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 ...

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Perspectives in Partial Differential Equations, Harmonic Analysis and Applications

Vladimir Gilelevich Maz'ya

0 IN

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45 s row

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2000 Mathematics Subject Classification. Primary O1A50, 26D10, 31B15, 34L40, 35J25, 35Q53, 42B25, 46-06, 46E35, 74J15.

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 Vladimir G. Maz'ya's 70th birthday / Dorina Mitrea, Marius Mitrea, editors. p. cm. - (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.

3. Harmonic analysis. I. Mitrea, Dorina,

QA377.P378 2008 515'.353-dc22

2008030028

Copying and reprinting. Material in this book may be reproduced by any means for educational 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 acknowledgment 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 Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permission®ams.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://vww.ams.org/

10987654321

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

vii

Capacity, Carleson measures, boundary convergence, and exceptional sets NICOLA ARcozzI, RICHARD ROCHBERG, and ERIC SAWYER

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 HAIM BREZIS and JEAN VON SCHAFTINGEN

33

Mutual absolute continuity of harmonic and surface measures for Hormander type operators LUCA CAPOGNA, NICOLA GAROFALO, and DUY-MINH NHIEU

49

Soviet-Russian and Swedish mathematical contacts after the war. A personal account. 101

LARS GARDING

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

105

A local Tb Theorem for square functions 175

STEVE HOFMANN

Partial differential equations, trigonometric series, and the concept of function around 1800: a brief story about Lagrange and Fourier JEAN-PIERRE KAHANE

187

Quantitative unique continuation, logarithmic convexity of Gaussian means and Hardy's uncertainty principle CARLOS E. KENIG

207

Boundary Harnack inequalities for operators of p-Laplace type in Reifenberg flat domains JOHN L. LEWIS, NIKLAS LUNDSTROM, and KAJ NYSTROM

229

Waves on a steady stream with vorticity MARKUS LILLI and JOHN F. TOLAND

V

267

CONTENTS

vi

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 FANS TRIEBEL

347

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

377

The mixed problem for harmonic functions in polyhedra of R3 MOISES VENOUZIOU and GREGORY C. VERCHOTA

407

On the scientific work of V.G. Maz'ya: a personalized account Vladimir Gilelevichl 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 "life is good for only two things: discovering mathematics and teaching mathematics". 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 prefigured 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: "Whatever he writes is beautiful, his love for art, music and literature seeming to feed his mathematical aesthetic feeling".

I. Rough childhood. Vladimir Maz'ya was born on December 31, 1937, 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 personal 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 '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 apartment. 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 desolate 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 problem for second order elliptic equations was published in Doklady Akad. Nauk SSSR. Upon finishing his undergraduate studies at Mathmech-LSU, Vladimir Maz'ya secured a position as a junior research fellow at the Research Institute of Mathematics and Mechanics of Leningrad State University. Two years later he successfully defended 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. According 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 dissertations 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


ix

of Sciences of the USSR, where he created and headed the Laboratory of Mathematical 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 recipient of a series of distinguished awards in relatively quick succession. In 1990 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, F4ance, 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.

o 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 C°, 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 Tricomi, 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

x


and important problem. These innovative ideas did not get instantaneous recognition 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 cell". 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 child 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 collaboration 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 C°, on a polyhedral surface. 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 logarithmic 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 80's, consists of establishing well-posedness results for certain auxiliary boundary value and transmission problems for the Laplace equation in weighted Sobolev spaces.

o 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 calculus 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 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 of solution is suitably extended?" have generated a large amount of attention and, in the second half of the 20th century, proofs were obtained in sufficient generality. 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

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

xi

the second order linear elliptic equations in divergence form with bounded measurable coefficients. Maz'ya showed that this property fails for higher-order equations which may admit variational solutions which are not locally bounded. The counterexamples 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.

o 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 r1 D P2 i . . D r8 with the property that the vector field is tangent to rk at points belonging to Pk+1, and is transversal to 18. 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 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 codimension greater than one in the boundary are negligible, in the sense that they play the same type of role as removable singularities.

oBoundary-value problems in domains with piecewise-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 C1 (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 1967 paper on elliptic boundary-value problems in domains with conic singularities, 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, asymptotic representations of solutions, well-posedness theorems, and methods for

xii


computing the coefficients in the asymptotics of solutions near boundary singularities. The theory thus developed, together with important applications to problems arising in mechanics, engineering and mathematical physics, is presented in the monographs [Ma4], [Ma-8], [Ma-15], and [Ma-16]. The aforementioned body of results complements the theory of elliptic boundary value problems in Lipschitz domains, as initiated by A. Calder6n, B. Dahlberg, E. Fabes, N. Riviere, 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 multipliers 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 [Marl] by the same authors reports on the more recent progress in this area. The obvious motivation for a thorough investigation of properties of multipliers stems from the study of partial differential equations of the type (1)

8a(aa,0(X)O u) = f

Lu :_

in

Q,

I«I,II I 1 - II*w(a-1) for p.p. E Ea and £a(v)

_ E(I*v)2 < E(I*w)2 =

£a(WIE.)

T.

T.

Consider the functions I*w in T and Iav = I*v in Ta. Define a new function 0 on T:

O(x) =

I*v(x) if x E Ta, I*w(x) if x E T \ Ta.

We have

I,0(6) > 1 q.e. on E, hence II0IILz > Cap(E). On the other hand, II1pIILz

= £a(v) + [E(w) - £.(W)] < £(W) = Cap(E),

and we have reached a contradiction. The measure WIE.

1 - II*w(a-1)' then, minimizes 9,,,(,u) over the set of the measures p such that II * p(l;) > 1 for p.p. in Ea, hence A = wa. The claim is proved. By the homogeneity of the energy, £a(W I

(1 - II*W(a-1))2£a(Wa)

_ (1 - II*w(a-1))2wa(Ea) _ (1 - II*w(a-1))w(Ea). As a consequence,

E_>a(I*w)P I*w

P1-P'

£a(WIEa)

V(w)(a-1))p -1
]. By definition of II II CM, £(p) A. Then, p(E) < p(E) = p(E)2 < Cap(E), (E µ £(µ) IIPIICM -

µ(E)

as wished.

0

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

CARLESON MEASURES AND EXCEPTIONAL SETS

9

5.2. On a Question of Maz'ya. Conditions 4 and 5 of Theorem 4 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 4 then so does any smaller positive measure. That fact follows from knowing that condition 4 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 It on T, let vµ = (I*p)2. THEOREM 5.3. Let p be a measure on T and let A be a measurable function on

T, 0 < A < 1. If I*Qµ < I*µ on T, then I*Qa,, < 2 I*(Ap). COROLLARY 5.4. If v < p and it, v are measures on T, then IIiiIICM(T) 2II pII cM(T).

µ-

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

M A(a) = max

I * (Ap) (y)

t) = Etvµ(S(aj)) 7

< EtI*p(aj) J

I*(AM)(a3)

< I*(AM)(o)

Inserting this estimate in the previous one, we have

I*vaµ(o) < 2. I*°µ(o) We now give a direct proof that the tree condition is equivalent to the capacitary condition, Theorem 5.6 below. That the capacitary condition implies the tree

ARCOZZI, ROCHBERG, AND SAWYER

10

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. LEMMA 5.5. We have

CapT(S(E)) < 4CapT(E),

for E = U8S(aj), where S(E) = US(aj). Note that CapT(S(E)) = CapT({aj}). Obviously, CapT(S(E)) > CapT(E). PROOF. Let cp be the extremal function for CapT(E):

Icp(() = 1 on E, CapT(E) _

cp2.

We show that it is near extremal for S(E). The function cp can be recovered from the equilibrium measure, cp = I*µ, and µ is clearly constant on each 8S(aj): there is Fj > 0 such that W = rj2-d(l3) V0 E S(ay)

Now, for allCEOS(aj), 1 - Icp(aj) = Icp(C) - IV(a)

1: (43) /3 E [aj,S]

= r,

2-d(f3) [a3,C]

= r,2-d(ai),(a9)-

Hence, 1 - cp(aj) = Icp(aj). Note that cp, the extremal function, is monotone increasing with respect to the partial ordering in T, thus c'(aj) > 1/d(ad). Hence,

Ico(aj) > 1 - 1/d(ad) > 1/2. This means that 2cp is an admissible function for E:

CapT(S(E)) < 4CapT(E).

0 THEOREM 5.6. Let p be a positive, Borel measure on T. Then µ satisfies sup I*[I*µ]2(a) < C1(µ). I*/b(a)

(5.5)

aET

if and only if p satisfies, for all sets E = UjBS(aj) in 8T,

µ (UjS(aj)) < C2(µ) CapT(UJBS(aj)). Moreover, C2(p) - C1(µ).

(5.6)


11

PROOF. Suppose that µ satisfies (5.5) with C1(µ) = 1. Recall that S(E) = 2 by Theorem 5.3. Hence,

US(a3 ). Then, µE = µI S(E) < µ satisfies II µE II CM(T)

CapT(S(E)) =

sup sup(v)CS(E)

-

v(S(E)) II

lIICM(T)

µ(S(E)) II µE II CM(T)

µ(S(E)) 2

i.e., A(S(E)) < 2 CapT(S(E)) < 8 CapT(E), by Lemma 5.5. As mentioned, the opposite inequality is already known.

6. Boundary Behavior and Exceptional Sets In this section we give a number of results about boundary behavior and exceptional 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 5.2, 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.

6.1. Boundary Values. It is not clear at first glance that functions in Vd must have boundary values on a large subset of 8T. 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 : d(a) < n} . Given F E Vd, set, for each n

Fn* = I(IDFI Xn)-

It is immediate that the sequence of functions {Fn *l is increasing in n, that each function extends by continuity to all of 8T, and that each function is in Dd and has norm at most I I F I I By monotonicity the extended limit Jim Fn* = F* is defined everywhere. It then follows using Fatou's lemma that IF* I2 has finite integral with respect to any Carleson measure µ for Dd. Hence F* is finite µ - 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 p - a.e. Hence, by Corollary 5.2, F has finite boundary limits q.e.


12

6.2. Beurling's Theorem. The prototypical result in this area is Beurling's theorem that any f E V has radial boundary values q.e. He did this by showing that the boundary values of the radial variation, V(f) (e$t) were finite q.e. and noting that at such points f 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 Dd we measure its radial variation by VT(F)(a) = I(IDFI)(a). THEOREM 6.1 (Discrete Beurling's Theorem). For F E Vd,

CapT({r E 8T: VT(F)(r) = +oo}) = 0. Hence

CapT({r E 8T : lim F(a) does not exist}) = 0. aEr

PROOF. Start with F E Dd. Hence DF E 12(T), and hence IDF) E 12(T), and

thus VT(F) E Dd . This insures that the set on which VT(F) = oo is a null set for any Dd Carleson measure and hence, by Corollary 5.2, has vanishing capacity. Finally, one easily checks that if VT (F) (r) < oo then limaEr F(a) exists.

6.3. Algebraic Approach Regions. We continue to focus on Dd 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 F defined on T by lim F(/3) _ S

ms

F(/3).

OE

d(Q)-'oo

At the level of metaphor, convergence along the geodesics r 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 r. The analog of nontangential convergence with wider wedges is obtained by also including points that are at most a fixed distance, k, from F. For a E T, n E N let

{a + n} = {(/3 E T : 0 >- a, d(a, /3) = n)} and we define

r1(k) = U {a + k} . aEr

THEOREM 6.2. Fix k > 1. For F E Vd, F has r1(k) limits q.e.; that is CapT({r E 8T : lim F(a) does not exist}) = 0. ri(k)

PROOF. For F E Dd we define DF(1) by

DF(1)(a) _ IDF(a)I + E IDF(/3)I. (3E{a+k}

Because DF E 12(T) we also have DF(1) E 12(T), the reason is that each value I DF(8)I shows up as part of DF(1)(6) and as part of at most one other Hence G = I(DF(1)) E Dd and thus G has boundary values q.e. Suppose now that r is a geodesic representing a point in the boundary at which G has boundary limit G(r). We claim that it follows that F has a boundary limit through the approach region r1(k). We are assuming limG(a) - G(r) = 0, (6.1) DF(1)(b').


13

Now pick a in r and let 8, /3' be two points of r1 (k) which are further from the root than a. Let a-k be the direct ancestor of a of order k. Now we have that J F(3)

JF03) - F(ay) I

- F(a) I + I F(a) - F(a') I

< E IDF('Y)I + E IDF(y')I ryE[a,,3]

4 (G(r) - G(a-k)) .

(6.2)

To see this last inequality we concentrate on the first sum. The geodesic segment [a, /3] has two regions, the first consisting of 's in r, the second part consisting

of the y"s not in r. In the first case I DF('y) I < DG('y). In the second case set E [a-k, a] and thus /3(y') = y'-k, the ancestor of y' of order k. We have IDF(y')I < DG(,3(y')). Thus the sum of the IDFI along [a,/3] is dominated by twice the sum of the DG 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 r1(k). We now consider larger regions. Fix an integer k and set

r2(k) = U {a + kd(a)}. aEr

We will consider limits over the sets r2(k). To do this we start with F and construct a majorant for its variation. Define k-1

DF(2)(a) = IDF(a)l +

max {IDF(/3)I :,3 E {a + kd(a) + j}} j=0

If F E Dd then DF E 12(T). We now claim that DF(2) E 12(T). Each DF(2)(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 DF(/3) I does appear in DF(2) (/3). The only other time that term can be used as a summand is in DF(2) (a) for the unique a which satisfies the two conditions a E [o, /3] and (k + 1) d(a) < 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)(a) shows that DF(2) E 12(T). A minor modification of the argument we used before now shows that F converges along r2(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 [o, a] with (k + 1) d(a') > d(a).

THEOREM 6.3. Fix k > 1. For F E Dd, F has F2(k) limits q.e.; that is CapT({F E aT : lim F(a) does not exist}) = 0. rz(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 > IxI. The thickened geodesics F1(k) correspond to the wider, but still non-tangential, approach regions y > 2k IxI. The regions r2(k) correspond to regions which are tangent to the boundary with the tangency being of finite order; roughly, r2(k)


14

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

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 different capacity. The capacities will be those associated with the spaces Dd,e, the Hilbert space of F functions on T for which IIFIIDd,

= IF(o)l2 + ET IDF(a)I2

2-Ed(a) < oo.

(6.3)

The approach regions of interest are these. For 0 < e < 1 and r a geodesic in T which defines an element of OT we set

U {a+

r3(e)

[2 ed(a)

D aEr (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 Theorem 6.3. We start with F E Dd and construct a majorant for its variation. Define 7=2.2ed(d)

DF(3) (a) = IDF(a)I + E max {IDF(a)I : ,Q E {a + j}}

(6.4)

.1=Zed(°)

If F E Dd then DF E 12(T), but now it need not hold that DF(3) E 12(T). For each

a let {$(a)j} be the vertices of T which appear on the right hand side of (6.4); that is, a and the selected elements where max I DF(3)I is attained. Thus

DF(3)(a) = E IDF(3(a)j)I 3

Hence

DF(3)(a)

< (E IDF(,3(a)j)I2) . (number of j's) 7

< C2ed(") (J: I DF(/3(a)i) I2).

Hence the sequence of numbers {2-Ed(a) I DF(3) (a) 12 } is summable because, again,

no vertex shows up as a,3(a)j more than a few times. Thus DF(3) E 12(T, 2-ed(a)) We now use the same arguments as before. Set G = I (DF(3) ); G will have finite 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 Dd,e. Also as in the previous proof, any boundary point r at which I(DF(3))(F) < 00 will be a boundary point where we have good convergence of F; in this case the good convergence meaning convergence over r3(e). The description of the Carleson measures for these spaces is given in [AR]. Here is the description for 0 < e < 1.

THEOREM 6.4. Suppose 0 < e < 1. Let µ be a positive Borel measure on T. Then, the following are equivalent:


15

(1) , is a Carleson measure for Dd,E; i.e. there is a constant C so that V f E Dd,e

JF(a)12 µ(a) < C JIF II vd,e

(6.5)

QET

(2) p satisfies the --tree condition. There is a constant C so that Va E T I*

(a) < CI*tt(a).

Hence we have

THEOREM 6.5. Fix E, 0 < e < 1. For F E Vd, F has r3 (e) limits for all r E OT with the possible exception of a set which is a null set for every measure it which satisfies the condition (6.6). In this case the approach regions have infinite order tangency, in fact subexponential contact. The Euclidean analogs of these regions shaped like the part of the upper halfplane where y > exp(- JxJ-E) Finally if a variant of Theorem 5.1 is available in 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 e = 0 case considered earlier; details will be in [ARSp]

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 measure 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 y > exp(- 1x1-1). Further work in that direction is in [NS], [DB], and [Tw]. It would be interesting to know if an analogous result holds for Dd and we leave that as a question. The proof just given shows that any F E Dd has limits along regions F3(1) with an exceptional set that is a null set for all the measures which satisfy (6.6) with e = 1. However the full boundary is, in fact, such a set. The quickest way to see that is to note G = Ig with g(a) = d(a)-1 has boundary values identically +oo. We could consider the subspace of Dd,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 F E Dd on T are not determined by the boundary values.

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

X= If ED:

I f'(z)12 dxdy is a Carleson measure for D}

.

The discrete analog is

Xd= IF E Dd : pp = IDF I2 is a Carleson measure for Dd }

.


16

As discussed briefly in [AR], the space X has a relation to V similar to the relation the space BMO has to the Hardy space H2. In fact one of the characterizations of functions in BMO is that f E BMO if and only if I f'(z)I2 (1 - I z12)dxdy is a Carleson measure for the Hardy space. The functions in BMO are both smaller than and smoother than generic functions in H2. Similarly functions X are smaller than generic functions in D. Specifically if f E D then for some small e f it holds that exp(e f I f I2) has integrable boundary values; however f E X insures that the boundary values of exp (exp(e f If I)) 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 Dd. For comparison recall that Theorem 6.1 states that

VF E Dd lim F(a) = E DF(a) exists for quasi-every r. aEr

(6.7)

' Er

Suppose now that F E Xd is fixed µ(a) = /F(a) = IDF(a)I2. Recall that the tree condition for µ is that there is a C so that for all a

I*(I*µ)2(a) < CI*p(a). THEOREM 6.6. Suppose p is a Carleson measure Dd then

E p(S(a)) < oo for quasi-every r. «Er

Equivalently

J

log+ Iz 1eaoI dµ (z) < oo for quasi-every

In particular, if F E Xd then

E (I*(DF)2)(a) < oo for quasi-every r. QEr

PROOF. The argument proving Theorem 6.1 applies to Ih for any square summable function h defined on T.The tree condition evaluated at the origin insures that h(a) = I*µ(a) = µ(S(a)) is such a sequence. That gives the first statement. The second follows from the first by estimating how often each value µ(Q) occurs

in the sum. By writing out all the terms in (I*(DF)2)(a) and then discarding those corresponding to vertices not on r we obtain a weaker, but more transparent, corollary.

COROLLARY 6.7. If F E Xd then

E d(a) I DFI2 (a) < oo for quasi-every F. aEr

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

li II*(IDFI2)(fl)and lim f log+

lwzl dp (z) . 11


17

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

7.1. Other Function Spaces on Trees. Various function spaces on T 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], [ARS1], [ARS2] where in addition to l2(T) study is also made of various weighted lP(T) 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 lP spaces and the arguments are more complicated. That work will be presented in [ARSp]

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]. THEOREM 7.1 (Beurling, 1940). For all f E D

Capp({e't : lim f (reie) fails to exist) = 0. r First select and fix f E D. Also select R so large that the hyperbolic disks of radius R centered at points of the tree, {D(a, R) : a E T} is a cover for D and so that for all a E T we have {a', a,., all C D(a, R). For each a E T we measure the local oscillation of f by

Osc(a) = Osc(a, f) = sup {If (z) - f (z') I : z, z' E D(a.R)} . Straightforward considerations of the geometry of the placement of T in D show

that the disks {D(a, 5R) : a E TI, have the property that there is an M so that no point is in more than M disks. This insures that {Osc(a)} E 12(T) because function theoretic estimates yield

Osc(a)2 < C

f

J D(n,5R)

2dxdy.

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

DT capacity zero. This insures that f has a limit along the path connecting the vertices in r and that in turn is enough to insure that f also has a limit along the radius which terminates at the point of the circle corresponding to the boundary element determined by P.

18


This outline gives convergence off of an exceptional set is of VT capacity zero

rather than D 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 VT 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 V(f) (eae) . However 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 VT 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 1P(T), then the argument shows that limits exist along F with an exceptional set of r 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

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 boundary convergence, they were all of the same sort, a geodesic r 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 approach to the results of Nagel, Rudin, and Shapiro given by Nagel and Stein [NS]. 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 [DB]. Particularly interesting to us is that a substantial part of the work there proceeds through analysis on model spaces defined on dyadic trees.


19

References [Ad] [AH] [AC] [A]

[AE]

D. Adams, On the existence of capacitary strong type estimates in Rn, Ark. Mat. 14 (1976), no. 1, 125-140. D. Adams, L. Hedberg, Function spaces and potential theory, Grundlehren der Mathematischen Wissenschaften, 314, Springer-Verlag, Berlin, 1996. 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. Shimane Univ. Ser. B Math. Sci. 30 (1997), 1-21 H. Aikawa, M. Essen, Potential theory- selected topics, Lecture Notes in Mathematics, vol. 1633. Springer-Verlag, Berlin, 1996.

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 [Ar]

[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 Publ. 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) Acts, 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. 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 operators, Ann. Inst Fourier (Grenoble) 36 (1986), no. 4, 207-228. [Ca]

[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 functions, 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). [M] 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 Dirichlettype spaces, Ann. of Math. (2) 116 (1982), no. 2, 331-360. [NS] A. Nagel, E. Stein, On certain maximal functions and approach regions, Adv. in Math. 54 (1984), no. 1, 83-106. D. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math. 24 (1980), no. 1, 113-139. [St] [Tw] J. Twomey, Tangential boundary behaviour of harmonic and holomorphic functions, J. London Math. Soc. (2) 65 (2002), no. 1, 68-84.


20

I. Verbitsky, Nonlinear potentials and trace inequalities, The Maz'ya anniversary collection, Vol. 2 (Rostock, 1998), 323-343, Oper. Theory Adv. Appl., 110, 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: [email protected] (Rochberg) DEPARTMENT OF MATHEMATICS, WASHINGTON UNIVERSITY, ST.

Louis, MO

63130, U.S.A. E-mail address: [email protected] (Sawyer) DEPARTMENT OF MATHEMATICS & STATISTICS, MCMASTER UNIVERSITY, HAMILTON,

ONTAIRO, L8S 4K1, CANANDA E-mail address: Saw6453CDN@ao1. com

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

H. = A + V.

(1.1)

where 0 is the lattice Laplacian on Zd, i.e. (1.2)

(n, n')

-{ 1 if In'-nil+...+Ind-nal=1 0 otherwise

and Vu, is a random potential, which we take of the form (1.3)

VV,,(n) = wnv,

or

V .(n) = w,vn, + vi,

(1.3')

with {w, In E Zd } independent Bernoulli or normalized Gaussian random variables and {v , I n E Zd}, {w n I n E Zd} decaying deterministic sequences. More precisely, (1.4)

vn

_

s;

for some a > 0

(InI + 1)«

and w, will satisfy (1.5)

IwnI

0. In the 1-dimensional case (d = 1), the spectral theory of H,' has been extensively studied and is well-understood.

Rewriting the equation Hb = E as V)n+l + On-1 + (Via)nOn = E'b (n E Z)

(1.6)

the main tool is the transfer matrix formulation of (1.6) ON+1

(1.7)

MN (E)

(001

o)

where (1.8)

MN(E) _ (E 1Vn

Vi

01) ... (E

01)

1

and the Lyapounov exponent

L(E)= ra mo N Elog IIMN(E)11.

(1.9)

For all rc > 0, H,,, 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)teR and means that for any given exponent A > 0 sup I E InI2A1(e

(1.10)

00

nEZ whenever

decays rapidly forlnl - oo.

Recall that (DL) implies pure point spectrum but not conversely. Despite quite precise and widely believed conjectures, the higher dimensional case d > 1 is still poorly understood. The belief is that for d = 2, the spectrum remains pure point with localized eigenfunctions (but for small rc a different behaviour of the localization length at the edge and in the bulk of the spectrum) and for d > 3 and small rc, 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 rc 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 rc. For d = 1, the behaviour is for rc -* 0 (i.e. the reciprocal of the Lyapounov exponent) and [ESY] establishes in d = 3 localization lengths at least 1 , 6 > 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 wnlnl-a, then again transfer matrix methods, it is shown in [DSS] that if V,,,(n) = (see [DSS], Theorem 1.1). If 0 < a < 2, H,,, satisfies a.s. (AL) and (DL). In fact the eigenfunctions OE have a decay rate (1.11)

IE(n)l < C(E)

If a > 2, HH,, has pure a.c.-spectrum, a. s.

exp{-c'lnll-2«}.

23

DYNAMICAL LOCALIZATION

The situation a = 2 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 d > 5 in fact) regarding (DL) with a random decaying potential. We will sketch a proof of the following THEOREM. Let d > 5 and H,, be given by (1.1) with

V,,(n)_

(1.12)

(nE7Gd)

where a > 0 is arbitrary, is is sufficiently small and wn is some deterministic potential of size Iwnl = O(r.

(1.13)

2InI-2«).

Then, with high probability in w, H,, fails (DL). More precisely sup InIAIeitH(0, n)I = oo if A > d.

(1.14)

tER nEZd

For d = 1, this behaviour is indeed impossible if a < 2.

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 expansion. 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 A by A + w with w 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 Hu, = A+ V,, with V,, as in (1.12) and show that at a specific energy E (the lower edge of the spectrum of A), H,, has an extended state and the Green's function G(E) of HH, behaves like the free Green's function G0(E), i.e.

IG(E)(n n')I -

(115)

1 In-n'Id-2.

'

Statement (1.14) is then easily deduced from (1.15). The analysis in [B] is actually only carried out for a > s but may be generalized to arbitrary a > 0 (it requires considering more terms in the multi-linear expansion

of G(E)). 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 E in the bulk of the spectrum,

IGo(E)(n,n')I - - (with an essential phase factor), while at the In-n'

The latter is square inteedge E_, we have IGo(E_)(n,n')I ' In 2 grable for d > 5, which explains our assumption. The renormalization for E in the bulk is quite different and much harder (cf. [ESY]).

JEAN BOURGAIN

24

(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 an infinite multilinear Bohr extension in is is unlikely to be tractable (even after renormalization). (6). Instead of lattice models, one could also consider a model on R', of the (n)co( - n) where 0 is the usual Laplacian, Vu, form Hw = -A + F,nEZd V,,, a random potential as above and W(x) a localizing bumpfunction on Rd. 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).

§2. The Green's function estimate (i) Redefine the Laplacian by subtracting 2d from the lattice Laplacian, thus d

(2.1)

j=1

and consider the Green's function Go = (-0 + io)-1 at 0-energy. Hence Gc(n'

n)

r e-27ci(n-n').f

J

1

In -

-,&(o

n'ld-2

(ii) Write (2.3)

with 0 as above and where

V=Vu,+w

(2.4)

with (2.5)

V (n) = V (n) --

K

(1 + InD)a

wn

(wn Bernoulli)

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) (2.6)

G=Go - GVGo

we obtain

(2.7) G = Go - GoVGo + GoVGoVGo - GoVGoVGoVGo +

± GVGo ... VGo.

Our aim is to bound IIGoVGoII, IIGoVGoVGoII,... using the randomness of V = Vu,

and probabilistic considerations. In an ideal situation of square cancellations, the

25

DYNAMICAL LOCALIZATION

square-summable for d > 5), together estimate I Go(n, n') I < n_ (with with some decay of Vn, would easily imply bounds of the form fin-

(2.8)

I(GoVGo...VGo)(n,n')I
« and taking in (2.7) an expansion of length 3, one derives that G is a perturbation of Go, IG(n, n') - Go(n, n') I
2 (o)

11

(2.12)

(1)

(8)

L r Wn1 ... Wn8 an n1 anln2 ... ane n, 11L2 E*

W

n1,... ,ne

< C. I

I a(o) ne,'n1 '

a(8) ne,n , I2J 1/2.

LLL

Proof. We may clearly assume am)n > 0. Since in the E* summation no (nl,... , n8) cancels, there is some index n8, which is not repeated or repeated an odd number of times. Specifying a subset I of 11,... , s} of odd size (at most 28 possibilities), we consider now s-tuples of the form (v(1), m, v(2), m, v(3), m ... )

where m E Zd appears on the I-places and v(1), v(2), ... are admissible complexes indexed by sub-intervals of {1, ... , s} determined by I.

JEAN BOURGAIN

26

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

*

(8) (nWn1 ...Wnean, l ...ane n,

nl,... ,ne

Wnl ... wnel annn1 ... a(81) m]

Ell Y' Wm [ mEZd

L2

J

v(1) II

(nl,... ,ne1)

(2.13)

1:*Wn.,+2

.. W

nea(81+1)

... a(82)

t m,ns1+2ne2,m ]

IIL2

U(2) 11

(net+2.... net )

where, for fixed m,

{n,,... ,ns1,ns1+2i... ,ns2,n82+2,...}.

m

We used here that if (n1, ... , n8) is admissible, then so is (n81,

n81+i....

,

n82) for

allI<s1<S2<S.

Enlargement of the original sum F,* 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 1'2

(2.14)

(2.13)
s (and assuming a < ). 2 The convolution operator M is symmetric and even in each variable with beInI-3(d-2). Hence haviour M(n) "

B'MEL£for lal 0. We will show that A < d.

JEAN BOURGAIN

32

It follows from (3.4), (3.5) that (3.6)

I (µo,. * PP)^(t)I < Ce-EItIIxI

-A

for some constant C. Next, we have from the resolvent identity

G(0 + ie) = G(0) - ieG(O)G(0 + ie) implying by (3.1), (2.62) (3.7)

IG(0 + ie)(0, x) - G(O)(O, x) I < e

1

I x - yI

1

d_z

0 such that for every domain 1 C R2,

A'>a. PROOF. Simply take some compactly supported divergence-free measure µ E R2) such that M(1; R2), and some compactly supported vector field E f Ra #:A 0. By translation and dilation, one has that

Ao >

.lR3 Y

{

1191111V 411,-

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 Sn = SB(o,1) Remember that An does not have an upper bound independent of the geometry. If we allow 1 to be multiply connected, A' has no upper bound. On the other hand, we do not know whether A'0 has an upper bound independently of the geometry for simply connected domains. PROBLEM 5. Does one have

sup{A' : S1 C R2 is a simply connected domain} < oo?

3. Higher dimensions 3.1. Inequalities for curves. Throughout this section r C R' is a simple, closed, rectifiable curve. The optimal constant in Theorem 2 is

Ar = sup{ f cp" t : cp" E CO°(R"; Rn) and

r

Jivin 3, is

A = sup{Ar : F C Rn is a closed rectifiable curve} attained? By which curve?

CIRCULATION INTEGRALS AND CRITICAL SOBOLEV SPACES

45

The answer to Problem 9 is open even when t is a planar curve of Rn, n > 3. There are numerous variants of Problem 9. In particular, one can define

Ar = sup{J cp" t : cp" E £ °(R"; R") and f IV A cp"In < 11

r

^

or one could work with different norms on WI,n(Rn; Rn), or even on W8'P(Rn; Rn),

with 0 < s < n and sp = n. 3.2. Inequalities for measures. As in two dimensions, we can also consider the relaxed problem with measures. When i is a finite vector measure such that div µ = 0, define

Ag = Sup{ fE CC°(Rn; Rn) and a

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{ Aµ : Ti is a measure, div/c = 0 and 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 r was taken among closed curves, µ is taken in the vector space of divergence-free measures. 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 µ be a maximizing measure in Proposition 3.4 (assuming that the supremum is achived). Is µ an integral along a curve? A partial answer is given by PROPOSITION 3.5. If µ achieves the supremum of Proposition 3.4, then /:c is an

extremal point of the unit ball in M#(Rn;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 measures. 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 measure that are not circulation integrals: Consider for k > 2 a kdimensional torus Tk and a constant vector field v on Tk such that the equation x = v does not have periodic solutions. If ' : Tk --p Rn maps Tk on 8 and v on j, one has that it defined by

fR 6 is atomic but is clearly not a circulation integral.

HAIM BREZIS AND JEAN VAN SCHAFTINGEN

46

PROOF OF PROPOSITION 3.5. Since one has clearly II#II = 1, assume by con-

tradiction that it = aµl + (1 - A)µ2i where A E (0,1), µl,µ2 E M#(Rn; Rn), IIµiII =1 and µi 0 µ. Let 0 E Wl'n(Rn;Rn) such that IIV0IILn =1 and fRn

Because cp is a maximizer for Aµ, n i=1

Therefore, cp cannot be a maximizer for Aµ;, and fRµ 3, and X! denotes the formal adjoint of Xj. The sub-Laplacian associated with X is defined by m

(1.2)

Lu = > Xj*Xju . j=1

A distributional solution of Cu = 0 is called L-harmonic. Hormander's hypoellipticity theorem [H] guarantees that every f--harmonic function is C°°, hence it is a classical solution of Lu = 0. W e consider a bounded open set D C R , 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

50

LUCA CAPOGNA, NICOLA GAROFALO, AND DUY-MINH NHIEU

the Dirichlet problem (1.3)

Gu=O inD, u=q5 on OD

.

Using Bony's maximum principle [B] one can show that for any 0 E C(OD) there exists a unique Perron-Wiener-Brelot solution H. 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 ADPx domains (admissible for the Dirichlet problem), for which we prove the mutual absolute continuity of the G-harmonic measure dw' and of the so-called horizontal perimeter measure dox = PX (D; -) on 49D. The latter constitutes the appropriate replacement for the standard surface measure on OD 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 0 E LP(OD, dox), for

1 < p < oo. If instead the domain D belongs to the smaller class o - ADPx introduced in Definition 8.10 below, we prove that C-harmonic measure is mutually

absolutely continuous with respect to the standard surface measure, and we are able to solve the Dirichlet problem for (1.2) for boundary data 0 E L"(OD, do), for

1 0 for which the real analytic domain Qm = {(z, t) E 1H1" I t > -MIz12}

,

admits a Go harmonic function u such that u = 0 on 8QM and which belongs exactly to the Holder class ra (in the sense that it is not any smoother) in any neighborhood of the (characteristic) boundary point e = (0, 0). 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, ..., X2n. In fact, since the paraboloid Il is a scale invariant region with respect to the non-isotropic group dilations (z, t) -> (Az, A2t), the smooth domain 1M should be thought of as a nonconvex cone from the point of view of the intrinsic geometry of G° (for a discussion of Jerison's example see section 4). 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 XG of the Green function near the characteristic set. This intuition was proved successful in the papers [LU1], [CGN1], 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 tangent outer X -ball condition. 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 L. The metric balls are not smooth (see [CG1]) and therefore it would not be possible to have a notion of tangency based on these sets.

MUTUAL ABSOLUTE CONTINUITY

53

4) In Dahlberg's mentioned theorem on the mutual absolute continuity between harmonic and surface measure in a Lipschitz domain D C R" there is one important property which, although confined to the background, plays a central role. If we

denote by o = H"-l I aD the surface measure on the boundary, then there exists constants a, /3 > 0 depending on n and on the Lipschitz character of D such that

a r"-1 < u(8D n B(x, r)) < /3 rn-1 for any x E OD and any r > 0. A property like this is referred to as the 1-Ahlfors regularity of a, and thanks to it surface measure is the natural measure on OD. 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 Pt = (0, ± 11) of OB. 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

v(aB n B(Pt, r)) =

(1.6)

rQ-2

where Q = 2n + 2 is the so-called homogeneous dimension of H' relative to the non-isotropic dilations (z, t) --> (Az, A2t) associated with the grading of the Lie algebra of IHII". The latter equation shows that at the characteristic points Pt surface measure becomes quite singular and it does not scale correctly with respect 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

ax (aD n Bd(x, r)) def Px (D; Bd(x, r)) To motivate such appropriateness we recall that it was proved in [DGN1], [DGN2] that for every C2 bounded domain D C H" one has for every x E aD and every 0 < r < R0(D) a

rQ-1 < o'x(OD n Bd(x, r)) < /3

rQ-1

.

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, ..., X,"} of smooth vector fields satisfying (1.1), we say that a connected bounded open set D C R" is admissible for the Dirichlet problem (1.3) with respect to the system X, or simply ADPx, if:

i) D is of class Cl; ii) D is non-tangentially accessible (NTAx) with respect to the Carnot-Caratheodory metric associated to the system {X1, ...,X91.} (see Definition 8.1); iii) D satisfies a uniform tangent outer X -ball condition (see Definition 6.2); iv) The horizontal perimeter measure is upper 1-Ahlfors regular. This means that there exist A, R° > 0 depending on X and D such that for every x E OD and

0 < r < R° one has Qx(OD n Bd(x, r)) < A I Bd(x, r)I

r


54

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, each one playing a different role. Let G(x, y) = GD (X, Y) = G(y, x) indicate the Green function for the sub-Laplacian (1.2) and for an ADPx domain D 1. By Hormander's theorem [H] and the results in [KN1], see Theorem 3.12 below, for any fixed x E D the function y --> G(x, y) is CO° up to the boundary in a suitably small neighborhood of any non-characteristic point ya E 8D. Let v(y) indicate the outer unit normal in y E 8D. At every point y E OD we denote by Nx(y) the vector defined by

NX(y) _ (< v(y), XI(y) >,..., < v(y), Xm(y) >) We also set

v(y), X l (y) >2 +...+ < v(y), Xm (y) >2 W (y) = I Nx (y) I We note explicitly that it was proved in [CDG2] that on 8D

dax = W do-. Denoting with E the characteristic set of D, we remark that the vector NX (y) _ 0 if and only if y c E. For y E 8D \ E we define the horizontal Gauss map at y by letting Nx (y)

vx () =

INx (y) L

DEFINITION 1.2. Given a C°° bounded open set D C R", for every (x, y) E D x (OD \ E) we define the subelliptic Poisson kernels as follows

P(x, y) = < XG(x, y), NX (y) >

,

K(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 8D \ E 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 E. Since as we have observed the function W vanishes on E, 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" a. Hereafter, for x E 8D it will be convenient to indicate with A(x, r) = 8D n 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. Let D C R' be a ADPx domain. For every p > 1 and any fixed x1 E D there exist positive constants C, R1, depending on p, M, Ro, x1, and on

the ADPx parameters, such that for xo E OD and 0 < r < R1 one has 3

P

1

A,

(x0,r)

K(x1, y)pdox(y)

j

< ox((x0,r)) C AJ

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

(x0r)

K(xl, y)dax(y)


55

Moreover, the measures dwxl and dax are mutually absolutely continuous. By combining Theorem 1.3 with the results if [CG1] we can solve the Dirichlet

problem for boundary data in LP with respect to the perimeter measure dvx. To state the relevant results we need to introduce a definition. Given D as in Theorem 1.3, for any y E OD and a > 0 a nontangential region at y is defined by Pa(y) = {x E D I d(x, y) < (1 + a)d(x, OD)} . Given a function u E C(D), the a-nontangential maximal function of u at y is defined by

N.(u)(y) =

sup Iu(x)I

xEr« (y)

THEOREM 1.4. Let D C Rn 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, dox), then Hf

(x) = 18D

f(y) K(x, y) do x (y) ,

and

II N«(Hf )II LP(aD,dax) C C IIf II LP(aD,dox)

Furthermore, HD converges nontangentially ox-a. e. to f on 8D. 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 C1,1 domain D C Rn which is NTAx the horizontal perimeter measure is lower 1-Ahlfors (this is a basic consequence of the isoperimetric inequality in [GN1]). Combining this result with the assumption iv) in Definition 1.1, we conclude that for any ADPX domain the measure ax is 1-Ahlfors. In particular, ax 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 o 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 LP with respect to do. To address this question in Definition 8.10 we introduce the

class of o - ADPx domains. The latter differs from that of ADPx domains for the fact that the assumption iv) is replaced by the following balanced-degeneracy assumption on v: there exist B, Ra > 0 depending on X and D such that for every x0 E OD and 0< r< R0 one has

max

( yEA(xo,r)

W(y)) o-(A(xa,r)) 1 there exists a constant C > 0 depending on D, X and p such that if f E LP (8D, do), then

Hf (x) -

18D f (y) P(x,y) da(y) ,

and

II N« (Hf ) II LP(BD,da) C C IIf II LP (BD,da

Furthermore, Hf converges nontangentially a-a. e. to f on 8D. 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 [LU1] for domains satisfying assumption iii) in Definition 1.1. The discussion of these examples is taken up in section 9. In closing we briefly describe the organization of the paper. In section 2 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 4 we discuss Jerison's mentioned example. Section 5 is devoted to proving some new interior a priori estimates of CauchySchauder type. Such estimates are obtained by means of a family of subelliptic mollifiers which were introduced by Danielli and two of us in [CDG1], 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 gradient of the Green function G is bounded up to the boundary, hence, in particular,

near E, 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 E, we are still able to obtain the boundedness of the horizontal gradient of G up to the characteristic set, although we now loose the uniformity in the estimates, see Theorem 6.9, 6.10 and Corollary 6.11. In section 7 we establish a Poisson type representation formula for domains which satisfy the uniform outer X-ball condition in a neighborhood of the characteristic set. This result generalizes a similar Poisson type formula in the Heisenberg group H"' obtained by Lanconelli and Uguzzoni in [LU1], and extended in [CGN2]


57

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 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 XG 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) I

,

x E D, y E OD ,

see (7.7), we prove the boundedness of the Poisson kernel y -+ K(x, y) on D. The main result in section 7 is Theorem 7.10. This representation formula with the estimates of the Green function in sections 5 and 6 lead to a priori estimates in LP for the solution to (1.3) when the datum ' E C(OD). 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 [CG1] enter the picture. In the opening of section 8 we recall the definition of NTAX-domain along with those results from [CG1] 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, 1.5 and 1.6. Finally, section 9 is devoted to the discussion of examples of ADPX and v ADPX domains and of some open problems.

2. Preliminaries In R', with n > 3, we consider a system X = {X1, ..., X,,,.} of C°° vector fields satisfying Hormander's finite rank condition (1.1). A piecewise C1 curve 7 : [0, T] -+ R'" is called sub-unitary [FP] if whenever 7'(t) exists one has for every

E]Rn m

< Xj(7(t)),S >2

7 (t),e >2 < j=1

We note explicitly that the above inequality forces 7'(t) to belong to the span of {X1(7(t)),..., X,,,,(7(t))}. The sub-unit length of 7 is by definition 18(7) = T. Given x, y E Rn, denote by So (x, y) the collection of all sub-unitary 7 : [0, T] -+ SZ which join x to y. The accessibility theorem of Chow and Rashevsky, [Ra], [Ch],

states that, given a connected open set 11 C Rn, for every x, y c Il there exists 7 E SS(x,y). As a consequence, if we pose

do(x,y) =

1 1, ,

we obtain a distance on 11, called the Carnot-Caratheodory distance on Il, associated with the system X. When 11=1Rn, we write d(x, y) instead of dR. (x, y). It is clear


58

that d(x, y) < do (x, y), x, y E Sl, for every connected open set Q C IRn. In [NSW] it was proved that for every connected Il CC R" there exist C, e > 0 such that

x,yEQ.

C Ix - yI r 1 . The following definition plays al key role in this paper.


59

DEFINITION 2.2. For every x E R', and r > 0, the set {1

B(x, r)

E(x, r) }

will be called the X-ball, centered at x with radius r.

We note explicitly that

B(x,r) = SZ(x, E(x, r)),

and that

1l(x, r) = B(x, F(x, r)).

One of the main geometric properties of the X-balls, is that they are equivalent to the Carnot-Caratheodory balls. To see this, we recall the following important result, established independently in [NSW], [SC]. Hereafter, the notation Xu = (Xlu,..., X,,,.u) indicates the sub-gradient of a function u, whereas XuI = will denote its length. THEOREM 2.3. Given a bounded set U C ]R1, there exists R0, depending on U and on X, such that for x E U, 0 < d(x, y) < R0, one has for s E N U {0}, and for some constant C = C(U, X, s) > 0 (2.7)

IX;1X,2...X,ar(x, y)I :5 C-1

r(x, y) ? C

d( x,

y)2_8

IBd(x, d(x, y))I'

d(x, y)2 IBd(x, d(x, y))

In the first inequality in (2.7), one has 7Z E {1, ..., m} for i = 1, ..., s, and Xj; is allowed to act on either x or y.

In view of (2.5), (2.7), it is now easy to recognize that, given a bounded set U C Rn, there exists a > 1, depending on U and X, such that (2.8)

Bd(x, a-'r) C B(x,r) C Bd(x,ar),

for x E U, 0 < r < R0. We observe that, as a consequence of (2.4), and of (2.7), one has (2.9)

C d(x, y) _< F (X,

r(x, y) /

C-1 d(x, y),

for all x E U,0 0 such that the closed balls B(xo, R), with xo E U and 0 < R < R0, are compact.


60

REMARK 2.5. Compactness of balls of large radii may fail in general, see [GN1]. However, there are important cases in which Proposition 2.4 holds globally, in the sense that one can take U to coincide with the whole ambient space and Ro = oo. One example is that of Carnot groups. Another interesting case is that when the vector fields Xj have coefficients which are globally Lipschitz, see [GN1], [GN2]. Henceforth, for any given bounded set U C Rn we will always assume that the local parameter Ro has been chosen so to accommodate Proposition 2.4.

3. The Dirichlet problem In what follows, given a system X = {X1, ..., X,,.,.} of Coo vector fields in R" satisfying (1.1), and an open set D C R' , for 1 < p < oo we denote by LlP(D) the Banach space If E LP(D) I Xj f E LP (D), j = 1, ..., m} endowed with its natural norm m

IIfIIC1,P(D) = IIfIILP(D) + E IIXjfIILP D) =1

The local space Li (D) has the usual meaning, whereas for 1 X juX, q dx = 0, JD j=I i.e., a harmonic function is a weak solution to the equation Lu = >'1 Xj*Xju = 0.

By Hormander's hypoellipticity theorem [H], if u is harmonic in D, then u E C°O(D). Given a bounded open set D C R", and a function 0 E L1"2(D), the Dirichlet problem consists in finding u E Li 2(D) such that Lu = 0

u-

(3.1)

in D ,

E La'2(D) .

By adapting classical arguments, see for instance [GT], one can show that there exists a unique solution u E L1"2(D) to (3.1). If we assume, in addition, that q5 E C(D), in general we cannot say that the function u takes up the boundary value 0 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 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 p = 2 of linear equations.

THEOREM 3.1. Let 0 E L1'2(D) fl C(D). Consider the solution u to (3.1). There exist C = C(X) > 0, and Ro = RO(D, X) > 0, such that given xo E OD, and

0 < r < R < Ro/3, one has osc {u, D fl Bd(xo, r)} < osc 1 0 , 0 D fl Bd(xo, 2R)}

+ osc (0, OD) exp

-C

IR rcapx (DO flBd(xo, t), Bd(xo, 2t))1 L

capx (Bd(xo, t), Bd(xo, 2t))

J

dt

t


61

In Theorem 3.1, given a condenser (K, 1), we have denoted by capx (K, Il) its Dirichlet capacity with respect to the subelliptic energy Ex (u) = fn I Xul2dx associated with the system X = {X1i ..., For the relevant properties of such capacity we refer the reader to [D], [CDG4]. A point xo E OD is called regular if, for any 0 E G1"2(D) fl C(D), one has

lim u(x) = O(xo)

(3.2)

x-xo

If every xo E aD is regular, we say that D is regular. Similarly to the classical 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 work of Bony [B], where the author also proved that sub-Laplacians satisfy an elliptic type strong maximum principle. We state next one of the main results in [B] in a form which is suitable for our purposes. THEOREM 3.2. Let D C R' be a connected, bounded open set, and 0 E C(aD).

There exists a unique harmonic function HD which solves (1.3) in the sense of Perron- Wiener-Brelot. Moreover, HD satisfies SUP 101

(3.3)

D

aD

Theorem 3.2 allows to define the harmonic measure dwx for D evaluated at x E D as the unique probability measure on OD such that for every 0 E C(OD)

H0 (x) = ISD 0(y) dwx(y),

x E D.

A uniform Harnack inequality was established, independently, by several authors, see [X], [CGL], [L]: If u is G-harmonic in D C RT 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)

B(x,r)

B(x,r)

Using such Harnack principle one sees that for any x, y E D, the measures dwx and dwy are mutually absolutely continuous. For the basic properties of the harmonic

measure we refer the reader to the paper [CG1]. Here, it is important to recall that, thanks to the results in [B], [CG1], the following result of Brelot type holds. THEOREM 3.3. A function 0 is resolutive if and only if ¢ E L1(aD,dwx), for 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, Q), we denote by cap(K,1) the sub-elliptic capacity of K with respect to 1, see [D]. DEFINITION 3.4. An open set D C Rn is called thin at X. E OD, if (3.5)

li r

O

r--.0

capx(D°n Bd(xo, r), Bd(xo, 2r)) > 0. capx(Bd(xo, r), Bd(xo, 2r))

THEOREM 3.5. If a bounded open set D C Rn is thin at xo E aD, then xo is regular for the Dirichlet problem.


62

Proof. If D is thin at X. E 8D, then fR rcapx(Dc n Bd(xo, t), Bd(xo, 2t))1 dt 00. L

capx(Bd(xo, t), Bd(xo12t))

t

J

Thanks to Theorem 3.1, the divergence of the above integral implies for 0
MIz12},

MER.

Since cZ 1 is scale invariant with respect to {8A}a>o we might think of fZM as the analogue of a convex cone (M > 0), or a concave cone (M < 0). Introduce the variable

T = 7-(Z, t) =

, Nz

(z, t) 0 e

.


66

It is clear that r is homogeneous of degree zero and therefore

LET = 0

.

Moreover, with 8 as in (4.1) , one easily checks that

Br=0. It is important to observe the level sets jr = ry} are constituted by the t-axis when ry = 1, and by the paraboloids

t=

'' 4

1-ry 2IzI2

if 1-yj < 1. Furthermore, the function r takes the constant value

4M

T 1

116M2 '

on 80m. We now consider a function of the form (4.5)

v = v(z, t) = Na u(r)

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

Gov = 4I,Na-2{(1-T2)u"(T)- 2ru'(T)+a(a+Q-2)u(T)} = 4,ONa-2{ (1 - r2)u"(r) - (n + 1)ru'(T) +

a(a 4+ 2n)u(T)

}

Using Proposition 4.1 we can now construct a positive harmonic function in 11M which vanishes on the boundary (this function is a Green function with pole at an interior point).

PROPOSITION 4.2. For any a E (0,1) there exists a number M = M(a) < 0 such that the nonconvex cone QM admits a positive solution of Gov = 0 of the form (4.5) which vanishes on 81ZM.

Proof. From Proposition 4.1 we see that if v of the form (4.5) has to solve the equation C°v = 0, then the function u must be a solution of the Jacobi equation (4.6)

(1 - T2)u'(T) - (n + 1)Tu(T) +

a(a

2n)

u(T) = 0 .

4 As we have observed the level {T = 1} is degenerate and corresponds to the t-axis {z = 0}. One solution of (4.6) which is smooth as T --+ 1 (remember, the t-axis is inside 11M and thus we want our function v to be smooth around the t-axis since by hypoellipticity v has to be in C°°(CIM)) is the hypergeometric function

a a n+1 1-r ga(T) = F -2,n+2' 2 ' 2

.

When 0 < a < 2 one can varify that

ga(1)=1, and that ga(r)- -ooasr---1+


67

Therefore, ga has a zero ra. One can check (see Erdelyi, Magnus, Oberhettinger and Tricomi, vol.1, p.110 (14)), that as a -+ 0+, then Ta -+ -1+. We infer that for a > 0 sufficiently close to 0 there exists -1 < Ta < 0 such that ga(la) = 0 .

If we choose

M=M(a)=

Ta T.2

then it is clear that on BSZM we have r

0,

1Xj1Xj2...Xj, JR u(x)I < R F(x,R)2+8

J

2(x,R)

Iu(y)I dy.

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 IX JR u(x) I

I u(y) I XKR(x, y) I dy.

0, depending on U and X, such that for every xo E U, and x, y E Rn \ Bd(Xo, r), one has

Ir(xo, x) - r(xo, y)l

(''

Bd(xo, r) I

d(x,

y)

Proof. We distinguish two cases: (i) d(x, y) > Or; (ii) d(x, y) < Or. Here, 0 E (0, 1) is to be suitably chosen. Case (i) is easy. Using (2.7) we find r(xo, xr) - r(xo, y) I

d(P, xo) - d(z, P) > 11- 3 9 I

9 r= r - 4 9 r = (1- 49 - 40)

2.

4 This proves (6.2). The above considerations allow to apply Theorem 3.9, which, presently gives keeping in mind that r(xo, ) E C°°(Bd(P, 4p)),

(6.3)

I r(xo, x)

- r(xo, y)) r/2, the latter estimate, combined with the increasingness of E(xo, ), leads to the conclusion C

sup £EBd(P,; p)

1 rE(xo, r)

Inserting this inequality in (6.3), and observing that rE ao,r < C lBd(xo r)l , we find

I r(xo, x) - r(xo, y) I C C

I Bd(xo, r) I

d(x, y)

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 R" is said to possess an outer X-ball tangent at xo E 8D if for some r > 0 there exists a X-ball B(xl, r) such that: (6.4)

B(xl, r) n D = 0.

xo E 8B(xl, r),

We say that D possesses the uniform outer X-ball if one can find Ro > 0 such that

for every xo E OD, and any 0 < r < Ro, there exists a X-ball B(xl,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 x0 E 8D implies that D is thin at x0 (the reverse implication is not necessarily true). Therefore, thanks to Theorem 3.5, xo is regular for the Dirichlet problem. Secondly, when X = { as , ..., as }, then the distance d(x, y) is just the ordinary Euclidean distance Ix - yI. 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 Cl,l domain and every convex domain possess the uniform outer X-ball condition. When we abandon the Euclidean setting, the construction of examples is technically 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 C lR' be a connected open set, and suppose that for some

r > 0, D has an outer X -ball B(xl, r) tangent at xo E 8D. There exists C > 0, depending only on D and on X, such that if q5 E C(OD), 0 - 0 in B(xl, 2r)nOD, then we have for every x E D I H0

(x) I

E(xl, t) is defined as in (2.6). Clearly, f is L-harmonic in R' \ {xl }.

Since r(xl, ) < E(xl, r)-1 outside B(xl, r), we see that f > 0 in R'' \ B(xl, r),


73

hence in particular in D. Moreover, f =_ 1 on 8B(xl, 2r) fl D, whereas f > 1 in (]R \ B(xi, 2r)) fl D . By Theorem 3.2 we infer

< f (x)

I H (x) I

for every

x E D.

The proof will be completed if we show that (6.6)

f (x) < C

d(x, xo)

r

for every

x E D.

Consider the function h(t) = E(xi, t)-1. We have for 0 < s < t < R0, E'(xi, T) h(s) - h(t) = (t - s) E(xi,T)2 for some s < T < t . Using the increasingness of the function r -> rE(xl, r), which follows from that of E(xi, ), and the crucial estimate

C < rE'(x1, r) < C-I E(xl,r)

-

which is readily obtained from the definition of A(xi, r) in (2.2), we find (6.7)

C tE(xlst) < h(s) - h(t)
0 such that G(x, y) 5 C

for each x, y E D, with x 0 y.

d(x, y) d(y, 8D) lBd(x, d(x, y))I


74

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, 8D) > ad 3+a '

or d(y, 8D) > R0, then the conclusion follows from (6.8). We may thus assume that

a d(y, 8D) < d(x' y) , a(a + 3)

(6.9)

and d(y, 8D) < R,,.

We now choose

r = min

d(x, y) aRo 2a(a + 3)' 2

One easily verifies from (6.9) that ad(y, 8D) < 2r. Let x,, be the point in 8D such that d(y, 8D) = d(y, x0) and consider the outer X-ball B(xi, r/a) tangent to the boundary of D in xa. We claim that y E D fl B(x1i(a + 3)r).

To see this observe that by (2.8) xa E B(x1i 11) C Bd(x1,r), and therefore

d(y, x1) < d(y, x0) + d(xo, x1) = d(y, OD) + d(x,,, x1)
d(x, y) - d(x1, y) > d(x, y)

-a

3r > d(x, y)(1- 212 ), a

and consequently

x E R" \ Bd(x1, (1 - 21 )d(x, y)) On the other hand (2.8) implies R"\Bd(x1, (1-

1

Ya2

n

)d(x, y)) C R"\B(xl, a (1_2a2)d(x,y)) C R \B(x1, (a+3)r),

the last inclusion being true since a > 1. We now consider the Perron-Wiener-Brelot solution v to the Dirichlet problem Lv = 0 in B(x1 i (a+3)r)flD, with boundary datum a function ¢ E C(8(B(x1, (a + 3)r) fl D)), such that 0 < 0:5 1, 0 = 1 on OB(xi, (3+a)r)f1D, and 0 = 0 on 9DnB(xl, (1+a)r). We observe in passing that, thanks to the assumptions on D, we can only say that v is continuous up to the boundary in that portion of 8(B(x1, (a + 3)r) fl D) that is common to 8D. However such continuity is not needed to implement Theorem 3.2 and deduce that 0 < v < 1. We observe that D fl B(x1, (a + 3)r) satisfies the outer ,C-ball condition at the point xo E OD. Applying Theorem 6.3 one infers for every y E D fl B(xi, (a + 3)r)

Iv(y)I < C d(y,OD) r

(6.10)

Let CD be as in (6.8) and define w(z) = CD1E(x 3d(x, y))G(x, z), where

(1 - 2a - 2a) Since x

B(x1i (a + 3)r), then Lw = 0 in B(xi, (a + 3)r) fl D.

Observe that if z E 8B(x1i (a + 3)r), then

d(x,z) > d(x, x1) - d(z, x1) > (1- 22) - (a + 3)r > Qd(x, y),


75

from our choice of r and,3. Consequently, in view of the monotonicity of r - E(x, r) and (6.8), we have that w < CD'E(x, d(x, z))G(x, z) < 1 on O(B(xi, (a + 3)r) fl D). By Theorem 3.2 one concludes that w(y) < v(y) in Df1B(xi, (a+3)r). The estimate of v established above, along with (2.1), completes the proof.

0 It was observed in [LU2, Theorem 50] that in a Carnot group, by exploiting the symmetry of the Green function G(y, x) = G(x, y), one can actually improve the estimate in Theorem 6.4 as follows G(x, y) < C d(x, y)-Qd(x, OD)d(y, 8D)

x, y E D , x # y ,

,

where Q represents the homogeneous dimension of the group. An analogous improvement can be obtained in the more general setting of this paper. To see this, note that the symmetry of G and the estimate in Theorem 6.4 give for every x, y E D (6.11)

G(y, x) = G(x, y) < C

d(x, y) I Bd(x, d(x, y)) - d(y, 8D)

where C > 0 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

w(z) = C-ld(x, 8D)-I

I Bd(x, d(x, y)) I

d(x, y)

G(z, x)

,

z E B(xI, (a + 3)r) fl D .

Using (6.11) instead of (6.8) we reach the conclusion that

w(z) < 1

,

for every z E 8(B(xi, (a + 3)r) fl D)

.

Since Lw = 0 in B(xl, (a + 3)r) fl D, by Theorem 3.2 we conclude as before that w(y) < v(y) in D n B(xi, (a + 3)r). Combining this estimate with (6.10) we have proved the following result.

COROLLARY 6.5. Suppose that D C R' satisfy the uniform outer X-ball condition. There exists a constant C = C(X, D) > 0 such that

G(x, y) < C d(x, OD)d(y, 8D) I Bd(x, d(x, y))I

for each x, y E D, with x

y.

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. THEOREM 6.6. Assume the uniform outer X-ball condition for D C R. There exists a constant C = C(X, D) > 0 such that

IXG(x,y)I 0 the set D possesses the uniform outer X-ball along Bd(P, 2e)nOD. There exists a constant C = C(X, D) > 0 such that IXG(x, y) I 5 C

d(x, y) lBd(x, d(x, y))

for each yEBd(P,Zc)nD, and x E D, with x 34 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, le) n D and assume that d(y, 8D) _< d(x, y). Choose 2r = d y, D and observe that if z E B(y, r) then d(z, y) < ar < e/2. 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.

0 COROLLARY 6.11. Let D C ]R" be a CO° domain. If D satisfies the uniform

outer X-ball condition in a neighborhood V of E, then for any x0 E D and every open neighborhood U of OD, such that x0 0 U, one has IIG(xof )IIc"oo(u) < C(xO7D,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 E is a consequence of the uniform outer X-ball condition and of the cited results in [Ci],[D], [NSJ and [CDG3]. Denote by V the neighborhood of E where the uniform outer X-ball condition holds. In view of the compactness of E, we have that W = UPEE B(P, 2e) C V, for some e > 0. We will consider also the set A = UPEE B(P, 2e) C W. In view of Theorem 3.12, we have that G(xO7 ) E C°°(D \ {A U {xo}}). In particular, G(x0f ) is smooth in U \ A. This implies the estimate JIG(xO1 )IIcl -(USA) < Co = Co(x0f D, V, X). To complete the

proof of the corollary we consider y E A and observe that there must be a P E E such that y E B(P, 2e). 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 IXG(x0, y)I < Cd(y, x0)1-Q < C1, with C1 depending only on X, D and U. At this point we choose C(x0, D, V, U, X) = min{Co, Cl}, and the proof is concluded. 0

7. The subelliptic Poisson kernel and a representation formula for L-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


78

for the Heisenberg group H" obtained by Lanconelli and Uguzzoni in [LU1] and extended in [CGN2] to groups of Heisenberg type. Consider a domain D which is regular for the Dirichlet problem. For a fixed point x° E D we respectively denote

by r(x) = r(x, x°) and G(x) = G(x, x°), the fundamental solution of L, and the Green function for D and C with pole at x°. Recall that G(x) = r(x) - h(x), where h is the unique L-harmonic function with boundary values F. We also note that due to the assumption that D be regular, G, h are continuous in any relatively compact subdomain of D \ {x°}. We next consider a CO° domain 1 C St C D containing the point x°. For any u, v E C°°(D) we obtain from the divergence theorem

L [u Lv - v Lu] dx =a=1I: f

[v Xju - u Xiv] < X1, v > d7

,

cz

where v denotes the outer unit normal and do the surface measure on 852. By Hormander's hypoellipticity theorem [H] the function x --p F(x°, x) is in C°° (D \

{x°}). By Sard's theorem there exists a sequence sk

oo such that the sets

{x E Rn I F(x°, x) = Sk} are C°° manifolds. Since by (2.7) the fundamental solution has a singularity at x°, we can assume without restriction that such manifolds are strictly contained in Q. Set ek = F(x°, sk 1), where F(x°, ) is the inverse function of E(x°, ) introduced in section two. The sets B(ek) = B(x°, ek) C B(x°, ek) C S2 are a sequence of smooth X-balls shrinking to the point x°. We note explicitly that

the outer unit normal on 8B ek is v = - Dr ,_, Applying (7.1) with v(x) = G(x), and St replaced by one has LG = 0, we find

Jslek

GLudx =

Jan

S2 \ B(ek), where

[uXG-GXu]<X,,v>do

+ j=1 E faB(ek) [GXju-uXjG]<XX,v>du. Again the divergence theorem gives

m

(7.2)

J

Lu dx = (ek)

-

j=1 B(ek)

X;, v > dv .


79

Using (7.2), and the fact that G = r - h, we find (7.3)

[G Xju - u XjG] < Xj, v > do, B(Ek)

m

E(xo7 Ek) j=1

f

B(Ek)

Xu<X3,v>do j=1 - J8B(ek) hXju<X3,v> do, J

z"8B(Ek) u Xjr < Xj, v > do, + 9=1

2

Gu dx + fm,Ek) u IXrI IDrI

1

E(xo, Ek) 'M

+E 7=1

u Xjh < Xj, v > do

8B(ek)

do,

EJ hXju<X3,v> do, j=1 8B(ek)

fu X h<Xj,v> dv

'm

B(ek)

Using (5.3) we find 2

u IXrI do, = u(xo) IDrI JOB(ek)

-

'Cu

JB(ek)

r-

1

E(xo, Ek) I

dx .

Keeping in mind that u, h E Coo (SZ), from the estimates (2.7) and the fact that IB(Ek)I E(xo7 Ek)

< C E2k7

letting k -+ oo, so that efk - 0, we conclude from (7.2), (7.3), (7.4)

[G Xju - u XjG] < Xj, v > do, + J G Cu dx.

u(xo) j=1

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

DEFINITION 7.1. Given a bounded open set 1 C Il C R" of class C', at every point y E 80 we let q X.(q v(y)7 X1(p') >7 ..., < v(y), ')

NX

>)

where v(y) is the outer unit normal to 1 in y. We also set m

W (y) = I NX (y) I

E < v(y)7Xj(y) >2 j=1

If y E OSZ \ E, we set

vX One has I vX (y) I

(y) =

NX I NX (y) I

for every y E OD \ E.

We note explicitly from Definitions 3.11 and 7.1 that one has for the characteristic set E of SZ

E = {y E OSZ I W(y) = 0} . Using the quantities introduced in this definition we can express (7.4) in the following suggestive way.


80

PROPOSITION 7.2. Let D C IR'2 be a bounded open set with (positive) Green function G of the sub-Laplacian (1.2) and consider a C2 domain S2 C S2 c D. For any u E COO (D) and every x E S2 one has

u(x) =

< Xu(y), NX (y) > dc(y) sI

+

fu(y) < XG(x, y), NX(y) > dv(y) sp

fG(x,Y)Iu()dy. z

If moreover £u-0 in D, then

u(x) = JG(x,Y) < Xu(y), NX (y) > do-(y) -fan u(y) < XG(x, y), NX (y) > d(y) sp

In particular, the latter equality gives for every x E St

fa

1. sz

REMARK 7.3. If U E C°°(D), then we can weaken the hypothesis on SZ and require only ci C D rather than St C D.

We consider next a C°° domain D C IR2 satisfying the uniform outer X-ball condition in a neighborhood of E. Our purpose is to pass from the interior representation formula in Proposition 7.2 to one on the boundary of OD. The presence of characteristic points becomes important now. The following result due to Derridj [Del, Theorem 11 will be important in the sequel. THEOREM 7.4. Let D C R''2 be a C°° domain. If E denotes its characteristic set, then o(E) = 0. We now define two functions on D x (8D \ E) which play a central role in the results of this paper. They constitutes subelliptic versions of the Poisson kernel from classical potential theory. The former function P(x, y) is the Poisson kernel for D and the sub-Laplacian (1.2) with respect to surface measure or. The latter K(x, y) is instead the Poisson kernel with respect to the perimeter measure ox. This comment will be clear after we prove Theorem 7.10 below. DEFINITION 7.5 (Subelliptic Poisson kernels). With the notation of Definition 7.1, for every (x, y) E D x (OD \ E) we let

P(x, y)

(7.5)

< XG(x, y), NX (y) >

We also define

K(x, y) = - < XG(x, y), vX (y) > W(y)) We extend the definition of P and K to all D x OD by letting P(x, y) = K(x, y) = 0 for any x E D and y E E. According to Theorem 7.4 the extended functions coincide o-a.e. with the kernels in (7.5), (7.6). (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 8D \ E. 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 x (8D \ E) we have (7.7)

P(x,y) < W(y) IXG(x, y)l

,

K(x,y) < IXG(x, y)l

.


81

We now introduce a new measure on OD by letting

dvx = W do .

(7.8)

We observe that since we are assuming that D E C°° the density W is smooth and bounded on OD and therefore (7.8) implies that dcx 0. By Proposition 7.2 (and the remark following it) we obtain for every k E N (7.12)

- 1 = f < XG(x, y), NX(y) > dc (y) nk

= frk < XG(x, y), NX (y) > do-(y) + f r< XG(x, y), NX (y) > dc(y) k

We now pass to the limit as k -+ oo in the above integrals. Using Corollary 6.11 and a(r2) - 0, we infer that lim

k-.oo

f < XG(x, y), NX(y) > da(y) = 0 . rek


82

Theorem 3.12, Corollary 6.11, and the fact that r / OD, allow to use dominated convergence and obtain

lim J < XG(x, y), NX(y) > da(y) =

k-woo

rk

f

< XG(x, y), Nx(y) > dv(y) . D

In conclusion, we have found

- 1 = JOD < X G(x, y), NX(y) > d(y)

arrk ark

,

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

1=-

J

< XG(x, y), vx (y) > dax (y) - f < XG(x, y), Nx (y) >

f K(x, y) dax (y) - f < XG(x, y), Nx (y) > da(y) Since as we have observed dax do(y) - f H (y) < XG(x, y), Nx (y) > ma(y) Mk

nk

At this point the conclusion follows along the lines of the proof of Proposition 7.7.

0 PROPOSITION 7.9. Let D be a C°° domain. i) If D satisfies the uniform outer X-ball condition in a neighborhood of E, then P(x, y) > 0 and K(x, y) > 0 for each (x, y) E D x OD; ii) If D satisfies the uniform outer X-ball condition, then there exists a constant CD > 0 such that for (x, y) E D x OD

0 < K(x, y) < CD 0 < P(x, y) _< CD W(y) Bd(x(d(, Bd(x d(x) y))I y)) II In particular, if we fix x E D, then for any open set U containing OD, such that x U, one has K(x, ) E L°°(D fl U). I

I

Proof. We start with the proof of part (i). We argue by contradiction. If

for some x E D and x,, E OD we had P(x, x°) = a < 0, then x° V E. By Theorem 3.12 there exists a sufficiently small r > 0 such that P(x, x') < a/2 for every x' E B(xO7 2r) fl OD. We can also assume that d(x°, E) > 2r. We now choose

' E C°° (OD) such that 0 < 0 < 1, ¢ - 1 on B(x°, r) fl OD and 0 - 0 outside B(xa, 3r/2) fl OD. Theorem 3.2 implies HD > 0 in D. By the Harnack inequality we must have HD (x) > 0. On the other hand, Theorem 7.8 gives

HD (x)
0 for which: (i) (Interior corkscrew condition) For any xo E OD and r < ro there exists

Ar(xo) E D such that M < d(Ar(xo),xo) < r and d(Ar(xo),OD) > M (This implies that Bd(Ar.(xo), ZM) is (3M, X) -nontangential.) (ii) (Exterior corkscrew condition) Do = R" \ D satisfies property (i). (iii) (Harnack chain condition) There exists C(M) > 0 such that for any e > 0 and x, y E D such that d(x, OD) > e, d(y, OD) > e, and d(x, y) < Ce, there exists a Harnack chain joining x to y whose length depends on C but not on E.

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 NTAX domain, then there exist constants C, R1 depending on the NTAx parameters of D such that for every y E OD and every 0 < r < Rl one has

C IBd(y,r)I 0 we denote by

A(y, r) = OD n Bd(y, r) the surface metric ball centered at y with radius r. We next prove a basic nondegeneracy property of the horizontal perimeter measure dorx in (7.8).

THEOREM 8.3. Let D C R" be a NTAx 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 OD and every 0 < r < RI

ax (A(y, r)) > C*

I Bd(y, r) l

r

In particular, vx is lower 1-Ahlfors according to [DGN2] and ax(A(y, r)) > 0.


86

Proof. According to (I) in Theorem 1.15 in [GN1] every metric ball Bd(y, r) is a PSx (Poincare-Sobolev) domain with respect to the system X. We can thus apply the isoperimetric inequality Theorem 1.18 in [GN1] to infer the existence of R1 > 0 such that for every y E 8D and every 0 < r < R1 min{I D f Bd(y, r)I , I D° n Bd(y, r)I }

< Ci9O

diam Bd(y, r)

Px(D; Bd(y, r))

,

I Bd(y, r)I $

where Q is the homogeneous dimension of a fixed bounded set U containing D. On the other hand, every NTAx domain is a PSx domain. We can thus combine the latter inequality with (7.11) and Lemma 8.2 to finally obtain ux (A(y, r)) >_ C* I Bd(y, r) l

r

This proves the theorem.

0 COROLLARY 8.4. Let D C RT be a NTAx domain of class C2 satisfying the upper 1-Ahlfors assumption in iv) of Definition 1.1. Then the measure ax is 1Ahlfors, in the sense that there exist A, Ri > 0 depending on the NTAx parameters of D and on A > 0 in iv), such that for every y E 8D, and every 0 < r < R1, one has (8.1)

A IBd(y,r)I

< ux(A(y,r)) < A_i

IBd(y,r)I

In particular, the measure ax is doubling, i.e., there exists C > 0 depending on A and on the constant C1 in (2.5), such that (8.2)

o-x(A(y,2r)) < C ax(A(y,r))

for every y E 8D and 0 < r < Ri. Proof. According to Theorem 8.3 the measure ax is lower 1-Ahlfors. Since by iv) of Definition 1.1 it is also upper 1-Ahlfors, the conclusion (8.1) follows. From the latter and the doubling condition (2.5) for the metric balls, we reach the desired conclusion (8.2).

0 The following results from [CG1] play a fundamental role in this paper. THEOREM 8.5. Let D C R" be a NTAx domain with relative parameters M, ro. There exists a constant C > 0, depending only on X and on the NTAx parameters of D, M and r0, such that for every xo E 8D one

C.. wArlx0l(A(xo,r)) > V THEOREM 8.6 (Doubling condition for L-harmonic measure). Consider a NTAx

domain D C lR" with relative parameters M, ro. Let xo E 8D and r < ro. There exist C > 0, depending on X, M and ro, such that wx(A(xo, 2r)) < Cwx(I(xa, r))

for any x E D \ Bd(xa, Mr).


87

THEOREM 8.7 (Comparison theorem). Let D C ]R' be a X - NTA domain with relative parameters M, r0. Let x0 E 8D and 0 < r < M . If u, v are Lharmonic functions in D, which vanish continuously on 8D \ 0(x07 2r), then for every x E D \ Bd(x0, Mr) one has u(x) < C- 1 u(Ar(xo)) C u(Ar(xo)) < v(x) v(Ar(xo)) v(Ar(xo)) for some constant C > 0 depending only on X, M and r0.

For any y E80 and a > 0 a nontangential region at y is defined by ra(y) _ {x E S2 I d(x, y) < (1 + a)d(x, ast)}

.

Given a function u the a-nontangential maximal function of u at y c 8D is defined by

N« (u) (y) = sup

xEr, (y)

l u(x) I

THEOREM 8.8. Let D C ]R" be a NTAX domain. Given a point x1 E D, let f E L1 (8D, dwx1) and define u(x) =

JOD

f(y)dwx(y),

x E D .

Then, u is L-harmonic in D, and:

(i) N.(u)(y) < CM.-, (f)(y), y E 8D; (ii) u converges non-tangentially a. e. (dwx1) to f . Theorem 8.7 has the following important consequence.

THEOREM 8.9. Let D C IRn be a ADPx domain, and let K(., ) be the Poisson Kernel defined in (7.6). There exists r1 > 0, depending on M and r0 , and a constant C = C(X, M, r0, Ro) > 0, such that given x0 E 8D, for every x E D \ Bd(xo, Mr) and every 0 < r < r1 one can find Exo,x,r C 0(x0, r), with QX (Exo,x,r) = 0, for which

K(x,y) < C K(Ar(xo), y) wx(A(xo, r)) for every y E A(xo, r) \ Exo,x,r.

Proof. Let x0 E OD. For each y E A(x0, r) and 0 < s < r/2 set

u(x) = wx(A(y,s)),

v(x) = wx(A(xo,r/2))

The functions u and v are L-harmonic in D and vanish continuously on OD A(x0f 2r). Theorem 8.7 gives

wx(A(y,s)) < C, wAr(xo)(A(y,s)) wAr(xo)(A(xr/2)) wx(A(xo,r/2)) for every x E D \ B(xO1 Mr). Applying (8.3) we thus find (8.3)

(8.4)

wx(A(y, s)) < C, wAr(xo)(A(y, s)) wx(A(xo, r/2)) wAr(xo) (A(x0, r/2))

\


88

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

(8.5)

wx(A(y,s)) 0'x(A(y,8))

MRI. Let 0 < r < R1. If Ar (xo) is a corkscrew for x0, then by the definition of a corkscrew, the triangle inequality and (2.3) it is easy to see that we have for all y E A(xo, r)

(8.7)

d(Ar(xo), y) ,,, Cr and I Bd(xo, r) I 0 depending on X and D such that for every xo E 8D and 0 < r < Ro one has max

(vEA(xo,r)

W(y)) a(A(xo,r)) < B lBd(xo,r)I

r

We note that Definitions 1.1 and 8.10 differ only in part iv). Also, (7.8) gives

ax (A(xo, r)) = f(xo,*) W (y)da(y)

(maYEA(x

r) W(Y)

)

a(A(xo, r))

This observation shows that

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

Proof of Theorem 1.5. The relevant reverse Holder inequality for P(xl, ) is proved starting from the second identity dwxl = P(xi, .)da in (7.16) and then arguing in a similar fashion as in the proof of Theorem 1.3 but using the non-degeneracy estimate in iv) of Definition 8.10 instead of the upper 1-Ahlfors assumption in Definition 1.1. We leave the details to the interested reader.

0

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 A(y, r) are the metric ones. Another comment is that away from the characteristic set the next result would be already contained in those in [MM]. THEOREM 8.11. Let D C ]R'n 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 0 and R. = RO(D, xo) > 0, depending continuously on xo, such that for any 0 < r < Ro one has (9.2)

ax (A(xo, r))

(YEA(max

r) W(y)

)

i(A(xo, r)) 0 and Ro = RO(D, xo) > 0, depending continuously on xo, such that for any 0 < r < R0, one has (9.3)

vx(A(xo,r)) > A-'

IBd(xo,r))

r

We mention that in Carnot groups of step r = 2 the upper 1-Ahlfors estimate

(9.2) was first proved in [DGN1], 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 or 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 = {X,i..., X,,,} be a set of CO° vector fields in 1Rn satisfying Hormander's condition with rank two, i.e. such that span{XI, ..., Xm, [X,, X2]....., [Xm-I, Xm] } = R" at every point. For every bounded C1,1 domain D C 1Rn the horizontal perimeter measure ax is a 1-Ah1fors measure. Moreover the stronger estimate (9.2) holds.

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

THEOREM 9.5. Let G be a Carrot group of Heisenberg type and denote by X = {X,, ..., Xm} a set of generators of its Lie algebra. Every C°° convex bounded domain D C G is a ADPx and also a o - ADPx domain. In particular, the gauge balls in G are ADPx and also o - ADPX domains. To conclude our review of Ahlfors type estimates, we bring up an interesting connection between 1-Ahlfors regularity of the X-perimeter ax and the Dirichlet problem for the sub-Laplacian, see [CG2]: THEOREM 9.6. Let D be a bounded domain in a Carrot group G. If the perimeter measure oX is 1 -Ahlfors regular, then every xo E 8D is regular for the Dirichlet problem.

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


95

COROLLARY 9.7. If r > 3 and m1 > 3, or m1 = 2 and r > 4, then there exist Carnot groups G of step r E N, with dim VI = m1, and bounded, C°° domains D C G, whose perimeter measure QX is not 1-Ah1fors regular.

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 9.2 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 Cannot 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. DEFINITION 9.8. Let G be a Carnot group and denote by g its Lie algebra. We say that a family .Fof smooth open subsets of g is a T-family if it satisfies (i) For any F c .F, the manifold OF is diffeomorphic to the unit sphere in the Lie algebra.

(ii) The family F is left-invariant, i.e. for any x E G and F E F we have log(xexp(F)) E .F. If D c g is a smooth subset and .F is a T-family, then we say that D is tangent

to F if for every x E aD there exists F E F such that x E OF and the tangent hyperplanes to OF and al) at x are identical, i.e. TXOF = TTOD. THEOREM 9.9. Let g be the Lie algebra of a Carnot group of odd dimension.

If D C g 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 x,, E al) 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 aD at xo. Let n = dim(G) = dim(g) be odd, and denote by S'rt-1 the unit (Euclidean) sphere of dimension n -1. Define the map N : OF --+ as follows: For each point x E OF set b = xxo 1D and observe that this is a smooth open set with x E aD fl OF. Set Spa-1

N(x) = the outer unit normal to the boundary of the translated set aD at the point x. This amounts to left-translating the point xo 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 G implies that N is a smooth vector field in OF. In order to prove the theorem we need to show that for some point x E OF the vector N(x)

is orthogonal to T.OF. In fact in that case the set F would be tangent to the translated set b at the point x°, and its left translation xox-1F could be chosen as

96


the element of F tangent to D at the point x0. 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 S". Consequently the vector fields obtained by projecting N(x) on TXBF must vanish for some point x E OF.

0 COROLLARY 9.10. Let G be a Carnot group of step two with odd-dimensional Lie algebra g and D C g be a smooth convex subset. If F is a T-family, composed of convex subsets, and invariant by the transformation x -p x-1 then for any x c OD there exists F E F such that F C Dc, and X E OF.

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 xo E 8D there will be a convex manifold F E F tangent to D at x0. Being D convex as well then D and F must either be on the same side or lay at different sides of the common tangent plane TxoBD. By translating x0 to the origin and considering either F or F-1 we can pick the manifold lying on the opposite side of D and hence disjoint from it. Choosing appropriate T-families of convex sets we can now prove our two main results concerning the uniform outer gauge pesudo-ball and X-ball conditions. COROLLARY 9.11. Let G be a Carnot group of step two with odd-dimensional Lie algebra g. Given a convex set D C G, for every xo E OD and every r > 0

there exists a gauge pseudo-ball B(xl, r) which is tangent to 8D in xo from the outside, i.e., such that (6.4) is satisfied. Furthermore, every bounded convex set in G 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 xo E 8D

and a new domain b obtained as the half space including D and with boundary TxoBD. Since b is a smooth convex domain then we can apply to it the previous theorem and find an outer tangent gauge ball at the point xo with radius r > 0. Clearly this ball will also be tangent to the original domain D at x0, and will be contained entirely in the complement of D. COROLLARY 9.12. Let G be a Carnot group of step two with odd-dimensional Lie algebra g. If for every x E 1Rn and for r sufficiently small the X-balls B(x, r) are convex, and B(x-1, r) = B(x, r)-1 then every bounded convex set in G satisfies 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 homogeneous dilations in the Carnot group. In particular, one has the estimate

x), z) > o on 8B(x, r) where we have denoted by Z the generator of the homogeneous dilations. This inequality, coupled with Hormander's hypoellipticity result, implies that 8B(x, r) is a smooth manifold, while the starlike property immediately implies that 8B (x, r) is diffeomorphic to the unit ball.


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-1x) and r(x-1) = I't(x), where rt is the fundamental solution of the transpose of the subLaplacian G. However, a sub-Laplacian on a Carnot group is self-adjoint, hence G* = -G and F(x) = r(x-1). Let us denote by II II the group gauge, if we assume -

that for all x,y E G one has r(xy-1) = r(yx-1) (this happens for instance if r(x) = r(IIxJI)), and set B(x,r) = {yi F(y-lx) > c} then B(x, r)-1 = {y-1I r(x-1y) > 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 nonlinear equations which arise in connection with the case p # 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

Gpu = EXj(IXUjp-2Xju) = 0,

(9.4)

j=1

in the Heisenberg group Hn. Here is the relevant result. THEOREM 9.13. Let D C Hn be a bounded domain satisfying the uniform outer X -ball condition. Given 1 0 such that 1/(p-1)

i

GD,p (9 , 9) S C

d(9, 9)

(lB('))l)

d(9', aD)

,

g, 9 E D , g' 54 g

(ii) If p = Q, then there exists C = C(G, D) > 0 such that GD,p(9 , 9)

C log

diam

\

d(9, 9

)) ) dd(9, 9)) ,

919f

ED, g 1 g

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 (9.4). For recent progress in this direction see the paper [MZZ].

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, Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathdodory spaces, J. d'Analyse Math., 74 (1998), 67-97. [GNg] N. Garofalo & Nguyen C. P., Boundary estimates for p-harmonic functions in the Heisenberg group, preprint, 2007. [GV] N. Garofalo & D. Vassilev, Regularity near the characteristic set in the non-linear Dirich-

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[NSW] A. Nagel, E.M. Stein & S. Wainger, Balls and metrics defined by vector fields I: basic properties, Acts, 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.

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83-94.

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[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. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ARKANSAS, FAYETTEVILLE, AR 72701

E-mail address, Luca Capogna: lcapognamuark.edu DEPARTMENT OF MATHEMATICS, PURDUE UNIVERSITY, WEST LAFAYETTE IN 47907-1968

E-mail address, Nicola Garofalo: garofalo®math.purdue.edu DEPARTMENT OF MATHEMATICS, SAN DIEGO CHRISTIAN COLLEGE, 2100 GREENFIELD DR, EL

CAJON CA 92019

E-mail address, Duy-Minh Nhieu: dnhieu®sdcc.edu


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 communications 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 Stalin'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 2000 Mathematics Subject Classification. Primary 01A60.

101

102

LARS GARDING

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 baltfi 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 motherin-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

103

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 INSTITUTIONEN Box 118 221 00 LUND

E-mail address: lays, garding®math.lu. se


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-toDirichlet maps, and Krein-type resolvent formulas for Schrodinger operators

on bounded Lipschitz domains in ]R°, n > 2. We also discuss the case of bounded Cl,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 Cl,''-domains

1 C I8", n E N, n > 2, with compact boundaries 852, (1/2) < r < 1 (including unbounded domains, i.e., exterior domains) with Dirichlet and Neumann boundary conditions on 852. Our results also applied to convex domains 1 and to domains

satisfying a uniform exterior ball condition. In addition, a careful discussion of locally singular potentials V with close to optimal local behavior of V 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 V E L°° (1; dnx) 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 852, we now consider generalized Robin and again Dirichlet boundary conditions, but under minimal smoothness conditions on the domain 52, that is, we now consider Lipschitz domains Q. Additionally, to reduce some technicalities, we will assume that 52 is bounded throughout this paper. Occasionally we also discuss

the case of bounded

Cl,r-domains,

(1/2) < r < 1. The principal new result in

2000 Mathematics Subject Classification. Primary: 35J10, 35J25, 35Q40; Secondary: 35P05, 47A10, 47F05.

Key words and phrases. Multi-dimensional Schrodinger operators, bounded Lipschitz domains, 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

106

F. GESZTESY AND M. MITREA

this paper is a derivation of Krein-type resolvent formulas for Schrodinger operators on bounded Lipschitz domains 52 in connection with the case of Dirichlet and generalized Robin boundary conditions on 852. In Section 2 we provide a detailed discussion of self-adjoint Laplacians with generalized Robin (and Dirichlet) boundary conditions on 0Q. 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 Robinto-Dirichlet map. Appendix A collects useful material on Sobolev spaces and trace maps for Cl,' and Lipschitz domains. Appendix B summarizes pertinent facts on sesquilinear forms and their associated linear operators. Estimates on the fundamental solution of the Helmholtz equation in R', n > 2, are recalled in Appendix C. Finally, certain results on Calderon-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 generalized Robin Laplacians and Dirichlet Laplacians, we emphasize that all results in this paper immediately extend to closed Schrodinger operators He,o = -Ae,ct + V, dom(He,n) = dom( - De,n) in L2(52; dnx) for (not necessarily real-valued) potentials V satisfying V E L°°(1 ; dnx), by consistently replacing -A by -A + V, etc. More generally, all results extend directly to Kato-Rellich bounded potentials V relative to -Ae,o with bound less than one. Next, we briefly list most of the notational conventions used throughout this paper. Let 1-l be a separable complex Hilbert space, ( , )% the scalar product in 1-1 (linear in the second factor), and I% the identity operator in 1-l. 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 1-l will be denoted by o (resp., vese( )). The Banach spaces of bounded and compact linear operators in 1-l are denoted by 5(l) and 13,(7-1), respectively. Similarly, 5(1-11,1-12) and Boo (f1,1-12) will be used for bounded and compact operators between two Hilbert spaces 1-1, and 112. Moreover, X1 --+ X2 denotes the continuous embedding of the Banach space X1 into

the Banach space X2. Throughout this manuscript, if X denotes a Banach space, X'` denotes the adjoint space of continuous conjugate linear functionals on X, that is, the conjugate dual space of X (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 in connection with quantum mechanical applications, we will, with a slight abuse of notation, dub -A (rather than A) as the "Laplacian" in this paper.

2. Laplace Operators with Generalized Robin Boundary Conditions In this section we primarily focus on various properties of general Laplacians -De,o in L2(52; dnx) including Dirichlet, -OD,O, and Neumann, -AN,n, Laplar cians, generalized Robin-type Laplacians, and Laplacians corresponding to classical

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

107

Robin boundary conditions associated with open sets 1 C R', n E N, n > 2, introduced in Hypothesis 2.1 below.

We start with introducing our assumptions on the set 1 and the boundary operator a which subsequently will be employed in defining the boundary condition

on an: HYPOTHESIS 2.1. Let n E N, n > 2, and assume that S2 C R'n is an open, bounded, nonempty Lipschitz domain.

We refer to Appendix A for more details on Lipschitz domains. For simplicity of notation we will denote the identity operators in L2(1; dnx) and LZ(a52; do-'w) by In and Ian, respectively. Also, we refer to Appendix A for our notation in connection with Sobolev spaces.

HYPOTHESIS 2.2. Assume Hypothesis 2.1 and suppose that ae is a closed sesquilinear form in L2 (01l; d'n-'w) with domain H112 (an) x H112 (an), bounded from below by ce E JR (hence, in particular, ae is symmetric). Denote by ®> celasz the self-adjoint operator in L2(00; do-1w) uniquely associated with ae (cf. (B.27)) and by ® E 13(H1/2(aS2),H-1/2(cS2)) the extension of o as discussed in (B.26) and (B.32). Thus one has

(f'09)1/2 = (9,®f)1/2, f,9 E H1/2(aSt). ceIIfIIL2(ac;dn-lw),

(2.1)

f E H1/2(oSl).

(2.2)

Here the sesquilinear form

)e = xs(asz)(, )x e(asz) : H8(012) x H-8(012) -> C,

(

s E [0,1],

(2.3)

(antilinear in the first, linear in the second factor), denotes the duality pairing between H'(80) and H-8(0 2) _ (H8(012))*,

(2.4)

s E [0,1],

such that

(f,9)8 =

fd'1w(e)J((),

f E H8(1912), 9 E L2(0S2; do-1w) s E [0, 1],

H-8(0 2),

(2.5)

and do-1w denotes the surface measure on an. Hypothesis 2.1 on 12 is used throughout this paper. Similarly, Hypothesis 2.2 is assumed whenever the boundary operator o is involved. (Later in this section, and the next, we will occasionally strengthen our hypotheses.) We introduce the boundary trace operator ryD (the Dirichlet trace) by -YD O:

C(S2) - C(04 'y,U = Ulan.

(2.6)

Then there exists a bounded, linear operator ryD (cf., e.g., [69, Theorem 3.38]), 'YD: H8(12) -+ H8-(112)(x11) y L2(a1; do-1w), -YD :

H3/2(S2) -.. Hl-E(011) y L2(a2; do-1w),

1/2 < s < 3/2, E E (0,1),

(2.7)


108

whose action is compatible with that of y . That is, the two Dirichlet trace operators coincide on the intersection of their domains. Moreover, we recall that

ryD : H8(f) ,

H8-(1/2) (812) is

onto for 1/2 < s < 3/2.

(2.8)

While, in the class of bounded Lipschitz subdomains in R", the end-point cases s = 1/2 and s = 3/2 of -ID E B(H8(12), H8-(1 /2)(812)) fail, we nonetheless have C3(H(3/2)+`(O), H1(012)),

'YD E

e > 0.

(2.9)

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 (2.10)

'YD : {u E H1"2(12) I Au E H8(12)} -+ L2(852;d"-1w),

xs compatible with (2.7), and is bounded when {u E H112(12) I Au E H8(0)} is equipped with the natural graph norm u' -. IIuIIH1/2(n) + I[ouIIH.(n) 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)

'YD : {u E H3"2(12) I Au E H1+8(12)} - H1(812),

which is again compatible with (2.7), and is bounded when {u E H3/2(12) I Au E H1+8(12)} is equipped with the natural graph norm u i-. IIuIIH8/2(n) +

Once again, this operator has a linear, bounded right-inverse (hence, in particular, it is onto). PROOF. For each s E IR set Ho(12) = {u E H5(12) I Du = 0 in 0} and observe that this is a closed subspace of H8(12). In particular, HA(12) is a Banach space when equipped with the norm inherited from H8(12). Next we recall the nontangential maximal operator M defined in (D.9). According to [39], or Corollary 5.7 in [51], one has

H1 2(12) = {u harmonic in 01 M(u) E L2(812;d"-lw)}

and u -

IIM(u)IIL2(0n;dn-1w)

(2.12)

is an equivalent norm on Ho 2(12). To continue,

fix some x > 0 and set d(y) = dist (y, 812) for y E Q. According to [28], the nontangential trace operator ('Yn.t.u)(x) _

Zli8m

u(y)

(2.13)

x-yi -3/2 we may then attempt to define 1/2 (Q)

'YD : H

+H .+2(Q) - L2(812; dr-1w)

(2.15)

by setting '7D (u + v) = '7n.t. u +'YDV,

U E Ho 2 (S2), v E

He+2(f2).

(2.16)


109

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 7n t.u ='yDU in L2(OQ; d"-1w) whenever u E H(112)+E(1), s > 0.

(2.17)

in the case when S is a bounded Lipschitz domain which is star-like with respect to the origin in 1R (cf. (A.6)). H(1/2)+- ,(n) for some e > 0, and for Assuming that this is the case, pick u E each t E (0, 1) set Ut(x) = u(tx), x E Q. We claim that

ut -+ u in H(1/2)+e(1) as t --+ 1.

(2.18)

To justify this, it suffices to prove that this is the case when u E C°°(12) as the result in its full generality then follows from a standard density argument. However,

for every u E C°° (1l) one trivially has ut -+ u as t -+ 1 in H' (1), hence in H(1/2)+E(9) Having disposed of (2.18), we may then conclude that ryDUt --+ ryDu in L2(85l; di-1w) as t -+ 1. Since for each t E (0, 1) we have ut E C(1l), it follows that 7Dut = 7DUt = utlen. Thus, altogether,

utloD -+ 'DU in L2(a 1;dr-1w) as t -+ 1.

(2.19)

On the other hand, for almost every x E an, and every t E (0, 1), we have that y = tx belongs to S2, converges to x as t -+ 1, and Ix - yI < (1 + ic) dist (y, 811) for some sufficiently large tc = ic(1) > 0 (independent of x and t). This implies that

Ut(x) -+ (7n.t.u)(x) pointwise, for a.e. x E an, as t -i 1.

(2.20)

Combining (2.19), (2.20) we therefore conclude that the functions yn.t.u, 'YDu E L2(852; d"-1w) 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 1/2(n) +H8+2 (n) with the canonical norm

wH

wlnf

IIUIIH; a(o) + IIVIIH-+a(o).

(2.21)

uEHH 2(0), vEH1+2(0)

The same type of argument as above (i.e., restricting attention to pieces of S2 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(S2) can be decomposed as u + v with u E H 1/2 (q) and v E H-+2(Q) for some s > -3/2, then wlac = Yn.t.u + 'DV. In other words, the action of the trace operator 'YD in (2.15), (2.16) is compatible with that of (2.6). This completes the study of the nature and properties of 'YD 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 {u E HI/2(Sl) I Au E He(n)} '-+ Ho 2(12) + H8+2(SZ),

s > -3/2,

(2.22)

continuously. To see this, pick u E H1/2(S2) such that AU E H8(n) and extend (cf. [87]) Au to a compactly supported distribution w E HS(1R"). Next, set

v(x) =

d"y En(x - y)w(y), n

x E S2,

(2.23)


110

where

En(x) =

n = 2,

2-n

(2.24)

is the standard fundamental solution for the Laplacian in Rn (cf. (C.1) for z = 0). Here wn-1 = 27rn/2/r(n/2) the Gamma function, cf. [1, Sect. 6.1]) represents the area of the unit sphere Sn-1 in R. Then v E H,+2(1) and Av = Au in D. As a consequence, the function w = u - v is harmonic and belongs to H1/2(0), that is, u = w + v with w E H1 2(f2), v E Furthermore, the estimate H.+2(0).

IIWIIH1/2(n) + IIvIIH.+2(n) 0 is implicit in the above construction. Thus, the inclusion (2.22) is well-defined and continuous, so that the claims about 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 'YD 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]). We now digress momentarily for the purpose of developing an integration by parts formula which will play a significant role shortly. First, if Sl is a bounded starlike Lipschitz domain in Rn and G is a vector field with components in H1/2(9)+

H'+2(Sl), s > -3/2, such that div(G) E LI(fl), then

f dx' div(G) = J d"-lw v YDG.

(2.26)

asa

Indeed, if as before Gt(x) = G(tx), x E Sl, t E (0, 1), then

div(Gt) = t(div(G))t in the sense of distributions in Q.

(2.27)

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 ft is a bounded star-like Lipschitz domain in lRn and G is a vector field with components in {u E H1/2([l) I Au E H8(S2)}, s > -3/2, such that div(e) E L'(fl). Since the latter space is a module over Co (Rn) and any Lipschitz domain is locally star-like, a simple argument based on a smooth partition of unity shows that the star-likeness condition on 0 can be eliminated. More precisely, Hypothesis 2.1,

G E {u E H1/2(fl) I Au E H8(fl)}n, s > -3/2, div(G) E L' (n; dx)

(2.26) holds.

(2.28)

Moving on, consider the operator (2.11). To get started, we fix s > -3/2 and assume that the function u E H3/2(fl) is such that Au E H1+8(Sl). Then, by the second line in (2.7),

for every e > 0.

ryDU E

(2.29)

To continue, we recall the discussion (results and notation) in the paragraph containing (A.11)-(A.16) in Appendix A. For every j, k E {1, ..., n}, we now claim

that a('YDu)

_

-3

(1q) ku - vcYD

(; u)

.

(2.30)


111

Since the functions aju, aku belong to the space {w E H112 (n) I Aw E H8(11)}, we may then conclude from (2.30) and (2.10) that a('YDU)

E L2(OSt; do-lw),

arj,k

(2.31)

and, in addition, a(7DU)

< C(IIuIIH3/2(sp) + IIouIIH1+e(2)),

(2.32)

11

for every j, k E {1, ..., n}. In concert with (2.32) and (2.29), the characterization in (A.16) then entails that 7DU E H'(OSl) and II7DUIIH1(aa2) < C(IIuIIH3/z(n) + IIDuIIH1+e(sa)). In summary, the proof of the claims made about (2.11) is finished, modulo establishing (2.30). To deal with (2.30), let 0 E Co (Rn) and fix j, k E {1, ..., n}. Consider next the vector fields Fj,k = (0, ..., 0, uak'O, 0..., 0, -uaj0, 0..., 0),

(2.33)

Gj,k = (0, ..., 0, Oaku, 0.... 0, -0ju, 0.... 0),

with the nonzero components on the j-th and k-th slots. Then Fj,k, Gj,k have components in the space {u E H1/2(fl) I Au E H8(1l)} with s > -3/2 and satisfy

div(Fj,k) = -div(Gj,k) _ (ajuak'0 - akuaj ) E L2(1l; dnx),

(2.34)

in the sense of distributions. Also,

v 7D(F'j,k) = (7DU) (vkaj'O - vjak'),

(2.35)

V 7D (Gj,k) _ b (vk7D (aj u) - vj7D (aku)) Hence, using (2.28), we obtain fOil

(7DU) (vkajO - vjak'b) _ f dlw v ' 7D(Fj,k) sz

_ f d"x div(Fj,k)

r

f dnx div(G,,k)

dry-lw0(vk7D(ajU) -Vj7D(aku)) asp

(2.36)

This justifies (2.30) and shows that the operator (2.11) 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 H3/2 (1l),

Au = 0 in Q,

7D (u) = f E Hl (all),

(2.37)

0

a result which appears in [101].

Next, we introduce the operator 7N (the strong Neumann trace) by 7N = v ' 7DV : H8+1(1) -, L2(an; d"-lw),

1/2<s -1/2, we set (with c as in (2.39)) (0, 7NU)1/2 =

jdnxV(x)

Vu(x) + H1(n) ((D, t(AU))(H1(n))-,

(2.41)

for all 0 E HI/2(8f2) and 4i E H1(52) such that YD 4) = 0. We note that this definition is independent of the particular extension 4i of 0, and that ryN is a bounded extension of the Neumann trace operator yN 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 trace operator (2.38) also extends to ryN : {u c H3/2(5l) I Au E L2(ft;d"x)} -+ L2(BSt;d"-1w) (2.42) in a bounded fashion when the space {u E H3/2(52) I Au E L2(f2; dnx)} is equipped This extension is with the natural graph norm u r-+ IIfhjH3/2(O) + compatible with (2.40) and has a linear, bounded, right-inverse (hence, as a consequence, it is onto). Moreover, the Neumann trace operator (2.38) further extends to IIoUIIL2(n;dny).

yN : {u E H1/2(ft) I Du E L2(52; d"x)} -' H-1(852)

(2.43)

in a bounded fashion when the space {u E H1/2(52) j Au E L2(f2; dx)} is equipped with the natural graph norm u '-4 IIUIIHl/2(n) + Once again, this extension is compatible with (2.40) and has a linear, bounded, right-inverse (thus, in particular, it is onto). PRooF.. Fix 0 E COO (ii). Applying (2.28) to the vector field G = z/'Vu yields

fd'_1wv n

y(Vu) =

Jn

dxV Vu+ fdxLu.

(2.44)

Consider now ¢ E H1/2(852) and 4; E H1(f2) such that yD4? = 0. Since C' (fl) --+ H1 (0) is dense, it is possible to select a sequence ij E COO (_D), j E N, such that ba --+ 4o in H1(52) as j -+ oo. This entails 0Oj -+ 04; in L2 (f2; d"x) and Oj Ian -+ 0 in H1/2((9f2) as j __+ 00. Writing (2.44) for oj in place of ip and passing to the limit j -+ oo then yields

Ian

r

r

n

In,

dx"

Du.

(2.45)

This shows that the Neumann trace of u in the sense of (2.40), (2.41) is actually v 'D (Vu). In addition, ffNUIIL2(an;dn-1w) = II v I'D(Vu)II L2(an;d^-=w)

C I'D(Vu)IIL2(0n;d1-1ao)1

< C(IIVuIIH1/2(n)n + IIo(ou)IIH-1(0)n)


= C(IIVUIIH1/a(n)n +

113

IIV(ou)IIH-1(O)n)

< C(IIuIIH312(O) + II LuII L2(r;dnx)),

(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 (2.47) 7NU = v 7D(VU) 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

ryn : {u E H1/2(1) I Au E L2(1;dnx)} - H-1(852) = (H1(8S2))*,

(2.48)

by setting (0, 7.U)1 = ffN('0), 7DU)o +

I U)L2(O;dnx) - (0t, U)L2(O;d,.x),

(2.49)

for all 0 E H1(012), where (D E H3/2(1l) is such that 7D(P = 0 and A-D E L2(SZ; dnx).

That such a -t can be found (with the additional properties that the dependence 0 H linear, and that satisfies a natural estimate) is a consequence of the fact that the mapping (2.11) 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 -1b. For this purpose, we recall the following useful approximation result:

C' (fl)

{u E H8(1l) I Du E L2(S2;dnx)} densely, whenever s < 2,

(2.50)

where the latter space is equipped with the natural graph norm u -- IIuIIH-(O) + IIAUIIL2(O;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 AU)L2(O;dnx) - (Al), U)L2(O;dnx) = 0

(7N(C,7DU)O +

(2.51)

whenever 4P E H3/2(f2) is such that yD(P = 0, 0-t E L2([;dnx), and u E C°°(S2). Note, however, that by (2.41) the roles of and u reversed we have (7N(-t),7DU)0 =

JinI2

C xV (x) Vu(x) + (0I), U)L2(O;dnx),

(2.52)

so matters are reduce to showing that

fd2xV(x)

Vu(x) =

AU)L2(f;d).

(2.53)

z

Nonetheless, this is a consequence of Green's formula (2.28) written for the vector field G = TVu (which has the property that 7DG = 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), (2.41). After re-denoting 5 by ryN, then this becomes the extension of the weak Neumann trace

operator, considered in (2.43). To this end, assume that u E H' (I) has Au E L2(1; dnx). Our goal is to show that (0,7NU)1/2 = (0,7nu)1

(2.54)


114

for every 0 E H1(8Sl) or, equivalently,

L for 4' E H3/2(Q) such that 0$ E L2jcrxV4'(x) (fl; d"x). However, (2.56)

(5N(4'),-YDU)O = (yN(0),yDU)1/2

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 4' reversed. This justifies (2.55) and finishes the proof of the lemma.

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 (see, e.g., [51]), one has the natural identification

(Hl(fl))* - {u E H-1(R") I supp (u) C Sl}.

(2.57)

Note that the latter is a closed subspace of H-1(R"). In particular, if Ry,u = uIn denotes the operator of restriction to 12 (considered in the sense of distributions), then Rn : (Hl(1l))` -> H-1(fl) (2.58) is well-defined, linear and bounded. Furthermore, the composition of Rj in (2.58)

with c from (2.39) is the natural inclusion of HR(Sl) into H-1(0). Next, given z E C, set

Wz(fl) = {(u, f) E H'(Il) x (Hl(Sl))* I (-A - z)u = f in v'(1)},

(2.59)

equipped with the norm inherited from H1(fl) x (H( fl)) *. We then denote by

'YN: W:(Q) - H-1/2(8Sl) the ultra weak Neumann trace operator defined by \0, iM(u, J) )1/2 =

J

(

(2.60)

x 04' (2) VU(2)

- z j 1'2(D(x)u(x) - H'(f)(4',f)(H'(n))*, t

(u,.f) E Wz(2), (2.61)

for all 0 E Hl/2(812) and 4' E H'(SZ) such that yD4 _ . Once again, this definition is independent of the particular extension 4' of i. 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 jz : {u E Hl(fl) I Au E Hs(l)} --' W,z(S2)

(2.62)

the injection

j. (u) = (u, t(-Au - zu)),

u E H1 (fl), Au E H9(fl),

(2.63)


115

where t is as in (2.39). Then

'ArOj. =' N

(2.64)

Thus, from this perspective, ryN can also be regarded as a bounded extension of the Neumann trace operator ryN defined in (2.38). Moving on, we now wish to discuss generalized Robin Laplacians in Lipschitz subdomains of Rn. Before initiating this discussion in earnest, however, we formulate and prove the following useful result: LEMMA 2.5. Assume Hypothesis 2.1. 0(1/e)) such that /3(e) > 0 (Q(e)

Then for every e > 0 there exists a

=

CIO

IIyDUIIL2(e0;d^-1w) < EIIoUIIL2(n;d^x)^ + a(e)II uII La(n;d^x) for all u E H1 (Q). (2.65)

PROOF. Since SZ is a bounded Lipschitz domain, there exists an h E C0 (Rn)n with real-valued components and n > 0 such that (cf., [48, Lemma 1.5.1.9, p. 40] ) (v h)c^ >, n a.e. on DSZ. (2.66) Thus, Il7DUII 2(8n;dn-1w)
0, I2uVul < CIVU12 + (1/e)Iu12,

(2.68)

u E H1(SZ),

and h E Co (W ')n, one arrives at (2.65). Next we describe a family of self-adjoint Laplace operators -De,n in L2 (SZ; dnx)

indexed by the boundary operator ©. We will refer to -Ae,o as the generalized Robin Laplacian. THEOREM 2.6. Assume Hypothesis 2.2. Then the generalized Robin Laplacian, -De,cz, defined by

- De,sa = -0,

(2.69)

dom(-De,o) = {u E H'(SZ) I Du EL 2 (0; dnx); (ryN + 67D)U = 0 in H-1/2(81)}, is self-adjoint and bounded from below in L2 (1k; dnx). Moreover,

dom(I - Ae,oIl/2) = H'(SZ).

(2.70)

PROOF. We introduce the sesquilinear form a_6 ,n ( , .) with domain Hl (SZ) x

H'(Q) by a-De,n (u, v) = a-Do,n (u, v) + ('YDU, ®'YDV)1/2'

u, v E H' (1),

(2.71)

where a-oo,n ( , ) on Hl(Q) x Hl (1) denotes the Neumann Laplacian form a_A0 n (u,

v) = in z

(Vv) (x),

u, v

H' (l).

(2.72)


116

One verifies that a_oe n

(

) is well-defined on H1(n) x H1 (0) since

,

7D E B(H'(I)7 H1/2()), 4 E ri(H"/2(8Il), H-1/2(811)),

(2.73) (2.74) do-lw))

(2.75)

(6 + (1 - Ce)Ian)1/27D E B(H1(11), L2(01l; do-1w))

(2.76)

(6 + (1 - ce)Iao)1/2 E B(H1/2(811), L2(81;

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

Employing (2.1) and (2.2), a-oe,,, is symmetric and bounded from below by ce. Next, we intend to show that a_oe,n is a closed form in L2 (11; d"x) x L2 (11; d'nx). For this purpose we rewrite (7DU, 6_(DV)1/2 as (7DU, e7DV )1/2

_ ((6 + (1 - ce)Iao)I127DU, (e + (1 - ce)Iao)1/2'yDV)L2(ata;dn-1w)

- (1 -Ce)(7DU,7DV)L2(an;dn-lw),

U,v E Hl(11),

(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 (11; dnx) since -yD is nonclosable as an operator defined on a dense subspace from L2(11; dnx) into L2(811; do-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-,&,,, (u, v) _ (DU, VV)L2(S1;dnx)n

+ ((e + (1- ce)Iaa)1/27DU, (6 + (1 - ce)Iasa)1/27DV)L2(aSt;dn-14,) +db(U,V)L2(fa;dnx),

u,V E H1(S1),

(2.78)

for db > 0. Then due to the nonnegativity of the second form on the right-hand side in (2.78), b_oe,s, is H'(ft)-coercive, that is, for some c1 > 0, (2.79)

b-oe.,,(u,u) %CIIIUI1i31(sa),

where

IIVUIIL2((a;dnx)n +

IIUIIL2((I;dnx)

Next, we note that by (2.76),

((6 + (1 - ce)Iao)1/27DU, (6 + (1 -

Ce)Iasa)1/27DV)L2(asa;dn-lfa,)

(2.80)

II(e+(1-Ce)Iasa)1/27DIIB(H1(

U, V E H1(St).

cIIuIIHI(ca) for some c > 0, one L2(Sa;dnx) infers that b_A9,,, is also Hl(11)-bounded, that is, for some C2 > 0, Since trivially, IIVuIIL2(n;dnx) + dbII UII

b-A.,. (u,u) 0, z E C\[0, oo). Then, 7D(-De,st - zIi)-(1+E)/4 E 13(L2(12; dnx), L2(BSl; do-lw)).

(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. LEMMA 2.16. Assume Hypothesis 2.11 in connection with -AD,II and Hypoth-

esis 2.12 in connection with -De,n, and let e > 0, z E C\[0, oo). Then, 'YN(-AD,SZ - zlcy)-(3+e)/4 E 13(L2(S2; dnx), L2(012; do-iw))'

(2.113)

'YD(-Ae,D - zIi)-(1+E)/4 E 13(L2(12; dnx), L2 (852; do-lw))

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 (2.113) for -AD,n under the weaker Hypothesis 2.1. The analog of Theorem 2.6 for smoother domains reads as follows: THEOREM 2.17. Assume Hypothesis 2.12. Then the generalized Robin Lapla-

cian, -De,n, defined by

-Ae,c = -A, dom(-De,o) = {u E H2(f2) I ('YN + 67D)u = 0 in H'12 (on)), (2.114)

is self-adjoint and bounded from below in L2(52; dnx). Moreover,

dom( - Ae,c11/2) = Hl(52).

(2.115)

PROOF. We adapt the proof of [43, Lemma A.1], dealing with the special case of Neumann boundary conditions (i.e., in the case 6 = 0), to the present situation. For convenience of the reader we produce a complete proof below. By Theorem 2.6, the operator Te,o in L2 (52; dnx), defined by

Te,n = -A,

(2.116)

dom(Te,n) = {u E H1(52) I Au E L2(52; dnx); (ryN + 6,yD)u = 0 in H-1/2(852)},

is self-adjoint and bounded from below, and dom(ITe,ciIl/2) = H1(52)

(2.117)


122

holds. Thus, we need to prove that dom(Te o) C H2(52).

Consider u E dom(Te,o) and set f = -Du + u e L2(11; d' x). Viewing f as an element in (Hl (52)) *, the classical Lax-Milgram Lemma implies that u is the unique solution of the boundary-value problem

(-O+Ici)u = f E L2(12) u c H1(1l),

(HI(Sl))*, (2.118)

('YN+OyD)u=0. One convenient way to actually show that u e H2(12),

(2.119)

is to use layer potentials. Specifically, let E,,(z; x) be the fundamental solution of the Helmholtz differential expression (-A - z) in Ht", n E N, n > 2, that is, IxJ/z1/2)(2-n)/2H(1)

(i/4) (21

2i/2(z1/2IxJ),

En(z; x) = a ln(JxJ),

n > 2, z E C\{0},

n=2, z = 0, n>3, z = 0,

(n 2)w _,. Ix12-",

(2.120)

Im(z1/2) > 0, x E 1

\{0}.

Here denotes the Hankel function of the first kind with index v > 0 (cf. [1, Sect. 9.1]). We also define the associated single layer potential

(89)(x) = j d"-lw(y) E. (Z; x - y)9(y),

x E S2, z E C,

(2.121)

s

where g is an arbitrary measurable function on 852. As is well-known (the interested reader may consult, e.g., [73], [101] for jump relations in the context of Lipschitz domains), if

(K#g)(x) = p.v.

f

n d"w(y).%.E,, (z; x - y)g(y),

x E 85l, z E C,

(2.122)

stands for the so-called adjoint double layer on 852, the following jump formula holds

NSz9 = (- 2Iao + K#)9-

(2.123)

It should be noted that

K# E B(L2(81l; d`w)),

x E C,

(2.124)

whenever fl is a bounded Lipschitz domain. See Lemma D.3.

Now, if we denote by w the convolution of f E L2(f2; d"x) with En(-1; ) in St, then w E H2 (52) and the solution u of (2.118) is given by

u = w + S-Ig

(2.125)

for a suitably chosen function g on 812. Concretely, we shall then require that ('YN + e'YD) S-19 = - (`YN + eYD) W,

(2.126)

2Ian + K# 1)g + e'YDS-19 = -('IN + eryD)w E H1/2(811).

(2.127)

or equivalently,

(-


123

By hypothesis, 6 E B (H3/2 (S2), H1/2 (011)) and hence 6yDS_1 E B. (H'/2(1), H'12(011))

(2.128)

as soon as one proves that S_, satisfies S_1 E B(H1/2(80),H2(1Z)).

(2.129)

To prove this, as a preliminary step we note (cf. [74]) that

S_1: H_' (812) -- H

+3/2 (S2)

(2.130)

is well-defined and bounded for each s E [0,1], even when 52 is only a bounded Lipschitz domain. For a fixed, arbitrary j E {1, ..., n}, consider next the operator 0,S_, whose integral kernel is 0 En(-1; x - y) = -00.. En(-1; x - y). We write ,.

0y3 = Evk(Y)vk(Y)0U; = Evk(y)0

+vj (y)v(y)'Vs

(y)

(2.131)

k=1

k=1

where the tangential derivative operators 8/0Tk,; = vk8; - vj 8k, j, k = 1, ... , n, satisfy (A.17). Using the boundary integration by parts formula (A.24) it follows

that n

83S-,h = D_1(v3h) +

S_1 k=1

h E Hl/2(8S),

8Tk / \ 8(vkh)

(2.132)

where, for z E C, DZh(x)

=j

do-lw(y) v(y) ' V [En(z; x - y)]h(y),

x E 1Z,

(2.133)

st

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 D_1: H'(81Z) -> H'+1/2(12),

0 < s < 1,

(2.134)

requires only that 1Z is Lipschitz. Assuming that multiplication by (the components of) v preserves the space H1/2(852) (which is the case if, e.g., 52 is of class C1,' for

some (1/2) < r < 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

K#, E B,,. (H1/2 (on)),

(2.135)

so -218 + K#1 + 6yDS_1 is a Fredholm operator in Hl/2(01) with index zero. This finishes the proof of (2.119). Hence, the fact that dom(Te,a) C H2(1Z) has been established.

Again we isolate the Neumann Laplacian -AN,n, that is, the special case 6 = 0 in (2.114), under Hypothesis 2.11,

-AN,n = -0, dom(-AN,o) _ {u E H2(52) I ryNU = 0 in H'/2(011)}.

(2.136)

Similarly, one can now treat the case of the Dirichlet Laplacian. This has originally been done under more general conditions on SZ (assuming the boundary of S2 to be compact rather than 1Z bounded) in [43, Lemmas A.1]. For completeness we repeat the short argument below:


124

THEOREM 2.18. Assume Hypothesis 2.11. Then the Dirichlet Laplacian, -AD,n, defined by

-AD,n = -A,

dom(-AD,n) = {u E H2(ft) I YDU = 0 in H3t2(Oft)},

is self-adjoint and strictly positive in L2 (Cl; d1zx). Moreover, dom((-AD,n)1/2) = H0'(fl).

(2.137)

(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 TD,n in L2 (Q; d"x), defined by

TD,n = -A,

(2.139)

dom(TD,O) = {u E Ho' (ft) I Au E L2 (f2; d"x); yDU = 0 in L2 (OS2; d"-lw)

is self-adjoint and strictly positive, and dom((TD,n)1/2) = Ho (S2)

(2.140)

holds. Thus, we need to prove that dom(TD;n) C H2(0). To achieve this, we

follow the proof of Theorem 2.17, starting with the same representation (2.125). This time, the requirement on g is that yDS_lg = h = yDw e H3/2 (an). Thus, it suffices to know that (2.141) yDS_1: H1/2(Ofl) -, H3/2(Oft) is an isomorphism. When Oft is of class C°°, it has been proved in [97, Proposition 7.9] that 'yDS_1: H8(OC) -, H8+l(Oil) is an isomorphism for each s E R and, if Cl is of class C',' 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's original proof of [97, Proposition 7.9]. Here we just note that the only significant difference is that if Oft is of class C"°' (instead of class C°°), 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,

0

Chs. 1, 2].

REMARK 2.19. We emphasize that all results in this section extend to closed Schrodinger operators

He,n =

V,

dom(He,n) = dom(- De,a)

(2.142)

for (not necessarily real-valued) potentials V satisfying V E L°°(Cl;d"x), consis tently replacing -A by -A + V, etc. More generally, all results extend to KatoRellich bounded potentials V relative to -De,sl with bound less than one. Extensions 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 problems associated with the Helmholtz differential expression -A-z in connection with the open set Cl. In addition, we provide a detailed discussion of Robin-to-Dirichlet maps, Me(OD) n, in L2(8ft; d"-1w).


125

In this section we strengthen Hypothesis 2.2 by adding assumption (3.1) below: HYPOTHESIS 3.1. In addition to Hypothesis 2.2 suppose that 6 E 13. (H1(OI ), L2(O t; di-1w)).

(3.1)

We note that (3.1) is satisfied whenever there exists some e > 0 such that

6E

13(H1-6(852), L2(,9

;

do-1w)).

(3.2)

We recall the definition of the weak Neumann trace operator ryN in (2.40), (2.41) and start with the Helmholtz Robin boundary value problems: THEOREM 3.2. Assume Hypothesis 3.1 and suppose that z E

\Q(-oe 0).

Then for every g E L2(852; dr-1w), the following generalized Robin boundary value problem,

(-A - z)u = 0 in 0,

U E H3!2 (12), (3.3)

1 ('YN + &yD) u = g on 852,

has a unique solution u = U. This solution ue satisfies 7Due E H1(91), 'yNue E L2(852; d"-1w), 3.4)

II-yDUeIIH'(00) + II'YNUeII L2(al;d^-lw), 5 CIIgJIL2(ao;dn-1w)

and II4IH3/2(n) 5 CIIgJIL2(ao;dn-1w),

(3.5)

for some constant constant C = C(6,12, z) > 0. Finally, ['1'D(-De,o - zIO)-1] E 13(L2(852; d"-1w), H3/2(n))

(3.6)

and the solution ue is given by the formula

Ue = (?'D(-De,o -

zIn)-1)"g.

(3.7)

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 As for existence, as in the proof of Theorem 2.17, we look for a zE candidate expressed as

u(x) _ (Sxh)(x), x E S2 (3.8) for some h E L2(852;d"-1w). This ensures that u E H3!2(12) and (-A - z)u = 0 in 0. Above, the single layer potential Sz has been defined in (2.121). The boundary condition (5N + 6'yD)u = g on 012 is then equivalent to ('yN + 6'yD) (Szh) = g,

(3.9)

respectively, to

(-2I,gn + K#)h + 6,yDSzh = g. Here K# has been defined in (2.122).

(3.10)

To obtain unique solvability of equation (3.10) for h E L2 (0 2;d'x-1w), given

g E L2(B1;d7L-1w), at least when z E C\D, where D C C is a discrete set, we proceed 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

(-a Ian+K#) = (-2Ian+Ko)+(K#-KO ),

(3.11)


126

and recall that (K# - Ko) E

(cf. Lemma D.3) and that

- 2lan + Ko is a Fredholm operator in L2 (852; dii-1w) with Fredholm index equal to zero as proven by Verchota [101]. In addition, we note that A7DSz E B.(L2(85Z;d2

w)),

(3.12)

which follows from Hypothesis 3.1 and the fact that the following operators are bounded Sz

L2(852; dra-1w) -4 J U E H 3/2(11) I Au E L2(52; dnx)},

7D : {u E H3/2 (f2) I Au E L2(52; dnx)} -i H1(852),

(3.13)

(where the space {u E H3/2 (f2) I Du E L2(fl; dnx)} is equipped with the natural graph norm u' -r IIUIIH3/2(sa) + IIDUIIL2(f;d^x)). See Lemma 2.3 and Theorem D.7.

Thus, (-1Iao + K#) + 67DSz is a Fredhohn operator in L2(85Z;dn-1w) with Fredhohn 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 (-!Ion + K#) + OyDSz on L2(852; dr-1w). For this purpose we now suppose that

(-2 Ian + K#)k + (&7DSzk = 0 for some k E L2(852; di-1 w).

(3.14)

Introducing w = Szk in 52 one then infers that w satisfies

(-A - z)w = 0 in 52,

w E H3/2(12), 1 ('YN + ®'yD)w = 0 on 852.

(3.15)

Thus one obtains, n

05

J

dnx IVwI2 = E J dnx 8jw8,w

j=1 n

S2

dnx Oww +

z

f do-1w (7D8jw vj-YDW

dnx J W12 + ('YDw,' Nw)L2(8n;dn-1&,)

J = T)o d"x IwI2 + (7DW,7`NW)1/2 =zf

d"x Iw12 - ('YDW, e7DW )1/2'

(3.16)

JJJSz

At this point we will first consider the case when z E C\R (so that, in particular, Im(z) # 0). In this scenario, recalling (2.1) and taking the imaginary parts of the two most extreme sides of (3.16) imply that ff dnx Iwl2 = 0 and, hence, w = 0 in

0. To continue, let yN e and -y p e denote, respectively, the Neumann and Dirichlet traces for the exterior domain Rn\S2. Also, parallel to (2.121), set

(Seat,z9)(x) _

fn

d"-1w(y)

En(z; x - y)g(y),

x E R'\SZ, z E C,

(3.17)


127

where g is an arbitrary measurable function on 852. Then, due to the weak singularity in the integral kernel of SZ, 'yDSZg = 7D tSext,Zg

for every g E L2(852; d`w),

(3.18)

whereas the counterpart of (2.123) becomes

for every g E L2(852;d"-lw).

7ritSext,Zg = (!last +K#)g

(3.19)

Compared with (2.123), the change in sign is due to the fact that the outward unit

normal to R\fZ is -v. Moving on, if we set wext(x) = (Sext,Zk)(x) for X E R'\52, then from what we have proved so far. 0 = 'YDW = YDSZk = ,yD

t,wext

in L2 (85Z; (r-lw).

(3.20)

Fix now a sufficiently large R > 0 such that S2 C B(0; R) and write the analogue of (3.16) for the restriction of wext to B(0; R) \11: dnx IVwextI2 = z (O;R)\St

f

d"x IwextI2 - (79t wext,,yNtwext)1/2

JBf (O;R)\1

-

i=R

do-lw(S) wext( ) ICI

.

Vwext(S).

(3.21)

In view of (3.20), the above identity reduces to dnx Iv,wextl2

J

B(O;R)\O

=

dnx Iwext12

z B(O;R)\S2

(3.22)

-

do-1w(e)

wext(S) IcI . Vwext( )-

Recall that we are assuming z E C\IR. Given that, by (C.17) (and the comment following right after it), the integral kernel of Sext,Zk has exponential decay at infinity, it follows that wext decays exponentially at infinity. Thus, after passing to limit R --> oo, we arrive at

crx Iowext I2 = fRn\a dx Iwext I2.

J]Rn\st

(3.23)

Consequently, taking the imaginary parts of both sides we arrive at the conclusion that wext = 0 in IRn\SZ. With this in hand, we may then invoke (2.123), (3.19) to deduce that

k=, tSZk-ryNSZk=5 twext-5Nw=0,

(3.24)

given that w, wext vanish in SZ and Rn\SZ, respectively. Hence, one concludes that k = 0 in L2 (852; do-lw). This proves that the operator (- !last + K#) + &yDSZ is injective, hence, invertible on L2(85Z; do-lw) whenever z E C\R. In the third step, the goal is to extend this result to other values of the parameter z. To this end, fix some zo E C\IR, and for z E C, consider

Az = [(-!last + Kt) +

6'YDSzo]-1

[(K#

- Kt) + 6ryD(SZ - &j].

(3.25)

Observe that the operator-valued mapping z -4 AZ E ,B(L2(852; do-lw)) is analytic

and, thanks to Lemma D.3, AZ E B ,, (L2(85Z;dn-lw)). The analytic Fredholm


128

theorem then yields invertibility of I + A. except for z in a discrete set D C D. Thus,

(-'11.90 + K#) + 67DS = E(-!Ian + KO) + A'YDSz0J[I +A.] 2 2

(3.26)

is invertible for z E C\D where D is a discrete set which, by the invertibility result proved in the previous paragraph, is contained in R. The above argument proves unique solvability of (3.3) for z E C\D, where D is a discrete subset of R. The representation (3.8) and the fact that 7DSz : L2 (L9n; d-1w)

H'()) boundedly,

(3.27)

then yields yDUe E H1 (80). Moreover, by (2.123) and (3.8), (3.28) 'YNUe ='YNSxh = (-!Ian +K#)h E L2(BQ;d"-1w) since by (2.124), K# E B(L2(OSl;d"-'w)). This proves (3.4) when z E C\D. For z E C\D, the natural estimate (3.5) is a consequence of the integral representation formula (3.8) and (D.28). NEW, fix a complex number z c C\(D U a(-Ae,n)) along with two functions, v E L2 (fl; d'°x) and g E L2(8 ; dn-'w). Also, let ue solve (3.3). One computes

(ue) v)V(n;d,.2;) _ (Ue, (-A - z)(-De,o - zIn)-ly)L2(n;dn=)

+ (7NUe,7D(-De,n - zIn)-1v)L2(an;dn-'w)

- (7DUe,7N(- De,n - zID)_1v)112 1 _ (YNUe, 'YD(-D9,n - zIn) v)L2(an;dn-lw)

+ (7DUe, a YD(-De,n - zIr2)-1v).1/2 7l)LZ(en'dn-lw)

+ 7D(-De,t2 - zIn)-ly, 6'YDUe)1/2

(7NUe, yD(-Le,o - zIn)YD(-fig n - zIo)-lv,,1/2 = ( (7N + 67D)Ue, YD(-Ae,n

- zIn)-l v)1/2

= ((7N +67D)Ue,7D(-Ae,n - EIn)_1V)LZ(an:dn-',) _ (9,7D(-O9,n - xIn) v)LZ{an;dn_lw)

_ ((YD (-Ae,n -- zIn)-1)*9,V)L2(n;dn.,)

(3.29)

Since v E L2 (Sl; dnx) was arbitrary, this yields

Ue = (7D(-Oe,o - zIn)-1)*g in L2(fl; dnx),

for z E C\(D U (3.30)

which proves (3.7) for z E C\(DUaFrom this and (3.5), the membership (3.6) also follows when z E C\(D U

The extension to the more general case when z E C\a(-De,n) is then done by resorting to analytic continuation with respect to z. More specifically, fix zo E


129

C\o(De,n). Then there exists r > 0 such that (B(zo, r)\{zo}) fl D = 0,

B(zo, r) fl o (-Ae,n) = 0,

(3.31)

since D is discrete and o (-De,n) is closed. We may then write

['YD(-Oe,n -xoIn)-1]* =

1j 27ri

dz (z - xo)-1['YD(-De,n - zln)-1]* (3.32)

C(zo;r)

as operators in B(H-1/2(a!Z), L2 (Q; d"x)), where C(zo; r) C C denotes the counterclockwise oriented circle with center zo and radius r. (This follows from dualizing the fact ryD(-Ae,n - xoIn)-1 E B(L2(SZ; d"x), H1/2(afl)), which in turn follows from the mapping properties (-De,n E 13(L2(9Z; d"x), H1(OSl)) and 'YD E B(H1(00), H1/2(OSl)).) However, granted (3.31), what we have shown so far yields that [yD(-De,n - zln)-1]* E B(L2(OSl; do-1w), H3/2(SZ)) whenever I z - zo I = r, with a bound xoIn)-1

II ['YD(-D8>0 - zIn)-1]* II B(L2(8n;dn-1w),H3/2(n)) < C = C(1l, e, zo, r)

(3.33)

independent of the complex parameter z E OB(zo, r). This estimate and Cauchy's representation formula (3.32) then imply that [7D(-De,n - xoIn)-1]* E B(L2(a1Z; do-1w), H3/2(St)).

(3.34)

This further entails that u = ['yD(-De,n - xoIn)-1]*g solves (3.3), written with zo in place of z, and satisfies (3.5). Finally, the memberships in (3.4) (along with naturally 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\v(De,n). The special case ® = 0 of Theorem 3.2, corresponding to the Neumann Laplacian, deserves to be mentioned separately. COROLLARY 3.3. Assume Hypothesis 2.1 and suppose that z E C\Q(-ON,52) Then for every g E L2 (49Q; d"-1w), the following Neumann boundary value problem,

(-0 - z)u = 0 in SZ,

u E H3/2 (Q),

7NU = g on aSl,

(3.35)

has a unique solution u = UN. This solution UN satisfies -YDUN E H1(OSl) and

II7DUNIIH1(eo) + II5NUNIIL2(Oft;dn-lw) < CII9IIL2(en;an-1w)

(3.36)

as well as IIuNIIH3/2(n) < CII9IIL2(e0;dn-1w),

(3.37)

for some constant constant C = C(®, Sl, z) > 0. Finally, ['YD (-ON,O - zln)-1] E ,Ci(L2(aQ; d"-1w), Hs 2(Sl)),

(3.38)

and the solution UN is given by the formula

Ue = (7D(-IN,c2 - zIn)-1)

*g.

(3.39)

Next, we turn to the Dirichlet case originally treated in [46, Theorem 3.1] but under stronger regularity conditions on Q. In order to facilitate the subsequent considerations, we isolate a useful technical result in the lemma below.


130

LEMMA 3.4. Assume Hypothesis 2.1 and suppose that z E C\a(-OD,a). Then

(-OD,n -

zIO)-'

: L2 (1Z; dnx) - {u E H31'(11) I Du E L2 (1l; dnx) }

(3.40)

is a well-defined bounded operator, where the space {u E H3/2(Il) I Du E L2(SZ; dnx)} is equipped with the natural graph norm u H IIuIIH3/2(n) + II1UIIL2(0;dnx)

PROOF. Consider Z E C\U(-AD,i ), f E L2 (12; dnx) and set w = (-OD,O zIn)-'f. It follows that u is the unique solution of the problem

(-0 - z)w = f in 12,

(3.41)

w E Ho (SZ).

The strategy is to devise an alternative representation for w_ from which it is clear

that w has the claimed regularity in f. To this end, let f denote the extension of f by zero to II8n and denote by E the operator of convolution by EE(z; ). Since the latter is smoothing of order 2, it follows that v = (Ef )In E H2(Q) and (-0 - z)v = f in 1Z. In particular, g = -yDV E H1(51Z). We now claim that the problem

(-0 - z)u = 0 in Q, u E H3/2(f2),

7DU = g on 812,

(3.42)

has a solution (satisfying natural estimates). To see this, we look for a solution in the form (3.8) for some h E L2(On; do-'w). This guarantees that u E H'/2 (n) by Theorem D.7, and (-A - z)u = 0 in ft Ensuring that the boundary condition holds comes down to solving yDSZh = g. In this regard, we recall that yDSo: L2(On;dn-1w) -> H1(O1) is invertible

(3.43)

(cf. [101]). With this in hand, by relying on Theorem D.7 and arguing as in the

proof of Theorem 3.2e can show that there exists a discrete set D C C such that yDSz: L2(81Z;'

_4w) -p H'(Ofl) is invertible for z e C\D.

(3.44)

Thus, a solution of (3.42) is given by

u = S.((YDS2)-'(yDV)) if z E C\D.

(3.45)

Moreover, by Theorem D.7, this satisfies IIUIIH3; Z(n) < C(1l, z) IIgjIHI(an) H1(852),

,fl(z)

9

7DU6,

z E C\o(-A6,52)

(3.58)

where u6 is the unique solution of

(-A - z)u = 0 in Q, u E H312(U),

on 852.

('YN + ®'YD)u

(3.59)

We note that Robin-to-Dirichlet maps h 'pive also been studied in [10]. We conclude with the following theorem, one of the main results of this paper:

THEOREM 3.7. Assume Hypothesis.l.,Then MDC8 sa(z) E 13(H'(80), L2(852; di-1w)),

z E C\o(-AD,n),

(3.60)

and Zin)-11

MD,6,0(x) _ (7N + e yD) [(7N + e YD)

f

,

z E C\a(-AD:n). (3.61)

Moreover,

Z E C\o(-Ae,n),

(3.62)

d"-lw)),

x E C\o(-06 n).

(3.63)

Me Dsa(z) = 'YD[7D(-A6,n - zIn)-1]',

z E C\o(-06,n).

(3.64)

Me D O(Z) E B(L2(852; d"-1w), H1(8S2)),

and, in fact, Me D n(z) E

In addition,

Finally, let z E C\(o(-A.D,n) U o(-Ae,c)). Then Me(OD) 0(z)

_ -MD esz(z)-1

(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), 'YD(-AD,SZ -210) -' = 0, and hence

UD = -[7N(-AD,O - xIn)-1] if = -[(7N + ®7D)(-OD,S2 -

zIn)-11 * f

(3.66)


133

by (3.54). Moreover, applying -(ryN+e7D) to UD in (3.54) implies formula (3.61). Likewise, (3.62) follows from Theorem 3.2. In addition, since Hl (852) embeds compactly into L2(852; d"-1w) (cf. (A.10) and [72, Proposition 2.4]), Me°D Q(z), z E C\v(-Ae,n), are compact operators in L2(852; d"-lw), justifying (3.63). Applying "(D to ue in (3.7) implies formula (3.64). There remains to justify (3.65). To this end, let g E L2(80; d"-1w) be arbitrary. Then

MD,9,0(z)Me D,n(z)9 = MD e,si(z)7DUe = -(YN + e"(D) uD, f = 'YDUe E Hl (aIl).

(-3 67 )

Here ue is the unique solution of (-A - z)u = 0 with u E H312(1)) and (ryN + ©yD)u = g, and UD is the unique solution of (-A - z)u = 0 with u E H3/2(fl) and 7DU = f E H'(852). Since (UD - u9) E H3/2(52) and 7DUD = f = "(Due, one concludes

7D(UD - ue) = 0 and (-A - Z)(UD - Ue) = 0.

(3.68)

Uniqueness of the Dirichlet problem proved in Theorem 3.6 then yields UD = ue which further entails that - (yN + e7D) UD = - (TN + 67D) ue = -g. Thus, MD°e,st(z)MM D,n(z)9 = -(TN + e7D)UD = -9,

(3.69)

implying MD,8 0(z)Me°D 0(z) = -Ian. Conversely, let f E H'(8S2). Then Me D,n(z)MD e,0(z)f = Me°D,n(z)( _ and we set

- ("(N + e'YD)UD) _"(DUE),

9 = -(5N + e7D) uD E L2 (BSZ; d"-1w).

(3.70)

(3.71)

Here UD, ue E H3/2(1) are such that (-A - z)Ue = (-A - z)uD = 0 in 1 and 'YDUD =f, (7N + e"(D)ue =g. Thus ("(N + e7D) (ue + uD) =0, (-A - z)(ue + UD) = 0 and (UD + ue) E H3/2(52). Uniqueness of the generalized Robin problem

proved in Theorem 3.2 then yields ue = -UD and hence "(Due = -7DUD = - f . Thus,

Me,D,st(z)MD,e,sl(z)f ='YDUe = -f, implying Me D (z)MD°e n(z) C -Ian. The desired conclusion now follows.

(3.72)

0

REMARK 3.8. In the above considerations, the special case e = 0 represents the frequently studied Neumann-to-Dirichlet and Dirichlet-to-Neumann maps MN,D,n(z) and MD°N n(z), respectively. That is, Mr,°D n(z) = Mo°D a(z) and MD°r, (z) = MD°o Thus, as a corollary of Theorem 3.7 we have n(z).

MN,D,n(z) = -MD°ivst(z)-1, whenever Hypothesis 2.1 holds and z e C\(v(-AD,n) U v(-AN,n)).

(3.73)

REMARK 3.9. We emphasize again that all results in this section extend to

Schrodinger operators He,n = -De,n + V, dom(He,n) = dom( - Ae,n) in L2 (fl; d"x) for (not necessarily real-valued) potentials V satisfying V E L°° (1; d"x),

or more generally, for potentials V which are Kato-Rellich bounded with respect to -Ae,n with bound less than one. Denoting the corresponding M-operators by


134

MD,N,n(Z) and Me,n,n(z), respectively, we note, in particular, that (3.56)-(3.65) extend replacing -0 by -A + V and restricting z E C appropriately. Our 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], [12], [13], [18]-[22], [32]- [35], [42], [44], [47, Ch. 3], [49, Ch. 13], [65], [66], [71], [80], [81], [841, [85], [88], [89], [100].

4. Some Variants of Krein's Resolvent Formula In this section we present our principal new results, variants of Krein's formula 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. In addition to Hypothesis 2.2 suppose that (4.1)

E B. (H1/2(8S2),

We note that condition (4.1) is satisfied if there exls

6E

some e > 0 such that

13(H1/2-£(852), H-1'2(852)).

(4.2)

Before proceeding with the main topic of this section, we will comment to the

effect that Hypothesis 3.1 is indeed stronger than Hypothesis 4.1, as the latter follows from the former via duality and interpolation, implying

8 E B (He (8S2), Hs-1(852)), 0 < s < 1. To see this, one first employs the fact that (H'0 (852), H.'(011))0,2 = H'(852)

for s = (1 - O)so + 9s1, 0 < 0 < 1, 0 < so, Si < 1, and so # Si, where

(4.3) (4.4) -)0,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 compact for j = 0, then T E B.((Xo,X1)e,p,(Y0,Y1)9,p) for every 0 E (0, 1). This is a result due to Cwikel [27]:

THEOREM 4.2 ([27]). Let X3, Yj, j = 0,1, be two compatible Banach space couples and suppose that the linear operator T : Xj -+ Yj is bounded for j = 0 and compact for j = 1. Then T : (Xo, X1)e,p -+ (Yo, Y1)o,p is compact for all 0 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 Dirichlet and Robin boundary value problems with solution in the energy space H1(52).

THEOREM 4.3. Assume Hypothesis 4.1 and suppose that z E C\v(-De,n). Then for every g E H-1/2(852), the following generalized Robin boundary value problem,

(-O - z)u = 0 in 52, 1

u E H1(52),

eyD)u = 9 on 852,


135

has a unique solution u = ue. Moreover, there exists a constant C = C(®, Sl, z) > 0 such that IIueIIHI(12) 2, and on domains satisfying Hypothesis 2.11.

In this manuscript we use the following notation for the standard Sobolev Hilbert spaces (s G R), H8(lRn) = U E S(R")' I IIUIIH-(Den) = J

+

H8(Sl) = {u E D'(Q) I u = Urn for some U E H8(lRn)} Ha (1) = {u E H8 (1R') I supp (u) C S2}.

oo(A.1)

^ ,

(A.2)

(A.3)

Here Y(fl) denotes the usual set of distributions on S2 C_ Rn, Sl open and nonempty,

S(Rn)' is the space of tempered distributions on Rn, and U denotes the Fourier transform of U E S(IRn)'. It is then immediate that H8' (SZ) '-+ H80 (Q) for - oo < so < s, < +oo, (A.4) continuously and densely. Next, we recall the definition of a C','-domain S2 C Rn, S2 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 St C lRn of class N if there exists a finite open covering {O; }i<j Hs-1(all),

0

< 1,

(A.17)

is well-defined, linear and bounded. This is proved by interpolating the case s = 1 and its dual version. In fact, the following more general result (extending (A.14)) is true.

LEMMA A.1. Assume that ii C R' is a bounded Lipschitz domain. Then for every S E [0,1],

H8(81) = {f E L2(8U;di-1w) I8f/0Tj,k E H-'-1 (8Q), 1 < j,k < n}

(A.18)

149

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

n

IIf II H'(asz) ~

II!IIL2(8fl;d^-1w) +

j II j,k=1

IIH,-1(aP).

of/OTj,k

(A.19)

PROOF. The left-to-right inclusion in (A.18) along with the right-pointing inequality in (A.19) are consequences of the boundedness of (A.17). 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 ail coincides with the graph of a Lipschitz function cp : R' ' -i R. Then for each sufficiently nice function f : ail -> R the Chain Rule yields

f-) (x, po(x)) = Tj,n

a

1

Ga

1 + IVV(x)I2 axj

[f (x, co(x))],

1 < j < n - 1.

(A.20)

On account of this and (A.9), we then deduce (upon noticing that a/aTn,n = 0) that n-1

n

EIlaf/0Tj,nJJHa-1(ao)

EIlej[f(',(P('))]IIH_-1(Ra-1).

j=1

(A.21)

j=1

Furthermore, we also have Iif('M'))IIL2(Rn-1;d^-1x).

IIfIIL2(8c;d--1w)

Next, we recall the general Euclidean lifting result Ha-1(Rn-1), H8(Rn-1) I ajf E = {f E H8-1(Rn-1)

1 < j< n- 1},

(A.22)

s E R, (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 0 the estimate which naturally accompanies (A.23).

LEMMA A.2. Assume Hypothesis 2.1. Then for every s E [0, 1] and j, k E

{1,...,n} (af/(9Tj,k , 9)1-8 = (f ,

(A.24)

19g/OTk,j)1

for every f E H8(a0) and g E Hl-e(ail). PROOF. Since for every s E [0, 1] (A.25)

C°O (R n) 1,9n , He (ail) densely,

it suffices to prove (A.24) in the case when f = uI a and g = vlan for u, v E C°O(Rn). In this scenario, we need to establish that

do-1w (au/0r3,k)v = f do-lw u(av/OTkj), Jas

1 < j, k < n.

(A.26)

sz

To this end, we rely on Green's formula (valid for Lipschitz domains) to write d

1w (au/0Tj,k)v =

z

=

Ja

f

d1w (vjaku - vkaju)v dnx [aj(vaku) - ak(vaju)]

z

=

f

dnx [(ajv)(aku) - (akv)(aju)].

(A.27)


150

One observes that the right-most integrand above is an antisymmetric expression in the indices j, k. Consequently, so is the left-most integral in (A.27). This, however, is equivalent to (A.26). Moving on, we next consider the following bounded linear map

{(w, f) E L2(SZ; d' x)" x (HI(1))` div(w) = f in}

H-1/2(on) = (Hl/2(8S2))* (A.28)

by setting H1 2(an)(0,v'(w,1))(H112(OD) = f d"xV4'(x)'w(x)+H1(el)(4),f)(H1(n)). (A.29) whenever yh E H'/2 (49Q) and 4' E H1(Q) is such that ryD4' = 0. Here the pairing H1(n)(4', f)(H1(n)). in (A.29) is the natural one between functionals in (H1(fl))* and elements in H1 (1) (which, in turn, is compatible with the (bilinear) distributional pairing). It should he remarked that the above definition is independent of the particular extension 4) E HI (SZ) of 0. Going further, one can introduce the ultra weak Neumarm trace operator 'Y. -

as follows:

yN

(u., f) E H1 (n) x (H1(SZ))* Au = fln}

H-1/2(on)

u1--*yN(u,f)=v'(Vu,f) (A.30)

with the dot product understood in the sense of (A.28). We emphasize that the ultra weak Neumann trace operator ryN in (A.30) is a re-normalization of the operator

ryN introduced in (2.38) relative to the extension of Du E H-1(l) to an element f of the space (Hl(SZ))* = {g E H-'(R`) Isupp(g) C S2}. 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 ('YD4', NN(u, A1/2 = (0D, Du)L2(n;d..x)n + H1(n)(4', f)(111(a))-,

(A.31)

valid for any u E H1(1), f E (H1(SZ))* with Du = f In, and any 4 G H1 (Q). The pairing on the left-hand side of (A.31) is between functionals in (H1/2(8SZ))* and elements in H'12(Ofl), whereas the last pairing on the right-hand side is between functionals in (H1(SZ))* and elements in H'(12). For further use, we also note that the adjoint of (2.7) maps boundedly as follows yD: (H8-112(8Sl))* (H8(f2))*, 1/2 < s < 3/2. (A.32) REMARK A.3. While it is tempting to view yD as an unbounded but densely defined operator on L2 (Q; dnx) whose domain contains the space Co (Q), one should note that in this case its adjoint ryD is not densely defined: Indeed (cf. [43, Remark A.4]), dom(y,) = {0} and hence yD is not a closable linear operator in L2(SZ; d''x).

Next we recall the following result from [46] (and reproduce its proof for subsequent use in the proofs of Lemmas A.5 and D.3).


151

LEMMA A.4 (cf. [46], Lemma A.6). Suppose St C ]R", n > 2, is an open Lipschitz domain with a compact, nonempty boundary O t. Then the Dirichlet trace operator 7D (originally considered as in (2.7)) satisfies (2.9). PROOF. Let U E H(3/2)+E(SZ), v E Co (]Rn), and ue E C°O(S2) y H(3/2)+E(SZ),

£ E N, be a sequence of functions approximating u in H(3/2)+f(1 ). It follows from (2.7) and (A.4) that 7DU,'YD(Vu) E L2(Oil; do-1w). Utilizing (A.13), one computes

for all j, k = 1,...,n, 1

dn-lw'YDU

an

< c lim I

t-*oo

f

n

av j,k I= l

10

fail

d"-1wut

19V

m

j,k

f

Jfail

d"-1W j,k

do-1W V'D(VUt)I C C II'Y'D(Du)II L2(BSt;dn-1w) IIVIILz(a .d^-lw) . (A.33)

Thus, it follows from (A.16) and (A.33) that 7Du E H1(O Z). Next, we prove the following fact: LEMMA A.5. Suppose SZ C ]Rn, n > 2, is a bounded Lipschitz domain. Then for

each r E (1/2, 1), the space C"(Oi) is a module over H1/2(i9 Z). More precisely, if Mf denotes the operator of multiplication by f, then there exists C = C(SZ,r) > 0 such that Mf E B(H1/2(OIl)) and IIMf IIs(H1/z(af)) 0 such that a(u,u) > Cojjujjl,,

u E V,

(B.14)

respectively, then,

A: V -> V* is bounded, self-adjoint, and boundedly invertible.

(B.15)

Moreover, denoting by A the part of A in 7-l defined by dom(A) _ {u E V I Au E 7-l} c 7-l,

A = Aldom(A): dom(A) - 7-l,

(B.16)

then A is a (possibly unbounded) self-adjoint operator in 7-1 satisfying

A > CoII,

(B.17)

dom(A1/2) = V.

(B.18)

In particular, A-1

(B.19)

E B(7-l).

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, §VL3, §VII.1], [37, Ch. IV], and [63].

Next, consider a symmetric form b(-, ) : V x V -+ C and assume that b is bounded from below by cb E IR, that is, b(u,u) > cb11U11'2W,

(B.20)

U E V.

Introducing the scalar product ( , )v(b) : V x V --4C (with associated norm denoted by II

-

IIV(b)),

(u, V)V(b) = b(u,v) + (1 - cb)(u,v)x,

u, v E V,

(B.21)

turns V into a pre-Hilbert space (V; ( , )v(b)), which we denote by V(b). The form b is called closed if V(b) is actually complete, and hence a Hilbert space. The form b is called closable if it has a closed extension. If b is closed, then Ib(u,v)+(1-cb)(u,v)rcl, H,

(B.27)

satisfying the following properties: B > Cblrc,

(B.28)

dom(IBI1/2) = dom((B

- CbIn)112) = V.

(B.29)

b(u, v) = (IBI'/2u, UBIBI1/2v)W

(B.30)

- ((B - Cbl1.t)1/2u, (B - CbI?)1/2V)?t +Cb(u, V) -H = V(b) (u, BV)v(b)»,

(B.31) (B.32)

u, v E V,

u E V, V E dom(B), dom(B) = {v E V I there exists an f E 1-l such that b(w, v) = (w, for all w c V}, Bu = fu, u E dom(B), dom(B) is dense in 9-l and in V(b). b(u, v) = (u, Bv)?-i,

(B.33)

(B.34) (B.35)

Properties (B.34) and (B.35) uniquely determine B. Here UB in (B.31) is the partial isometry in the polar decomposition of B, that is, B = UBIBI,

IBI = (B*B)1/2.

(B.36)

The operator B is called the operator associated with the form b. The norm in the Hilbertt space V(b)* is given by I IIuIIV(b) c 1},

with associated scalar product,

;

(t1, t2)v(b)* = V(b)((B + (1- Cb)11_1k1, t2)v(b)*,

$ E V(b)*,

(B.37)

£1,t2 E V(b)*.

(B.38)

v E V,

(B.39)

Since

II(B + (1- cb)I)vIIV(b)* = IIVIIv(b),

the Riesz representation theorem yields

(B + (1

- cb)I) E B(V(b), V(b)*) and (B + (1 - cb)I) : V(b) -+ V(b)* is unitary. (B.40)

In addition,

V(b)(u, (B + (1 - cb)' Iv)v(b), = ((B + (1 - cb)In)1/2u, (B + (1 = (u,vV(b), u,v E V(b).

- cb)I7j)1`2v)7. (B.41)


155

In particular, JI(B + (1- Cb)I7{)112UII ?{ =

IIUIIV(b),

(B.42)

U E V(b),

and hence

(B + (1

- cb)Ix)1/2 E X3(V(b), f) and (B + (1- cb)Ix)1/2: V(b)

l 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 follows: Let N,,, j = 1, 2, be complex separable Hilbert spaces, and T : dom(T) --> N2, dom(T) C }{1, a densely defined operator. Consider the nonnegative form aT : dom(T) x dom(T) --+ C defined by

aT(u, v) = (Tu, Tv)N2,

u, v E dom(T).

(B.44)

Then the form aT is closed (resp., closable) if and only if T is. If T is closed, the unique nonnegative self-adjoint operator associated with aT in Ni, whose existence is guaranteed by the second representation theorem for forms, then equals T*T. In particular, one obtains aT(u, v) = (jTju, jTjv)x1,

u, v E dom(T) = dom(jTj).

(B.45)

In addition, since b(u, v) + (1

- Cb)(u, v)r[ = ((B + (1 - COIN) 1/2U, (B + (1 - Cb)IN)1/2V) u, v E dom(b) = dom(IB1112) = V, (B.46)

and (B + (1 - COIN) 1/2 is self-adjoint (and hence closed) in N, a symmetric, Vbounded, and V-coercive form is densely defined in N x N 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(a,) x dom(aj), j = 1, 2, bounded from below and closed, then also

(a1 + a2):

(dom(a1) fl dom(a2)) x (dom(a1) fl dom(a2)) -+ C, (u, v) i--> (al + a2) (u, v) = a1(u, v) + a2 (u, v)

(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) D V x V, and that there exist 0 0 such that l b(u, u) I 3, z = 0 ,

_L(2_n)'2 I x 12-n ,

H(')

Im(z1/2) > 0, x E 1[8n\{0},

denotes the Hankel function of the first kind with index v > 0 (cf. [1, Sect. 9.1]) and wn-1 = 27tH/2/r(n/2) (r(.) the Gamma function, cf. [1, Sect. 6.1]) represents the area of the unit sphere S'-1 in As z - 0, En(z, x), x E Rn\{0} is continuous for n > 3, where

Rn.

.T

x12-n,

x E R'\{0}, n > 3,

ln(zl/21x1/2) [1 + O(x1 x12)]

+ -! (1) + O(zIxI2),

En(z, x) z- 0 En(0, x)

(n - 2)(i1n-1

but discontinuous for n = 2 as E2(z, x) z=oo

(C.2)

(C.3)

xE1R2\{0}, n = 2.

Here 1'(w) = r(w)/r(w) denotes the digamma function (cf. [1, Sect. 6.31). Thus, we simply define E2(0;x) = 2,I1n(Ix1), x E R2\{0} as in (Cl). estimate En we recall that (cf. [1, Sect. 9.1]) H(n-

2)/2(.) = J()l (.)+ ( - )/ with J and Y the regular and irregular Bessel functions, respectively.

(C.4)

We start considering small values of IxI and for this purpose recall the following absolutely convergent expansions (cf. [1, Sect - 9. 1]): W _1 k 2k SEC\(-oo,0], v E R\(-N),

J-1 (0 = (2) F-4kk P(v)+ k+ 1),

(C.5)

k=0

J-,.(S) _(-1)mJm((), Y"(C) =

C E C, M E No,

JJ(C) cos(y7r) -

7

Se C\(-oo, 0], v E (0, oo)\N,

sin(vir)

(m - k - 1)1 (2k S

2m

-

k-o

S2m- 00

k1=:0 [,O(k+

k!

(C.6)

4

+2 a Jm(() ln((/2)

(- I

1) +y'1(m+k= 1)]4kk!(m+k)!' S E C\(-oc, 0], m E No.

(C.7)


157

We note that all functions in (C.5), (C.7), and (C.8) are analytic in C\(-00,0] and that is entire for m E Z. In addition, all functions in (C.5)-(C.8) have continuous nontangential limits as C -+ 77 < 0, with generally different values on either side of the cut (-oo, 0] due to the presence of the functions (' and In((). (We chose v E R and subsequently usually v > 0 for simplicity only; complex values of v are discussed in [1, Ch. 9].) Due to the presence of the logarithmic term for even dimensions we next distinguish even and odd space dimensions n: (i) n = 2m + 2, m E No, and z E C\{0} fixed: m

i

2lrIxI

i

(i7)

E2m+2(z; x) = 4 (z1/2)

=

H,il) zl/2IxI)

[J.

(zl/2IXI)]

(zl/2IxI) + m{0(Ixlm) iln(z 1/2

i4 (z1/2I) (z - i 2IxI )

+

)O(lxlm)

(C.9)

2IxI

7r

(1 - 5m,o) [(m - 1)! + (1 - 6m,1)(m - 2)! (z14I2

+ O(Ixi4)]

.

(ii) n = 2m + 1, m E N, and z E C\{0} fixed: (1/2)-m i 2ir x) (zl/2IxI) Elm+1(z, x) = 4 (z1/2I H,(n (1/2) J 4zm/2ir-m21-mlxll mh(m)

1(z112IxI)

=

2zi /

2

(21ril x1)-me%z li2 1x

1

r: (m + k - 1)! k!(m -1 k k _o

[(2m -

(- 2iz1/2IxI)- k

.

(41rjxj)-1 [1 + iz1/2IxI + O(IxI2)], IxI-o

1)!

m=

1)W2m]-11XI1-2m [1 + O(Ix12)],

1,

m > 2,

(C.10)

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 IxI - 0:

LEMMA C.1. Fix z E C\{0}. Then the fundamental solution En(z; ) of the Helmholtz equation (-0 - z)' (z; - ) = 0 and its derivatives up to the second order satisfy the following estimates for 0 < IxI < R, with R > 0 fixed:

IEn(z;x)-En(0;x)I
5,

+ 1], n >, 4,

(C.12)


158

I8j8kEn(z; x) - 8j8kEn(0; x)I
0 is a fixed parameter. Next, at every boundary point the nontangential maximal function of a mapping u (defined in either SZ+ or St_) is given by

(Mu)(x) = sup {Iu(y)I I y E r (x)} (D.9) (with the choice of sign depending on whether u is defined in Q+, or Sl_) and, for u defined in St±, we set (D.10) (yn t.u)(x) = lim u(y) for a.e. x E 8II. yEr (x)

For future reference, let us record here a useful estimate proved in [36], valid for any Lipschitz domain f C 1R1 which is either bounded or has an unbounded boundary. In this setting, for any p E (0, oo) and any function u defined in 11, IIuIIL-P/(n-1)(I;d"x)

< C(Sl, n, p) 11 Mull

LP(BSI;dn-lw).

(D.11)


160

THEOREM D.2. There exists a positive integer N = N(n) with the following significance. Let 52 C R" be a Lipschitz domain with compact boundary, and assume

that

k E CN(R"\{0}) with k(-x) = -k(x)

(D.12)

and k(Ax) = A-("-1)k(x), A > 0; x E ]E8"\101. Define the singular integral operator

(Tf)(x) = Ion d-lw(y) k(x - y)f(y),

x E R"\Bc.

(D.13)

Then for each p E (1, oo) there exists a finite constant C = C(p, n, (9i) > 0 such that 7fT (D.14) IIlYl (` f)II LP(a0;dn-14,)

CIIkls"-1 II CN Hill LP(8O;dn-1w).

Furthermore, for each p E (1, oo), f E LP (&2; d't-1w), the limit

(Tf)(x) = p.v.

f

d"-lw(y)

k(x - y)f (y) = lim -, o+

f

>g

d"-'w(y)

k(x - y)f (y)

----Y-1

(D.15)

exists for a.e. x E 852, and the jump formula

1'n.t.(Tf)(x) =

lim

(Tf)(z) = ±-Li(v(x))f (x) + (Tf)(x)

(D.16)

2ErK (x)

is valid at a.e. x E 852, where v denotes the unit normal pointing outwardly relative to 52 (recall that `hat' denotes the Fourier transform in R"). Finally, IIT!IIH1/z(n) C C I

I f IIL2(OO;dn-(D.17)

See the discussion in [24], [25], [73]. LEMMA D.3. Whenever S1 is a Lipschitz domain with compact boundary in IIt",

K# E B(L2(8il; d"-1w)),

z E C,

(D.18)

and

(K#

- K#) E B (L2(852; d"-lw)),

'YDSz E 13(L2(852; do-1w), H1(8il)),

z1, z2 E C,

(D.19)

z E C.

(D.20)

PROOF. We recall the fundamental solution E"(z; ) for the Helmholtz equation (-A - z)iO(z; ) = 0 in ]R' introduced in (2.120). Then the integral kernel of the operator K# - Ko is given by k(x, y) = v(x) (VE"(z; x - y) - VEn(0; x - y)), x, y E Oil. (D.21) By (C.12) we therefore have I k(x, y) J < CI x-yl2-", hence (D.1) holds with fi(t) = t. Note that (D.2) is satisfied for this choice of 0, so (D.19) is a consequence of Lemma D.1. In addition, (D.18) follows from (D.19) and Theorem D.2, according to which Ko E B(L2(852;&`w)) . Finally, the reasoning for (D.20) is similar (here (A.15) is useful).

LEMMA D.4. If 52 is a Cl,', r > 1/2, domain in R' with compact boundary, then

(K# - K#)

E B. (H1I2(8S2)),

z1, z2 E C.

(D.22)


161

PROOF. The integral kernel of the operator K* - Ko is given by (D.21). By Lemma A.5, the operator of multiplication by components of v E [Cr(t9 Z)]n belongs

to B(H1/2(8SZ)). Hence, it suffices to show that the boundary integral operators whose integral kernels are of the form ,9jEn(z; x - y) - 8jEn(0; x - y)), x, y E On, j E {1, ..., n}, (D.23) belong to 13(L2(BSt;dn-1w),H1(8SZ)). This, however, is a consequence of (A.18), (A.19) (with s = 1), (C.13), and Lemma D.1 (with V )(t) = t).

LEMMA D.S. Let 0 < a < (n - 1) and 1 < p < q < oo be related by 1

1

p Then the the operator J3, defined by q

J«.f (x) = fRn-I `-l y is bounded from

Ix -

- (a+ 1)p n1.

yIn-1-c f (y)'

(D.24)

x E R+, f E LP(Rn-lid"-1x), (D.25)

L"(Rn-1; do-lx) to Lq(R+; dnx), that is, for some constant C«,p,q >

0,

II J«f IIL9(R+;dnx) 0, the following equivalence of norms holds: IIUIIH-+1()

(D.27)

II'IIL2(n) + IIVuIIH'(r).

THEOREM D.7. Let 1 C Rn be a bounded Lipschitz domain. Then for every z E C, (D.28) Sz E 13(L2(81; do-1w), H3/2(f )), and

Sz E 13(H-1(80), H1/2(c )).

In particular, Sz E B(Ha-1(8S2),HB+(1/2)(cl)),

(D.29)

0 0, Ix-zI , then, in the current context,

Iw - xl< Iz - xI< ict= (al at< (a) Ix - vi,

(D.59)

and 1w - xl + I w - yj > Ix - y l. Hence, 1W-Y1>

(1-a) lx - yI,

(D.60)

and, therefore,

IKt(z,y)-K(x,y)I 5 Ctlx-yl-"-1.

(D.61)

Next, we split the domain of integration of II (appearing in (D.51)) into dyadic annuli of the f o r m 2' at at

d"yiKt(z,y) -K(x,y)IIf(y)I


E

III = .f (x)

dny Kt (z, y) [.f (y)

1>Ix-y1>E

- .f (x)],

(D.64)

dny Kt(z, y).

Consequently, lim

lim

iT-ZI Ix-3/I

whereas lim lim III = lim E0 I-IIx-yj>E

yK(x, y).

(D.67)

Now, this last limit is known to exists at a.e. x E Rr (see, e.g., [76]). Once the pointwise definition of the operator T has been shown to be meaningful, the boundedness of this operator on LP(IRn; dnx), 1 < p < oo, is implied by that of

0

T**.

THEOREM D.11. There exists a positive integer N = N(n) with the following significance: Let 11 C ]Rr be a Lipschitz domain with compact boundary, and assume

that

k E CN(Rn\{0}) with k(-x) = -k(x) and k(Ax) =

A-(n-')k(x),

A > 0, x E IRn\{0}.

(D.68)


166

Fix q E CN (1R) and

the singular integral operator

(Tf)(x) = p.v. J d"-'w(y) (rl(x) - 7)(y))k(x - y)f(y), as2 fan

X E 852.

(D.69)

Then

(D.70)

T E 13(L2 (852; d'°-1w), H' (852)).

PROOF. Fix an arbitrary f r= L2 (Oil;dn-'w) and consider u(x)

=f

d"-lw(y) (77(x) - v7(y))k(x - y)f(y),

xE

52.

(D.71)

sa

Since T f = uI an, it suffices to show that IIM(ou)IIL2(ac;dn-1w) < CIIfIIL2(asl:dn-1w),

(D.72)

for some finite constant C = C(f2) > 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 (83u)(x) = u1(x) + u2(x), x E 52, (D.73) where

ul(x) = (8377)(x) j

do(y) k(x - y)f (y)

(D.74)

and

U2 (X) = f d' 1w(y) ((x) -(y))(83k)(x - y)f(y).

(D.75)

sp

Theorem D.2 immediately gives that II A'ru1IILa(a0.dn-1,,,) < CIIIIIL2(ata;dn-1w),

(D.76)

so it remains to prove a similar estimate with u2 in place of u1. To this end, we note that the problem localizes, so we may assume that 77 is compactly supported and 52 is the domain above the graph of a Lipschitz function V : RI-1 -+ R. In this scenario, by passing to Euclidean coordinates and denoting g(y') = f (y', 1/2. In particular, v E [CT(8S2)j". Thanks to Lemma D.4 it suffices to show that (D.38) holds for z = 0. To this end, we write

Cl,'-

Ko =Ka+(K0 -K0)

(D.89)

and observe that the integral kernel of the operator R = Ko - Ko is given by

(v(x) - v(y)) VEn(0; x - y), x, y E 8f2.

(D.90)

Let %, E [C°°(R96)]", a E N, be a sequence of vector-valued functions with the property that Tj,,j& 2

v in ["''8S2)]' as a -* 00,

(D.91)

and denote by Ra the integral operator with kernel

(rta(x) - da(y)) VEn(0; x - y),

x, y E O

.

(D.92)

From (2.129) we know that 'YDVSO E !3(H1/2(852), H1/2(a1)n),

(D.93)

which implies that for each j E {1, ..., n}, the principal-value boundary integral operator with kernel BjE,,(0;x - y) maps H1/2(8Sl) boundedly into itself. From this, (D.90), (D.91), and Lemma A.5 we may then conclude that

Ra -' R in B(H1/2(852)) as a -+ oo.

(D.94)

Also, from Theorem D.11 we have

Ra E B(L2(BSl;d' 'w),H1(8S2))' Bc,(H1"2(8S2)) for every a E N. (D.95) From (D.94) and (D.95) we may then conclude that R E Ba (H1/2(852)),

(D.96)

Ko E Bp3(H1/2(852)),

(D.97)

hence, ultimately,

by (D.89), (D.96) and (D.37).

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.


169

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ROBIN-TO-DIRICHLET MAPS AND KREIN-TYPE RESOLVENT FORMULAS DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSOURI, COLUMBIA, MO 65211, USA

E-mail address: [email protected] URL:http://ww.math.missouri.edu/personnel/faculty/gesztesyf.html

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSOURI, COLUMBIA, MO 65211, USA

E-mail address: marius®math.missouri.edu

URL:http://wwv.math.missouri.edu/personnel/faculty/mitream.html

173


A local Tb Theorem for square functions Steve Hofmann Dedicated to Prof. V. Maz'ya on the occasion of his 70th birthday. ABSTRACT. We prove a "local" Tb Theorem for square functions, in which we

assume only L4 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 L2 boundedness of a singular integral operator T may be deduced from sufficiently good behavior of T on some suitable non-degenerate test function b. 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 b, is replaced by local control, on each dyadic cube Q, of the action of T on a test function bQ, which satisfies some uniform, scale invariant L7' bound along with the non-degeneracy condition Co

< 1Q1-1

jbQ,

(1.1)

for some uniform constant Co. A collection of 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 L°°, 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 L°° control of the test functions is relaxed to L4 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 2000 Mathematics Subject Classification. Primary 42B25; Secondary 35J25 . S. Hofmann was supported by the National Science Foundation. ©2008 American Mathematical Society 175

STEVE HOFMANN

176

Kato problem [HMc], [HLMc], [AHLMcT] (see also [AT] and [S] for related results), and to variable coefficient layer potentials [AAAHK]. In this note, we consider the square function estimate /lotf(X)12 tdt < CII!IIL2(RI)I

1 n+1d

(1.2)

where Otf(x):= f .ten

t(x,y)f(y)dy

and {tt(x, y)}tE(o,-), satisfies, for some exponent a > 0, to Ilt (x, y) I 1 and a constant B such that k ' and 10

kL,± (the Poisson kernels for L and L*, respectively) satisfy (3.2) for every cube

Q C Rn. 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. We 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

A LOCAL Tb THEOREM

183

and to St*. By Lemma 5.2 of [AAAHK], it suffices to prove slipIIatStfIILz(R.) 1). For the Anderson-Bernoulli model this was known for n = 1 ([CKM]; [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 207

CARLOS E. KENIG

208

THEOREM 1.1 ([BK]). There exists S > 0 s.t. for 0 < E < S, HE displays Anderson localization a.s., n > 1. In establishing this result we were lead to the following deterministic quantitative unique continuation theorem: Suppose that u is a solution to Au + Vu = 0 in Rn, where I V I < 1, and Iul < Co, u(0) = 1. For R large, define

M(R) = inf

sup 1744.

IxoI=R B(xo,1)

Note that by unique continuation, SUPB(x0,l) Iu(x)I > 0. How small can M(R) be? TIIEORLM 1.2 ([BK]).

M(R) > Ccxp(-CR413 log R).

REMARK 1.3. In order for our argument to give the desired application to Anderson localization for the Bernoulli model, we would need an estimate of the form M(R) > C, exp(-C,Ro), with ,0 < 1+15 1.35. Note that 4/3 = 1.333.... As it turns out, this is a quantitative version of a conjecture of E.M. Landis. He conjectured (late 60's) that if L u+Vu = 0 in R1, where IVI < 1, IuI _< Co, and Iu(x)I < Cexp(-CIxll+), then u - 0. This conjecture of Landis was disproved by Meshkov ([M]), who constructed such a V, u -0 0, with Iu(x)I < Cexp(-CIxI4/3) This example also shows the sharpness of our lower bound on 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?

Let us now turn our attention to parabolic equations. Thus, consider solutions to

Btu- Au+W(x,t) Vu+V(x,t)u= 0 in Rn x (0,1], with I W I < N, IVI < M. Then, as is well-known, the following backward uniqueness result holds: If Iu(x, t) I < Co 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 assume that u is a solution on 1R x (0, 1], where R+ = {x = (x',x,,) : x,,, > 0}, without any assumption on ulaa+x[0,1] This was a crucial ingredient in their proof that weak (Leray-Hopf) solutions of the NavierStokes system in R3 x [0, 1), which have uniformly bounded L3 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 that, instead of u(x, 1) = 0, we assume that Iu(x,1)1 < Cexp(-CIxI2+E), for some e > 0. Is then u = 0? Clearly, the exponent 2 is optimal here.

THEOREM 1.4 ([EKPV1]). The Landis-Oleinik conjecture holds. More precisely, if IIu(-,1)I1L2(B(0,1)) > S, there exists Ro = Ro(5,M, N, n) > 0 s.t. for yI > R0, we have IIu(-,1)IIL2(B(0,1)) ? Cexp(-CIyI2log Iyl) Moreover, an analogous result holds for u only defined in IR+ x (0, 1].

QUANTITATIVE UNIQUE CONTINUATION ...

209

The proof of this result uses space-time rescalings and parabolic Carleman estimates, in the 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

i0tu+Au+F(u,u)u=0, in 1'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 iatu+Lu+Vu=O, in R' x [0, 1], and deal with suitable V(x, t) so that we can, in the end, set V(x,t) = F(u (x, t), %x, t)).

In order to motivate our work, I will first recall the following version of Heisenberg's

uncertainty principle, due to Hardy, [SSJ: if f : ]R -+ C, and we have f (x) = 0(e*A.2) and f O(e-''rBt'), A, B > 0, if A-B > 1, then f - 0. For instance, if CEeXp(-CElfl2 ), If(x)I < then f - 0. This can easily be translated into an equivalent formulation for solutions to the free Schrodinger equation. For, if v solves CEexp(-CEIxI2+E),

i8ty+8xv=0 inlRx [0,1], with v(x,0) = vo(x),then

v(x t) =

C

eil=-vl/4tvo(y)dy,

_Vft

so that v(x, 1) = Ce'1=I2/4 f e-ixy/2ei1312/4vo(y)dy.

If we then apply the corollary to Hardy's uncertainty principle to f (y) = eiVa/4vo(y),

we se that if Iv(x, 0)I 0, and aj belong to suitable function spaces, then u - 0 This is clearly optimal for Btu + 88u = 0. The same result holds for e"x9"2dx. 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 estimates". A typical application of Theorem 1.11 is: THEOREM 1.12 ([EKPV3]). Let

ul, u2 E C([0,1]; H3(R)) n L2(I xI2dx), solve

atu+O u+ukOxu=0 onR x [0, 1]. A ssume th a t

u1(-, 0) - u2(', 0), u1(-,1) - u2(.,1) E HI(eaxs+ Zdx)

for any a > 0. Then ul

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- 0, a solution of

Btu+ON +uk0xu=0 in Rx [0,1] S. t.

Iu(x, 0) 1 + Iu(x,1)1 5 Cexp(-Cx3+2).

QUANTITATIVE UNIQUE CONTINUATION

213

.

2. Convexity properties of Gaussian means of solutions to Schrodinger equations As mentioned before, [EKPV2] proved that if it E C([0,1]; H1(IRTB)) solves

iatu + L u + V (x, t)u = 0

x [0,1]

in II

U(0) = uo

u(1) = u1 (ealxl8

and ui E L2

dx) for some a > 0, 0 > 1, then We > 0, b > 0 s.t.

sup O (Stf + [S, A]f, f)/H - ilatf - Af - Sf Il2/(2H)

iii) Moreover, if

latf-Af-SfI <MiIfI+F inR' x[o,1], st+[s,A]>-M0 and

M2 = supo0, Iatf-Sf-Af1
0)

t(1 - t)e,Ix12VuhIL2(itXl0,1]) + II

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 2.3 (Energy method). Assume that u E L°°([0, 1]; L2) fl L2([0,1]; H1) satisfies atu = (a + ib) (Au + V (x, t)u) + F(x, t) in R" x [0,1],

a > 0, b E R. Then, for 0 < T < 1, e-MT

IIe7a1xl'/(a+47(a2+b2)T),j(T) 1
0 where a > 0, b E R, I I V I I ao < M1. Then, 3N., s. t. sup

Ilellxl2u(t)II

0 in [-1,1]. 1\ Hence, for s < t we have a(t)et log Ha(s) < a(s)8t log Ha(t). Integrating between [-1, 0] and [0, 1] and using the evenness of a, we conclude

Ha(0) < Ha(-1)1'2Ha(1)1/2. Now, if a solves 32x3 + a" - 2

a

=0

a(O) = 1,a`(1) = 0

a is positive, even, and limR,c,. Ra(R) = 0. Also, aR(t) = Ra(Rt) also solves the equation. If the formal calculation holds for HaR, 11

e

Zu(a)II2
0 we have "log convexity" in this last problem. Moreover, by the Appel Lemma and our definitions, we have 1 e7alxl2ua(0)11

R)) = 0. Let u e C([0,1]; L2) solve

atu = i(/u + V (x, t)u) in Rn x [0, 1]. Assume in addition that V E L"°(R"-1), and that 11elx12/p2u(0) 11 < 00,

eIxI2/a2u(1)

R)) = 0. Assume that u E C([0,1]; L2) is a

solution of Btu = i(,Lu + V (x, t)u) such that eI1I2//3

2u(0)

E L2,

e1x12/`t2u(1)

in Wn x [0, 1],

E L2, and a3 < 2. Then u - 0.

Preliminaries: Let -y = 1/aO. 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: 2u(t)II L2 +supII

(4.1)

[01I

[0,1]11

t(1-t)eyjx'2Vu(t)II L2(R,"X[o,1]) 0, set f = eµlx+Reit(1-t)12u, where 0 -R2/4µ, so that

H(t)e-R2t(1-t)/sµ is log

convex in [0, 1] and so

H(1/2) < II(0)1/2H(1)1/2eR2/32PP.

Letting p t -y we see that e2ryl.+

J

°

12

-

Ju(1/2)12
1/2). The path from the formal argument to the rigorous one is not easy. We will do it instead with the Carleman inequality:

LEMMA 4.2. Let q5(t), fi(t) be smooth functions on [0, 1], g(x, t) E Co (W' x [0, 1]), el = (1, 0, ... , 0). Then, for p > 0, we have (for R > 0),

if

3R [

n(t)

"(t)]2]e2`'(t)eLr`l*-0(t)ejj21g12

[10

0, recall that u solves i8tu + Au = Vu, and that the estimates (4.1) hold. Choose then ra(t), 0 < r) < 1, i - 1 where t(1 - t) > 11R, 71 - 0 near t = 1; 0, so that supp 77' C {t(1 - t) < 1/R},

177 '1 < CR.

Choose also M >> R, 0 E Co (W'), and now set g(x, t) = rj(t)6(x/M)u(x, t), which is compactly supported in R'2 x (0, 1), so that our estimate holds.

(i8t +A)g = V9 + in'(t)0 (n,) u + ( 317 Ao (n&) u+ I + II + III. Finally, let µ = (1 + E)-37R2. Our inequality then gives: E (1 + E)3 R2

ff

e2w(t)e2al

n+¢(t)el 121912

1/R 'y.e

CARLOS E. KENIG

226

We are now going to restrict to integration over the region I -K I < d, it - 1/21 where d is small, to be chosen. Then,

-2b=(4-2d)

R + q5(t)el so that

> 16 - 26 2 - 25

R + 0(t)el ve(t) _

['w(t)

(2) +

- (2)]

I .

(2) - I'b'(B)IS > 16ju,

16114

1d

so that, in our region of integration, I2

2µ I R

+

q5(t)el

+ 2 (t) >

2µ

2(1+e)R4

16

16µ4

-

(1 - ) R4 4dµ - 2 2S - d(1 +E)16µ =

2 (1 + 6)4 R2 - CsR2 = 2 R2 16 y4 [(1 +e)3

since p = (1)3R. But, if y > 1/2, y

(1 + e)4

(1 + E)3

4y

2

^(R 2

16 (1 + e)3

- (1 +

E)4

4-y

-C

]

> 0,

for some e small, and so, for b smaller than that we get a lower bound of C,,6R2. We thus have CE,6R

r

It-1/2I 0, and cl > 1, all depending only on p, n, a, 0, ry, such that if 0 < b < b, then Ilog

u(y2) g v(y2)

i(yl) _ to V(y1)

0, such that {B(xi, ri)} constitutes a covering of an open neighbourhood of 812 and such that, for each i, (1.9)

12 fl B(xi, ri) 812 fl B(xi, ri)

= {x = (x', x,`) E R" : x" > oi(x')} fl B(xi, ri), = {x = (x', x,,) E R" : x, = oi(x,)} fl B(xi, ri),

BOUNDARY HARNACK INEQUALITIES FOR OPERATORS OF p-LAPLACE TYPE

233

in an appropriate coordinate system and for a Lipschitz function 0,. The Lipschitz If it is Lipschitz then Il is NTA constant of Il is defined to be M = m a x i I I I V of III with ro = minri/c, where c = c(p, n, M) > 1. Moreover, if each 0i : Ri-1-*R can be chosen to be either CI- or C',°-regular, then fZ is a bounded C1- or C'.°-domain. We say that Sl is a quasi-ball provided St = f (B(0,1)), where f = (fl, f2,..., f")

R" -+ R" is a K > 1 quasi-conformal mapping of R' onto R. That is, fi E W1,"(B(0, p)), 0 < p < 00,1 < i < n, and for almost every x E R" with respect to Lebesgue n-measure the following hold, (i)

(ii)

IDf(x)I" = sup Ih1=1 Jf(x) ? 0 or Jf(x) < 0.

IDf(x)hI" < KI Jf(x)I,

In this display we have written D f (x) = (a ) for the Jacobian matrix of f and Jf(x) for the Jacobian determinant of f at x. Remark 1.10. Let Sl C R" be a bounded Lipschitz domain with constant M. If M is small enough then 11 is (5, ro)-Reifenberg flat for some 5 = S(M), ro > 0 with 5(M) -+ 0 as M --+ 0. Hence, Theorems 1-2 apply to any bounded Lipschitz domain with sufficiently small Lipschitz constant. Also, if Il = f(B(0,1)) where f is a K quasi-conformal mapping of R" onto R", then one can show that ail is 5-Reifenberg fiat, with ro = 1, where 5-+0 as K--+1 (see JR, Theorems 12.5 -12.7)).

Thus Theorems 1, 2, apply when i2 is a quasi-ball and if K = K(p, n) is close enough to 1.

To state corollaries to Theorems 1-2 we next introduce the notion of Reifenberg flat domains with vanishing constant.

Definition 1.11. Let fl c: R" be a (5, ro) -Reifenberg flat domain for some 0 < 5 < d, ro > 0, and let w E 852, 0 < r < ro. We say that 81Z fl B(w, r) is Reifenberg flat with vanishing constant, if for each e > 0, there exists r' = i (e) > 0 with the following property. If x E Oil f1 B(w, r) and 0 < p < r-, then there is a plane

P' = P'(x, p) containing x such that

h(OinB(x,p),P'f1 B(x,p)) The following corollaries are immediate consequences of Theorems 1-2.

Corollary 1. Let Il C R" be a domain which is Reifenberg flat with vanishing constant. Let p, 1 < p < oc, be given and assume that A E Mp(a, 0, y) for some (a,#, y). Let W E 852, 0 < r < ro. Assume that u, v are positive A-harmonic functions in 52 fl B(w, 4r), u, v are continuous in Sl fl B(w, 4r) and u = 0 = v on 8il fl B(w, 4r). There exist ri = ri (p, n, a, ,13, y) < r and c2 = c2 (P, n, a,,8, y) > such that if w' E Oil fl B (w, r) and 0 < r' < r*, then Ilog U(Y1) _log u(y2) < c2 v(yl) v(y2) whenever yl, y2 E fl fl B(w', r').

\

Iy1- y2I l

1

0

r'

Corollary 2. Let Q C R", p, a, 8, y and A be as in the statement of Corollary 1. Let to E 852 and suppose that u, ii are positive A-harmonic functions in 52 with

JOHN L. LEWIS, NIKLAS LUNDSTROM, AND KAJ NYSTROM

234

u = 0 = v continuously on 812 \ {w}. Then u(y) = AO(y) for all y E 12 and for some constant A.

Remark 1.12. We note that if 1t is a bounded Cl-domain in the sense of (1.9) then 12 is also Reifenberg flat with vanishing constant. Hence Corollaries 1-2 apply to any bounded C'-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 MM (a), 12 is equal to a truncated cylinder and w is the center on the bottom of 1Z.

Step A. (Uniform non-degeneracy of IDul - the `fundamental inequality'). There exist b1 = 5i (p, n, a, /3,,y), cl = cj (p, n, a, 8, y) and A =)(p, n, a, such that if 0 < 6 < 61, then .1-1 d(y(8S2) < jVu(y)I < Ad(y(e12) whenever y E 1t fl B(w, r/6i).

(1.13)

If (1.13) holds then we say that IVul satisfies the `fundamental inequality' in 12 fl B(w, r/cl).

Step B. (Extension of IVuIP-2 to an A2-weight). There exist S2 = 62(p, n, y) and e2 = c2 (p, n, a, 0, y) such that if 0 < 6 < b2, then I Vuln-2 extends to an A2(B(w, r/(c1c2))-weight with constant depending only on p, n, a, j3, y.

For the definition of an A2-weight, see section 4. The `fundamental inequality' 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 constructions we let 12 C Rn, 6, re,, p, a, ,l3, y, A, to, r, u and v be as in the statement of Theorem 1. Let b = min{dl, S2} where 61 and 62 are given in Step A and Step B respectively. We extend u and v to B(w, 4r) by defining u - 0 - v on B(w, 4r)\S2.

Step C. (Deformation of A-harmonic functions). Let r* = r/c and assume that (a) (b)

(1,14) (c)

0 < u < v/2 in S2 fl B(w, 4r*), 1 < u(ar- (u')), v(ar. (w)) c 1, whenever r E (0,1].

(1.25) is a consequence of (1.16) and (1.14) (b). 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 state 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), S2 is equal to a truncated cylinder (see (2.7) and Lemma 2.8), and w is the center of the bottom of f (Step 0). In 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 IVuIP-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] .

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 fl C R'. We then prove the boundary Harnack inequality for non-negative A-harmonic functions, A E MP(a), vanishing on a portion of a hyperplane. Throughout this section we will assume

that A E Mp(a, 0,,y) or A E Mp (a) for some (a, 0, 7) and 1 1, not necessarily the same at each occurrence, depending only on p, n, M, a, 13, y where M denotes the NTA-constant for SZ C R. In general, c(a1, ... , a,) denotes a positive


237

constant > 1, which may depend only on p, n, M, a,)3, ,y and al, ... , af6, not necessarily the same at each occurrence. If A ;:e B then A/B is bounded from above and below by constants which, unless otherwise stated, only depend on p, n, M, a,'3, y. 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 R" and whenever u is defined on B(z, s). We put 0(w, r) = OlI fl B(w, r) whenever w E Oft, 0 < r. Finally, e171 < i < n, denotes the point in R' with one in the i th coordinate position and zeroes elsewhere.

Lemma 2.1. Given p,1 < p < oo, assume that A E M,, (a, /3, y) for some Let u be a positive A-harmonic function in B(w, 2r). Then

f

rp-n

(i)

y).

IVurpdx < c( max u)p, B(w,r)

B(w,r/2)

max u < c min u.

(ii)

B(w,r)

B(w,r)

Furthermore, there exists Q = &(p, n, a,,8, y) E (0, 1) such that if x, y E B(w, r), then

(...I (iii)

(x

< u ) - (y)I -c

(it)

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]).

Lemma 2.2. Let Q C Rn be a bounded NTA-domain, suppose that p, 1 < p < oo, Let W E 8), 0 < r < ro, and is given and that A E Mp(a,,0, y) for some suppose that u is a non-negative continuous A-harmonic function in fl fl B(w, 2r) and that u = 0 on A(w,2r). Then (i)

P-n

Furthermore, there exists B(w, r), then

J Df1B(w,r/2)

lVu(pdx < c( max u)p. SIfB(w,r)

6(p, n, M, a, 0) ry) E (0, 1) such that if x, y E fl fl 1a

(ii)

lu(x)-u(y)I

c 0. ry

1

WIB(w,2r)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].

Lemma 2.3. Let ) C Rn be a bounded NTA-domain, suppose that p, 1 < p < 00, is given and that A E M,, (a, 0, y) for some (a, 0, y). Let w E Of, 0 < r < ro, and suppose that u is a non-negative continuous A-harmonic function in fl fl B(w, 2r) and that u = 0 on A(w, 2r). There exists c = c(p, n, M, Q,)31 y), 1 < c < oo, such

that if i = r/c, then

max u < cu(ae(w)).

ON1B(w,r)

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

238


particular, the proof also applies in our situation.

Lemma 2.4. Let 1 C R' be a bounded NTA-domain, suppose that p, 1 < p < 00, is given and that A E MP (a,,3, y) for some (a, 0, y). Let w e Oft, 0 < r < ro, and suppose that u is a non-negative continuous A-harmonic function in n fl B(w, 4r) and that u = 0 on 0(w, 4r). Extend u to B(w, 4r) by defining u - 0 on B(w, 4r)\S2. Then u has a representative in W1'P(B(w, 4r)) with Holder continuous

partial derivatives of first order in St fl B(w, 4r). In particular, there exists & E (0,1], depending only on p, n, a,,(3, y such that if x, y E B(zu, r"/2), B(tD, 4r) C [l rl B(w, 4r), then

max U. c 1 IVu(x) - Vu(y)I 0, c > 1, depending on a, p, n, that (2.14)

osc (s) < c (s/t)A osc (t), 0 < s < t < 1/4.

Letting a-'0 it follows as in the proof of Lemma 2.4 that u on compact subsets of Q X2(0). Thus (2.11), (2.12) and (2.14) also hold for u. Moreover, (2.11), (2.12), (2.14), arbitrariness of x, and interior Harna.ck - Holder continuity of u are easily shown to be equivalent to the conclusion of Lemma 2.8 when v(y) = y,,.

We 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 R" be an open set, suppose that p, 1 < p < oo, is given and let A E y) for some (a, 3, y). Let F : R" --+ R" be the composition of a translation, a rotation and a dilation z --+ 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 u is Aharmonic in F-1 (G) and A E Mp(a,/3,'y).

Proof. Suppose that F(z) = z + w for some w E R", i.e., F is a translation. In this case the conclusion follows immediately with A(z, q) = A(z + w,,rl) arid

Suppose that F(z) = rz, where IF is an orthogonal matrix AE with det r = 1. In this case the conclusion follows with A(z, rl) = A(rz, rn) and A E Mp(a, l3, y). Finally, suppose that F(z) = rz for some r E (0,1]. Then u is A-harmonic in F-1(G) with A(z, n) = rr'-1A(rz, r-1rl). Moreover, property (i), (ii) and (iv) in Definition 1.1 follow readily. To prove (iii) in Definition 1.1 we see

that IA(z,n) - A(y,n)I 0, Q,.(Y)CQ,,2(0)} I Qr(y) I

f

If(z)Idz.

Let (3.14)

G={yEQ1/2(0): M(XF)(y)CE?}

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 cE-77E-P??Ea = CE 17 (3.15) I1/2(0) \ GI < cE-"IFI
1,y E 0 and assume that 1 ut(y) < Ioul(y)I 8Js} C {y E W' :

0}.

From (3.38) we find that if we define

A' =

{(y', 0) + 20Jse,, y' E W_'), c' = {y E R' : y,, > 20Js},

then

12' fl B(z, 2s) C S2 fl B(z, 2s).

(3.39)

Let v be a A-harmonic function in 52' fl B(z, 2s) with continuous boundary values on 8(11' fl B(z, 2s)) and such that v 406s, = 0 whenever y E a[52' fl B(z, 2s)] and y,a < 30Js.

Existence of v follows once again from the Wiener criteria of [GZ], the maximum principle for A-harmonic functions, and the fact that the W1,P-Dirichlet problem for these functions in Sl' fl B(z, 2s) always has a solution. By construction and the maximum principle for A-harmonic functions we have v < u in 52' fl B(z, 2s). Also, since each point of 8[Sl' fl B(z, 2s)] where u v lies within 806s of a point where u is zero, it follows from (3.36) and Lemmas 2.2, 2.3 that u < v+cSQu(y) on 0[52' fl B(z, 2s)]. In particular, again using the maximum principle for p-harmonic functions we conclude that v < u < v + cS°u(y) in 52' fl B(z, 2s).


247

Thus, using the last inequality and (3.36) we see that (3.40)

provided

1 < v(y) < (1 - c8°)-1 whenever y E a' n B(y, 2d(y, on)) is small enough. Using Lemma 3.25 and the construction we also have a-l d(V,

(3.41)

and assume (325)k4]. With 8(p, n, a, 0, y, E*) now fixed, it follows from Harnack's inequality for A-harmonic functions that

that i (4.22)

(325)1+E*

E

c-1(325)k(1+E*)u(a£/8(0))

> c-lr(1+E*). u(ar(0)) ? c lu(a(32J)1C(0)) ? In (4.22) we have also used the fact that u(at/s(0)) > for some c = c(p, n, a,)3, 1/c+ (p, n, a, 0) y, e*), for some c+ > 1, which follows from the definition of 6 in

JOHN L. LEWIS, NIKLAS LUNUSTROM, AND KAJ NYSTROM

252

terms of a*, Harnack's inequality, and the fact that u(ai(0)) = 1. (4.22) completes the proof of (4.11) and hence the proof of Lemma 4.8.

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(xj, r?), j = 1, 2,... be a Whitney decomposition of R'a \ S2 into open cubes with center at x j and sidelength

rj. Then Uj0(xj,rj) = Rn \ S2 and Q(xj,rj) fl Q(xi,,r;) = 0 when i # j. We furthermore construct the Whitney cubes in such a way that 10-4nd(Qj, 852) _< rj < 10-2'd(Qj, 812). Let r" = r/e2, where 6 = c(p, n, 1 < c < oo, is so large that the `fundamental inequality' in Lemma 3.35 holds in 52 f) B(w, r/c6). From the NTA property of fZ we may also suppose 6 is so large that if Q j fl B(w, 50T) # 0, then there is a wj E 12f1B(w, cr") for which d(wj, 812) - d(wj, xj) - d(xj, (952). Here A ^, B means that A/B is bounded from above and below by constants which only depend on n. Next we define A(x) Vu(x) Ip-2 whenever x E 12 fl B(w, 50r") and we let r

be the set of all j such that if j e F then Qj fl B(w, 50r) # 0. Moreover, if j E F then we choose wj E 12 fl B(w, &) as above and define A(x) = A(wj) when x E Qj. This defines A almost everywhere on B(w, 50r) with respect to Lebesgue n measure, since it follows from (4.27) that for 6 small enough, 812 fl B(w, r) has Lebesgue n measure zero. From the definition of A, Lemma 3.35, and the Harnack inequality for A-harmonic functions we see that (4.23)

A(x) = A(wj)

A(z) whenever x E Qj and z E B(wj, d(wj, 852)/2).

Let A = A if p > 2 and A = 1/A if 1 < p:5 2. If w E B(w, r) and d(iv-, 8i)/2 < f < f, then from Lemmas 2.1 - 2.3, (4.23), and Holder's inequality it follows that

J jdx
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(zv, 852)/2 < r- < r", then (4.26)

f

A-ldx
0 be A-harmonic in SZ \ B (w, r'),

continuous in R'' \ B(w, r'), with fz = v =- 0 on R" \ [fl U B(w, r')]. Suppose for some rI, r' < rI < i0, and b > 1, that b-I d(y(8S1) < IVu(y)I < bd(y,On)

whenever y E f fl [B(w, rl) \ B(w, r')]. There exists b* > 0, A, c >_ 1, depending on p, n, a, b, such that if 0 < S < S* < S (b as in Theorem 1), then

A-' d(y,ei) < IVv(y)I 24, so that u(z,ek)

_

u(y,bk) u(y)

u(z)

< -

u(z,Sk) 17

u(z)

whenever z, y E i fl [B(w, pk) \ B(w, sk)]. Moreover, fix z as in the last display and choose q > 0 so small that (1 - Ep) u(z, k) < u(y, t) < (1 + Ep) u(z, W

(5.14)

u(z) u(y) u(z) whenever y e S2 fl [B(w, pk) \ B(w, sk)] and t E [ k, k+1]. To estimate the size of 77 observe, for t E [ck, Sk+1 ], that u(y,t)

=

u(y)

u(y,t)

u(y,Lk)

u(y, k)

u(Y)

< (1 + eo/2)(1 +

,)u(x,tk) 11(z)

Thus if 17 = Eo/4 (eo small), then the right hand inequality in (5.14) is valid. A similar argument gives the left hand inequality in (5.14) when q = co/4. Also since k < 2/e0, and e', or depend only on p, n, a, b, we deduce from (5.13) that one can take 81k = cask for c3 = c(p, n, a, b) large enough. From (5.14) we first find that (5.7) holds with L = u(Z, k in 11 fl [B(w, Pk) \ B(w, sk)] and thereupon that (5.8) also holds. From (5.8) we now get, as in (5.12), that (5.11) is valid for l = k in Slfl [B(w, -) \ B(w, 2c2sk)]. Let sk+1 = 2c3c2sk and Pk+1 = . Using (5.11) and the induction hypothesis we have log

(5.15)

u(z,k+1) -log u(z)

u(y)

10 g

U(z, ekt+1)

U(z,Sk)

_ log u(y, G+1) u(y,Sk)

+ log u(z,ek) - log

G kc1

U(Y, k) u(y)

u(z)

Sk+1

\ min(Iz - wI, ly - wl)


257

whenever z, y e it fl [B(w, pk+r) \ B(w, sk+r)]. From (5.15) and induction we get (5.13) with k = m. Since cr', p,n > ri/c, for some large m) = v and t) replaced c = c(p, n, a), we can now argue as in (5.14) to first get (5.7) with by v and then (5.8) for v. We conclude that Lemma 5.4 is valid for z, y E fZ fl [B(w, ri/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 H \ B(w, ri/c). O

5.1. Proof of Theorem 2 when A E Mp (a). Let Q C Rn, w E 0Q, b, p, ro, a, 13, y, be as in Theorem 2. Let A C Mp(a), and suppose that u, v, are minimal positive A-harmonic functions relative to w E 811. If (5.1) holds for u in n fl B(w,ri), then we can apply Lemma 5.4 to u, v and let r'-40. 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 81 and 0 < ri < r-0 for which the `fundamental inequality' in (5.1) holds in On B(w, ri). Moreover, it suffices to show that (5.1) holds for some ri = ri (p, n, a), 0 < ri < r-0, A = A(p, n, a) > 1,

in Sl(w, i) n B(w, ri) where n = 4(p, n, a) is as in (5.2). To this end we show there exists c = c(p, n, a) > 1 such that if c2r' < r 0 is A-harmonic in Il \ B(w, r') with continuous boundary values and u. __ 0 on 89 \ B(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 Oil. With this game plan in mind, observe from Lemma 2.15 and (1.7), that we may assume r = 1, w = 0, and (5.16) B(0, 4n) fl {y : y,,, > EL} C f2, B(0, 4n) fl {y : y,,, < -p} C Rn \ 0,

where µ = 500nb*, 0 < p < 10-10° and r' < (5*)2. Here b* is temporarily allowed to vary but will be fixed after the proof of Lemma 5.19. Extend a to be continuous on Rn \ B(0, r'), by putting u =_ 0 on Rn \ (Cl U B(0, r')). Using the notation in (2.7), let Q = Qi i_µ( ien) \ B(0, ,,fj_z) and let vi be the A-harmonic function in Q with the following continuous boundary values, vi (y)

= u(y),

v i ( y)

=

(yn

- P),u(

y c- 8Q fl {y : 2p < yn}, ?! )

, y E 8Q fl { y : y < yn < 2 µ } .

Comparing boundary values and using the maximum principle for A-harmonic functions, it follows that (5.17)

yr r', as we see from the maximum principle for A-harmonic functions. Using these facts and Lemmas 2.1- 2.3 we find that

u < vi + cµ°/2 u(/ien),

(5.20)

on 8Q. By the maximum principle this inequality also holds in Q. Here & is the exponent of Holder continuity in Lemma 2.2. Using Harnack's inequality, we also find that there exist r = 'r(p, n, a) > 1 and c = c(p, n, a) > I such that (5.21)

max{2b(z), ?b(y)} < c(d(z, 8Q)/d(y, 8Q))T min{V(z), '(y)}

whenever z E Q, y E Q f1 B(z, 4d(z, 8Q)) and 0 = u or vI. Also from Lemmas 2.12.3 applied to vI, we get (5.22)

vi(2vrp-en) ? c 1 u(V N'en)

Let p, 0 be as in Lemma 5.18. Using (5.20) - (5.22), we see that if y E !5 (0, rl/4) n

[B(0, p) \ B(0, 2 f)], then fi(y) < vi(y) + cµo/2 f'(V Wen) s > 2r'. Using (5.24) with t = 1, s = 2,/t-i, and Lemmas 2.1 - 2.3 we see that (5.25)

v1 < c s 2w(fµen) on aQ \ B(0, VIA-),


259

where c depends only on p, n, a. Let v be the A-harmonic function in Q with continuous boundary values v = 0 on 8Q \ B(0, and v = v1 on 8B(0, .\'fA-). Then from (5.25) and the maximum principle, we see that (5.26)

< v1 < v + cµ,1/2u( jden) in Q.

Let p = pl/2-29, 9 small, and a = j z-0 be as in Lemma 5.19. Using (5.21) applied toz/J= 3wefind (5.27) v" > c 1(p,1/2/ap)T u(/en) = c 1µ397u(\en) on fl(0, 2ap)\B(0, p/(2a))], where 'r is as in (5,21) and the nontangential approach region f2 was defined above (5.2) relative to w, i . Also, since depends

only on p, n, a, it follows that c = c(p, n, a) in (5.27). If we define 9 by 0 = min{5/(127), 0/4}, then from (5.26), (5.27) we get 1
1 that v < 1-1/c' on 8B(2 jen, 3vr/A-/2). If y E B(2vfp-en, 3VrtI/2) \ B(2/2en, set eNIY_ZI2/w

(5.32)

b (y) =

e9N/4

- eN

- eN


260

where z = 20.u-e,,.. Then S = 0 on 8B(2 fe,s, vfu-), and (__ 1 on 8B(2 fen, 3Vfj-Z/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() > 0 in B(2Vilizen, 3 ji/2) \ B(2 fen, ). Moreover, using these facts and the maximum principle we deduce

1 - v(y) > (c+ f)-ld(y,OB(2 Ecen, vfjL))

(5.33)

in B(2 Jen, 3vrA-/2) \ B(2v,p-en, Next for fixed t > 1 put

i) provided c+ = c+ (p, n, a) is large enough.

0 = {yEQ':2\en+t(y-2/en)EQ'}, v(2 fen + t(y F(y) = F(y, t) = v(y) t-1

whenever y E O.

From (5.33) for t > 1 fixed, t near 1, and basic geometry it follows that

F > c-1 v on 80.

(5.34)

We note that (iv) of Definition 1.1 and A E M,(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 functions 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 > c -1v in D. Letting t->1, using Lemma 2.4 and the chain rule, we get claim (5.29). The proof of Lemma 5.19 is finished.

As mentioned earlier, Lemmas 5.18, 5.19 together with Lemma 5.4 imply Theorem 2 when A E MM (a).

5.2. Proof of Theorem 2. We are now ready to prove Theorem 2 in the general case.

Lemma 5.35. Let 11 be a bounded (6, ro) -Reifenberg fiat domain and let w E 00. Let A E MM(a, 03, 'y) for some (a, /3,'y) and 1 0 be A-harmonic in 0 \ B(w, r'), continuous in Rn \ B(w, r'), with fi = 6 = 0 on Rn \ IQ U B(w, r')]. Then there exists b., a > 0, c+ > 1, depending on p, n, a, /3, ry, such that if 0 < b < S,, < h (b as in Theorem 1) and r1 = ro/c+, then Ilog

\ ({ z) 0(-`

r'

(y)y)

log (60

c+

min(rl,Iz

- wI, Iy -'wl) /

whenever z, y E 11 \ B(w, c+r').

Proof: Once again we assume that r'/rl c 'a-Tu(pen/a, t) on SZ(w, /2) n (B(w, 2p) \ P (w, p/2)). Thus, for some c' = c'(p, n, a, /3, y) > 1, (5.40)

t)
0 fixed, such that 1l+E' < h(a.(w)) < !s\ I (6.2) `rJ h(a*(w)) p. (6.1)

I

(S)

whenever y E Q fl B(w, r"), h E {u, v}, where 0 < s < 4r. Observe again, for x, A E Rn, t; E Rn \ {0}, that 1

A.,(x, A)

- Ai(x, e) =

dtA1(x, to + (1- t)e)dt

J n

(x, to + (1j=1

0

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 LC = 0 in 0 fl B(w, r"), where n

L((x) = E

axi(aij(x)C.;)

i:9=1 1

and aij(x) = J

!A (tVu(x) + (1 - t)Vv(x))dt,

0

for 1 < i, j < n. Moreover, from the structure assumptions on A, see Definition 1.1, we find that n

(6.5)

c+1 (IDu(x)I +

IVv(x)I)p-2 ItI2

< i,j=1 E

tt

aij(x)Sjttj

2 and A = 1/A for 1 0, small enough, that (6.8)

J

adx
1 to vary in (6.11). A, c, are then fixed near the end of the argument. Put A u(y)

u(ar. (w)) - v(ar. (w))' A v(y)

v(y)

v (y) = u(ar. (w)) - v(ar* (w)) + f)(ar. (w)) Observe from (6.11) that e = u' - v'. Using Definition 1.1 (iv) we see that u', v' are A-harmonic functions. Let L be defined as in (6.4) using u', v', instead of u, v,

264

JOHN L_ LEWIS, NIKLAS LUNDSTROM, AND KAJ NYSTROM

and let e1, e2 be the solutions to Lei = 0, i = 1, 2, in Sl fl B(w, r*), with continuous boundary values: (6.13)

-v(ar.(w)) e2 (y) = v(ar (w))

PI(y) =

whenever y E 8(Sl fl B(w, r*)). From Lemma 6.6 we see that Lemma 4.7 can be applied and we get, for some c+ > 1 and r+ = r*/c+, that (6.14)

u e1(ar+ (w))

c+ e2(ar+(w))

e1(y) e2(y)

ei (ar+(w)) c+e2(ar-(w))

whenever y E 1 fl B(w, 2r+). We now put

c=c+, r

r+, A=cel(a,-,(w))'

and observe from (6.14) that (6.15)

Ael(y) - e2 (Y) > 0 whenever y E fI fl B(w, 2r').

Let e = A el - e2 and note from linearity of L that e, e, both satisfy the same linear locally uniformly elliptic pde in 11 fl B(w, r*) and also that these functions have the same continuous boundary values on 8(1 fl 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) > 0 in f2f1B(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. O

We note that in [LN5] Lemma 6.10 was proved under the assumptions that u and v are non-negative p-harmonic functions in S2 fl B(w, 2r) and that f2 C R" is a Lipschitz domain. In this case the constants in Lemma 6.10 depend only on p, n and the Lipschitz constant of Q. 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, Sl, w, r be as in Theorem 1 and let u, v be the A-harmonic functions in fl fl B(w, 2r) with

u = max{u, v} and v = min{u, v} on 8[1 fl B(w, 2r)].

From the maximum principle for A-harmonic functions we have u > v and hence we can apply Lemma 6.10 to conclude that u(a*(w))

u(y)

u(a' (w))

v(aT(w)) - v(y) - v(aT(w))

whenever y E Slf1B(w, r"). Moreover, using the definition of il, v, and the inequalities

in the last display we can conclude that (6.16)

v(y) < cv(z) whenever y, z E S2 fl B(w, r").

Next if x E 811 fl B(w, f/8), then we let

M(P) = sup v and m(p) = B(x,p)

s(n p)

V


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

M(p) - m(p) < c*(m(p) - m(p)).

(6.17)

Likewise, we can apply Lemma 6.10 with fi = M (p) v and

u to conclude

(M(p) v - u)/u ~ constant on H n B(w, p). Using this inequality together with (6.16) it follows that

(M(p)v - u)/v N constant on H fl 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 (iv). Thus if c* is large enough, then (6.18)

M(P) - m(P) < c*(M(P) - MOD-

If osc (t) = M(t) - m(t), then we can add (6.17), (6.18) and we get, after some arithmetic, that c* - 1 osc (p")
0 Pc - d (u(y) + c)2 and consider the eigenvalue problem

f" = uf,

(5.3a) (5.3b)

f (0) = 0,

f 0 0.

gf(PC) = f ` (PC,),

It is easy to see that there exists a solution with p = v2 > 0 if and only if 9P, > 1, in which case f (z) = a sinh vz for some a # 0, where v is uniquely determined by tanhvPc 1 vPP

gPP,

When gP,, > 1 all the other eigenvalues p of (5.3) (there are infinitely many) are negative and determined by tan vP, = 1 p = -v2 and f (z) = sin vz where VPC

gPc

By a similar calculation, every eigenvalue of (5.3) is negative when gP,, < 1, and when gPP = 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

4(f, c) =

fP' f '(z)' dz - gf f0 ` f (Z)2 dz

over the class of non-zero functions f E observations we infer that when gP, < 1, (5.4a)

W1,2(0, Pc,)

with f (0) = 0. From the above

inf {q(f, c) : f E W1'2[0, Pr], f # 0, f (o)) = 0} > 0.

However, when gPc > 1, (5.4b)

inf {q(f, c) : f E 46'1'2[0, P], f # 0, f (0)) = 0 } < 0

WAVES ON A STEADY STREAM WITH VORTICITY

275

and, for all j > 2, (5.4c)

inf

dim(F)=j

{sup q(f, c) : f E F C W1"2 [O, p,:], f # 0, f (0) = 0} > 0.

We return now to our study of (5.1). In addition our basic assumption that u E C2(-d, 0) fl Cl,'O[-d,O] with u(O) = 0, we now assume that

u(y) < 0, y E [-d,0).

(5.5)

When f : [0, P,] -p R is smooth and f (0) = 0, let dt

y

v(y)

_ fd

(u(t) + c)2

Then v(-d) = 0 and, when substituted in (5.2), we infer from (5.4a) that Al (c) > 0 when gP,, < 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 AI (c) -+ -00 as c / 0 and A, (c) > 0 for all c < 0 with Icl sufficiently large. Hence, for each k E N, there exists ck < 0 such that -k2 = AI (ck ) PROOF. Note first that Pc _

(5.7)

T-dY_

= oo as c/o,

d u(y)2

since u(0) = 0 and Iu' (0) I < oo. Let f : [0, P J --+ R be such that f (0) = 0 and substitute y

v(y) =

fd

dt (u(t) + c)2.

Then v(-d) = 0 and, when substituted in (5.2), we find that PO - 9P,

AI(c)

J

(

d(u(y) + c)2

=

(5.8)

° d(u(y) + c)2

J

J-d (u(t) + c)2 }2 dy P` I - 9 1

y

dt

12

p fd (u(t) + C)2)

dy

Now, 1

1>_

dt

P-Jd(u(t)+c)2 '0foryE[-d,l)ascJO.

by (5.7), and hence AI (c) --> -oo as c / 0, by (5.7) and the dominated convergence theorem. It follows from (5.4) that AI (c) < 0 if and only if Pg > 1, which is true for all c < 0 with jej sufficiently small, by (5.7). Finally note from (5.6) that AI (c) > 0 for all Icl sufficiently large, since gPP -> 0 as Icl -- oo. Since AI (c) obviously depends continuously on c < 0, the result follows.

M. LILLI AND J. F. TOLAND

276

To consider the behaviour of \1(c) for positive c suppose that

u(-d) < u(y) for all y E (-d, 0] and let u = -u(-d) > 0.

(5.9)

Note that

Pe-4ooas c\, u.

(5.10)

We now restrict attention to c E (u, oo). LEMMA 5.3. Suppose that (5.9) holds. (a)

-g

lim inf AI (c) < cNn IIu+!IIL2(-d.0)

(5.11)

(b) Al (c) > 0 for all c > 0 sufficiently large.

(c) Suppose that -k2 > lim infer XI (c), k E N. There exists ck > u such that

a1(ck) = -k2.

PROOF. Since Pr-4oo asc\u, 1>

J_d(u(t)+c)2

-' 1 as c \, u for all y E (-d,0],

and (5.11) follows from (5.8). As in the preceding proof, Ai(c) > 0 for all c sufficiently 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) = woy, wo E R. When wo > 0 the hypotheses of Lemma 5.2 are satisfied. Moreover, in Lemma 5.3,

u = wo and In + of l .2(_l,0) = wo/3. Hence -k2 = \1(c) for some c > u > 0 is an eigenvalue if wo'k' < 3g.

In fact, for this example we have seen from an explicit calculation that -k2 = A1(c) for some c > u if and only if wo (1- k-1 tanh k) < gk-1 tank k. (In particular, -1 = A1(c) for some c > wo if and only if wa < 3.194g (which may be compared 0 with wo < 3g, the criterion in the Lemma). THEOREM 5.4. The parameter values ck in the preceding two lemmas are bifurcation points for problem (2.7) when U(z ; c) = u(z) + c. PROOF. We have shown that (4.2a) is satisfied when c = ck , and (4.2c) follows by the self-adjointness of (4.3) in an L2 setting. Since 8U/8c(0; cl) = 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 ck are bifurcation points for the problem of waves on 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 (1949), 425 - 441. [2] H. W. 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 -116. [4] B. Buffoni and J. F. Toland, Analytic Theory of Global Bifurcation - An Introduction. Princeton University Press, Princeton, N. J., 2003.

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. A. 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, Bifurcation from a simple eigenvalue, J. Funct. Anal. 8 (1971), 321-340. [9] R. A. Dalrymple and J. C. Cox, Symmetric finite-amplitude rotational water waves, J. Physical Oceanography, 6 (1976), 847-852. [10] I. I. Daniliuk, On integral functionals with a variable domain of integration, Proc. Steklov Inst. Math. 118 (1972). In English, Amer. Math. Soc. (1976). [11] E. B. Davies, Spectral Theory and Differential Operators, Cambridge University Press, Cambridge, 1995. [12] M.-L. Dubreil-Jacotin, Sur Is. determination rigoureuse des ondes permanentes periodiques dampleur finie. J. Math. Purses Appl. 13 (1934), 217291. [13] D. Gilbarg and N. S. 'Itudinger, 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 0708:0541V1 [Math.AP] 3rd Aug 2007, to appear. UNIVERSITAT AUCSBURG, INSTITUT FOR MATHEMATIK, UNIVERSITTSSTRASSE 14, 86159 AUGsBURG, GERMANY

Current address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK

E-mail address: lillitmath.uni-augsburg.de DEPARTMENT OF MATHEMATICAL SCIENCES, UNIVERSITY OF BATH, CLAVERTON DOWN, BATH

BA2 7AY, UK

E-mail address: jft9maths.bath. ac.uk


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

ABSTRACT. We give an answer to an old question of T. Murai concerning the characterization of the boundednesss of the Cauchy integral operator on arbitrary sets 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).

1. Introduction Takafumi Murai asked in [Mu] the following question:

given a compact set E C C such that its Hausdorff 1-dimensional measure satisfies 0 < H' (E) < oo, is that true that Cauchy integral operator is bounded in L2(E, Hl I E) if and only if H' (E n Q) < C y(E n Q). Here y 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 < H'(E) < Co. Then the Cauchy integral operator T (and also T*) is bounded on L2 (E, Hl) if and only if there exists a constant C such that for every square Q on the plane (1.1) Hl (E n Q) < C y(E n Q). 2000 Mathematics Subject Classification. Primary 47B36; Secondary 42C05. Key words and phrases. Analytic capacity, Hausdorff content, nonhomogeneous harmonic analysis, secretive functions. The first author was supported in part by NSF Grant 0501067. The second author was supported in part by NSF Grant 0501067.

279

280

F. Nazarov, A. Volberg

Definition. Let K be a compact set in C. y(K) := sup{zlim Iz f (z) l : f E Hol(C \ K), If (z) < 1 Vz E C \ K, f (oo) = 0},

7+(K) .= sup{ liar I z f (z) I: f (z) _ J

du(), ,uE M+(K), I f(z) I< 1 `dz E C\ K} .

By definition, (1.2)

-t+(E)

'Y(E)

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 us recall that there is another criterion of the boundedness of Cauchy integral. It is obtained in [NTV2], [T1] 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 points z1, z2, z3 E C we call R(zl, z2, z3) the radius of the circle (may be

oo) passing through those points. Then Merger's curvature of a positive measure It is by definition C2 (A)

dp(z1)du(z2)dp(z3) 1

(II f

R2(zI, z2, z3)

)

If p = H1 I E for a certain compact E, 0 < Hr (E) < oo, we will use the following notations: c2(E) := c2(H'IE). We are ready to quote the criterion proved by Nazarov-Treil-Volberg in [NTV2] and Tolsa in [Ti]. THEOREM 1.2. 1) Cauchy integral operator is bounded in L2 (p) if and only

if there exists a finite constant C such that for every square Q (1.3)

µ(Q) < CAQ),

where f(Q) is the length of the side of Q, and (1.4)

C2(/IQ)2 < Cp(Q).

2) In particular, if it = H' I E, for a certain compact E, 0 < H' (E) < oo, then the boundedness of the Cauchy integral operator in L2(E, Hl) is equivalent to (1.5)

H1(E n Q)!5 C I(Q),

and (1.6)

c2(E n Q)2 < C H1(E n Q).

On analytic capacity of portions of continuum

281

Remark. Actually one can sometimes skip assumption (1.3) as Tolsa has shown in Lemma 5.2 of [T5]. This is the case for measures it such that their upper density lim sup p(B(x, r)) /r

is uniformly bounded. We are grateful to the referee for this remark. We will show 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 efforts. 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 multidimensional 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,H1IE) of the Cauchy integral operator. We want to use Theorem 1.1. So for our goals it is sufficient to prove the following implication (Q is always a square) H1(E n Q) < C2y(E n Q). (1.7) VQ c.2 (E n Q)2 < C1 H1(E n Q) < C2t(Q) To prove (1.7) we need the trivial inequality y > y+ and the following characterization of y+ due to Melnikov (see e.g. [Tl], it can be found in [Vol also): (1.8)

-f+(K) x

sup

4411,

,u:,u(B(x,r)) - y +

H1(EnQ)2 > a c2(E n Q)2 + H1(E n Q) > aH'(EnQ).

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 describes all such curves on the plane for which the Cauchy integral operator is bounded on L2 (F, d H'). This is the class of curves 1' (called Ahifors-David curves) satisfying

H1(D(x, R) n t)) < C R for any 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 (1.9)

even for chord-arc curves) there exists a purely analytic proof of the boundedness of the Cauchy operator on L2 (I', d Hl) (see [CJS], [Chr]), but all the proofs of the theorem of David are the mixtures of analytic and geometric arguments, [Dal],

282


[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, b) a simple computation in geometric terms of y(I' 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 T1 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 be measured in terms of simple geometric characteristics of the compact. The result of Tolsa [T4] (see also the exposition of this result in [Vol together with its multi-dimensional analogs) only confirms this because the analytic capacity is proved to be 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 H1 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. We will use the symbol hl (E) to denote 1-dimensional Hausdorff content of E, namely,

hl(E) := inf{E rj : E C UjD(xj, rj)} . It is clear that H1 is larger than hl, 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 y(I'). Thus, for a continuum r

a hl(I') < y(r) < A h'(r) . We are going to prove (1.10) for sets r n II, r n Q, and r n D. We restrict

(1.10)

ourselves to the the case of half-planes.

THEOREM 1.3. Let Il be a closed half-plane, and let r be a continuum. Then (1.11)

ahl(rnII) 1

(5.1)

iE.F

for all collections c = {ci}iE.F E l , 114" S 1, EiEF ciryi > la L. Now we are going to find such a collection c, for which (5.1) fails. This collection will consist only of l's and 0's. Define for X E R := sup lvI (D(x,r))

(Miv)(x)

r

r>o

Let us define another maximal function on I = UiEyli. Namely, if x E Ii, i E .F v* (x) := sup I vI (D(x, r) )

r

r>3 r;

The segment Ii is well inside D (yi,10 ri) and v is outside of it, therefore

sup v*(x) < inf (Mlv)(x). xEl;

xElc

Consider .To

d

{i E F : 3x E Ii, v*(x) > 1L

}.

On 10 := UiE.Fuli we have (this is just (4.2))

Mlv > 100

(5.2)

On the other hand,

{x:(Mlv)(x)>

1001,

L }. The segment 1, lies well inside D(yi, 10r,), and v is supported outside of it, therefore, :

i E F2

And so

i rif 2:Eli

f

10

P(z, x) dI vl (z) > L

f fP(zx)dIvRz)dx > L

(5.7) iEF2

Ili I .

iEF2

On the other hand, iE.F2

and, hence,

f f P(z, x) dluI (z)dx = f(f

P(z, x) dx) dlvl (z) , {Ef 2 I{

290


(5.8)

iE.F2

f j'

r

/

P(z, x) dlvl (z)dx =

J

(J P(z, x)Xu, Fj1; dx) djvj (z) < fi djvJ(z) < 1.

Now (5.7) and (5.8) give us that II'I 0 is an absolute constant.

We can apply Theorem 6.1 to the proof of Theorem 1.1. We will meet the situation, where IIv11 = Hl(E n Q), Iv(K)I = y(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], [Ti]): (6.2)

y+ (E n Q) > ay(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, it 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], [NTV5J, [Vo] in the proof of Theorem 1.1. We avoided curvature criterion from [NTV2], [Ti] as well. 4) This, in particular, shows that their exists also a multi-dimensional analog of our Theorem 1.1.

Let us be in R" now. Let E be a compact subset of R" such that 0 < Hn-1(E) < oo. Let T denote the vector Riesz transform operator with kernel R(x - y), where

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 y as before, the reader can get acquainted with it by reading [Vo]. We have

THEOREM 6.2. Operator T is bounded in L2(E, H"-1I E) if and only if there exists a finite constant C such that for every cube Q in R11 we have (6.3)

Hn-1(E n Q) < C y(E n Q) .

292


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 homogLnes", Lecture Notes in Math. 242, Springer-Verlag, Berlin, 1971. [CJS] R. R. COWMAN, 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.

[Dai] G. DAVID, Op6rateurs integraux singuliera 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, Revista Mat. Iberoamericana, v. 14 (1998), 369-479. [DaJ] G. DAVID, J.-L. JouRNE, A boundedness criterion for generalized Calder6n-Zygmund operators, Ann. of Math., 120 (1984), 371-397. [DaJS] G. DAVID, J.-L. JOURNE; S. SEMMES, Operateurs de Calderdn-Zygmund, fonctaons panaace.retive et interpolation, Rev. Mat. Iberoamer., 1 (1985), 1-56. [Faj 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, 99 114. [Ma] P. MATTILA "Geometry of Sets and Measures in Euclidean Spaces". Cambridge Univ. Press, 1995.

[Mel] M. MELNIKOV An estimate of the Cauchy integral over an analytic are. Sbornik: Mathematics, 71 (1966), NO 4, 5030514. [Me2] M. MELNIKOV Analytic capacity: discrete approach and curvature of a measure. Sbornik; Mathematics, 186 (1995), 827-846. [Mu] T. MURAL "A Real Variable Method for the Cauchy Transform, and Analytic Capacity", Lecture Notes in Math., 1307, Springer Verlag, Berlin-Heidelberg-New York, 1988. [MMV] P. MATTILA, M. MELNIKOV, 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. NAZAROV, S. TREIL, A. VOLBERG, Weak type estimates and Collar inequalities for Calder6n-Zygmund operators on nonhomogeneous spaces. IMRN International Math. Res. Notices, 1998, no. 9, 463-487. [NTV2] F. NAZAROV, S. TREIL, A. VOLBERG, Cauchy integral and Calderdn-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, Acts,

Math., 190 (2003), 151-239. , Nonhomogeneous Tb theorem which proves Vitushkin's conjecture, Preprint No. 519, CRM, Barcelona, 2002, 1-84. [St] E. STEIN, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals", with the assistance of Timothy S. Murphy, Princeton Math. Ser. 43, Monographs in Harmonic analysis, iii, Princeton Univ. Press, Princeton, 1993. [Tl] X. TOLSA, L2-boundedness of the Cauchy integral operator for continuous measures, Duke Math. J., 98 (1999), no. 2, 269-304. [T2] X. TOLSA, Cotlar'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, Painleve's problem and the semiadditivity of analytic capacity, Acts, Math., 190 (2003), no. 1, 105-149. [NTV5]

On analytic capacity of portions of continuum

293

[T5] X. TOLSA, On the analytic capacity ry+, Indiana Univ. Math. J., 51 (2), (2002), 317-344. [Vo] A. 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. 53706

E-mail address: nazarov®math.visc.edu DEPARTMENT OF MATHEMATICS, MICHIGAN STATE UNIVERSITY, EAST LANSING, MICHIGAN 48823, USA AND SCHOOL OF MATHEMATICS, EDINBURGH UNIVERSITY, EH9 3JZ, UK

E-mail address: [email protected]


The Christoffel-Darboux Kernel Barry Simon* ABSTRACT. A review of the uses of the CD kernel in the spectral theory of orthogonal polynomials, concentrating on recent results.

CONTENTS 1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

Introduction The ABC Theorem The Christoffel-Darboux Formula Zeros of OPRL: Basics Via CD The CD Kernel and Formula for MOPS Gaussian Quadrature Markov-Stieltjes Inequalities Mixed CD Kernels Variational Principle: Basics The Nevai Class: An Aside Delta Function Limits of Trial Polynomials Regularity: An Aside Weak Limits Variational Principle: Mate-Nevai Upper Bounds Criteria for A.C. Spectrum Variational Principle: Nevai Trial Polynomial Variational Principle: Mate-Nevai-Totik Lower Bound Variational Principle: Polynomial Maps Floquet-Jost Solutions for Periodic Jacobi Matrices Lubinsky's Inequality and Bulk Universality Derivatives of CD Kernels Lubinsky's Second Approach Zeros: The Freud-Levin-Lubinsky Argument Adding Point Masses

References

296

298 299 302 303 304 306 308 309 311 312 315 316

317 319 320 321 322 323 323

324 325 328

329 331

2000 Mathematics Subject Classification. 34L40, 47-02, 42C05. Key words and phrases. 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. ©2008 Barry Simon 295

B. SIMON

296

1. Introduction This article reviews a particular tool of the spectral theory of orthogonal polynomials. Let µ be a measure on C with finite moments, that is, I Izln di (z) < oo

(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, {zn}n 0 are independent in L2(C, dp), so by GramSchmidt, one can define monic orthogonal polynomials, Xn(z; dµ), and orthonormal polynomials, xn = X/ l l Xn l l L2 . Thus,

f3Xt(z;d,4)dP(z)=O

j = 0, ... , n -1

(1.2)

Xn(z) = zn + lower order

(1.3)

fXn(Z)Xm(Z)d/A = Snm

(1.4)

We will often be interested in the special cases where µ is supported on R (especially with support compact), in which case we use Pn, pn rather than Xn, xn, and where it is supported on 8IlD (D = {z l lzl < 1}), in which case we use 4'n, cpn. 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}°O and Verblunsky coefficients {an}' o with recursion relations (p_1 = 0;

Po=4'o=1): zPn(z) = an+1Pn+1(z) + bn+iPn(z) + anPn-1(z)

n+1(z) = z4n (z) - 6n4 e(z) 46n(z)

= zn 4n(1/2)

We will sometimes need the monic OPRL and normalized OPUC recursion relations:

zPn(z) = Pn+1(z) + bn+1Pn(z) + a,,Pn-1(z)

(1.8)

zcpn(z) = Pncn+1(z) + anWn(z)

(1.9)

P. = (1 - la-l21/2

(1.10)

Of course, the use of pn implies IanI < 1 and all sets of {an}n o obeying this occur. Similarly, bn E R, an E (0, oo) and all such sets occur. In the OPUC case, {an}°O_o determine dµ, while in the OPRL case, they do if sup(lan[ + lbn[) < 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]. We will use nn (or ,n(dic)) for the leading coefficient of xn, pn, or pn, so Kn = l[Xn[IZ1'(dµ)

The Christofel-Darboux kernel (named after [23, 28]) is defined by n K. (Z' C) _ E xi (z) x:i W .i=o

(1.11)

(1.12)

THE CHRISTOFFEL-DARBOUX KERNEL

297

We sometimes use Kn (z, (;,u) if we need to make the measure explicit. Note that if c > 0, Kn ('z, C; ci) = c-1Kn(z, (;,u)

(1.13)

since xn(z; cdji) = c-1/2x. (z; dµ). By the Schwarz inequality, we have IKn(z;()I2

(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 Szegd [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, S real, the complex conjugate is irrelevant-and some authors leave it off even for complex z and C. 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) = E 1Pi x)12

(1.15)

j=0

Continuum analogs of the ratios of this object for the first and second kind polynomials 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 n K.,, (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 M. on L2(C, dp) of multiplication by z: (Mz f) (z) = Z f (z)

(1.16)

If supp(dy) is compact, MZ is a bounded operator defined on all of L'(C, dg). 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]). With this in mind, we use o(dµ) for the spectrum of M,,f that is, the support of dµ, and a,... (dA) for the essential spectrum. When dealing with OPRL of compact support (where M,, is bounded selfadjoint) or OPUC (where M,x is unitary), we will sometimes use

o8(di), o.(dp), opp(dic) for the spectral theory components. (We will discuss a'ess(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 depends on its interpretation as the integral kernel of a projection. In L2(C, dµ), the set of polynomials of degree at most n is an n + 1-dimensional space. We will use

B. SIMON

298

irn for the operator of orthogonal projection onto this space. Note that

(inf)(S) = f K. (z, () f (z) dµ(z)

(1.17)

The order of z and S is 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 (() =

fK(z)f(z)dP(z)

(1.18)

In particular, since K7, is a polynomial in (of degree n, we have K,, (z, w) =

f

Kn(z, ()Kn((, w) dIL(()

(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 commutators. Another theme, motivated by Lubinsky [60, 61], is the study of asymptotics of K,, (x, y) on diagonal (x = y) and slightly off diagonal ((x - y) = O(n)). Sections 2, 3, and 6 discuss very basic formulae, and Sections 4 and 7 simple applications. Sections 5 and 8 discuss extensions 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. While 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 GramSchmidt, 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 kk,n, 0 < j, m < n, by

K.

n

kjm

(z, j,'m=o

One also has the moment matrix

(z', zk) = f

z,zk du (z)

(2.1)


299

0 < j, k < n. For OPRL, this is a function of j + k, so M(n) is a Hankel matrix. For OPUC, this is a function of j - k, so m(n) is a Toeplitz matrix. THEOREM 2.1 (ABC Theorem).

(m(n))-i = k(n)

(2.3)

PROOF. By (1.18) for a =r0, ... , n,

J

K,,.(z, ()zt d i(z)

(2.4)

Plugging (2.1) in for K, using (2.2) to do the integrals leads to n k(Q)mq,1 (3 _ Ce

(2.5)

j,4=0

which says that kjq mqt) = b 4 which is (2.3).

Here is a second way to see this result in a more general context: Write

j xj(z) = > ajkzk k=0

so we can define an (n + 1) x (n + 1) triangular matrix a(n) by ajk

Then (the Cholesky factorization of k) k(n) = a(n)(a(n))*

(2.9)

with * Hermitean adjoint. The condition

(xj,xt) = bje

(2.10)

(a(n))*m(n) (a (n)) = 1

(2.11)

says that the identity matrix. Multiplying by (a('))* on the right and [(a(n))*]-1 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, C) = an+i

Cpn+i(z)pn(S) -pn(z)pn+1(S)

(3.1)

and for OPUC that Kn(z, 0 _

`Pn+1(z) 0. Let S = d2/(d+ yam). Then at least one of pn and pn_1 has no zeros in (xo - 5, xo + 5).

They also have results about zeros near isolated points of v(dp).

5. The CD Kernel and Formula for MOPs Given an t x f matrix-valued measure, there is a rich structure of matrix OPs (MOPRL 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 products," maps from L2 matrix-valued functions to matrices, (( , )) R and ((- , ))L. The R for right comes from the form of scalar homogeneity, for example, ((.f,9A))R = ((.f,9))RA

(5.1)

but ((f, Ag)) R is not related to ((f, 9))R. There are two normalized OPs, pR(x) and p;'(x), orthonormal in (( , ))R and ((', '))L, respectively, but a single CD kernel (for z, w real and t is matrix adjoint),

(z, w) _ Epk (z)pR (w)t k=0

=

pk (z)tPk (w) k=0

One has that ((Kn (- , z), f (' )))R = (Jrn.f) (z)

where irn is the projection in the

))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 Gohberg-Scmencul formula; see [10, 35, 39, 40, 48, 100, 101].

B. SIMON

304

6. Gaussian Quadrature Orthogonal polynomials allow one to approximate integrals over a measure dµ 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 Jn;F(b) be the n x n matrix

Jn;F(b) =

b1

al

0

a1

b2

a2

0

a2

b3

(i.e., we truncate the infinite Jacobi matrix and change only the corner matrix element bn to biz + b). Let (b), j = 1,..., n, be the eigenvalues of Jn;F (b) labelled by xl < 12
Q(x1+1) = 0, Q(y) < 0 on (xei xe+1). Tracking where Q' changes sign, one sees

0

that (7.2) holds. THEOREM 7.2. Suppose dp is a measure on R with finite moments. Then 1

{iIx' (xo)<xo}

An-1(xjn) (xo),xj n) (xo))

14(-00, x0J) (7.4)

1

A((-00, X0)) !

E

/ Kn-1(x(n)(x0),x(n)(x0))

3

7

REMARKS. 1. The two bounds differ by Kn_1(xo, 2. These imply µ({xo}) < Kn-1(x0,x0)-1

In fact, one knows (see (9.21) below) A({xo}) = oKn-1(xo,xo)li

If µ({xo}) = 0, then the bounds are exact as n --> oo.

xo)-1

(7.5)

B. SIMON

308

PROOF. Suppose 0. Let f be such that xj(")(x0) = x0. Let Q be the polynomial of Lemma 7.1. By (7.2),

FW-x, xo]) < f Q(x) dp, and, by (7.1) and Theorem 6.5, the integral is the sum on the left of (7.4). Clearly, this implies 1

114(X

00)) >

xjn) V (x0)) {jlxfn) (xo)>xo) Kn-1(xjn)(xo),

which, by x -; -x symmetry, implies the last inequality in (7.4). COROLLARY 7.3. If f < k - 1, then k-1

1

E

(Xe)])

j=C} I K(x(n)(x0), x(n)(x0))
£+ 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' are distinct zeros of P,, (x), E = z (E + E') and b > 1I E - E'I, then 62

]E - E' >

- (11E - E'12)2 3n

Kn(E, E) suply-91 ajxj (z) j=o

with x j the orthonormal polynomials for a measure d11, then E ajxj (zo) = 1 and Thus the lemma implies: IIQnIIL2(C,dµ) = >j olajl2.

THEOREM 9.2 (Christoffel Variational Principle). Let p, be a measure on C with finite moments. Then for xo E C,

min(JIQn(z)I2doIQn(zo)=1,deg(Qn) 1, we use Kn(z, b) = z S Kn

\z

(9.13) S

which implies, for z = eiw, zo = reie, r > 1, IQn(z, zo)I2 dp _ Pr-1(0, oo. Thus, aj approaches a set of Verblunsky coefficients rather than a fixed one.

For any finite gap set e of the form (10.2)/(10.3), there is a natural torus, J, of almost periodic Jacobi matrics with oess(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 J7e are periodic if and only if each [al, j3j] has rational harmonic measure. In this case, we say e is periodic. DEFINITION.

I ti., bn} 1, {[L.,

dm({an, bn}

1) = E e-j (l am+j - am+j l + I bm+j - bm+j l)

(10.5)

j=0 J (10.6) dm({an, bn}, Je) = min dm,({an, bn}, J) JEJ, DEFINITION. The Nevai class for e, N(e), is the set of all Jacobi matrices, J, with (10.7) dm(J,Fe) - 0

as m - 00. This definition is implicit in Simon [81]; the metric dm is from [26]. Notice the isospectral torus is the set of {an}°O_0 with that in case of a single gap e in an = aei9 for all n where a is a dependent and fixed and 0 is arbitrary. The above definition is the Lopez class. That this is the "right" definition is seen by the following pair of theorems: THEOREM 10.1 (Last-Simon [56]). If J E N(e), then aess(J) = C

(10.8)

THEOREM 10.2 ([26] for periodic e's; [75] in general). If QeM(J) = vac(J) = C

then J E N(e).

11. Delta Function Limits of Trial Polynomials Intuitively, the minimizer, Qn (x, xe), in the Christoffel variational principle must be 1 at zo and should try to be small on the rest of o(dµ). As the degree gets larger and larger, one expects it can do this better and better. So one might guess that for every S > 0, sup I Qn(x, xo)I -+ 0 (11.1) Ix-xoI>5 XEQ(dµ)


313

While this happens in many cases, it is too much to hope for. If X1 E o(dµ) but µ has very small weight near x1i then it may be a better strategy for Qn not to be small very near x1. Indeed, we will see (Example 11.3) that the sup in (11.1) can go to infinity. What is more likely is to expect that I Q,,(x, xo)12 dµ will be concentrated near xo. We normalize this to define I Qn(x, xo)I2 dµ(x) f I Q.(x, xo)12 dµ(x)

d?lnxo) (x) =

(11.2)

so, by (9.6)/(9.7), in the OPRL case, d??nxo)(x)

- IKn(x,xo)I2 K. (x, xo) dµ(x)

(11.3)

We say µ obeys the Nevai S-convergence criterion if and only if, in the sense of weak (aka vague) convergence of measures, d?7nx0)(x)

(11.4)

a.,o

the point mass at x0. In this section, we will explore when this holds. Clearly, if xo 0 a(dµ), (11.4) cannot hold. We saw, for OPUC with dµ = d9/21r and z 0 BIID, the limit was a Poisson measure, and similar results should hold for suitable OPRL. But we will see below (Example 11.2) that even ono (dµ), (11.4) can fail. The major result below is that for Nevai class on e'nt, it does hold. We begin with an equivalent criterion: DEFINITION. We say Nevai's lemma holds if Ipn(x0)I2 = 0

line

(11.5)

-

n-.x Kn(xo,xo)

THEOREM 11.1. If dµ is a measure on R with bounded support and

inf an > 0 n

(11.6)

then for any fixed xo E R,

(11.4)q(11.5) REMARK. That (11.5) = (11.4) is in Nevai [67]. The equivalence is a result of Breuer-Last-Simon [14]. PROOF. Since

Kn-1(xo,x0)

_

Ipn(x0)I2

(11.7)

1 - Kn (x0, x0)

K. (x0, x0) Kn-1(xo,x0) 1 Kn(x0, xo)

(11.5) so

(11.5)

Ipn+1(x0) I2

Kn (x0, xo)

-

Ipn+1(xo)

I2

Kn+1(x0, x0)

K,,+1(xo, xo) Kn (xo, xo)

(11.8)

0

We thus conclude (11.5)

By the CD formula and

f

Ipn(xo)I2 + Ipn+1(xo)I2 K. (xo, xo ) po(x),

-, 0

Ix - x0I2IKn(x,xo)12 dµ = an+1U'n(xo)2 +pn+1(x0)2]

(11.9)

B. SIMON

314 so, by (11.6) and (11.10),

fix - xoI2 dnn"°)(x)

0

(11.5)

when a,,, is uniformly bounded above and away from zero. But since dr7n, have support in a fixed interval,

(11.4)4. f Ix-xol2d71( x0),0 EXAMPLE 11.2. Suppose at some point x0, we have lim (Ipn(xo)I2 + 1pn+1(x0)12)11" - A > 1

(11.11)

limsup 1p-(x0)12 > 0

(11.12)

n-+oo

We claim that Kn(xo, xo)

for if (11.12) fails, then (11.5) holds and, by (11.7), for any e, we can find No so for n > No, Kn+1(xo,xo) < (1+e)Kn(xo,xo) (11.13) so

x0)1/n < 1 So, by (11.5), (11.11) fails. Thus, (11.11) implies that (11.5) fails, and so (11.4) lim

fails.

REMARK. As the proof shows, rather than a limit in (11.12), we can have a lim inf > 1.

The first example of this type was found by Szwarc [94]. He has a dµ 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 (11.11) holds. The Anderson model (see [20]) provides a more dramatic example. The spectrum is an interval [a, b] and (11.11) 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 11.3. A Jacobi weight has the form

d4(x) = Ca.,5(1 - x)a(1 + x)° dx

(11.14)

with a, b > -1. In general, one can show [93] (11.15) p,(1) ". cna+1/2 so if x0 E (-1,1) where jpn(xo)j2 + lpn_1(x0)12 is bounded above and below, one has na+1/2 K,(xo, 1)I 0-1/2 n K.(xo, xo)

-

so if a > 2,

--r oc. Since djt(x) is small for x near 1, one can (and, as

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

THEOREM 11.4 (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 (11.5) and so (11.4) holds uniformly on e. THEOREM 11.5 (Zhang [108], Breuer-Last-Simon [14]). Let e be a periodic finite gap set and let p lie in the Nevai class for e. Then (11.5) and so (11.4) holds uniformly on e. THEOREM 11.6 (Breuer-Last-Simon [141). Let e be a general finite gap set and

let p lie in the Nevai class for e. Then (11.5) and so (11.4) holds uniformly on compact subsets of

eint.

REMARKS. 1. Nevai [67] proved (10.4)/(10.5) for the classical Nevai class for every energy in e but only uniformly on compacts of e"'t Uniformity on all of e using a beautiful lemma is from [69]. 2. Zhang [108] proved Theorem 11.5 for any p whose Jacobi parameters approached a fixed periodic Jacobi matrix. Breuer-Last-Simon [14] noted that without 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 11.7 ([14]). In the next section, we will discuss regular measures. They have zero Lyapunov exponent on veB6(µ), so one might expect Nevai's lemma

could hold-and it will in many regular cases. However, [14] prove that if b" - 0 and an is alternately 1 and 1 on successive very long blocks (1 on blocks of size 32

and 1_2,21 1 on blocks of size 2"2), then dp is regular for c(dp) = [-2,2]. But for a.e. \ [-1, 1], (10.4) and (10.3) fail.

xE

0

CONJECTURE 11.8 ([14]). The following is extensively discussed in [14]: For general OPRL of compact support and a.e. x with respect to p, (10.4) and so (10.3) holds.

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 [91, 85] for the theory in the context of orthogonal polynomials.

DEFINITION. Let p be a measure with compact support and let e = oee8(/i). We say µ is regular for a if and only if

lim (al ... a,,)11" = C(e)

n-cc

(12.1)

the capacity of e.

For e = [-1, 1], C(c) = 2 and the class of regular measures was singled out initially by Erdos-Turan [32] and extensively studied by Ullman [103]. The general theory was developed by Stahl-Totik [91].

Recall that any set of positive capacity has an equilibrium measure, p, and Green's function, Get defined by requiring Ge is harmonic on tC \ e, Ge (z) = log IzI + 0(1) near infinity, and for quasi-every x E C,

lim G. (zn) = 0

Zn- x

(12.2)

B. SIMON

316

(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; THEOREM 12.1. Let e C R be compact and regular for the Dirichlet problem. Let p be a measure regular for c. Then for any e, there is b > 0 and CE so that sup

Ipn(z,di.)I dpe

(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 7+1 K,, (x, y; du) and its diagonal. Therefore, it is interesting that. one can say something about n+l+ Kn(x, x; dµ) dµ(x) without pointwise asymptotics. Notice that

dpn(x) ° n + 1 K. (X, x; dg) dµ(x)

(13.1)

is a probability measure. Recall the density of zeros, vn, defined after (12.5).

THEOREM 13.1. Let p have compact support. Let v" be the density of zeros and An given by (13.1). Then f o r any f = 0,1, 2, ... ,

f xe dvn+l - f Xt dpn I

0

(13.2)

In particular, dpn(.7) and dv"(j)+1 have the same weak limits for any subsequence

n(j).

THE CHRISTOFFEL DAR13OUX KERNEL

317

PROOF. By Theorem 6.2, the zeros of Pn+1 are eigenvalues of irnMxirn, so

J x" dvn+l

n+1 is a basis for ran(irn),

On the other hand, since

(13.3)

n

fXdin n+1 > f xelpj(x)I2dµ(x) j=0 n+1Tr(1r MxTn)

(13.4)

It is easy to see that (7rnMxlrl,)t - 7rnMxe7rn is rank at most t, so

LHS of (13.2) < n+1 IIMxIie goes to 0 as n -+ oo for

fixed.

REMARK. This theorem is due to Simon [88] although the basic fact goes back to Avron-Simon [7].

See Simon [88] for an interesting application to comparison theorems for limits of density of states. We immediately have: COROLLARY 13.2. Suppose that

dj.c = w(x) dx + dµ8

(13.5)

with dps Lebesgue singular, and on some open interval I C e = ve8s(dtt) we have dvn -+ dvv and

dv,,. rI=v,,. (x) dx

(13.6)

and suppose that uniformly on I, lim 1 Kn(x, x) = g(x)

n

(13.7)

and w(x) # 0 on I. Then g(x)

0((

)

xj

PROOF. The theorem implies dv, r I = w(x)g(x). Thus, in the regular case, we expect that "usually" n K., (x, x) --+ wetx)

(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(zp), 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. SIMON

318

DEFINITION. Let dp be a measure on R of the form

dp = w(x) dx + dps

(14.1)

where dp,, is singular with respect to Lebesgue measure. We call x0 a Lebesgue point of p if and only if

n

Xo

2 P. L! n 0+n 2

n, xo +

n

1) --- 0

lu(x) -wo(xo)l dx

(14.2) 0

(14.3)

J."o-1

0

n

It is a fundamental fact of harmonic analysis ([76]) that for any p Lebesguea.e., x0 in ]R is a Lebesgue point for A. Here is the most general version of the MN upper bound: THEOREM 14.1. Let e C K be an arbitrary compact set which is regular for the Dirichlet problem. Let I C e be a closed interval. Let dp be a measure with compact support in K with o-e8S(dp) C e. Then for any Lebesgue point, x in I, lim inf 1 Kn, (x, x) > p` (x) n-oo n w(x)

(14.4)

where dp, t I = pe(x) dx. If w is continuous on I (including at the endpoints as a function in a neighborhood of I) and nonvanishing, then (14.4) holds uniformly on

I. If xn -> x E I and A = supra n1xn - xI < oo and x is a Lebesgue, then (14.4) holds with Kn(x, x) replaced by Kn(xn, xn). If w is continuous and nonvanishing on I, then this extended convergence is uniform in x E I and xn's with A < A0 < oc. REMARKS. 1. If I C e is a nontrivial interval, the measure dpe [ I is purely

absolutely continuous (see, e.g., [85, 89]).

2. For OPUC, this is a result of Mate-Nevai [64]. The translation to 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 Szeg6 condition, but that is only needed for lower bounds on )' (see Section 17). They also don't state the x,, -> x,,. 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 based on Jost-Floquet solutions for periodic problems; see Section 19 (and also [87, 891).

One can combine (14.4) with weak convergence and regularity to get

THEOREM 14.2 (Simon [88]). Let e c R be an arbitrary compact set, regular for the Dirichlet problem. Let dp be a measure with compact support in K with Qess(dp) = e and with dp regular for e. Let I C e be an interval so w(x) > 0 a.e. on I. Then (1)

J

1 Kn(x, x) w(x) - pe (x) dx -+ 0

(14.5)

I

(ii)

1 K. (x, x) dp9 (x) -+ 0

JI n

(14.6)


319

PROOF. By Theorems 12.2 and 13.1, n

K. (x, x) dy -f dpe

(14.7)

Let ill be a limit point of n Kn (x, x) dµ8 and

dv2 = dpe - dill

(14.8)

If f > 0, by Fatou's lemma and (14.4), >-

fPe(x)f(x)dx

(14.9)

that is, dv2 r I > pe(x) dx r I. By (14.8), dv2 r I < pe(x) dx. It follows dill I is 0 and dv2 r I = dPe r I. By compactness, IKn(x, x) dy8 r I -* 0 weakly, implying (14.6). By a simple argument [88], weak convergence of 1-nKn(x, x)w(x) dx -' pe(x) dx and (14.4) imply (14.5).

15. Criteria for A.C. Spectrum Define

lim inf

1 Kn (x, x) < oo } n

so that

][8\N=

lim

n

(15.1)

J

Kn(x,x)

=}

(15.2)

Theorem 14.1 implies

THEOREM 15.1. Let e C R be an arbitrary compact set and dµ = w(x) dx+dji,,

a measure with v(µ) = e. Let E8 = {x I w(x) > 0}. Then N \ E,,, has Lebesgue measure zero.

PROOF. If xo E R \ Ea,, and is a Lebesgue point of µ, then w(xo) = 0 and, by Theorem 14.1, xo E R \ N. Thus,

(R\Ear)\(R\N)=N\Ea., has Lebesgue measure zero. REMARK. This is a direct but not explicit consequence of the Mate-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

I 1 K.(x, x) dµ(x) = 1

(15.3)

f lim inf 1 K. (x, x) dµ(x) < 1

(15.4)

ln implies

n

so

THEOREM 15.2 ([55]). Ea, \ N has Lebesgue measure zero.

320

B. SIMON

Thus, up to sets of measure zero, Eac = 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 = A cos(ran + 6), 1Al < 2, A # 0, a irrational). This operator has been heavily studied; see Last [54].

16. Variational Principle: Nevai Trial Polynomial A basic idea is that if dµ1 and dµ2 look alike near xo, there is a good chance that K.(xo, xo; dµ1) and K.(xo, x0; dµ2) are similar for n large. The expectation (13.8) says 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 An(x0i dpi) in the Christoffel variational principle for An(x0i dµ2), but Example 11.3 shows this will not work in general. If dµ1 has a strong zero near some other xl, the trial polynomial for dµ1 may be large near xl and be problematical for dµ2 if it does not have a zero there. Nevai [67] had the idea of using a localizing factor to overcome this. Suppose e C R, a compact set which, for now, we suppose contains v(dpl) and a(d)U2). Pick A = diam(e) and consider (with [ - ] = integral part)

1-

(x - x0)21 [en] A2

C

J1

= N2[En](x)

(16.1)

Then for any 5, sup

N2[En] (x) < -c(J,e)n

(16.2)

I x-xu l >h xEe

so if Q,,-2[,,,](X) is the minimizer for pi and e is regular for the Dirichlet problem and µl is regular for e, then the Nevai trial function N2[en] (x)Qn-2[en] (x)

will be exponentially small away from x0. For this to work to compare An(x0, dµ1) and ) (xo, dµ2), we need two additional properties of )n(xo, dpi):

(a) an(xo, dµ1) > Cee-en for each E < 0. This is needed for the exponential contributions away from x0 not to matter. (b)

lim lim sup e10

A.(-To, dµ1)

=1

n-.oo An-2[en] lx+ dill) -

so that the change from Qn to Qn_2[en] does not matter. Notice that both (a) and (b) hold if

lim n)n(xo, dµ) = c > 0

n-.oo

(16.3)

If one only has e = a ..(dµ2), 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.

THE CHRISTOFFEL-DARIOUX KERNEL

321

17. Variational Principle: Mate-Nevai-Ibtik Lower Bound In [66], Mate-Nevai-Totik proved: THEOREM 17.1. Let dp be a measure on 8IID

dp = 2 e) d9 + dµ8

(17.1)

which obeys the Szeg6 conditions

Then for a.e. O,

J

E 0,

log(w(9))

21r

> -oo

liminf n)n(O ) > w(9

(17.2)

(17.3) This remains true zf an(9 ) is replaced by An(On) with 0, -+ 6,,, obeying sup nl Bn 8". 1 < 0C. )

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 On = O. The general On result is due to Findley [33].

3. The B,,. for which this is proven have to be Lebesgue points for dp 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 u8(I) 0, (17.3) holds uniformly if 9. E I. By combining this lower bound with the Mate-Nevai upper bound, we get the result of Mite-Nevai-Totik [66]: THEOREM 17.2. Under the hypothesis of Theorem 17.1, for a.e. B0" COlD, lim nan(B,,c,) = w(9

n--.oo

(17.4)

)

This remains true if an(9 ) is replaced by an(9n) with On -> B"', obeying supnJBn-

6.. ;r < oo. If I is an interval with w continuous on I and µ8(I) = 0, then these results hold uniformly in I. REMARK. It is possible (see remarks in Section 4.6 of [68]) that (17.4) holds if a Szegd condition is replaced by w(O) > 0 for a.e. 9. Indeed, under that hypothesis, Simon [88] proved that 27r

J

Iw(9)(nan(6))-1-11

dO

--* 0

There have been significant extensions of Theorem 17.2 to OPRL on fairly general sets:

1. [66] used the idea of Nevai trial functions (Section 16) to prove the Szegd condition could be replaced by regularity plus a local Szegd condition. 2. [66] used the Szeg6 mapping to get a result for [-1, 1]. 3. Using polynomial mappings (see Section 18) 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 u(dp) C e to ae8s(dp) C e):

B. SIMON

322

THEOREM 17.3 (Totik 196, 99]). Let e be a compact subset of R. Let I C e be an interval. Let du have a,.8(du) = e be regular for e with

I, log(w) dx > -oo

(17.5)

Then for a. e. X,,, E I, lim 1 n-oo n

K n(xoo,xoo ) = Pe(xoo) w(xo,,)

(17.6)

The same limit holds for nKn(xn,xn) if supnnlxn 00. If U.(1) = 0 and w is continuous and nonvanishing on I, then those limits are uniform on x., E I and on all xn's with sup, nIxn - x,,,, 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 e without an assumption of regularity for the Dirichlet problem is in [99].

18. Variational Principle: Polynomial Maps In passing from [-1, 1] to fairly general sets, one uses a three-step process. A finite gap set is an e of the form e = [al, 011 U [a2, 02] U ... U [Cye+1, /3e+1]

(18.1)

a1 < 01 < a2 < N2 < ... < at+1 < 0e+1

(18.2)

where

Ef will denote the family of finite gap sets. We write e = e1 U . . . U e1+1 in this case with the ej closed disjoint intervals. Ep will denote the set of what we called periodic finite gap sets in Section 10-ones where each ej has rational harmonic measure. Here are the three steps: (1) Extend to e E E. using the methods discussed briefly below. (2) Prove that given any e E Ef, there is e(n) E Ep, each with the same number

of bands so ej C

C eon-'1 and rlne(n) = e... 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 e(') = {x I dist(x, e) m }, then e(m) is a finite gap set and e = Ante(1). Step (1) is the subtle step in extending theorems: Given the BogatyrevPeherstorfer-Totik theorem, the extensions are simple approximation. The key to e E Ep is that there is a polynomial A: C --> C, so,&-1([-1, 1]) = e and so that e3 is a finite union of intervals ik with disjoint interiors so that A is a bijection from each ek 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 Ep from the results for [-1, 1] using this polynomial mapping.

For spectral theorists, the polynomial 0 = !A where A is the discriminant for the associated periodic problem (see [43, 53, 104, 95, 89]). There is a direct construction of ,& by Aptekarev [4] and Peherstorfer [70, 71, 72].

THE CHR.ISTOFFEL-DARBOUX KERNEL

323

19. Floquet-Jost Solutions for Periodic Jacobi Matrices As we saw in Section 16, models with appropriate behavior are useful input for comparison theorems. Periodic Jacobi matrices have OPs for which one can study the CD kernel and its asymptotics. The two main results concern diagonal and just off-diagonal behavior:

THEOREM 19.1. Let p be the spectral measure associated to a periodic Jacobi

matrix with essential spectrum, e, a finite gap set. Let du = w(x) dx on e (there can also be up to one eigenvalue in each gap). Then uniformly for x in compact subsets of e'nt,

Kn ( x,x ) ---,

w` (x ) and uniformly for such x and a, b in R with (al < A, Jbi < B,

(19 . 1)

Kn.(x + n, x + sin(irpe(x)(b - a)) (19.2) 7rp,(x)(b - a) Kn(x, x) REMARKS. 1. (19.2) is often called bulk universality. On bounded intervals, it goes back to random matrix theory. The best results using Riemann-Hilbert methods 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]. 2. For [-1,1], Lubinsky [601 used Legendre polynomials as his model. The references for the proofs here are Simon [87, 89].

The key to the proof of Theorem 19.1 is to use Floquet-Jost solutions, that is, solutions of (19.3) anon+I + bnun + an-lUn-1 = xun for n E Z where {an, bn } are extended periodically to all of Z. These solutions obey Un+p = eio(x)Un

(19.4)

For x E elnt, un and un are linearly independent, and so one can write p.-1 in terms of u. and U.. Using d0

1

Pe (x) _

I

dx l

(19.5)

one can prove (19.1) and (19.2). The details are in [87, 89].

20. Lubinsky's Inequality and Bulk Universality Lubinsky [60] found a powerful tool for going from diagonal control of the CD kernel to slightly off-diagonal control-a simple inequality.

THEOREM 20.1. Let p < it* and let Kn, Kn be their CD kernels. Then for any x, (,

I KK(z, () - K; (x, ()I2 < K. (Z' z)[K.((, ) - K, ((, ()]

(20.1)

REMARK. Recall (Theorem 9.3) that Kn(C,() > Kn{(, (). PROOF. Since Kn - Kn is a polynomial 2 of degree n:

K. (z, () - KK(z, 0 = f K,(z, w)[Kn(w, () - K.* (w, ()] dµ(w)

(20.2)

B. SIMON

324

By the reproducing kernel formula (1.19), we, get (20.1) from the Schwarz inequality if we show

f

I Kn(w, C) - KK(w, ()12 dp(w) < ,,,(C, C) -

(20.3)

Expanding the square, the Kn term is Kn(C, () by (1.19) and the K,, K,*,, cross term is -2Kn(C, () by the reproducing property of K,, for du integrals. Thus, (20.3) is equivalent to f jKn(w, c) I2 dp(w) < K;ti(C, ()

(20.4)

This in turn follows from I t< p* and (1.19) for p*! This result lets one go from diagonal control on measures to off-diagonal. Given any pair of measures, It and v, there is a unique measure p. V v which is their least upper bound (see, e.g., Doob [30]). It is known (see [85]) that if It, v are regular for the same set, so is p V v. (20.1) immediately implies that (go from µ to u* and

then j* to v): COROLLARY 20.2. Let it, v be two measures and p.* = It V v. Suppose for some zn --> zoo, w,+ - zoo, we have for 17 = it, v, p* that lim

Kn (zn, zn;17)

n-+oo Kn(zoo, zoo; 71)

= lim kn(wn, wn; r)) = 1 n-'oo K,, (zoo, z. -M)

and that lim

Kn (zoo) zoo; p')

n ,oo Kn(zoc, zoo; p*)

Kn (zoo, zoo; y) = 1 n-,oo Kn zoo, Z.; A

= lim

Then (20.5) lim Kn(zn, wn;1i) = 1 n-+oo Kn(zn, wn; v) REMARK. It is for use with xn = xoo + °-n or x,,, + pan that we added xn --+ x" ,

to the various diagonal kernel results. This "wiggle" in x,,. was introduced by Lubinsky [60], so we dub it the "Lubinsky wiggle." Given Totik's theorem (Theorem 17.3) and bulk universality for suitable models, one thus gets: THEOREM 20.3. Under the hypotheses of Theorem 17.3, for a.e. x,,,, in I, we have uniformly for lal, IbI 0,

where 8 is the unit normal to 80. We take (1.2) f E Ho(0), g r= L2(0) This Dirichlet problem for A2 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 Mathematics Subject Classification. 35J40, 35B40. This work was supported by NSF grant DMS-0456861. ©2008 American Mathematical Society 337

338

MICHAEL E. TAYLOR

for specific 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 analysis and semigroups. The plan of the rest of this paper is as follows. In §2 we replace SI by OR = [0, R] x 0, and study the solution to

02u = 0 on SIR, (13)

u(0, x) = f (x), c0yu(0, x) = 9(x), u(y, x) = 0, 8u(y, x) = 0, x E 80, y E [0, R], u(R, x) = 0, 8yu(R, x) = 0, x E 0,

which is a Dirichlet problem for A2u = 0 on the Lipschitz domain SIR. Results of [2] yield a unique solution to (1.3), with non-tangential maximal function bounds on u and Vu in £2(Of R). We 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

"of 0

u(y,x)I2dxdy oo. Following the Lax program, we study a semigroup of operators. In this case, we study the semigroup Sy on Ho (0) ® L'(0), defined by (1.5)

/ fl =

Sy I

( u(y)

9 8yu(y) where u is the solution to (1.1) constructed in §3. In §5 we look at another semigroup, V 8 : Y -+ Y, where

Y={uEH2(1?):A2u=0, uand 8.vu=0on l[8+x80}, (1.6)

78u(y, x) = u(y -I- s, x).

This set-up is more closely parallel to that of [3] than the consideration of Sy. We note parallel results for 78, also making use of the results of §§2-3.

Remark. It is interesting to compare (1.1) with the Dirichlet problem for A: Au = 0 on S2, u(0, x) = f (x), x E 0, (1.7)

u(y,x)=0, xEa0, y>0. The method of separation of variables represents the non-exploding solution to (1.7) as

(1.8)

u(y, x) =

>f k=1

(k)e-akycok(x),

A SAINT-VENANT PRINCIPLE FOR LIPSCHITZ CYLINDERS

339

where {cOk : k > 1} is an orthonormal basis of L2(0) consisting of Dirichlet eigen-

functions of Ox, cpk E NIP), Oxck = -Ak k, 0 < Al < A2 < A3 / oo, and J(k) _ (f, cpk)L2(o). In such a case, .1k - Ck1/('i-1), k AkI f (k)I2 < oo, and (1.8) is both asymptotic and convergent, in H01(0). Such a separation of variables approach would work for solutions to &2u = 0 if 0 were replaced by a compact manifold without boundary, or if the lateral boundary conditions in (1.1) were replaced by (1.9)

u(y, x) = 0,

x E 8C, y > 0,

Oxu(y, x) = 0,

but for (1.1) this method fails.

2. The Dirichlet problem on fiR Here we fix R E (0, oc) and discuss solutions to (1.3), which can be rephrased as (2.1)

A2U = 0 On S2R,

2GIOQR = .fR,

0NUIan, = 9R, where 8N is the unit normal to 8S2R. Here fR = f on {0} x 0, fR = 0 on the rest of BSZR, while gR = g on {0} x 0, gR = 0 on the rest of 8S2R. The hypothesis (1.2) yields (2.2)

fR E H'(O R),

9R E L2(8IR) Work of [2] yields a unique solution to (2.1), smooth in the interior of f2R, and satisfying (2.3)

IIu*IIL2(eoR) + II(Vu)*IILZ(8QR) 0 is the smallest eigenvalue of -A,; on L2 (0), with the Dirichlet boundary condition. Since (-Aw, w) < IIowIIL2IIwIIL2, we deduce that (3.7)

IIAWII22(s2) > A'IIWIIL2(O),

dw E HH(fl).

MICHAEL E. TAYLOR

342

Now the F iedrichs method produces a positive, self-adjoint operator G, with D(G1/2) = H0 2(n),

(3.8)

and £ is the self-adjoint extension of A2 defined by the Dirichlet boundary condition. From (3.5) and (3.7) we have

L - A2I > 0.

(3.9)

In particular, G is invertible on L2(1l), so we have (3.2), with

w = G-1F E D(.C) C D(G1/2) = H02(9). Once we have u in (3.3), the local regularity results of Propositions 2.2-2.3 readily extend. We get (3.10)

cp8y u E

(3.11)

H512-E(S2),

`de > 0, m E Z+,

whenever V = cp(y) E Co' ((0, oo)).

4. The semigroup Sy and asymptotic behavior We defuse the semigroup

Sy : X - X, X = Ho (O) ® L2 (O),

(4.1)

by Sy

(4.2)

(I)

9u(y))),

ayu(y where u = u(y, x) is the solution to (1.1) constructed in §3. Given that i9, 'u E H512-E([a,b]

x 0),

de > 0, m E Z+, whenever 0 < a < b < co, we see that Sy is strongly continuous in y E (0, oo). Whether Sy is strongly continuous at y = 0 requires further investigation, but (4.3)

since our focus is on the behavior as y -; oo, we leave this point. From (4.3) we obtain

S1 : X ---> H5/2-e(O) e H5/2-E(0), for all e > 0, whenever y > 0. In particular, S' is compact on X. (4.5) y>0 In particular, the spectrum of 91 has only 0 as an accumulation point. (4.4)

PROPOSITION 4.1. If ( E SpecS', then ICI < I. PROOF. To begin, since S1 is compact, 0 # C E spec S' => 3 nonzero (9f) E X such that Sl (9f) (4.6)

Sk (9) =

(k (f

u(y + k, x) = Sku(y, x), where u is the solution to (1.1) constructed in §3. In such a case, (4.7)

J m J l u(y, x) I2 dx dy = Ic12k ZOO f l u(y, x)I2 dx dy.

0

0

=

((9f

A SAINT-VENANT PRINCIPLE FOR LIPSCHITZ CYLINDERS

343

But by (3.2)-(3.3) we have u E L2(R+ x 0), so the left side of (4.7) tends to 0 as k oo. This forces I(I < 1. 0

To proceed, fix e = e-K E (0, 1), and let Pe be the projection of X onto the part of Spec S1 lying inside De _ {( E C : I(I < c}, i.e., (4.8)

Pe=

27ri

f((1-S)-'d(, 7E

where /E = ODE or, if a point in Spec S' lies on ODE, take ye = ODe_v for appropriate 77 < < E. Then the range of Q = I - PE is a finite dimensional subspace

X. of X, on which Sy acts as a semigroup. One has (4.9)

SyQE = e-yAE

where AE is a linear operator on the finite dimensional space XE, with (4.10)

Spec A. _ {A1, ... , AN(E) },

Re ai > B > 0,

where B > 0 is taken so that SpecS' C {( E C : I(I < e-B}.

(4.11)

One can decrease a and increase the list {A , ... , AN(E) }; note that

as j - oc.

Re A3 -> +oo,

(4.12)

Now for each Aj there are associated eigenvectors of Ae i and also possibly generalized eigenvectors, so we have (possibly with repetitions of some of the eigenvalues A3)

(4.13)

SyQ

f)

N(e) M(9)

= F

aie

yee a:y

Pae(x)

for certain functions Wje(x) and' he(x). It remains to analyze, as y oo, ZE = SyPE = PESyP£7

(4.14)

a semigroup on the range of Pe. Note that Spec ZE C {( E C : I(I < e}.

(4.15)

The spectral radius formula implies (4.16)

limsup II ZS-1 I,C n-,00 X)

< e,

hence

(4.17)

IIZE IIc(x)
v(e), sufficiently large.

Now if n+1 < y < n+2, using the fact that {Sy : y c [1, 2]} has uniformly bounded operator norm on X, (4.18)

IIZE Ile(x) 0, then Bpq(Rn) are the classical Besov spaces. Otherwise we refer to 129, 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. We 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 1.71. We give a brief description of some basic notation. As usual C"(R) with u E N collects all (complex-valued) continuous functions on IR having continuous bounded derivatives up to order u (inclusively). Let 'OF E C"(R),

(2.12)

u E N,

VGM E C"(R),

be real compactly supported Daubechies wavelets with W M (x) x'dx = 0

(2.13)

for all v E No with v < u.

JR

Recall that 1GF is called the scaling function (father wavelet) and 0M the associated

wavelet (mother wavelet). We extend these wavelets from IR to R" by the usual tensor procedure. Let u E N and

G = (G1,...,Gn) E G° _ {F,M}n which means that Gr is either F or M. Let j E N,

G = (G1 i ... , Gn) E G3 _ {F, M}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 G° has 21 elements, whereas Gj with j E N has 2n - I elements. Let L E No and n

(2.14)

YG,,m(x) = 2U+L)n/211 OG,. (21+Lxr - mr)

,

G E Gi, M E Zn,

r=1

where j E No. We always assume thatF and "il'M in (2.12) have L2-norm 1. PROPOSITION 2.3. Let L E No and is. E N. Then

jENo, GEG3, rnEDn}

(2.15)

is an orthonormal basis in L2(1R' ).

REMARK 2.4. This is a cornerstone of (inhomogeneous) wavelet theory. We refer to the above-mentioned books. In IRn one may choose L = 0. But later on when it comes to domains then it will be important that one can choose L E No large. The L2-normalisation of VG is natural. But for our later purposes it is convenient to switch to an Lam-normalisation. To prepare what follows we remark that any f E L2 (IR") can be represented as ao

(2.16)

f=

E

j=0 GEGj TnEZ

am 2-jn/2 qG m

WAVELETS IN FUNCTION SPACES

351

with (2.17)

Aj, = Ain (f) = 2jn/2 f f (x) `7E ,m(x) dx = 25n/2 (f,'f`G,m) n

and (2.18)

2-jet. Iim,Gl2

IIf IL2(Rn)ii = j,G,m

(Recall that the *4 m's are real).

2.3. Wavelet bases in Apq(Rn). We extend the wavelet representation according to Proposition 2.3 and (2.16)-(2.18) from L2(Rn) to Aa,q(Rn) where A = B

or A = F and s E ]R, 0 < p, q < oo (with p < co for the F-spaces). First we need a substitute of the sequence spaces f2 in (2.18). Let Xjm be the characteristic function of a cube Qjm in Rn with sides parallel to the axes of coordinates, centred at 2-gym and with side-length 2-j+1 where m E Zn and j E No. DEFINITION 2.5. Let s E R, 0 < p < oe, 0 < q < oo. Then bpq is the collection of all sequences (2.19)

A={A,n EO: jENo, GEG', mEV}

such that 00

(2j(s_)q E

Ira IbP4ll =

j=o

q/ p\ 1/q

E

GEG!

IP

< 00

mEZ°

and fpq is the collection of all sequences (2.19) such that 1/q

Ila lfPgll =

2J3Q IAI. xjm(')h

ILP(Rn)

< 00

(j'G'M

with the usual modifications if p = oo and/or q = oo. REMARK 2.6. One has bbp = fPP. With s = 0, p = q = 2, one gets the right-hand side of (2.18).

We use standard notation naturally extended from Banach spaces to quasiBanach spaces. In particular {bj} 1 C B in a complex quasi-Banach space is called a basis if any b E B can be uniquely represented as 00

(2.20)

b = E Aj bj,

Aj E C

(convergence in B).

j=1

A basis {bj}j' I is called an unconditional basis if for any rearrangement a of N (one-to-one map of N onto itself)

is again a basis and

00

(2.21)

b = E Aa(j) bo(j)

(convergence in B)

j=1

for any b E B with (2.20). Standard bases of separable sequence spaces as considered in this paper are always unconditional. We refer to [1] for details about bases in Banach (sequence) spaces. Similarly as in (2.20), (2.21) we speak about

352

HANS TRIEBEL

unconditional convergence in S'(Rn) and in Apl,q(R' ). Local convergence in Apq(]li;n) means convergence in A?,q(K) for any ball K in l[8n. Let apq be either bq or fq. In any case u E N in (2.12) and hence in (2.14) will be chosen in such a way that (independently of L) 00

(2.22)

>2 >2 >2 )m 2-9n/2G m

E apq,

j=O GEGj mEZ^

converges unconditionally in S'(lRn) and locally in any A,;(R') with o < s. This justifies to abbreviate (2.22) by (2.23)

AG 2-i"/2 g'G m,

A E apq,

j,G,m

We use the nowadays standard abbreviations (2.24)

Qp=nl 1--1 J+

and

\

1

vnq=n(min(p,q) -11

)+

where 0 < p, q < oo and b+ = inax(b, 0) if b E R. THEOREM 2.7.

(i) Let 0 max(s, o, - s). Let f E S'(Rn). Then f E BPq(lR") if, and only if, it can be represented as (2.25)

AJ,G 2-jn/2 qi

f=

m

G,m,

A E bsPq,

j,G,m

unconditional convergence being in S'(Rn) and locally in any space BPq(l8n) with a < s. The representation (2.25) is unique, Aj,G = 2jn/2 (f,

(2.26)

'PG,m)

and

I: f F a {23n/2 (f, G,m)l 1

(2.27)

J

is an isomorphic map of BPq(Rn) onto bpq. If, in addition, p < oo, q < oc, then { 4' ,m } is an unconditional basis in Bpq (R L) .

(ii) Let 0 max(s, o ,q - s).

Let f E S'(]l8n). Then f E Fb(I8n) if, and only if, it can be represented as (2.28)

f=

AriG 2-jnl2 qC j+G'M

A

Ef(

,

unconditional convergence being in S'(Rn) and locally in any space Fpq(R') with v < s. The representation (2.28) is unique with (2.26). Furthermore, I in (2.27) is an isomorphic map of FFq(Rn) onto fpq. If, in addition, q < oo, then ITi'm} is an unconditional basis in Frq(Rn).


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 V m may serve simultaneously as atoms and as kernels of local means. Atomic representations in function spaces

as used nowadays go essentially back to [14, 151. 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

a=

(2.29)

2-2(s-P) 2-jn/2

W:; m,

G E G'.

As far as the required cancellations for the atoms are concerned we remind of (2.30)

J

+If0 m(x) dx = 0

if j E N and 1,31 < u,

nen

as a consequence of (2.13) and (2.14). Then it follows from the atomic representation theorem that f E S'(1Rn), given by (2.25), belongs to Bnq(lR") and (2.31)

If IBpq(Rn)II 0, a>0,andallmEZ".

The theory of periodic distributions and related periodic spaces BPI, (TI) and Fpq(T") has some history which is not the subject of this survey. We rely on 124, Chapter 3] and [27, Chapter 91 where one finds also further references. Let be the same dyadic resolution of unity in R' as in (2.2), (2.3) and in Definition 2.1. Let f E D'(T"), given by (2.34), be extended periodically to R. Then f E S'(II8") (using the same letter f) and ((pji)u(x) = E a,,, tpj (21rm) e i21rmx mEZ'

are trigonometrical polynomials. This justifies the following periodic counterpart of Definition 2.1.

be the above resolution of unity in ]R'

DEFINITION 2.10. Let cp =

.

(i) Let 0 < p c3 2-1,

r=1

where r = Ofl stands for the (non-empty) boundary. It is always assumed that the positive numbers c1, r2, c3 are sufficiently small in dependence on Sl such that for any j E No there are points with (3.2)-(3.4). Based on (2.12), (2.13) we extend now (2.14) to n

(3.5)

'G m(x) = 2(j+L)n/2 11 1)G,. (2j+Lxr - mr) ,

m E 7Ln,

r=1

for all j E No, G E IF, 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)

supp TG ,n C B (2-j-Lm, e 2-j) ,

j E No, m E V.

Let F = {F..... F} E {F, M}n. DEFINITION 3.3. Let SI be an arbitrary domain in Rn with f # IRn and let 7L0

be as in (3.2)-(3.4). Let L E N and u E N be as in (2.12), (2.13) and (3.5), (3.6).

Let K E N, D >0 and e4 >0. Then (3.7) (3.8)

j E No; r = 1,...,Nj} where Nj ENT, supp 4 i . C B (x;., r,2 2-'') , j E Nlo, r=1. ... , Nj, {4

:

;

with B (xi, c22-3) as in (3.4), is called a u-wavelet system (with respect to l) if it consists of the following three types of functions. (i) Basic wavelets (3.9)

4i° = JYG,n,

for some G E {F,1't-T}n,

M E Z.n.

(ii) Interior wavelets (3.10)

j E N,

(I)T = `pG m+

dist (xi, r) > c4 2-j,

for some G E {F, M}n*, M E 7L'. (iii) Boundary wavelets (3.11)

d;,,.,m, qIj

j E N,

disc (xi, r) < c4 2-',

lm-m,l apq.

Let Qtr be the (roughly) indicated Whitney cubes in (3.17) and let 0 _< 91, E D(Q) be a related resolution of unity, say, (3.21)

Q1r(x) = 1

supp pir c 2Qir C Cl, lr

if x E Cl.


363

Then the refined localisation space Rp'q,"oc(SZ) consists of all f E L"'(0) such that 1/p

(3.22)

Ilf

IFn4T1oc(n)IIe

apq in (3.14) is a wavelet basis in FP4''1oc(1) with the expansions (3.15), (3.16) now with

A E f q(z) in place of f102(Z0). This theory started in 133] (where we denoted FFs4r1' (12) as Fpq(12)). There we got para-bases. As for final results in terms of the above orthogonal basis with the related sequence spaces ffq(Z0) according to Definition 3.9 we refer to 1351.

4. Spaces on E-thick domains 4.1. Classes of domains. So far we described wavelet bases for function spaces Apq on R" and T. In case of arbitrary domains 1? we have a satisfactory wavelet representation for Lp(12), 1 < p < oo, according to Theorem 3.11 and we indicated in Discussion 3.13 an extension of these assertions to refined localisation spaces FP9rioc(c2) But in general these spaces do not coincide with corresponding spaces F ( 12 ) or F (12) according to Definition 3.1. All spaces Apq(12), Apq(f1) introduced there originate from the related spaces Apq(llt") 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). in R' (and also in Tn) playing a double This is well reflected by the wavelets role as atoms and kernels of local means as outlined in Discussion 2.8. In case

of domains the basic wavelets 1)° in (3.9) and the interior wavelets r in (3.10) fit in this scheme, but not the boundary wavelets in (3.11). To compensate this shortcoming we are going to introduce the natural class of E-thick domains covering as special cases the better known class of Lipschitz domains. Tacitly we always assume that a domain (= open set) n in 1R' is not empty, 1 # 0. Let 1(Q) be the side-length of a (finite) cube Q in >I8" with sides parallel to

the axes of coordinates. Let I be an (arbitrary) index set. Then ai-

b2

for

iEI

(equivalence)

for two sets of positive numbers jai : i E I} and {bi two positive numbers c1 and c2 such that

c1ai < bi 0 such that (4.2)

jh(x') - h(y')j< c ix' - y'I for all i E Rr-1,

y' ER"-1

If h is infinitely differentiable and all derivatives are bounded then h is called a C°° function. The distance dist (I'1, r') between two sets in lR' has the same meaning as in (3.1). DEFINITION 4.1. (i) Let n E N. A domain (= open set) in Rn with Q # I8" and r = on is said to be E-thick (exterior thick) if one finds for any interior cube Qi C U with (4.3) l(Qi) - 2-', dist (Q', r) - 2-j, j > jo E N,

a complementing cube Qe C St° = IIYn\U with

l(Qe) N 2-', dist (Qe, P) N dist (Qi, Qe) ^' 2-j, j>joEN. (ii) Let 2 < n E N. A Lipschitz graph domain (COO graph domain) in 1R' is the collection of all points (x', xn) with x' E ]18n-1 and h(am) < xn < oo, where h(x) is a Lipschitz function according to (4.1), (4.2) (a C°° function). (iii) Let 2 < n E N. A bounded Lipschitz domain (bounded C°° domain) in ]Rn is a bounded domain U in IIt" where the boundary r can be covered by finitely many open balls B j in R" with j = 1, ... , J, centred at P such that (4.4)

B;nU=B,n11, where f2. .

for j=I,...,J,

are rotations of suitable Lipschitz graph domains (C°° graph domains) in

R". REMARK 4.2. The equivalence constants in (4.3), (4.4) are independent of j. In other words, a domain Q is called E-thick if for any choice of positive numbers cl, c2i c3, c4 and jo E N there are positive numbers c5, c6, c7, c8 such that one finds for each interior cube Qi c: 0 with cl 2-3 < l(Q') c2 2-3, c3 2-7 < dist (Q2, r) < c4 2-3,

j > jo, an exterior cube Qe C St` with (4.5) c52 -j < l (Qe) < cs 2-j, c72' < dist (Qe, F) < dist (Qi, Qe) < cs 2-'j,

j > jo. One checks easily that n is E-thick if, and only if, one has (4.5) for some c5, ... , c8 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 R" is E-thick. On the other hand, E-thick domains may be rather irregular. It may happen that iFl > 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 I-thick (interior thick) domains, where, roughly speaking, the above cubes Qi and Qe change their roles. Conditions of similar types (E-thick and I-thick) have been used several times in literature in connection with function spaces and PDE'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 119, pp. 4,8]. In connection with Sobolev


365

and Poincare inequalities in function spaces (preferably Sobolev spaces) conditions of the above type play a role resulting in John domains and plump domains. Details, references, examples and discussions may be found in [12, Section 4.31. In connection with atomic representations of function spaces in rough domains we introduced 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 136, 13] we referred also to other classes of domains in the literature.

4.2. Wavelet bases in Fpq(1). As roughly indicated in Disussion 3.13 for the refined localisation spaces Fpgrl0C(S2) according to (3.20)-(3.22) in arbitrary domains tZ one has a counterpart of Theorem 3.11. If S2 is E-thick then these refined localisation spaces coincide with the spaces Fpq(S2) according to Definition 3.1. We formulate the outcome and discuss afterwards the key ideas. We incorporate

now F..(S2) = B1.(1). Let fpq(Z) be the sequence spaces as introduced in Definition 3.9 and let opq be as in (2.24).

PROPOSITION 4.3. Let St be an E-thick domain in R' according to Definition 4.1(i) and let Fpq (1) with 0 < P< oo,

(4.6)

s > apq,

0 < q < oo,

(q = oo if p = oc) be the spaces as introduced in Definition 3.1. Let for u E N with

u>s,

with Nj E { (D!.: j E No; r = 1, ... , Nj } be an orthonormal u-wavelet basis in L2(ft) according to Proposition 3.7 and Definition 3.5. Then n n (4.7)

max(1, p) < v < oo,

Fpq(S2)

s- >-

,

(what means v = oo if p = oo). Furthermore, f E L (1) is an element of Fpq (1) if, and only if, it can be represented as roo

(4.8)

f=L

L j=0 r=1 NN

2-jn,2

ti

A E f;q(Zu),

absolute (and hence unconditional) convergence being in the representation (4.8) is unique with A = A(f ), (4.9)

M ,(f) = 2jn12 (.f, c) =

2j"/2

in

and

I: f'-' A(f) =

If f E Fpq(0) then

{2,"/2

(f>4'r)}

is an isomorphic map of FPq(S2) onto fpq(Z0). If p < oo, q < oo then {4P;.} is an unconditional basis in Fpq(1). Discussion 4.4. First we remark that (4.7) is a continuous Sobolev embedding. It ensures that f E Fpq (0) admits an expansion of type (3.15) with f,2 (Zn) in place of fp°,2(Z0) (locally if p = oo). By Definition 3.1 one gets f E Fpq(S2) from (4.10)

f = gjS2

with 9 E FPq(S2) C Fpq(lR").

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366

This reduces (4.8) to corresponding expansions in R" as considered in Theorem 2.7 and Discussion 2.8. One does not need moment conditions for the atoms in (2.29) with 4, in place of %PG,.,.i. For the counterpart of the local means in (2.32), now given by (4.9), (4.11)

(f) =

jk(v)f(Y)dY,

kr = 2'"/2 T

in place of G ,n and u > 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 V, as in (3.11). This is the point where the assumption that S2 is E-thick comes in naturally. Then supp ki. C Q' for some interior cube Qi with (4.3). Let Qe be a related complementing cube with (4.4). Then there is a function kr E C '(R') with supp k;. C Qe such that k, (x) = k1, (x) + k1, (x),

(4.12)

x E R",

is an admitted R'-kernel having in particular the required moment conditions. Furthermore with g as in (4.10) (in particular supp g c Il) one gets (4.13)

n(

jP, (x) g(x) dx

=

f

k (J) f (y) dy = X,' (f }.

low 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 Proposition 4.3 and of Discussion

4.4 for the spaces 13pq (St) with 0 < p, q < oo and s > up. But all this will be covered by Theorem 4.8 below. The preference given to Fpq(1l) comes from the little history of the so-called refined localisation property which can be written as FP4r1oc(ci) =

for E-thick domains SZ,

with s, p, q as in (4.6) where we outlined in Discussion 3.13 what is meant by Fpgrloc(D) We proved this property first for bounded C°° domains in 130, Theorem 5.14, pp. 60/611 and then for bounded Lipschitz domains in [32, Proposition 4.20, pp. 208/2091.

4.3. Wavelet bases in Apq(cl). We are now ready for the main result of Section 4. Let Apq(c) and Apq(11) be the spaces introduced in Definition 3.1 for arbitrary domains S2 in 1R'y with fl # R", considered as subspaces of D'(S2). Let as before

ar=n G -11 +

and

upq=n(

1

min(p, q)

-1 )

// +

where b+ = max(b, 0) if b E R.

DEFINITION 4.6. Let 0 be an E-thick domain in Rn according to Definition 4.1(i). Then (4.14)

Fpq(SE) =

Fpq(Q) Fpq(1l)

if 0max(s) ap-s),

s#0.

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368

Then f E D'(S2) is an element of BP-'q(D) if, and only if it can be represented as oa

(4.19)

f=

Nj

aT 2-jn/2 $r,

A E bpq(ZQ),

j-O r=1

unconditional convergence being in D'(S2) and locally in any space BPq(f) with then the representation (4.19) is unique with a < s. Fhrtherrnore, if f E

A = A(f) as in (4.17), and I in (4.18) is an isomorphic map of Ba,q (S2) onto b;q(Zo). If, in addition, p < oo, q < 00, then {fir} is an unconditional basis in Rpq(S2).

Discussion 4.9. As indicated above, (4.19) with s > aa, and (4.16) with .s > Upq are atomic decompositions, no moment conditions are needed. But for the coefficients a;., interpreted as local means one needs moment conditions for the kernels k,j, in (4.11). Since n is E-thick one can circumvent this difficulty by (4.11)(4.13). If s < 0 then one does not need moment conditions for the kernels k,J., but for the atoms. However the necessary repair for the boundary wavelets -1>j in (3.11) can be done in the same way as in (4.12). Otherwise the above theorem 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 S2 must he 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.

COROLLARY 4.10. Let S2 be a Lipschitz domain in R71 with n > 2 according

to Definition 4.1 or an interval if n = 1. Then Theorem 4.8 remains valid for all spaces Fpq(Sl) in (4.14) and all spaces I3tq(Sl) in (4.15) (including s = 0). REMARK 4.11. As mentioned in Remark 4.2 Lipschitz domains are E-thick.

5. Spaces on C' domains 5.1. Preliminaries, spaces on manifolds. All constructions of wavelet bases so far in lR", T", arbitrary domains and F,-thick domains rely on dual pairings, say, (D(Sl), D'(S2)), 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 4ir in Definition 3.3, having compact supports in the underlying domain Q. This type of arguments cannot be extended to spaces Apq(S2) having boundary values

at I = aOS2. We discussed this point briefly in Remark 4.7. In particular, there is no duality for these spaces within (D(S2), D'(S2)). One can try to circumvent these difficulties by decomposing Apq(S2) into an interior part, say, A,,q(S2), to which the

above considerations can be applied, and trace spaces on F. In rough domains or for spaces Apq(0) with p < 1 such an approach does not work. For this reason we restrict ourselves to bounded C' domains S2 according to Definition 4.1 and to spaces Apq(52) 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 C°° manifolds r (without boundary), where d E N. Let {Vj, j } 1 with J E N be an atlas of r


369

consisting of a covering r = V 1 Vj and homeomorphic maps (5.1)

Uj =')j(V3) C Rd, onto open connected bounded sets in Rd, say in the unit ball U in Rd, with the usual compatibility conditions converting r into a compact C°° manifold. Then 7Vj:V-4-

D(r) = CO°(P) and D'(P) have the usual meaning. Let {Xj} i C D(F) be a resolution of unity with supp X j C V.7

i EXj(y)=1

and

if

yEl,.

j=1

If f E D'(r) then

(Xjf)oV)jlES'(Rd),

j=1,...,J.

DEFINITION 5.1. Let d E N and let r be a compact d-dimensional C°° manifold.

Let A E {B, F} and s E R, 0 < p, q < oo (p < oo for the F-spaces). Then

Apq(F)={f ED'(I'): (Xjf)-031EApq(Rd), j=1,...,.7} and

II(Xjf) o i' 1 IA-'q(Rd)II

1If IApq(I')EI _ j=1

REMARK 5.2. It follows by standard arguments (diffeomorphic maps, pointwise

multipliers) that these spaces are independent of compatible atlasses and related resolutions of unity. Basically one can transfer the wavelet expansions for the spaces Apq(Rd) according to Theorem 2.7 via (5.1) to Apq(T) using that Qj, 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.

REMARK 5.3. (Wavelet bases on curves). The boundary r of a bounded C°° domain Sl in the plane 1R2 (planar domain) consists of finitely many closed connected

C°° curves, which are diffeomorphic to the 1-torus 7 = T' 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 C°° distortion factors originating from the diffeomorphic maps can be incorporated in the wavelets. Then one gets orthonormal wavelet bases { 1pT.r } on F 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 Apq(F) of the same type as in Theorem 2.15 with n = 1. 5.2. Decompositions. Let fl be a bounded C°° domain in 11 and let r = 49Q be its boundary. Let Apq(il) and Apq(fl) be as in Definition 3.1 and let Apq(F) 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 Apq(1l)

with 1 l/p.

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370

has the usual meaning. Let K = [s - P ] - be the largest integer K E K < s - 1. Let v be the (outer) normal on r. Then (5.2)

trr' : 9 -' {trr

k

vk :

NO

with

K=is -I]-,

0 < k < K}

maps

B;q(fl)

fl BPq P -k(r)

onto

k=0 and

11

onto

FPq(ft)

B

-k(r).

k=0

Futhermore there are linear and bounded extension operators (5.3)

extr' :

{go,.. -,9K} - 9

mapping (5.4)

rlK B;q P-k(r)

into B q(1 )

k=0 and

K rIB;'-'(r)

(5.5)

into FPq(fl)

k=0 such that

K

trip o extrp = id,

identity in

_k (r). p_

JJ B; k=0

We refer to [27, p. 2001 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 g on r with

supports in an e-neighbourhood in r of some point ry E r into functions with supports in a corresponding e-neighbourhood of ry in Ti. This can be done but will be shifted to [35]. Its paves the way to clip together wavelet bases of the spaces APq(SZ) according to Theorem 4.8 with wavelet bases for BP,(r) (if exist). This procedure relies on the following assertion. Recall that Ap'Q(SZ) is the completion of D(fl) in Apq(ft), whereas APq(ft) has the same meaning as in Definition 3.1. PROPOSITION 5.4. Let f2 be a bounded C°" domain in ]fin according to Defini-

tion 4.1(iii) and let r = 8f2. Let

1 3 is much more complicated than in case of bounded C°° domains in the plane R2. If one has wavelet bases in suitable spaces B;q(r) with r = Oil then one can argue as indicated in Discussion 5.11. If IF (or one of its components) is diffeomorphic to an d-torus (d = n - 1), then one can apply Theorem 2.15. If

r=Sd={xERd+i: IxI=1}, 2

- 1,

01- ! ¢ NO,

for l = 1, ... , d-1. This can be done by induction with respect to dimensions starting from Theorem 5.10(ii) which causes the curious (and presumably unnatural) restrictions for a. COMMENT 6.4. By Definition 3.1 all spaces APOq(0) on arbitrary domains U are

defined by restriction of Apq(R") to U. As mentioned in Remark 3.2 one recovers in case of bounded Lipschitz domains ft the classical Sobolev spaces W. (0) =If E Lp(ft) : Da f E Lp(U), Jal < k} , with 1/p

Ilf IWp (I) II =

II Daf ILp(n)Ilp 1a1 0. For "indefinite weights" w, the following quadratic form inequality is equivalent to the fundamental notion of the relative form boundedness of the potential energy operator w with respect to the kinetic energy operator

Hp = -A: I(wu, 01= f Iu12wI 0. Here w is a locally integrable function which may change sign, or more generally, a real- or complex-valued distribution. This form boundedness problem, which is important to mathematical quantum mechanics, was recently solved by Maz'ya and Verhitsky [MV2]. It will be discussed below for general second order differential operators with distributional coefficients (not necessarily elliptic) in place of the Laplacian [MV6].

CONTENTS 1. 2. 3.

Introduction Basic trace inequalities Dyadic and radial trace inequalities and nonlinear potentials

378

382 387

1991 Mathematics Subject Classification. 31B15, 35J10, 35J60, 42B25. Key words and phrases. Weighted norm inequalities, indefinite weights, nonlinear potentials, form boundedness, Schrodinger operators, general second-order differential operators. The author was supported in part by NSF Grant DMS-0556309. ©2008 American Mathematical Society 377

I. E. VERBITSKY

378

Indefinite weights and Schrodinger operators Infinitesimal form boundedness, 'Irudinger's and Nash's inequalities Form boundedness of general second order differential operators References

4.

5. 6.

391 395

399 404

1. Introduction We will consider weighted norm inequalities of the type (1.1)

f E L"(do),

IITfIIL4(dw) 1. Here D(R") _ Co (R"), 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 = -A. 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) coincides with Q(Ho) 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 : W1,2(R") -* W-1"2(R"). When the form bound a > 0 in (1.10) can be arbitrarily small (with b depending on a) w is said to be infinitesimally form bounded relative to L. 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 IIVuIIL2(Rn), with its relativistic counterpart, II(-A + 1)1/4aII2 2(Rn) associated with the Sobolev space W112,2(W) of order 1/2.

The form boundedness problem for H = -A + w where w E D'(R), along with its infinitesimal version, was recently solved by Maz'ya and Verbitsky [MV2][MV6]. The main ideas discussed below can be expressed as follows: (1.9) holds for a distributional potential w if and only if it holds for I00-1wI2 in place of w. This nonlinear transformation w -p IVA-1wI2 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 = -A + w based on sharp estimates of equilibrium potentials and related weighted norm inequalities [MV2]. Similarly, the class of w E D' (R') associated with the relativistic form bound-

edness of f = (-A + 1)1/2 + w is invariant under the transformation w -+ I (1 A)-1/2w12. The relativistic problem can be reduced to a similar nonrelativistic one for the operator H = -A + c:' 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 Ho = -A on L2(R). Here w is an arbitrary realor 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 potentials w E D'(R") which are -A-form bounded with relative bound zero, i.e., for every e > 0, there exists C(e) > 0 such that (1.11)

2

2

I(wu, u)I 3,

52-" f

(If(x)12 + I'Y(x)I) ix = 0, 6(xo)

once (1.11) holds. Here B6(xo) is a Euclidean ball of radius 6 centered at x0.

WEIGHTED NORM INEQUALITIES

381

In the opposite direction, it follows from the results of [MV5] that (1.11) holds whenever (1.14)

lim

62r-nf

sup xoER"

(If(x)I2+ y(x)I)r dx = 0,

BS(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 between the positive and negative parts of w come into play. It includes functions with highly oscillatory behavior as well as singular measures, and contains properly the class of potentials based on the original Fefferman-Phong condition where I W I

is used in (1.14) in place of Il'I2 + Iyi Moreover, one can expand this class further using a sharp condition due to Chang, Wilson, and Wolff [ChWW] applied to If 12+IyI. 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 Ap weights, along with appropriate localization arguments, and good understanding of trace inequalities for nonnegative potentials w. In [MV5], we also study quadratic form inequalities of Trudinger type where C(e) in (1.11) has power growth, i.e., there exists co > 0 such that (1.15)

I (wu, u')I < E IIVuIIL2(R") +

CC-13

IIUIIL2(Rn),

Vu E Co (Rn).

for every e E (0, co), where 0 > 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 turns out, it is still possible to characterize (1.15) using only If I and IyH defined by (1.12), provided 13 > 1. In this case (1.15) holds if and only if both of the following conditions hold: (1.16)

Sup xoER"

52 + -n

0 0 and xo E R" ([M3], Sec. 1.4.7).

I. E. VERBITSKY

382

For general w, we obtain the following result: If p > 2, then (1.18) holds if and only if VA Lo lies in the Morrey space C2, A(Rn), Where A = n + 2 - 4p. For

p = 2, it holds if and only if V0-1w E BMO(Rn), and for 0 < p < 2, whenever V 1w E Lip1_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)

1IuII

I (wu, u) I
0. It is easy to see that the symmetric part of the matrix (a%j) must be uniformly bounded, and the skew-symmetric part reduces to the first-

order terms. The main problem here is to investigate 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. Applications 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], [MaZi], [PV1]-[PV3].

2. Basic trace inequalities We start with the following important theorem [M3]. THEOREM 2.1. (Mas'ya) Let 1 < p < oo. Let 12 be an arbitrary open set in Rn, and let w be a nonnegative locally finite Borel measure on Q. Then the inequality (2.1)

I IuI I L'(O,dw) 1 on E, U E C0 (1l) } .

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 in the case

1! = R' for Riesz potentials defined by

Iaf = (-A) 2 f = c(n, a) (I

Ia-n

* f),

0 < a < n,

where c(n, a) is a normalization constant. We also set f (y) dw(y), x E Rn, J Rn Ix - yI"-a for potentials with a Borel measure dw in place of dx, and Iaw = Ia(1dw) f f - 1 on R". The Riesz capacity offna measurable set E C Rh is defined by (2.4)

(2.5)

Ia (f dw) (x) = c(n, a)

Capa p(E) = inf

I

IgI5 dx : Iag(x) > 1 on E,

g E L+(R")

In the case a = 1 it is known that Capl,p(E) capl,p(E, R") for compact sets E, where R") 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

S2=R". THEOREM 2.2. (D. Adams-Dahlberg-Maz'ya) Let 1 < p < oo and let 0 < a < n. Let w be a nonnegative Borel measure on R". Then the following statements are equivalent. (i) The inequality (2.6)

IIIafIILP(d.) t}. Then (2.14) holds if and only if P=9 dt

< 001 000

An alternative noncapacitary characterization of (2.14) for 0 < q 1; Verbitsky q > 0) Let 1 < p < oo and 0 < q < p < oo. Let w be a positive Bored measure on R''. Then (2.14) holds if and only if (2.22)

9 P--1

Wa,p W E L P

(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

3. Dyadic and radial trace inequalities and nonlinear potentials In this section we concentrate on integral operators with general dyadic and radially nonincreasing kernels, as well as the corresponding nonlinear potentials, and two weight inequalities. Continuous versions for convolution operators with radial kernels 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 R", and K : D - R+. The kernel KD(x, y) on R" x R' is defined by (3.1)

KD(x, y) = E K(Q) XQ(x) XQ(y), QED

where XQ is the characteristic function of Q ED.

Let a be a locally finite positive Borel measure on R, and let f E Ll (da). We define the dyadic integral operator: (3.2)

f da.

K(Q)XQ(x)

TKD [f do] (_T) = fn KD(x, y)f (y) &(y) _ QED

IQ Q

In case f = 1, we set TKa [a](x) = E K(Q) a(Q) XQ(x) QED

If 0 < q, p < +oc, and a and w are locally finite Borel measures on R", the corresponding dyadic trace inequality is given by: (3.3)

fRn I TK,[f

dal

f E L"(&).

14 dw < c II!III,y(da),

Assume for a moment that q, p > 1. Duality then gives that (3.3) is equivalent to the inequality: (3.4)

fR' I TK-p [gdw]I r da 0 such that the trace inequality n

f E LP(da),

I TKa [fdC] I ° (x) dw(x) 0.

r

J

r k(s) s"-1 ds,

THEOREM 3.5. Let 0 < q < p < +oo and 1 < p < oo. Let w be a nonnegative Borel measure on R". Suppose k = k(I x - yI ), 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)

I

l k *fIILQ(d,,,) < C Of IILp(R"),

fE

LI(R7G),

holds if and only if

Wk[w] E La(w).

(3.15)

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 R". Suppose k = k(Ix - yI), where k(r) is a lower semicontinuous nonincreasing positive function on R+, and Wk[w] is defined by (3.12). Then there exist positive constants CI, C2 which depend onlyP on k, p and n such that (3.16)

C1 Iik*wIIL,(Rn)

3) is given by (1XI-1 JU(X)II + IVU(X)12 )dxl

3 a

IIUILa(Rn) = [JRn In this section, we assume that n > 3, since for the homogeneous space L1'2 (R")

our results become vacuous if n = 1 and n = 2. Analogous results for inhomogeneous Sobolev spaces hold for all n > 1. For V E D'(R"), consider the multiplier operator on D(R") defined by (4.1)

< V u. v >:=< V, u v >,

u, v E D(R"),

> represents the usual pairing between D(W) and JY(R"). L-1.2(R") = L1.2(R")* the dual Sobolev space. If the Let us denote by

where
is bounded on L12(R") x L1>2(R"): (4.2)

1 < VU, V > 1:5 C

IIVuIILa(Rn) IIVVIIL2(Rn),

where the constant c is independent of u, v E D(R), then V u E L-1"2(R"), and the multiplier operator can be extended by continuity to all of the energy space (As usual, this extension is also denoted by V.) We denote the class of multipliers V : L1'2(R") -, L-1'2(R") by M(L1'2(R") -, L-1'2(R"))Note that the least constant c in (4.2) is equal to the norm of the multiplier operator: IIVIIM(L1.a(Rn)-L-1.2(Rn)) = sup f IIVuIIL-1.a(RTM) :

IItIIL1.2(Rn) :5 1}.

For V E M(L1'2(R") -+ L-1'2(R")), we will need to extend the form < V, uv >

defined by the right-hand side of (4.1) to the case where both u and v are in L12(R"). This can be done by letting

= elm , N-oo where u = limjv UN, and v = hmN-oo VN, with UN, VN E D(R"). It is known that this extension is independent of the choice of UN and VN. We now define the Schrodinger operator H = Ho + V, where Ho = -A, on the

energy space L1'2(R"). Since Ho : L',2(R) -+ L-1-2(R") is bounded, it follows that H is a bounded operator acting from L1'2(R") to L-1"2(R") if and only if V E M(L112(R") -> L-1,2(R")). Clearly, (4.2) is equivalent to the boundedness of the corresponding quadratic form: < VU, U > I = I < V, IUI2 > I < C I IVUI IL2(Rn),

I. E. VERBITSKY

392

where the constant c is independent of u E D(R). If V is a (complex-valued) measure on R", then this inequality can be recast in the form: (4.3)

JR" [ u(x)I2 V

e II VU[IL2(R),

u E D(R").

For positive distributions (measures) V, this inequality is well studied (see Sections 1 and 2). V.

t

We now state the main result for arbitrary (complex-valued) distributions By L r (R")" = L ®C" we denote the space of vector-functions = (rl, ... , r") such that ri E Ll C(R"), i = 1, ... , n. THEOREM 4.1. Let V E D'(R"). Then V E M(L1'2(R") -+ L-1,2(R")), i.e.,

the inequality I < Vu, V > I < cIIuiiL1.2(Rn) IIvIIL1.2(Rn)

(4.4)

holds for all u, v E L1 2(R"), if and only if there is a vector field f E L210JR") such

that V = divand (4.5)

Iu(x)I2Ir(x)I2dx 0) and E > 0, (4.7)

f

I (x)I2 dx < C(n, E) R2IVII(L1'a(Rn)-L-1a(R», R (xo )

where R > max{1, Ixol}.

The following statements are concerned with sharp estimates of equilibrium potentials associated with a set of positive capacity.

PROPOSITION 4.4. Let 6 > z and let P = Pe be the equilibrium Newtonian potential of a compact set e C R" of positive capacity. Then (4.8)

IIVP1IIL2(Rn)

=

26

1

cap (e).

REMARK 4.5. For S < 2, it is easy to see that VP % L2(R").

PROPOSITION 4.6. Let 6 > 0, and let v be a real-valued function such that v E L1 2(R"). Then (4.9)

IIVvIIL2(Rn) < I IV(vP6)(x)12 P(X)215 26 1:5 By Proposition 4.6

IIVIIM(L1,2(Rn)-,La 1(R°))

IIVP5IIL2(Rn) IIVvIIL2(Rn).

Iow(x)I2 P(x)26
1 dx-a.e. on e. Thus,

fIi(x)I2dx

< C(n, b) cap (e)


395

for every compact set e c R', 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 (-0)-1/2V. THEOREM 4.11. Under the assumptions of Theorem 4.1, it follows that

V E M(Ll'2(Rn) -' L-1'2(Rn)) if and only if

(-A)-1/2V

E M(L1"2(Rn) -+ L2(Rn)).

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 W" 2 (Rn) the Sobolev space of weakly differentiable functions on

Rn (n > 1) such that IIuIIW- 2(R°) = IIuIIL2(R") + IIVUIIL2(R") < +00,

and by W-1.2(Rn) = WI,2(R")* the dual Sobolev space. For a compact set e C R", the capacity associated with W1,2(Rn) is defined by

u E Co (Rn),

cap (e) = inf { IluIt ..2(R") :

u(x) > 1 on e}

For 0 < r < oo, we denote by Lun;f(Rn) all f E

.

such that

IIfIILu.i, = sup IIXBI(xo) fIILi(R") < 00xoER'

By Lr(R")n = Lr(R") ® C" we denote the class of vector fields {r) 1 Rn -+ Cn, such that I'J 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 W. If V E 7Y(R") is nonnegative, i.e., coincides with w E M+(Rn), we write fR" Iu(x)I2 dw in place of (V, IuI2) = (Vu, u) for the quadratic form associated with the distribution V, if u E Co (R2). Sometimes we will use fR" Iu(x)12V(x)dx in

place of (Vu, u) even if V is not in Llo,(R). We set

mB(f)=

IBIIBf(x)dx

for a ball B C R', and denote byf BMO(Rn) the class of f E Lio,(Rn) for which sup 1 xuER",a>0 IB6(xo)I

6(xo)

If (x) - mlBa (xo) (f) 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(R") (the so-called local BMO) which we denote by bmo(R"). It can be defined in a similar way as the set of f E Lun;f(R") such that the preceding condition holds for 0 < J < 1 (see [St], p. 264). The Morrey space L'-\(W) (r > 0,A > 0) consists of f E Ll0 (R'1) such that sup

1

xoER^,6>0 IBo(xo)I

,

f

If Ir dx < +oc. 8(xo)

I. E. VERBITSKY

396

In the corresponding inhoinogeneous analogue, we set 0 < 6 < 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]. THEOREM 5.1. Let V E D'(R"), n > 2. Then the following statements hold. (i) Suppose that V is represented in the form: V = div f + y,

(5.1)

where f E L oC(RT)" and y E Ll,,(R°) satisfy respectively the conditions: 10

hm

(5. 2)

sup sup u

lim sup sup 6 +OxoER" U

(5.3)

fB

( moo )

I

t(x)I2 Iu(x)I2dx 2

=

0,

IIVuIIL'(B&(xo))

xI2dx fB6(xo) Iy(x)iulO IIVuII2

= 0,

L2(Ba(xo))

where u E Co (B6(xo)), u# 0. Then V is infinitesimally form bounded with respect to -A, z. e., for every E> 0 there exists C(e) > 0 such that (1.11) holds. (ii) Conversely, suppose V is infinitesimally form bounded with respect to -A.

Then V can be represented in the form (5.1) so that both (5.2) and (5.3) hold. (1-0)-1 V in the representation Moreover, one can set I' = -V(1-A)-'V and -y = (5.1).

REMARK 5.2. In the statement of Theorem 5.1 one can replace conditions (5.2)

and (5.3) with the equivalent condition where I(1-i)-' VI2 is used in place of Iil2 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 Irl2 and I.

THEOREM 5.3. Let V E D'(R' ), n > 2. The following statements are equivalent:

(i) V is infinitesimally form bounded with respect to -0.

(ii) V has the form (5.1) when r = -V(1 - A)-1 V, y = (1- 0)-1 V, and the measure w E M+ (R) defined by (5.4)

dw = (Ii(x) I2 + I y(x)I) dx

has the property that, for every E > 0, there exists C(E) > 0 such that (5.5)

C(e) I IVulli2(Rn),

-< e Ilaul12

f" lu(x)l2

Vu E Co (Rn).

(iii) For w defined by (5.4), W (p) a

(5.6)

lim

sup

1

6-+O Po:diamPo 3. In the onedimensional case, the infinitesimal form boundedness of the Sturm-Liouville operator H = - a + V on L2 (Rl) is actually a consequence of the form boundedness. THEOREM 5.4. Let V E 1Y(R1). Then the following statements are equivalent. z (i) V is infinitesimally form bounded with respect to

(ii) V is form bounded with respect to - s, i.e., I(V U. u)I < const II2I1u,l.z(R1),

`du E Co (RI).

(iii) V can be represented in the form V = 141 +'y, where rx+1

sup (II'(x)I2 + I y(x)I) dx < +oo. xER1Jx (iv) Condition (5.10) holds where

(5.10)

I'(x) =

sign (x - t) a-Ix-'l V (t) dt, l

'y(x) _

IR'

ex-tl V (t) dt

are understood in the distributional sense.

The statement (iii)=(i) in Theorem 5.4 is known ([Sch], Theorem 11.2.1), whereas (ii)*(iv) follows from [MV2]. We now state a characterization of the form subordination property (1.15). It was formulated originally in [Zru], in the form of the inequality: (5.11)

(Vu, u)I < E IIVuIIL2(R") +CE

IIuIIL=(R"),

Vu E Co (Rn),

for V > 0. Such V are called c"-compactly bounded in [Tru]. It follows from Nash's inequality that (1.15) yields (5.11) with v = n22Q + 11; the converse is also true, provided v > 2 , and is deduced using a localization argument. In the critical case 2(5.11) holds if and only if V E LOD(Rn), while for 0 < v < a, it holds only v=

ifV=0.

Necessary and sufficient conditions for (1.15), or equivalently (5.11) with v =

n2 6 + 2 (see [MV5]), can be formulated in terms of Morrey-Campanato spaces

I. E. VERBITSKY

398

using mean oscillations of the functions f and y which have appeared in Theorems 5.1-5.4.

THEOREM 5.5. Let V E D'(Rn), n > 2, and let 0 0 such that (1.15) holds for every e E (0, CO). Then V can be represented in the form

V=divT+y,

(5.12) where

-V(1 - A)-1V E L'c(R")' and y = (1 - A)-1V E Li c(R")

Moreover, there exists S0 > 0 such that (5.13)

B6(xo)

If(x) - MB,(..) (r) 12 dx < c JIy(x) I dx < c Sn- +1,

(5.14)

Sn-2 +i

,

0<S 1,

and f EL°'(Rn)n if0 0, (1.15) is equivalent to the following localized energy condition:

< (B6(xo))Cbn-20 3, and w = 0 if n = 1, 2 (see e.g. [M4], Sec. 2.4). A close sufficient condition on V E LIl°(R"), V > 0, which ensures that V E 9Yt+ 2 (R" ), is provided by the Fefferman-Pllong class V 1+` dy < const r"-2(1+E),

(6.9)

f1--V1 0, and the constant does not depend on r > 0, x E R".

A complete characterization of the class of admissible measures 912+1,2 (R') can be expressed in several equivalent forms discussed above. These criteria employ various degrees of localization of w, and each of them has its own advantages depending on the area of application.

We now state the main form boundedness criterion [MV6]. For A = (a;j), let At = (aji) denote the transposed matrix, and let Div: D'(R")""" --+ D'(Rn)" be the row divergence operator defined by n

(6.10)

Dir(a;j) _ (E 8j a;j) 1 j=1

THEOREM 6.1. Let G = div (A V-) + b' V + V, where A E D'(R")n"n, b E D' (R")" and V E D'(R"). n >_ 2. Then the following statements hold. (i) The sesquilinear form of C is bounded, i.e., (6.4) holds if and only if 1 (A + At) E L°O(R")nxn, and b and V can be represented respectively in the form b = F+ Div F, V = div where F is a skew-symmetric matrix field such that (6.11) (6.12)

F-

(A - At) E

BMO(R')nxn,

2

whereas c and h belong to L OC(R")", and obey the condition (6.13)

2(Rn). (ii) If the sesquilinear form of C is bounded, then c, F, and h in decomposition (6.11) can be determined explicitly by (6.14) (6.15)

ICI2 + Ih12 E 9)1

c = V (0-' div b),

h=V

(A-' V),

F = A-'curl [b - 1 Div (A - At)] + (A - At). 2

where

(6.16)

A-'curl [b - 1 Div (A - At)] E BMO(Rn)nxn,

and (6.17)

IV(A-ldivb)I2 + IV(t-1 V)12 E 9R+2(R").

I. E. VERBITSKY

402

REMARK 6.2. Condition (6.16) in statement (ii) of Theorem 6.1 may be replaced with 6-

(6.18)

Div (A - At) E BMO_I (Ra)n, 2

which ensures that decomposition (6.11) holds.

REMARK 6.3. In the case n = 2, we will show that (6.4) holds if and only if (A + At) E L°O(R2)2x2, b- 1 Div (A - At) E BMO_I (R2)2, and V = div b = 0.

REMARK 6.4. Expressions like V(A-'div b), Div(A-1curlb), and

V(0-' V)

used above which involve nonlocal operators are defined in the sense of distributions.

This is possible, as is shown in [MV6], since A-Idivb, A-1curlbb, and A-'V can be understood as the limits in the sense of the weak BMO-convergence (see [St], p. 166) of, respectively, A-1 div (t,bN b), L1-' curl (IbN b), and A-1 (IPN V) as N -> +oo. Here 1,bN is a smooth cut-off function supported on Ix: I xI < N}, and the limits above do not depend on the choice of N.

It follows from Theorem 6.1 that L is form bounded on L', 2 (R) x V. 2(R) if and only if the symmetric part of A is essentially bounded, i.e., (A + At) E 2 L°°(Rn)n ' , and bI . V + V is form bounded, where (6.19)

bl

= 6 - Div(A - At).

In particular, the principal part Pu2 = div(A Vu) is form bounded if and only if

(A + At) E L°O(R")nxn,

(6.20) 2

Div (A - At) E BMO-1(Rn)"

(6.21)

i

A simpler condition with (A - At) C. BMO(Rn)nxn in place of (6.21) is sufficient, 2 unless n < 2. but generally not necessary, Thus, the form boundedness problem for the general second order differential operator in the divergence form (6.3) is reduced to the special case (6.22)

C = b - V + V,

b E D'(R" )n,

V E D'(Rn).

As a corollary of Theorem 6.1, we deduce that, if b - V + V is form bounded, i.e., for all u, v E Ca (R"), (6.23)

JR"

(6-Vuv+ Vu v)dx

CIIuJILi.2(ft)IIvlILL 2(Rn),

then the Hodge decomposition

b' = V(0-'div b') + Div (A-'curl

(6.24)

holds, where O''curl6 E BMO(Rn)nxn, and (6.25)

[

I V (L-' div g)12 + I V (A-1 V) I2 ] dy < coast

rn-2,

Ix-vl 0, x E Rn, in the case n > 3; in two dimensions, it follows that

div6=V=0. We observe that condition (6.25) is generally stronger than A-'divb E BMO combined with A-1 V E BMO, while the divergence-free part of 6 is characterized

byA-1curlb'EBMO,forall n>2.


403

A close sufficient condition of the Fefferman-Phong type can be stated in the following form: (6.26)

JI x-yl j

do not a priori have better bounds than the right side of (2.7). However, (2.7) together with weak convergence in L2 P5t2-1) and pointwise convergence on the bases B(v, 2-i) shows that for each j and every X E 522-1 a subsequence of uk (X) =

fP

ukds =

f

Pxf ds + I L(V,2_J) P ukds

D2-

converges to u(X), perforce with Poisson representation that must be an extension from D2-1 of f. Here Pj is the Poisson kernel for the Lipschitz polyhedral domain 112-i and may be seen to be in L2(8522-1) by Dahlberg [Dah77]. Consequently u has nontangential limits f on OSt, and by (2.4) and (2.8) the theorem is proved.

2.2. The mixed problem with vanishing Dirichiet data. Theorem 2.11. Let S2 C R3 be a compact polyhedral domain with 2-manifold connected boundary. Then for any g E L2 (N) there is a unique solution a to the mixed problem (1.2) that vanishes on D and has Neumann data g on N. Further (Du*)2ds < C fN g2ds af0

THE MIXED PROBLEM FOR HARMONIC FUNCTIONS IN POLYHEDRA OF R3

415

Proof. Again by [Bro94] there exists a unique solution uj in S22-3 to the mixed problem so that g on an n N2-; , W W. Dud = 0 on the S(v, 2-') and u2 = 0 on D2-, . Lemma 2.7 and Remark 2.9 imply

J852na5221_;

g2ds+ J

(VuT)2ds < C I

IDuj 12dX

loft nN2_; 521 A Poincare inequality independent of j is also needed here and is supplied by the following lemma. Polyhedral domains are naturally described as simplicial complexes. See for definitions and notations [G1a70] [RS72] [VV03] [VV06] or others.

Lemma 2.12. Suppose u is harmonic in f22-,, with 8,u = g on 8S2f1N2-;, on the !3(v, 2-3) and u = 0 on D2-; . Then f 522_,

IVuI2dX

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