AMERICAN MATHEMATICAL SOCIETY

107

Harmonic Analysis and Partial

Differential Equations

Proceedings of a Conference held April 4-5, 1988

Titles in This Series

Volume

1 Markov random fields and their 19 Proceedings of the Northwestern applications, Ross Kindermann and homotopy theory conference, Haynes J. Laurie Snell R. Miller and Stewart B. Priddy, Editors

2 Proceedings of the conference on 20 Low dimensional topology, Samuel J. integration, topology, and geometry in Lomonaco, Jr., Editor linear spaces, William H. Graves, Editor 21 Topological methods in nonlinear

3 The closed graph and P-closed functional analysis, S. P. Singh, graph properties in general topology, S. Thomaier, and B. Watson, Editors T. R. Hamlett and L. L. Herrington 22 Factorizations of b" ± 1, b = 2,

4 Problems of elastic stability and 3, 5, 6, 7, 1 0, 11, 12 up to high vibrations, Vadim Komkov, Editor powers, John Brillhart, D. H. Lehmer,

5 Rational constructions of modules J. L. Selfridge, Bryant Tuckerman, and

for simple Lie algebras, George B. S. S. Wagstaff, Jr.

Seligman 23 Chapter 9 of Ramanujan's second

6 Umbral calculus and Hopf algebras, notebook-Infinite series identities, Robert Morris, Editor transformations, and evaluations,

7 Complex contour integral Bruce C. Berndt and Padmini T. Joshi

representation of cardinal spline 24 Central extensions, Galois groups, and functions, Walter Schempp ideal class groups of number fields,

8 Ordered fields and real algebraic A. Frohlich

geometry, D. W. Dubois and T. Recio, 25 Value distribution theory and its Editors applications, Chung-Chun Yang, Editor

9 Papers in algebra, analysis and 26 Conference in modern analysis statistics, R. Lidl, Editor and probability, Richard Beals,

10 Operator algebras and K-theory, Anatole Beck, Alexandra Bellow, and

Ronald G. Douglas and Claude Arshag Hajian, Editors

Schochet, Editors 27 Microlocal analysis, M. Salah Baouendi,

11 Plane ellipticity and related problems, Richard Beals, and Linda Preiss

Robert P. Gilbert, Editor Rothschild, Editors

12 Symposium on algebraic topology in 28 Fluids and plasmas: geometry and

honor of Jose Adem, Samuel Gitler, dynamics, Jerrold E. Marsden, Editor

Editor 29 Automated theorem proving, W. W. 13 Algebraists' homage: Papers in Bledsoe and Donald Loveland, Editors

ring theory and related topics, 30 Mathematical applications of category S. A. Amitsur, D. J. Saltman, and theory, J. W. Gray, Editor G. B. Seligman, Editors 31 Axiomatic set theory, James E.

14 Lectures on Nielsen fixed point theory, Baumgartner, Donald A. Martin, and Boju Jiang Saharon Shelah, Editors

15 Advanced analytic number theory. 32 Proceedings of the conference on Part 1: Ramification theoretic methods, Banach algebras and several complex Carlos J. Moreno variables, F. Greenleaf and D. Gulick,

16 Complex representations of GL(2, K) for Editors finite fields K, llya Piatetski-Shapiro 33 Contributions to group theory,

17 Nonlinear partial differential equations, Kenneth I. Appel, John G. Ratcliffe, and Joel A. Smeller, Editor Paul E. Schupp, Editors

18 Fixed points and nonexpansive 34 Combinatoric& and algebra, Curtis mappings, Robert C. Sine, Editor Greene, Editor

http://dx.doi.org/10.1090/conm/107

Titles in This Series

Volume

35 Four-manifold theory, Cameron Gordon 53 The Selberg trace formula and related and Robion Kirby, Editors topics, Dennis A. Hejhal, Peter Sarnak,

36 Group actions on manifolds, Reinhard and Audrey Anne Terras, Editors

Schultz, Editor 54 Differential analysis and infinite 37 Conference on algebraic topology in dimensional spaces, Kondagunta

honor of Peter Hilton, Renzo Piccinini Sundaresan and Srinivasa Swaminathan, and Denis Sjerve, Editors Editors

38 Topics in complex analysis, Dorothy 55 Applications of algebraic K-theory to Browne Shaffer, Editor algebraic geometry and number theory,

39 ErreH Bishop: Reflections on him and Spencer J. Bloch, R. Keith Dennis, Eric

his research, Murray Rosenblatt, Editor M. Friedlander, and Michael R. Stein,

40 Integral bases for affine Lie algebras Editors

and their universal enveloping algebras, 56 Multiparameter bifurcation theory, David Mitzman Martin Golubitsky and

41 Particle systems, random media and John Guckenheimer, Editors

large deviations, Richard Durrett, Editor 57 Combinatorics and ordered sets, 42 Classical real analysis, Daniel Ivan Rival, Editor

Waterman, Editor 58.1 The Lefschetz centennial conference. 43 Group actions on rings, Susan Proceedings on algebraic geometry,

Montgomery, Editor D. Sundararaman, Editor

44 Combinatorial methods in topology and 58.11 The Lefschetz centennial conference. algebraic geometry, John R. Harper Proceedings on algebraic topology, and Richard Mandelbaum, Editors S. Gitler, Editor

45 Finite groups-coming of age, John 58.111 The Lefschetz centennial conference. McKay, Editor Proceedings on differential equations,

46 Structure of the standard modules A. Verjovsky, Editor for the affine Lie algebra Ap>, 59 Function estimates, J. S. Marron, Editor James Lepowsky and Mirko Prime

60 Nonstrictly hyperbolic conservation 47 Linear algebra and its role in

systems theory, Richard A. Brualdi, laws, Barbara Lee Keyfitz and

David H. Carlson, Biswa Nath Datta, Herbert C. Kranzer, Editors

Charles R. Johnson, and Robert J. 61 Residues and traces of differential Plemmons, Editors forms via Hochschild homology,

48 Analytic functions of one complex Joseph Lipman

variable, Chung-chun Yang and 62 Operator algebras and mathematical Chi-tai Chuang, Editors physics, Palle E. T. Jorgensen and

49 Complex differential geometry and PaulS. Muhly, Editors nonlinear differential equations, 63 Integral geometry, Robert L. Bryant, Yum-Tong Siu, Editor Victor Guillemin, Sigurdur Helgason, and

50 Random matrices and their R. 0. Wells, Jr., Editors applications, Joel E. Cohen, 64 The legacy of Sonya Kovalevskaya, Harry Kesten, and Charles M. Newman, Linda Keen, Editor Editors

51 Nonlinear problems in geometry, 65 Logic and combinatorics, Stephen G.

Dennis M. DeTurck, Editor Simpson, Editor

52 Geometry of normed linear spaces, 66 Free group rings, Narian Gupta

R. G. Bartle, N. T. Peck, A. L. Peressini, 67 Current trends in arithmetical algebraic and J. J. Uhl, Editors geometry, Kenneth A. Ribet, Editor

Titles in This Series

Volume

68 Differential geometry: The interface 86 Representation theory and number between pure and applied mathematics, theory in connection with the local Mladen Luksic, Clyde Martin, and Langlands conjecture, J. Ritter, Editor William Shadwick, Editors 87 Abelian group theory, Laszlo Fuchs,

69 Methods and applications of Rudiger Gobel, and Phillip Schultz, mathematical logic, Walter A. Carnielli Editors and Luiz Paulo de Alcantara, Editors 88 Invariant theory, R. Fossum,

70 Index theory of elliptic operators, W. Haboush, M. Hochster, and

foliations, and operator algebras, V. Lakshmibai, Editors

Jerome Kaminker, Kenneth C. Millett, 89 Graphs and algorithms, R. Bruce and Claude Schochet, Editors Richter, Editor

71 Mathematics and general relativity, 90 Singularities, Richard Randell, Editor James A. Isenberg, Editor 91 Commutative harmonic analysis,

72 Fixed point theory and its applications, David Colella, Editor

R. F. Brown, Editor 92 Categories in. computer science and

73 Geometry of random motion, Rick logic, John W. Gray and Andre Scedrov,

Durrett and Mark A. Pinsky, Editors Editors

74 Geometry of group representations, 93 Representation theory, group rings, and

William M. Goldman and Andy R. Magid, coding theory, M. Isaacs, A. Lichtman,

Editors D. Passman, S. Sehgal, N.J. A. Sloane, and H. Zassenhaus, Editors

75 The finite calculus associated with 94 Measure and measurable dynamics, Bessel functions, Frank M. Cholewinski R. Daniel Mauldin, R. M. Shortt, and

76 The structure of finite algebras, Cesar E. Silva, Editors David C. Hobby and Ralph Mckenzie 95 Infinite algebraic extensions of finite

77 Number theory and its applications in fields, Joel V. Brawley and George E. China, Wang Yuan, Yang Chung-chun, Schnibben and Pan Chengbiao, Editors 96 Algebraic topology, Mark Mahowald

78 Braids, Joan S. Birman and Anatoly and Stewart Priddy, Editors Libgober, Editors 97 Dynamics and control of multibody

79 Regular differential forms, Ernst Kunz systems, J. E. Marsden, P. S.

and Rolf Waldi Krishnaprasad, and J. C. Simo, Editors

80 Statistical inference from stochastic 98 Every planar map is four colorable,

processes, N. U. Prabhu, Editor Kenneth Appel and Wolfgang Haken

81 Hamiltonian dynamical systems, 99 The connection between infinite

dimensional and finite dimensional Kenneth R. Meyer and Donald G. Saari,

dynamical systems, Basil Nicolaenko, Editors

Ciprian Foias, and Roger Temam, 82 Classical groups and related topics, Editors

Alexander J. Hahn, Donald G. James, 100 Current progress in hyperbolic systems: and Zhe-xian Wan, Editors Riemann problems and computations,

83 Algebraic K-theory and algebraic W. Brent Lindquist, Editor number theory, Michael R. Stein and 101 Recent developments in geometry, R. Keith Dennis, Editors S.-Y. Cheng, H. Choi, and Robert E.

84 Partition problems in topology, Greene, Editors Stevo Todorcevic 102 Primes associated to an ideal,

85 Banach space theory, Bor-Luh Lin, Stephen McAdam Editor 1 03 Coloring theories, Steve Fisk

Titles in This Series

Volume

104 Accessible categories: The foundations of categorical model theory, Michael Makkai and Robert Pare

105 Geometric and topological invariants of elliptic operators, Jerome Kaminker, Editor

106 Logic and computation, Wilfried Sieg, Editor

107 Harmonic Analysis and Partial Differential Equations, Mario Milman and Tomas Schonbek, Editors

Harmonic Analysis and Partial

Differential Equations

ONTEMPORARY ATHEMATICS

107

Harmonic Analysis and Partial

Differential Equations

Proceedings of a Conference held April 4-5, 1988

Mario Milman and Tomas Schonbek, Editors

AMERICAN MATHEMATICAL SOCIETY • PROVIDENCE, RHODE ISLAND

EDITORIAL BOARD

Richard W. Beals, managing editor Sylvain E. Cappel! Jan Mycielski Jonathan Goodman Linda Preiss Rothschild Craig Huneke Michael E. Taylor

The Conference on Harmonic Analysis and Partial Differential Equations was held at the Florida Atlantic University, Boca Raton, Florida on April 4-5, 1988.

1980 Mathematics Subject Classification {1985 Revision). Primary 35-06, 42-06.

Library of Congress Cataloging-in-Publication Data

Harmonic Analysis and Partial Differential Equations {1988: Florida Atlantic University) Harmonic analysis and partial differential equations: proceedings of the conference held

April 4-5, 1988/Mario Milman and Tomas Schonbek, editors. p. cm.-(Contemporary mathematics, ISSN 0271-4132; v. 107) "The Conference on Harmonic Analysis and Partial Differential Equations was held at the

Florida Atlantic University, Boca Raton, Florida on April 4-5, 1988""-T.p. verso. Includes bibliographical references. ISBN 0-8218-5113-6 (alk. paper) 1. Harmonic analysis-Congresses. 2. Differential equations, Partial-Congresses.

I. Milman, Mario. II. Schonbek, Tomas, 1944-. Ill. Title. IV. Series: Contemporary mathematics (American Mathematical Society); v. 107. QA403.H236 1988 5151 .2433-dc20

90-34635 CIP

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this pub-lication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Executive Director, Ameri-can Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248.

The appearance of the code on the first page of an article in this book indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that the fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 27 Congress Street, Salem, Massachusetts 01970. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale.

Copyright @1990 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted

to the United States Government. 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.§

This publication was typeset using A.MS -TEX. the American Mathematical Society's TEl< macro system.

10 9 8 7 6 54 3 2 1 95 94 93 92 91 90

Contents

Preface

Gradient Estimates at the Boundary for Solutions to Nondivergence Elliptic Equations

B. BARCELO, L. ESCAURIAZA, and E. F ABES

Interior Regularity of Solutions to Monge Ampere Equations

Xl

L. A. CAFFARELLI 13

The Helson-Szego Theorem in U of the Bidimensional Torus M. COTLAR and C. SADOSKY 19

Galerkin Methods for the Boundary Integral Equations of Elliptic Equations in Nonsmooth Domains

B. E. J. DAHLBERG and G. VERCHOTA 39

Some Applications of Hardy Spaces and BMO in Harmonic Analysis and Partial Differential Equations

R. FEFFERMAN 61

The Positive Cone in Triebel-Lizorkin Spaces and the Relation Among Potential and Maximal Operators

B. JAWERTH, C. PEREZ, and G. WELLAND 71

Changes of Variable and Ap Weights R. JOHNSON 93

Progress on Two Problems Posed by Riviere C. E. KENIG 101

Fredholm Theory for Periodic Solutions of Some Semilinear P.D.E.s with Homogeneous Nonlinearities

A. C. LAZER and P. J. MCKENNA 109

Stability of Solitary Waves W. A. STRAUSS 123

IX

Preface

In this volume we have collected, in final form, papers presented at the conference on Harmonic Analysis and Partial Differential Equations held April 4-5, 1988 at Florida Atlantic University.

We thank the participants for their contributions which reflect the quality and the substance of the conference. We are also grateful to Florida At-lantic University for the financial support that made the program possible. A number of people contributed their time and energy helping to make the conference a success, in particular we thank Beth Broer, Jack Freeman, and Yanda Milman.

XI

Mario Milman Tomas Schonbek

Boca Raton, Florida

.:· .-.

ISBN 978-0-8218-5113-5

9 780821 851135

CONM/107