Frazier and I.E. Verbitsky, Global exponential bounds for Green's function and the [FV] cond1tional gauge, preprint (2008). [H] K. Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand. , 45 (1979), 77-102. K. Hansson, V .G. Maz'ya and l.E. Verbitsky, Criteria of solvability for multidimensional [HMV] Riccati's equations, Arkiv for Matern. 37 (1999), 87-120. [HW] L.I. Hedberg and Th.H. Wolff, Thin set� in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) , 33 ( 1983), 161-187. N.J. Kalton and I.E. Verbitsky, Nonlinear equations and weighted norm inequalities, [KV] Trans. Amer. Math. Soc. 351 (1999), 3441-3497. R. Kerman and E.T. Sawyer, The trace inequality and eigenvalue estimates for [KS] Schrodinger operators, Ann. Inst. Fourier (Grenoble) 36 (1986), 207-228.

[AH]

WEIGHTED NORM INEQUALITIES

(KiM a) [KT)

[Lj

[M aZi) [Mlj [M2] [M3]

[M4] [MN] [MSh]

[MVl)

[MV2] [MV3]

[MV4] [MV5] [MV6]

[NTV) [PV1]

[PV2]

[PV 3) [RS ] [RSS]

[S ] [Sch] [Sim] [ StW]

T. Kilpel ainen and J.

405

Maly, The Wiener test and potential estimates for qua.siline.a.r

Acta Math., 172 ( 1994) , 137-161. H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. D.A. Labutin, Potential estimates for a class of f1LUy nonlinear elliptic equations, Duke Math. J. 1 1 1 (2002), 1-49. J. M aly and W.P. Ziemer, Fine Regularity of Solutwns of Elliptic Partial Differential Equations, Amer. Math. Soc., Providence, R.I . , 1997 V.G. Maz'ya, Classes of domains and embedding theorems for functional spaces, Dokl. Akad. Nauk SSSR, 133 (1960), 527-530. V.G. Maz'ya, On the theory of the n-dimensional Schrodinger ope.mtor, lzv. Akad. Nauk SSSR, ser. Matern ., 28 (1964), 1145-1172. V . G. Maz'ya, On certain integral inequalities for functions of many variables, Probl. Math. A nal. , 3, Leningrad Univ. (1972), 33-68. English trans!.: J. Soviet Math., 1 (1973), 205-234. V.G. Maz'ya, Sobolev Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1985. V.G. Maz'ya and Y. Netrusov, Some counterexamples for the theory of Sobolev spaces on bad domains, Potential Analysi s, 4 (1995), 47-65. V.G. Maz 'ya and T.O. Shaposhnikova, The Theory of Multipliers in Spaces of Dif ferentiable Functions. Monographs and Studies in Mathematics, 23, Pitman, Boston London, 1985. V.G. Maz'ya and I.E. Verbitsky, Capacitary inequalities for· fractional integrals, w'il.h applications to partial differential equations and Sobolev mult�pliers, A rkiv for M atern. , 33 (1995), 81-115 V.G. Maz'ya and I.E. Verbitsky, The Schrodinger operator on the energy space: buund edness and compactness criteria, Acta Math . , 188 (2002), 263-302 V.G. Maz'ya and I.E. Verbitsky, Boundedness and compactness criteria for the one dimensional Schriidinger operator, In: Function Spaces, Interpolation Theory and Re lated Topics. Proc. Jaak Peetre Conf. , Lund, Sweden, August 17-�2, 2000. Eds. M. Cwikel, A. Kufner. G. Sparr. De Gruyter, Berlin, 2002, 369-382 V.G. Maz'ya and I.E. Verbitsky, The form boundedness criterion for the relativistic Schrodinger operator, Ann. lnst. Fourier 54 (2004) , 317-339. V.G. Maz'ya and I.E. Verbitsky, Infinitesimal form boundedness and Trudinger 's sub ordination for the Schrodinger operator, Invent . Math. 162 (2005) , 81-136. V.G. Maz'ya and I. E. Verbitsky, Form boundedness of the general second order differ ential operator, Comm. Pure Appl. M ath . 59 (2006), 1286-1329. F. N azarov , S. Treil, and A. Vol berg, The Bellman functions and two-weight inequalities for Ilaar multipliers, J. Amer. Math. Soc. 12 (1999), 909-928. N.C. Phuc and I.E. Verbitsky, Local integral estimates and removable singularities for quasilinear and Hessian equations with nonlinear source terms, C omm . PDE 31 (2006), 1779-1791. N.C. Phuc and I.E. Ver b itsky, Quasilinear and Hessian equations of Lane-Emden type, to appear in Ann. Math. N.C. Phuc and I.E. Verbitsky, Singular quasilinear and Hessian equations and ·inequal ities, prepri nt ( 2008). M. Reed and B. Simon, Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjoininess, Academic Press, New York-London, 1975. G.V. Rozenblum, M.A. Shubin, and M.Z. Solomyak, Spectral Theory of Differential Operators, Encyclopaedia of Math. Sci., 64. Partial Differential Equations VII. Ed. M. A. Shub in. Springe r- Verl ag , Berlin-Heidelberg, 1994. E . T . Sawyer, Two weight norm mequalities for certain maximal and integral operators, Trans. Amer. M ath . Soc. , 308 (1988), 533-545 M . Schechter , Opemtor Methods in Quantum Mechanics, North-H olland , Amsterdam New York - Oxford, 1981. B. Simon, Schroding�r .9emigroup.9, Bull. Amer. Math. Soc. 7 (1982), 447 526. E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Prince ton University Press, Princeton, New Jersey, 1971. elliptic equations,

I. E. VERBITSKY

406

[St]

E.M. Stein, Harmonic Analysis: Real- Variable Methods, Orthogonality, and Osc'illatory

[T)

H.

Integrals, Princeton University Press, Princeton , NJ,

Verlag, Basel,

1992.

N.S. Trudinger,

[Tru]

Linear elliptic operators with measurable coefficients, Ann. Scuola

Norm. Sup. Pisa 27 [TW)

[V 1 ]

1993

Triebel, Theory of Function Spaces. II, Monographs in Mathematics, 84, Birkhii.user

N.S. Trudinger and

(1973), 265-308 . X.J. Wang, On the

weak continuity of elliptic operators and appli

cations to potential thMry, Amer. J. Math. 124

(2002), 369-410.

Verbitsky, Weighted norm inequalities for· maximal opemtors and Pisier's theorem

on factorization through LP• oc , Integr. Equat . Operator Theory,

I.E.

15 (1992), 121-153.

I.E. Verbitsky, Imbedding and multiplier theorF.ms for discrete Littlewood-Paley spaces,

[V2]

Pacific J. Math. , 176

(1996), 5 29-556 .

Verbitsky, Superlinear equations, potential theory, and weighted norm inequalities,

[V�J

I.E.

[V4]

5, Prague, May 31 - June 6, 1998, 1-47. I.E. Vcrbitsky, Nonlinear potentials and

Nonlinear Analysis, FUnction Spaces and Applications, Proceedings of the Spring School, tmce ineq·ualities, in: The Maz'ya anniversary

collection, Vol.

2.

1998. 1999, 323-343

Operator Theory: Advances and Applications, 110, Birkhauser Verlag,

Proceedings of the International Conference on Functional Analysis,

Partial Different ial Equations and Applications. Restock, Germany,

tember 4,

A ugust 31 -

Sep

[VW 1]

I . E . Verbitsky and

R.L.

Wheeden,

Weighted inequalities for fractional integrals and

(VW2]

I.E. Verbitsky and

R.L.

Wheeden,

Weighted norm inequalities for integral operators,

applications to semilinear equations, J. Funct. Anal . , 129

Trans. Arner. Math.Soc. 350

(1995), 22 1-241 .

(1998), 3371-3391.

SCHOOL OF �ATHEMATICS, UNIVERSITY OF BIRMINGHA M,

BIRMINGHA:-.1, B15 2TT,

UNITED

KINGDOM

E-m.n.il address : I . E . Verbitsky«lbham . ac . uk

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF M ISS OURI , COLUMBIA, �0

E-mail address: igorl!lmath .missour i . edu

6521 1,

USA

Proceedings of Symposia in Pure l\1athematics

Volume 79, 2008

THE MIXED PROBLEM FOR HARMONIC FUNCTIONS IN POLYHEDRA OF JR3 MOISES VENOUZIOU AND GREGORY C. VERCHOTA

ABSTRACT. R.

Brown's

theorem on mixed Dirichlet and Neumann bound special case of polyhedral do mains. A (1) more general partition of the boundary into Dirichlet and Neu mann sets is u�ed on (2) manifold boundaries that are not locally given as the graphs of functions. Examples are constructed to illustrate necessity and other implications of the geometric hypotheses . M.

ary conditions is extended in two ways for the

1 . INTRODUCTION In [Bro94] R. M . Brown initiated a study of the mixed boundary value problem for harmonic functions in crea.-;ed Lipschitz domains n with data in the Lebesgue and Sobolev spaces £2(00) and W1 •2 (an) (with respect to surface measure ds) taken in the strong pointwise sense of nontangential convergence. At the end of his article Brown poses a question concerning a certain topologic geometric difficulty not included in his solution: Can the mixed problem be solved in the (infinite) pyramid of JR3, \X1 \ + \X2 \ < Xa , when Neumann and Dirichlet data are chosen to alternate on the faces? In this article we avoid the geometric difficulties of what can be called Lipschitz faces or facets and provide answers in the cm;e of compact polyhedral domains of JR3 . Some other recent approaches to the mixed problem for second order operators and systems in polyhedra can be found in [MR07] [MR06] [MR05] [MR04] [MR03] [MR02] and [Dau92] . Consider a compact polyhedron of JR3 with the property that its interior n is connected. n will be termed a compact polyhedral domain. Suppose its boundary an is a connected 2-manifold . Such a domain n need not be a Lipschitz domain. Partition the boundary of n into two disjoint sets N and D, for Neumann data and Dirichlet data respectively, so that the following is satisfied. (1.1) (ii)

(i) ]\l is the union of a number (possibly

D = an \ N is nonempty.

zero

) of closed faces of 80.

(iii) Whenever a face of N and a face of D share a !-dimensional edge as boundary, the dihedral angle measured in n between the two faces it-> less than Date :

2000

7r.

June 9, 2008.

Mathematics Subject Classification.

35J30,35J40. second author gratefully acknowledges part ial support provided Foundation through award DMS-0401159 E-mail address: [email protected] The

407

by

the National Science

@2008 American

Ma.thematica.l Society

408

MOISES VENOUZIOU AND GREGORY C. VERCHOTA

The L2-polyhedral mixed problem for harmonic functions is (1 .2) Given f E Wl.2 (8D) and g E L2 (N) show there exists a solution to l:::,. u = 0 in n such that (i) u -tn.t. f a.e. on D. (ii) 8,u -+n.t. g a.e. o n N. (iii) 'Vu* E L2(8D ) . Here 'Vu* is the nontangential maximal function o f the gradient of u . Generally for a function w defined in a domain G For a choice of a > 0

by

( 1.3)

w* (P)

=

XH(P)

sup lw(X) J , P E 8G.

nontangential approach regions for each P E fJG are defined

r(P) = {X E G : IX - PI < (1 + a)dist(X, 8G)}

Varying the choice of a yields nontangential maximal functions with comparable LP(fJG) norms 1 < p � oo by an application of the Hardy-Littlewood maximal function. Therefore a is suppressed. In general when w * E LP(fJG) is written it is understood that the nontangential maximal function is with respect to cones determined by the domain G. The outer unit normal vector to n (or a domain G) is denoted v = Vp for a . e . P E an and the limit of (ii) is understood as r(P) 3X--+P

lim

vp

·

'Vu(X) = g(P)

and similarly for (i) . A consequence of solving ( 1 .2) is that the gradient of the solution has well defined nontangential limits at the boundary a.e. In addition, as Brown points out, solving the mixed problem yields extension op erators W1•2(D) --+ W1•2 (8D) by f �--+ u l an where u is a solution to the mixed prob lem with u iD = f . Consequently problem (1 .2) cannot be solved for all f E W1•2(D) when D and N are defined as on the boundary of the pyramid. For example, since the pyramid i::; Lipschitz at the origin so that Sobolev functions on its boundary project to Sobolev functions on the plane, solving (1 .2) implies that a local W1•2 function exists in IR2 that is identically 1 in the first quadrant and identically zero in the third. Such a function necessarily restricts to a local W ! •2 function on any straight line through the origin. But a step function is not locally in w! ·2 (JR). The boundary domain D (and its projection)
restrictions of W1•2(8D) functions f to D llfllb

=

f i n =!

_inf

lenf J2 + 1Vt fl 2ds

THE

MIXED PROBLEM FOR HARMONIC FUNCTIONS IN POLYHEDRA OF

IR3

409

Here J denotes all W 1 •2(80) functions that restrict to f on D, and \lt denotes the tangential gradient. That this is a norm follows by arguments such as: Given f E W1• 2 (80) and a real number a , the functions af form a subset of all extensions af of (af) I D so that lla/ I I D � I a i i i ! I l D , and thus likewise IIJIID � l a l - 1 ll af iiD when a # 0. This normed space is complete by using the standard completeness proof for Lebesgue spaces: Given a Cauchy sequence {fj } let jk be such that llh- IJ IID < 2-k for all i, j � Jk and define 9k = /jk+ l - fjk . Then there exists an extension §k such that l l §k l l wt.2(8fl) < 2-k . Extensions of fin+l may then be defined by h + L:�= l §k Cauchy in W1•2 (8D). Completeness will follow. The Banach space of restrictions to D is undoubtedly the generally smaller Sobolev space HI (D) (e.g. [Fol95] p . 220) , but this will not b e pursued further . A homogeneous Sobolev semi-norm on D is defined by

l l f llb· = _inf f l o =!

(1 .4)

f 1Vt jj2 ds lan

When 80 is connected the following scale invariant theorem is established in the Section 2.

Theorem 1.1. Let n

c IR3 be a compact polyhedral domain with connected 2manifold boundary an = D u N satisfying the conditions (1.1). Then given f E W 1 •2 (an) and g E L2 (N) there exists a unique solution u to the mixed problem (1.2). In addition there is a constant C independent of u such that

hn (\lu* )2 ds � C (1 1 ! 1 1· + IN

2 ds

g

)

In the following section it is proved that a change from Dirichlet to Neumann data on a single face is necessarily prohibited when the change takes place across the graph of a Lipschitz function. The strict convexity condition of (1.1) is also shuwn to be necessary. In the final section compact polyhedra are discussed for which the set N is necessarily empty. 2. PROOF OF THEOREM 1 . 1 The estimates that follow are scale invariant. Therefore to lighten the exposition a bit it will be assumed, when working near any vertex of the boundary of the compact polyhedron 0, that the vertex is at least a distance of 4 units from any other vertex. Because an is assumed to be a 2-manifold it will also be assumed that each edge that does not contain a given vertex v as an endpoint is at least 4 units from v and similarly each face. Consequenlly, by anoLher application of the manifold condition, the picture that emerges is that the truncated cones C(v, r) = {X E n : l v - X I � r} for any vertex v and 0 � r � 4 are homeomorphic to the closed ball JE3 while the cone bases !3(v , r ) {X E n : l v - X I r } =

are homeomorphic to the closed disc JE2 .

=

410

l\10ISES VENOUZIOU AND GREGORY C. VERCHOTA

Define

v

where the finite union is over all boundary vertices. Then each Or is a Lipschitz domain (see, for example, §12.1 of [VV06] and Theorem 6.1 of [VV03] for a proof and generalizations in dimensions n ;::: 3). Likewise the interiors of the arches defined by A(v, r, R) E n : r � lv - X I � R} , 0 < r < R < 4 (2.1) are Lipschitz domains. In general neither of these kinds of domains have a uniform Lipschitz nature as r -4 0. Therefore the following polyhedral Rellich identity of [VV06] will be of use. It is proved as in [JK81] by an application of the Gauss divergence theorem, but with respect to the vector field

= {X

w

:=

X

IXT ' x E JR3 \ {o}

when the origin is on the boundary of the domain.

Lemma 2.1. Let A be any arch (2. 1) of the polyhedral domain n c IP1.3 and suppose u is harmonic in A with 'Vu* E L2(&A) . Then, taking the vertex v to be at the

origin

2

( 2.2)

L (W

·

\7u)2

f� = faA

v·

W[\7u[ 2 - 28vuW \7uds

Lemma 2.2. With A = A ( v, r, R) and u as in Lemma 2. 1

(2.3) d 2 (W \7u )2 � { l l

1A

·

·

: JB(v,R) [\7u[ 2 ds + 2 JB(11, { { l8vu[[\7t'u[ds r) (W \7u)2ds + 2 JannA ·

Proof. The term v · W on the right of (2.2) is negative on B( v, r) and vanishes on 80. Likewise the second integrand on the right of (2.2) is a perfect square on B(v, r), the negative of a square on B(v, R) , and W \7u is a tangential derivative 0 on en. ·

The partition D U N = 80 induces a decomposition of the Lipschitz boundaries 8flr into a Dirichlet part, a Neumann part, and bases 13(v, r) of the cones removed from fl. Define 1)

and

Dr = Oflr \ Nr. This partition of 8flr satisfies the requirements of a creased domain in [Bro94] . See

[VV06] pp. 586-587. (Including the bases in the Dirichlet part would also satisfy the requirements.) It will therefore be possible to invoke Brown's existence results in the domains nr. Similarly, arches A = A (v, r, R) are creased Lipschitz doma·ins with

Nf"( v)

= (N n 8A(v, r, R)) u B(v , r) U l3(v , R)

THE MIXED PROBLEM FOR HARMONIC FUNCTIONS IN POLYHEDRA OF

IR3

411

and

v.

for each vertex Brown's estimate from (Bro94] Theorem 2. 1 is not scale invariant. However, the following special case is. c IR.n be a creased Lipschitz domain with 8G D U N. Then there exists a unique solution u to the mixed problem (1 .2) for data f identically zero and g E Furthermore there is a constant C determined only by the scale invariant geometry of G, D and N and independent of g such that

Theorem 2.3. (R. M. Brown) Let G

£2 (N).

=

As is c IR.n be a creased Lipschitz domain with 8G = D U N. Suppose that D is connected. Then there is a constant C such that for all harmonic functions u with 'Vu* E

Theorem 2.4. (R. M. Brown) Let G

L2 (8G) lac (Vu*)2ds � C (fv IY'tul2ds + JN (8�.�ufds )

Proof. Subtracting from u its mean value over D allows the Poincare inequality 0 over the connected set D. The conclusion still applies to u.

Lemma 2.5. Let 0

c

IR.3 be a compact polyhedral domain with 2-manifold boundary Let v be a vertex and let j be a natural number. Suppose u is harmonic in the arch 2-i , 2) with 'Vu* E and u vanishing on D�-; . Then there is a constant C independent of j so that

partitioned

as

{

8f1.

JannA(11, 2 - ; ,2) C

= DUN.

L2 (8A)

A(v,

1 Vul2ds �

(lannN2 -; {

(8vu)2ds +

{ Jl3(v, 2

_

.

')

(W · Vu)2ds +

JA(v,1, ) I Vul 2dX)

2 Proof- For natural numbers k � j and real numbers 1 � t � 2 the arches Ak ,t : = t2- k , t21- k ) are geometrically similar Lipschitz domains. Therefore by the scale invariance of Brown's Theorem 2.3 above

A(v,

lfannA

k,t

1 Vul2ds � C

}{N t21-k (8vu)2ds t2-k

with C independent of k. Take v to be the origin. For each k, integrating in 1 � t � 2 and observing that v = W or - W on any cone base B

412

tv101SES VENOUZIOU AND GREGORY C. VERCHOTA

Summing on k = 1, 2, . . . , j and using Lemma 2 . 2 on the arch A( v, 2-i , R) for each 1 � R � 2 together with the vanishing of u on D�- J again

� jrannA(v, - , ) I Y'u l 2ds 2

+2

J

2

{ (W Jl3(v,2-i )

·

s:; 4 C (

(8vu) 2ds +

r 1'7ul 2ds \7u)2ds + 2 { l 8v u l l 'lt u l ds + ! 1 A(v,l ,2) JannN: i r

j8f!nN2_i

j13(v,R)

'7 u l 2 dX)

An application of Young's inequality (2ab � � a2 + f.b2) allows the square of the tangential derivatives in the second to last term to be hidden on the left side and the normal derivatives to be incorporated in the first right side integral. Integrating D in 1 � R S:: 2 yields the final inequality. By the same arguments, but using Theorem 2.4 and then Young's inequality in suitable ways for the D portion and the N portion of the last integral of Lemma 2.2, the next lemma is proved. For a given vertex, D n C(v, R1 ) is connected if and only if any D n A( v, r, R2 ) is connected.

Lemma 2.6. Let n c IR3 be a compact polyhedral domain with 2-manifold boundary partitioned as an = D U N . Let v be a vertex and let j be a natural number. 8uppose D n C(v, 2) is connected and u is harmonic in the arch A(v, 2-j , 2) with \lu* E L2 (aA) . Then there is a constant C independent of j so that

{ l 8 u l 2 ds + . . Vu l 2 ds :5 C ( 1 Vt u l 2 ds + { _ JannA(v,'2 , .�) Jv2 ; JaonN2-j .

1

(

v ) (W . Vu)2ds + I c( r 1 '7u l 2dX) Jf3(v,2-J ) , .A(v,l ,2) .

Let v be a vertex of the compact polyhedral domain n and consider the collection of nontangential approach regions r(P) for G = D and parameter a (1.3) with P E an n C( v, 4). By scale invariance each approach region can be truncated to a region rT(P) = {X E r(P) : IX - PI < (1 + a)di.9t(X, aA(v, r/2, 2r))}, lv - PI = r so that the collections {rT(P) ; r ::; lv - PI :::; 2r} can be extended in a uniform way to systems of nontangential approach regions regular in the sense of Dahlberg [Dah79] for the arches A(v, r/2, 4r) . Denote b y wT the nontangential maximal function of w with respect to the trun cated cones rT . Denote the Hardy-Littlewood maximal operator on an by M . See, for example, [Ste70] pp.l0-11 or [VV03] pp.501-502 for polyhedra. For a large enough a geometric argument shows that there is a constant inde pendent of P and w such that (2.4)

w* (P) < CM (wT) (P) + max l w l , P E aD n C(v, 4) K

where K is a compactly contained set in the Lipschitz domain f12 • Using Theorems 2.3 and 2.4 to estimate the truncated maximal functions intro duces into the proofs of Lemmas 2.5 and 2.6 a doubling of the dyadic arches and therefore one dyadic term that is not immediately hidden by Young's inequality. Thus by the same proofs

THE

MIXED PROBLEM FOR HARMONIC FUNCTIONS

IN

POLYHEDRA OF

Lemma 2 . 7 . With the same hypotheses as Lemma 2. 5 there i s a constant pendent of j so that

41 3

IR3

C

inde

f ( �ur ) 2 ds - � f IY'ul 2 ds � 2 lannA(v,2-i,21-i) lannA(v,21-i ,2) c

(

r . (W . Vu)2ds + r (avu)2ds + { in1 lannN2_ ; .JB(v,2-J)

Lemma 2.8. With the same hypotheses as Lemma

pendent

of j

so that

2. 6 there is a constant

r ��u l2ds � (V'uT) 2 ds - � r 2 lannA(v,2 - i ,21 -i) lannA(v,21 -.i ,2) 2 -; (W · Vu)2ds + I V't ul2ds + ln'�.V, -i la u l ds + a

C (L2_i

l(v 2

)

I V'ul2dX

fo

C

inde

)

I V'ttl2dX 1 ) , Remark 2.9. Lemmas 2.5 and 2.6 apply to the negative terms of Lemmas 2.7 and 2.8. Consequently those terms may be removed from the inequalities. v

2 . 1 . The regularity problem. The regularity problem is the mixed problem for an = D.

Theorem 2.10. Let

0

c

R3 be a compact polyhedral domain with 2-manifold

connected boundary. Then for any f E W 1•2 (aO) the regularity problem is uniquely solvable and the estimate for the solution u

holds with

C

r I V' * l 2 ds � c r I Y'tfl2 dB loo Jan

independent of f .

u

Proof. For each 02-; there is a unique solution Uj to the mixed problem with uj = f on D2 -; and a,_.'Uj = 0 on S2-; by Brown's existence result [Bro94] . By definition of the truncated approach regions in each vertex cone C (v, 4) the regions may be extended to a regular system of truncated approach regions for the an nanl part of the boundary. Thus the truncated nontangential maximal function can be defined there. By Lemma 2.8 and Remark 2.9, summing over all vertices, using analogous estimates on the local Lipschitz boundary of 00 outside of the vertex cones and using W · Y'uj 0 on the bases B(v, 2-J ) , =

(2.5)

with C

fv2H (VuJ ) 2 ds � C (Lz-i I V't! l 2ds + l, IV'uj l2dX)

independent of j. Subtracting from uj the mean value mf of f over an does not change (2.5) . Thus Poincare (see [VV06}p.639 for polyhedral boundaries) can be applied over 80 with constant independent of j in (2.6)

r I V'uj l2dX � r (uj - m, )avujds ln1 jD2-j

r ( ! - ffiJ )a.,ujdS � j r I 'Vuj i 2 ds lran IY't f l 2 ds + JD2-;

D2 ; CE =

(

414

JV10ISES VENOUZIOU AND GREGORY C. VERCHOTA

Applying Lemma 2.6 to the part of the integral over the regions D��i and using W \luj = 0 again ·

$ r::C

+

f. 12_i j \lujl2ds (1z-i l \ltfl2ds + k, j \luj i2dX) 121 -J j \luj j2ds

so that (2.6) yields

� ln,{ j\lu1 i2dX

$ (C< + r:: C)

+ r::

r::

lao{ j\l tfl2 ds l{v2, _, j \luJ i 2ds .

for all r:: chosen small enough depending on C but not on j. Using this in (2.5) for chosen small enough gives

r::

(2.7) with the constant independent of j . Given any compact subset of n , (2.7) together with Uj = f on D2 -i for all j implies there exists a subsequence so that both Uj�c and \luj, converge uniformly on the compact set to a harmonic function u and its gradient respectively. A diagonalization argument gives pointwise convergence on all of 0. Intersecting a compact subset K with the truncated approach regions yields compactly contained regions and corresponding maximal functions \luJ: K ----. \luT,K uniformly. Thus by (2.7) and then monotone convergence, as f2 is exhausted by compact subsets K, (2.8)

fv

See [JK82] for these arguments. . (\l(uj - uk)T ) 2 ds for k > j A difficulty with the setup here is that the 21-J do not a priori have better bounds than the right side of (2.7). However, (2.7) together with weak convergence in £2 (80 2 - i ) and pointwise convergence on the bases B(v, 2-1) shows that for each j and every X E f2 -; a "ub"equcnce of

2

converges to u(X) , perforce with Poisson representation that must be an extension from D2 -j of f. Here Pf is the Poisson kernel for the Lipschitz polyhedral domain f22 -1 and may be seen to be in L 2 (8f22 -1) by Dahlberg [Dah77] . Consequently u has nontangential limits f on an, and by (2.4) and (2.8) the theorem is proved. 0 2.2. The mixed problem with vanishing Dirichlet data. Theorem 2.11. Let f2 c �3 be a compact polyhedral domain with 2-manifold connected boundary. Then for any g E L2 (N ) there is a unique solution u to the mixed problem ( 1 .2) that vanishes on D and has Neumann data g on N . Further s;

laor (\lu*)2ds lNr g2ds c

THE MIXED PROBLEM FOR HARMONIC FUNCTIONS IN POLYHEDRA OF JR.3

415

Proof. Again by [Bro94] there exists a unique solution Uj in 02 -i to the mixed problem so that a,_. uj = g on an n N2 -j ' w . \luj = 0 on the B(v, 2-j ) and Uj = 0 on D2 -j . Lemma 2.7 and Remark 2.9 imply

(

r g2ds + r l\luj l2dX (\luJ) 2 ds � c r lnl lannN2-j lann&fl2t-j

)

A Poincare inequality independent of j is also needed here and is supplied by the following lemma . Polyhedral domains are naturally described as simplicial com plexes. See for definition.s and notations [Gla70] [RS72] [VV03] [VV06] or others.

Lemma 2.12.

Suppose

on the B(v , 2-i ) and

u

u =

is harmonic in 0 -j with 8v u = g on 8D n N2 -j , 8v u = 0 2 0 on D2-J . Then

l2_) I "Vul2 dX � C hnnN2-i g2ds

with C independent of j .

Proof. By Green's first identity and Young's inequality

( 2 . 9)

fn2_j "V l

I u 2 dX =

foors2_i

u 8v uds � C<

lan'lN2-i i ds + E lannN2- J u2 ds

The polyhedron fi can be realized as a finite homogeneous simplicial 3-complex. A cone C(v, 1) is then the intersection of the ball l X I � 1 with the star St(v, 0) in the 3-complex n of t he ,·ertex v. Each 2-simplex a2 of St(v, 0) that is also contained in ]\l is contained in a unique 3-simplex a3 E St( v, 0) . Let B denote the unit vector in the direction from the barycenter of a3 to v. Then a2 n { IX I � 1 } may be projected into the sphere lXI = 1 along lines parallel to B by Q 1-+ Q + tqB onto a set contained in a3nB(e. 1 ) . The sets { Q + tB : Q E a2 nN:}__; (v) and 0 � t � tq } are contained in a3 n A( t· . 2- i , 1). Thus by the fundamental theorem of calculus for each Q E 80 n Ni_1 (r) and integrating ds (Q) (2. 10)

r u2ds Janrwi_1 (r)

� c

(1

A(v,2-i ,1)

l \l u l 2 dX +

2ds I Jl3(v,1) u

)

where the corn;taut depends only on the projections, i.e. only on the finite geometric properties of the complex that realizes n and not on j. By the fundamental t heorem, the connectedness of 01 and the vanishing of u on the fixed nonempty set D

{ u2 ds � C { j "Vu l 2dX ln1 Jan1

This together with (2.10) implies E

f lannN2-i

u2 ds < eC

/

ln2 -j

I 'Vu l 2 dX

and E can be chosen independently of j so that (2.9) yieldH the lemma. The lemma yields the analogue of (2.7) (2. 1 1 )

0

416

MOlSES VENOUZIOU A N D GREGORY C. VERCHOTA

Continuing to argue as in the proof of Theorem 2.10 , this and the vanishing of the on D2-i produces a harmonic function u defined in n that is the pointwise limit of a subsequence of the Uj· In addition u satisfie::;

Uj

Jran

2 ( "VuT ) ds � C

JrN

g2 ds

which in turn yields the maximal estimate of the theorem. To show that u assumes the correct data, ( 2 1 1 ) along with weak £2-convergence, pointwise convergence and the Poisson representation in each n2-j proves as before that u vanishes nontangentially on D. By constructing a Neumann function (pos sible by [JK81] ) in analogy to the Green function, or by using the invertibility of the classical layer potentials [Ver84] , a Neumann representation of u in each n2 j can be obtained so t.hat � = g nontangentially on N can be deduced by the same arguments. Uniqueness follows from Green's first identity valid in polyhedra when V'u* E £2 . 0 .

-

2.3. Proof of Theorem 1.1. Recall the definition of the homogeneous Sobolev semi-norm (1 .4) . Lemma 2. 13. When an is connected and il f il bo D.

=

0, f is identically constant on

The lemma says t.hat f equals the same constant value on each component of D. Proof Because the semi-norm equals zero there is a sequence of extensions J; of f and a ::>equenee of numbers mi so that by Poincare in the second inequality

0 Proof of Theorern 1 . 1 . Choose an extension 1 of f so that J80 I "Vt fl2ds � 2llfllbu · This is always possible by the lemma. Then from Theorem 2.10 there is a unique solution UD with regularity data 1 and f8nC'IluiJ )2ds ::::; Cllf llbo · From Theorem 2.11 there is a unique solution UN vanishing on D, with Neumann data g - avuD on N, and

The solution is u =

UD + UN.

Theorem 2.11 established uniqueness. 0

3. ON VIOLATIONS OF THE POSTU LATES FOR THE PARTITION an

=

DUN

When D is empty the mixed problem is the Neumann problem and solvable for any data that has mean value zero on the boundary [VerOl] . We consider the two remaining postulates.

THE MIXED PROBLEM FOR HARMONIC FUNCTIONS IN POLYHEDRA OF R3

4t7

3.1. N is the union of a number (possibly zero) of closed faces of an. Solving the mixed problem means that every W1• 2 (D) function has a W1•2(an) extension. This observation raises the possibility that the mixed problem might be solvable when a given (open) face F has nonempty intersection both with D and with N in such a way that D n F is an extension domain. Here we will only consider the possibility that this extension domain has a Lipschitz boundary [Ste70] and show that the mixed problem is never solvable when this condition on

the partition occurs. Let ¢ : IR --+ IR be a Lipschitz continuous function y = ¢(x) with II
directly below Po and M units from the bottom of the rectangle. Here it will be convenient to name the region N that is strictly below the graph and contained in the rectangle. Call its complement in the rectangle D. Let (x, y, t) be the rectangular coordinates of JR3 with origin coinciding with the origin of the plane. Let Z be the open right circular cylinder of IRa with center p0 that intersects the plane in precisely the (open) rectangle. The domain n = Z\D C JR3 is regular for the Dirichlet problem. This follows by the Wiener test applied to each of the points of an = az U D and the observation that the Newtonian capacity in JR3 of a disc from the plane is proportional to its radius . See, for example, [Lan72] p. 165. Here the Lipschitz (or NTA) condition is also used. Consequently the Green function, g = g0 for n with pole at the origin, is continuous in n \ {0}. Approximating Lipschitz domains to 0 are constructed as follows. For each T > 0 define Lipschitz surfaces with boundary (the graph of ¢) by

Dr

=

{p + s(p - re 3 ) : p is on the graph of ¢ and 0 :::; s} n

Z

Here e3 is the standard basis Yector perpendicular to the xy-plane. Denote by Hr the part of z between D and Dr · Then nT = n \ HT = z \ HT are Lipschitz domains. Denote by g.,. the Gr€€n function for Or with pole at Te3 .

Lemma 3.1. (i) -atg(x. y. t) for t > 0 has continuous boundary values avg - limt!O g(x, y, t)/t at every point of D for which y > ¢(x) .

:=

(ii) JD (avg)2ds = + x . (iii) atg (x, y, 0) = 0 at every point of N \ {0} . (iv) avg E L 2 (oZ).

Proof. (i) follows by Schwarz reflection while (iii) follows by the symmetry in t of 0 and g. The maximum principle shows that the Green function for Z dominates from below the Green function for 0, gz :::; g :::; 0. On az both Green functions vanish so that av9z 2: avg 2: 0 while av9Z is square integrable there, establishing (iv). D. S. Jerison and C. E. Kenig's Rellich identity for harmonic measure ([JK82] Lemma 3.3) is valid on any Lipschitz domain G that contains the origin. It is (n - 2 )wa (O ) =

{ (avga (Q))2v lac

·

Qds(Q)

with respect to the vector field X. Here ga(X) = F(X) + wc(X) is the Green function for G, and F is the fundamental solution for Laplace's equation. Denote

4 18

:tvlOISES VENOUZIOU AND GREGORY C. VERCHOTA

by w.,. , w and Wz the corresponding harmonic functions for the n.,. , n and z Green functions respectively. By Z ::> Z \ D 11 ::> fl.,. and the maximum principle =

8.,g.,.

S

av9Z on an.,. \ D \ D.,.

and (3.1)

For Q E D and 11 = 11Q the outer unit normal to 11.,., 11 (Q - u3 ) = r , while for 0. Formulating the Rellich identity with respect to the Q E Dr , v · (Q - T€3 ) vector field X - re3 and using these fact!> (n = 3) •

=

wT (Teg )

so that

=

r

r

lan�\D\D�

laz

(a,_,gT )2 v . (Q - T eg ) ds + T

( fJ.,gz ) 2 v (Q - Te3 )ds + T ·

r (8,_,gT )2ds

JD

$_

r (avg)2ds = Wz (re3 ) + T JDr (avg?ds

JD

w (re3 ) - Wz (T€3 ) < Wr (Te3 ) - Wz (Te3 ) $_ T T and (ii) follows from (3. 1 ) and r 1 0.

{ (8.., g)2ds

}D

D

For o > 0 define smooth subdomains of 0

Go = {g

<

-o}.

---+ an. uniformly. The Ov Y laa 6 ds are a collection of probability measures on IR3 that have harmonic measure for 11 at the origin as weak-* limit. By G8 j D., Green's first identity, and monotone convergence

8G5

(3.2)

{

Jn\Br

IV'g iz dX

< oo

for all balls centered at the origin. With ¢, N, D and Z aH above define the half-cylinder domain Z+ Z : t > 0}. Then D u N c 8Z+ n {t = 0}.

=

{(x, y, t) E

Lemma 3.2. . Suppose 6u = 0 in Z+, \7u* E L 2 (8Z+ ) , 8vu ---+ n.t. 0 a . e . on N , and u ___.n .t. 0 a .e . on D. Let Y c Z be a scaled cylinder centered at Po with dist(fJY, 8Z) > 0. Let Y+ be the corresponding half-cylinder. Then u E C(Y + ) ·

Proof. The hypothesis on \7u* implies u* E L2 (8Z+) so that u and V'u have non tangential limits a.e. on fJZ+ [Car62] [HW68] . Extend u to the bottom component of Z \ D \ N by u(x, y , t) = u(x, y, -t). By the vanishing of the Neumann data on N, D.u = 0 in the sense of distributions in the domain 11 = Z \ D and then classically. Fix d > 0 and suppose X E Y + is of the form X = (x, y, d) for y � ¢(x) - Md. Denote 3-balls of radius and distance to D comparable to d by Ed. Denote 2-discs in fJZ+ with radius comparable to d by b..d and let

f denote integral average. Then

THE MIXED PROBLEM FOR HARMONIC FUNCTIONS IN POLYHEDRA OF by the mean value theorem, the fundamental theorem of calculus, the of u on D and the geometry of the nontangential approach regions

l u (X)I �

f

Bd(X)

l u i � Cd (

f

R3

4 1 \J

a. e. vanishing

\lu*ds)

�d (x,y+2Md,O)

C depends only on M . By absolute continuity of the surface integrals and \lu* E £2 there i� a function TJ(d) ----? 0 as d ----? 0 so that J� J \lu*)2ds � TJ(d) for all �d c az+ . Consequently the Schwarz inequality now yields l u(X)I ::; C7J(d) . Suppose now X is of the form X = (x, (x) - Md, t) for 0 � t � d. Because u

where

has been extended

l u(X) I �

f

lui � (

B.t(X)

and

f

f

l ui ) + d(

Hd(x,.p ( x ) - Md,d)

\lu*ds)

�d(x,¢(x)-Md,O)

lu(X)I � 2Cry(d). The lemma follows.

Partition 8Z+ by N+ = N, D+ = 8Z+ \N and 8Z+ let zr be the scaled cylinder centered at p0 of width the corresponding half-cylinders Z+ with

0

= N+ uD+· For 3/4 > r > 0 2r and length 81vfr. Define

N� = N+ n az� (not a scaling of

N+) and D� = 8Z� \ N�

Z+ is called a split cylinder with Lipschitz crease. By (3.2) and the Fubini theorem, g E W1•2 (8Z+ \ {t = 0}) for a.e. r.

With this partition

3.3. Let 9 be the Green function for n Z\D For almost every > r > 0 there exists no solution u

Proposition

with pole at the to the L2 -mixed problem (1.2) in the split cylinder with Lipschitz crease Z+ 71Jith boundary values u ----? n .t. g E W1•2 (D+ ) and Ov U -+ Ov9 = 0 on N'; . origin.

�

=

Proof. Suppose instead that there is such a solution u with \lu* E £2(8Z+). Then the first paragraph of the proof of Lemma 3.2 applies and, in particular, u extends to zr \ D evenly and harmonically across N+ . The Dirichlet data that n takes a.e.

on D+

is a continuous function, as is the Dirichlet data that u t akes (continuously) u takes a. e. on 8Z+ will be shown to be a continuous function if it can be shown to be continuous across the boundary 8N� of the surface N� . Lemma 3.2, scaled to apply to the split cylinders here, shows that the Dirichlet data is continuous across the Lipschitz crease part of 8N.f- . The same argument used there works on the other parts: Suppose dist(X, azr) = d for X E N'; . Let be a disc approximately a distance d from X + de3. Then Ll d c azr n on

N� . The Dirichlet data

D+

ln(X)

-f gds l � I f �d

Bd( X )

u(Y) - u (Y + de3)dYI + I

f

Bd (X+ de3)

u(Y)

-f

gds l

� C17(d)

t:.. d

and the continuity across 8N'; follow� from the continuity of g and 7J(d) ----? 0. Thus t h e data u takes a . e . on 8Z� i s a continuous function. Since also u• E L2(8Z+) it follows t hat u E C(Z.f- ) . The evenly extended ·u is then continuous on zr , harmonic in zr \ D with the same Diriclllet data as g on 8( Z" \ D) . The maximum principle implies u = g.

420

lv!OISES VENOUZIOU AND GREGORY C. VERCHOTA

Let 9r denote the Green function for zr \ D with pole at a point {P} of N+ . Again 9r is cont inuous in zr \ {P}. Let B c B c zr \ D be a ball centered at P. Then by the maximum principle cg 2: 9r on zr \ B for some constant c. By this domination, the vanishing of both g and 9r on D+ n { t = 0} and ( ii ) of Lemma 3.1 applied to Yn it follows that o.,g which is not in L2 (D) can neither be square integrable over the smaller set D+ n { t = 0}. Since u = g this contradicts the 0 a..o:;sumpt.ion on the nontangent ial maximal function of the gradient . The nonsolvability of the L2-mixed problem in the split cylinders can be extended to nonsolvability in any polyhedron that has a Lipschitz graph crease on any face by a globalization argument. Let g and r be as in the Proposition. By using the approximating domains zr n Ga as 6 -+ 0, the Green s representat ion '

g(X)

=

{

lazr

o.,Fx g - Fx o.,gds -

{

_ Fx dft0, X E zr \ D

JDnzr

can be j �stified where J-L0 is harmonic measure for n = Z \ D at the origin and F is the fundamental solution for Laplace's equation. Let x E C0 (IR3) be a cut-off function that is supported in a ball contained in zr centered at Po, and is identically 1 in a concentric ball B r with smaller radius. Then define

u(X)

= -

f _ FxxdJ.lP JDnzr

harmonic in R3 outside supp(x) n D. Similarly g(X) - u(X) is harmonic inside Br. Consequently V'u* � L 2 (Br n D ) by applying a scaled (ii) of Lemma 3. 1 to g again. Also (3.3)

u(X) = -

{ _ Fx (Q) (x(Q) - x(X)) dp,0(Q)

lvnz··

- x(X) {

lazr

ovFx9 - pX Ovgds + x(X)g(X)

The last term has bounded Neumann data on N and vanishing Dirichlet data on D. The Cauchy data of the middle term is smooth and compactly supported on D U N . For any X � D the gr-adient of the first term is bounded by a constant , depending on X, times

l lnzr

I

�

Fx (Q ) dJ-to ( Q ) :::; - Fx (o) + g x (o) :::; 47r X I (negative ) Green function for D = Z \ D with pole

Here gx is the at X. Thus the first term is Lipschitz continuous on D+ U N+.. Altogether u has bounded Neumann data on N and Lipschitz continuous data on D while \i'u* � L 2 (D) . Finally V'u E Ltoc(lR3 ) by (3.3) since this is true for X9· Thus whenever a split cylinder Z+ can be contained in a polyhedral domain so that az� n {t = 0} is contained in a face and so that the Lipschitz crease is part of the boundary between the Dirichlet and Neumann parts of the polyhedral boundary, then the harmonic function u just constructed is defined in the entire polyhedra domain. Its properties suffice to compare it with any solution w in the class \i'w* E L2 by Green's first identity J I V'u - V'wl 2 dX = J(u - w) o,_ (u - w)ds. Regardless of the nature of the partition away from Z+. , when w has the same data as does u it must be concluded, as in Proposition 3.3, that it is identical to u . This establishes

THE MIXED PROBLEM FOR HARMONIC FUNCTIONS IN POLYHEDRA OF JR3

421

Theorem 3.4. Let n

c JR3 be a compact polyhedral domain with partition an = D u N. Let :F be an open face of an such that :F n D is a Lipschitz domain of :F with nonempty complement F n N. Then there exist mixed data (1.2) for which there are no solutions u in the class V'u* E L 2 (an ) .

3.2 . Whenever a face of N and a face of D share a !-dimensional edge boundary, the dihedral angle measured in n between the two faces is less than Continue to denote points of JR3 by X = (x, y, t ) . Define D to be

as

the upper half-plane of the xy-plane. Introduce polar coordinates y r cos 0 and t = r sin 0, let 1r � a < 21r and define N to be the half-plane e = a. The crease is now the x-axis. 1r.

=

Define

b(X) = r 2"c. sin( � e) 20' for X above D U N. These are Brown's counterexample solutions for nonconvex plane sectors (Bro94] . The Dirichlet data vanishes on D while the Neumann vanishes on N, and V'b* tfi L2. These solutiollB are globalized to a compact polyhedral domain with interior dihedral angle a : Denote by e the intersection of a (large) ball centered at the origin and the domain above D U N. Then b(X) is represented in e by b(X) =

{ OvFxbds - { Fxav bds lae\D lae\N

Let x E Ctf (IR3) be a cut-off function as before , but centered at the origin on the crease. Define u(X) = 8v Fx xbds Fx xovbds

i

fv

As before, u is harmonic everywhere outside supp(x) and V'u * rf. L2 (supp(x) n (D u N)). Also

(3.4) u(X) = {

JNne av F _

x (Q)

- lne

n

(D U N)

(x(Q) - x (X ) ) b (Q) ds (Q)

pX ( Q)

(x(Q) - x(X)) av b ( Q) ds ( Q)

- x(X) {

lae\ N\D avFxb - F

x 8v bds

+ x(X)b(X)

Again the boundary values around the support of x are the issue. The last two terms are described just as the middle and last after (3.3). The gradient of the second term is bounded because the integral over D can be no worse than, for example, J; dx J� v'xl+r2 � < oo for any /3 < 1 (e.g . /3 1 - : ) . 2 For a 8�. derivative define tangential derivatives (in Q) to any surface with unit J normal v by a; = ViOj - VjOi · Then by the harmonicity of F away from X and the derivative of the first integral equals the sum in divergence theorem in e, the .7 i of f ai Fxa; ( (x - x(X ) )b) ds =

at

}Nne

_

plus integrals over ae \ N \ D ( b vanishes on D) that will all be bounded since X is near the support of X · When the tangential derivative falls on b the integral is

42 2

J>,1QISES VENOUZIOU AND GREGORY C. VERCHOTA

bounded like the second term of (3.4) . The remaining integral has boundary values in every LP for p < oo by singular integral theory. (In fact, it too is bounded by a closer analysis, thus making it consistent with the example from Section 3.1.) Finally V'u E Lfoc(JH:3 ) by its now established properties and the corresponding property for b. The argument using Green's first identity as at the end of Section 3.1 is justified and

The solutions u can now be placed in polyhedral domains that have interior dihedral angles greater than oT equal to 1r and provide rnixed data for which no L2 -solution can exist.

4. POLYHEDRAL DOMAINS THAT ADMIT

0:-JLY TIIE TRIVIAL MIXED PROBLEM

Consider the L2-mixed problem for the unbounded domain exterior to a compact polyhedron. \Vhen the polyhedron is convex the requirement of postulate (iii) of (1.1) eliminates all but the trivial partition from the class of well posed mixed problems. In this C&':ie we will say that the exterior problem is rnonochrornatic. The mixed problem for a compact polyhedral domain can also be monochro matic for the interior problem. An example is provided by the regular compound polyhedron that is the union of 5 equal regular tetrahedra with a common center, a picture of which may be found as Number 6 on Plate III between pp.48-4U of H. S. M. Coxeter's book (Cox63] . An elementary arrangement of plane surfaces that elucidates the local element of this phenomenon is found upon considering the do main of ffi'.3 that is the union of the upper half-space together with all points (x, y, t) with (x, y) in the first quadrant of the plane, i.e. the union of a half-space and an infinite wedge. The boundary consists of 3 faces: the 4th quadrants of both the xt and yt-planes and the piece of the xy-plane outside of the 1st quadrant of the xy-plane. The requirement of postulate (iii) is met only by the negative t-axis. But no color change is possible there because any color on either of the 4th quadrants must be continued across the positive x or y-axis to the 3rd faee of the boundary. On the other hand, a color change is possible for the complementary domain and i::; possible for the exterior domain to the compound of 5 tetrahedra. Is there a polyhedral surface with a finite number of facp.-; for which both interior and exterior mixed problems are monochromatic? (Agm65] Shmuel A gmon , Lectures

REFERENCES

on elliptic boundary value p ro b lems, Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Math

ematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London , 1965. MR MR01 782·16 (31 #2504)

R.M. Brown, The mixed problem for Laplace 's equation in a cla.s.� of Lipschitz domains,

Comm. Part. Diff. Eq. 19 (1994), no. 7-8, 1217-1233. Lennart Carlcson, On the existence of boundary values for harmonic functions in several variables, Ark. Mat. 4 (1962), 393-399 (1962). MR MR0159013 (28 #2232) (Cox63) H. S . M. Coxeter, Regular polytopes, Second edition, The Macmillan Co . , New York, 1963. MR MR0151 873 (27 #1856) [Dah77] Bjorn E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), no. 3, 275-288. MR. 57 #6470 [Dah79) __ , On the Poisson integral for Lips chitz and C1 -domains, Studia Math. 66 (1979), no. 1 , 13-24. MR 81g:31007 [Dau92] Monique Dauge, Neumann and mixed problems on curvilinear polyhedra, Integral Equa tions Operator Theory 15 (1992), no. 2, 227-26 1 . MR. MR1 147281 (93e:35025)

[Bro94]

(Car62]

THE MIXED PROIJLEM FOR. HARMONIC FUNCTIONS IN POLYHEDRA OF IR3

[Fol95] [Gla70] [GT83]

[HW68] [JK81J (JK82] [Lan72]

[MR02]

[MR03]

[MR04]

[MR05j

[MR06]

[MR07]

423

Gerald B. Folland, Introduction to partial differential equations, second ed., Princeton University Press, Princeton, NJ, 1995. MR MR1357411 (96h:35001) L. C. G laser, Geometrical combinatorial topology, vol. 1, Van Nostrand Reinhold, 1970. David Gilbarg and Neil S . Tr udinger , FJlliptic partial differential equations of second order, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Prin ciples of Mathematical Sciences], vul. 224, S p1·inger-Verlag, Berlin, 1983. MR MR737190 (86c:35035) R. A . Hunt and R. L. Wheeden , On the boundary values of harmonic functions, Trans. Amer. Math. Soc. 132 (1968), 307-322. David S. Jerison and Carlos E. Keni g, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 203-207. MR 84a:35064 D. S. Jerison and C. E. Kenig, Boundary value problems on J,ip8r.hitz domains, MAA Studies in Mathematics, Studies in Partial Differential Equat ions 23 (1982), 1-68. N. S. Landkof, Foundations of modern potential theo·ry, Springer- Ver lag, New York, 1972, Translated from the Russian by A. P. Doo hovskoy, Die Grundlehren der mathcmatischen Wissenschaften, Band HlO. MR MR0350027 (50 #2520) V. G. Maz'ya and J. Rossmann, Point estimates for Green's matrix to boundary value problems for second order elliptic systems in a polyhedral cone, ZAMM z. Angew. Math. Mech. 82 (2002), no. 5, 291-316. MR MR1902258 (2003d:35002) Vladimir G. Maz'ya and J\irgen RoBmann, Weighted Lp estimates of solutions to bound ary value problems for second order elliptic systems in polyhedral domains, ZAMM Z. Angew. Math. Mech. 83 (2003), no. 7, 435-467. MR MR.1987897 (2004g: 35061) V. Mazya and J. Rossmann, Schauder estimates for solutions to boundary value problems for second order ell1ptic systems in polyhedral domains, Appl. Anal. 83 (2004) , no. 3, 271-308. MR �1R2033239 (2004k:35098) V. Maz'ya and J. Rossmann, Pointwise estima.tes for Green's kernel of a mixed boundary value problem to the Stokes system in a polyhedral cone, Math. Nachr. 278 (2005 ) , no. 15, 1 766-1810. MR }.fR2182091 (2007b:35269) V. G. Maz'ya and J. Rossmann, Schauder estimates for solutions to a mixed boundary value problem for the Stokes system in polyhedral domains, Mat h . Methods Appl. Sci . 29 (2006). no. 9. 965-- 1 017. MR MR2228352 (2007£:35220) V. Maz'ya and J. Rossmann, Lp estimates of solutions to mixed boundary value problems for the Stoke.� system in polyhedral domains, Math. Nachr. 280 (2007), no. 7, 751 793. MR MR2321139 C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Springer Verlag, 1972. Elias M. Stein. Singular integrals and differentiability properties of functions, Princeton Mathematical Series. No. 30, Princeton University Press, Princeton, N.J., 1970. MR 44 #7280 G. C. Verchota. Layer potentials and regularity for the Dirichlet problem for Laplace 's equation in Lipschitz domains, J. Funct. A nal. 59 (1984), no. 3, 572-611. G regory C. Verchota, The use of Rellich identities on certain nongraph boundaries, Harmonic analysis and boundary value problems (Fayetteville, AR , 2000), Amer. Math. Soc . , Providence, RI, 2001, pp. 127-138. MR 1 840 431 Gregory C. Verchota and Andrew L. Vogel, A multidirectional Dirichlet problem, J. Geom. Anal. 13 (2003), no. 3, 195-520. MR 2004b:35060 , The multidirectional Neumann problem in JR4, Math. Ann. 335 ( 2006 ) , no. 3, 571-{)44. MR MR222 1125 (2007£:35053) ·

[RS72j

[St e70) (Ver84]

(Ver01 j (VV03) [VV06]

___

215 CARNEGIE. SYRACUSE

UNIVERSITY,

E-mail address: gvercbotiDsyr. edu

SYRACUSE NY 13244

9 780821 844243

PSPUM/79

Perspectives in partial differential equations, harmonic analysis and applications: a volume in honor of Vladimir G. Maz'ya's 70th birthday

Read more

Perspectives in partial differential equations, harmonic analysis and applications: a volume in honor of Vladimir G. Maz'ya's 70th birthday

Read more

Perspectives in partial differential equations, harmonic analysis and applications: a volume in honor of Vladimir G. Maz'ya's 70th birthday

Read more

Harmonic analysis and partial differential equations: in honor of Calderon

Read more

Harmonic Analysis and Partial Differential Equations

Read more

Harmonic analysis and partial differential equations

Read more

Distributions, Partial Differential Equations, and Harmonic Analysis

Read more

Harmonic Analysis and Partial Differential Equations

Read more

Perspectives in Partial Differential Equations, Harmonic Analysis and Applications (Proceedings of Symposia in Pure Mathematics)

Read more

Harmonic Analysis and Partial Differential Equations

Read more

Advances in Phase Space Analysis of Partial Differential Equations: In Honor of Ferruccio Colombini's 60th Birthday (

Read more

Analysis, Partial Differential Equations and Applications: The Vladimir Maz'ya Anniversary Volume (Operator Theory: Advances and Applications)

Read more

Advances in Phase Space Analysis of Partial Differential Equations: In Honor of Ferruccio Colombini's 60th Birthday (Progress in Nonlinear Differential Equations and Their Applications, Volume 78)

Read more

Partial Differential Equations and Applications

Read more

Partial differential equations and applications

Read more

Elements and analysis of partial differential equations

Read more

Control of partial differential equations and applications

Read more

Geometry in partial differential equations

Read more

Partial differential equations in biology

Read more

Partial differential equations in mechanics

Read more

Partial differential equations in biology

Read more

Partial differential equations in physics

Read more

Differential equations in applications

Read more

Partial Differential Equations and Spectral Theory (Operator Theory: Advances and Applications Advances in Partial Differential Equations)

Read more

Advances in Partial Differential Equations)

Read more

Differential Equations in Applications

Read more

Differential equations in applications

Read more

Trends in Partial Differential Equations of Mathematical Physics (Progress in Nonlinear Differential Equations and Their Applications)

Read more

Harmonic Analysis and Partial Differential Equations: Essays in Honor of Alberto P. Calderon (Chicago Lectures in Mathematics)

Read more

Nonlinear partial differential equations in differential geometry

Read more

Recommend Documents

Perspectives in partial differential equations, harmonic analysis and applications: a volume in honor of Vladimir G. Maz'ya's 70th birthday

...

Perspectives in partial differential equations, harmonic analysis and applications: a volume in honor of Vladimir G. Maz'ya's 70th birthday

Perspectives in partial differential equations, harmonic analysis and applications: a volume in honor of Vladimir G. Maz'ya's 70th birthday

Harmonic analysis and partial differential equations: in honor of Calderon

Harmonic Analysis and Partial Differential Equations

Harmonic analysis and partial differential equations

HARMONIC ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS BJ RN E. J. DAHLBERG CARLOS E. KENIG ISSN 0347-2809 DEPARTMENT OF ...

Distributions, Partial Differential Equations, and Harmonic Analysis

Universitext Dorina Mitrea Distributions, Partial Differential Equations, and Harmonic Analysis Universitext Unive...

Harmonic Analysis and Partial Differential Equations

HARMONIC ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS BJÖRN E. J. DAHLBERG CARLOS E. KENIG ISSN 0347-2809 DEPARTMENT OF ...

Perspectives in Partial Differential Equations, Harmonic Analysis and Applications (Proceedings of Symposia in Pure Mathematics)

Perspectives in Partial Differential Equations, Harmonic Analysis and Applications Vladimir Gilelevich Maz'ya 0 IN ...

Harmonic Analysis and Partial Differential Equations