Christ M., Kenig C.E., Sadosky C. Harmonic Analysis and Partial Differential Equations: Essays in Honor of Alberto P. Calderon

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Harmonic Analysis and Partial Differential Equations

Harmonic Analysis

and Partial Differential

Equations

Essays in Honor of Alberto P. Calderon

EDITED BY

Michael Christ, Carlos E. Kenig, and Cora Sadosky

The University of Chicago Press

Chicago and London

The University of Chicago Press, Chicago 60637

The University of Chicago Press, Ltd., London

C 1999 by The University of Chicago

Paperback edition 2001

Printed in the United States of America

08 07 06 05 04 03 02 01

2 3 4 5

ISBN: 0-226-10456-7 (cloth)

ISBN: 0-226-10455-9 (paperback)

Library of Congress Cataloging-in-Publication Data

Harmonic analysis and partial differential equations : essays in honor of

Alberto P. Calderdn / edited by Michael Christ, Carlos E. Kenig, and

Cora Sadosky.

p. cm. - (Chicago lectures in mathematics series).

Includes bibliographic references.

ISBN: 0-226-10456-7 (cloth : alk. paper)

1. Harmonic analysis. 2.Differential equations, partial.

1. Calder6n, Alberto P. II. Christ, Francis Michael, 1955-

II1. Kenig, Carlos E., 1953- . IV. Sadosky, Cora, 1940-

V. Series: Chicago lectures in mathematics.

QA403.H225 1999

515'.2433-dc2l

99-20385

CIP

0 The paper used in this publication meets the minimum requirements of

the American National Standard for Information Sciences-Permanence of

Paper for Printed Library Materials, ANSI Z39.48-1992.

Chicago Lectures in Mathematics Series

Robert J. Zimmer, series editor

J. P. May, Spencer J. Bloch, Norman R. Lebovitz, and Carlos E. Kenig, editors

Other Chicago Lectures in Mathematics titles available from the University of

Chicago Press:

Simplicial Objects in Algebraic Topology, by J. Peter May (1967)

Fields and Rings, Second Edition, by Irving Kaplansky (1969, 1972)

Lie Algebras and Locally Compact Groups, by Irving Kaplansky (1971)

Several Complex Variables, by Raghavan Narasimhan (1971)

Torsion-Free Modules, by Eben Matlis (1973)

Stable Homotopy and Generalised Homology, by J. F. Adams (1974)

Rings with Involution, by I. N. Ilerstein (1976)

Theory of Unitary Group Representation, by George V. Mackey (1976)

Commutative Semigroup Rings, by Robert Gilmer (1984)

Infinite-Dimensional Optimization and Convexity, by Ivar Ekeland and

Thomas Turnbull (1983)

Navier-Stokes Equations, by Peter Constantin and Ciprian Foias (1988)

Essential Results of Functional Analysis, by Robert J. Zimmer (1990)

Fuchsian Groups, by Zvetlana Katok (1992)

Unstable Modules over the Steenrod Algebra and Sullivan's Fixed Point Set

Conjecture, by Lionel Schwartz (1994)

7bpological Classification of Stratified Spaces, by Shmuel \Veinberger (1994)

Lectures on Exceptional Lie Groups, by J. F. Adams (1996)

Geometry of Nonpositively Curved Manifolds, by Patrick B. Eberlein (1996)

Dimension Theory in Dynamical Systems: Contemporary Views and

Applications, by Yakov B. Pesin (1997)

A Concise Course in Algebraic 7bpology, by J. P. May (1999)

Contents

Preface vii

Introduction

Michael Christ, Carlos E. Kenig, and Cora Sadosky

1. Calder6n and Zygmund's Theory of Singular Integrals 1

Elias M. Stein

2. Transference Principles in Ergodic Theory 27

Alexandra Bellow

3. A Weak-Type Inequality for Convolution Products 41

Alexandra Bellow and Alberto P Calder6n

4. Transference Couples and Weighted Maximal Estimates 49

Earl Berkson, Maciej Paluszyfiski, and Guido Weiss

5. Periodic Solutions and Nonlinear Wave Equations 69

Jean Bourgain

6. Some Extremal Problems in Martingale Theory and Harmonic

Analysis 99

Donald L. Burkholder

7. The Monge Ampere Equation, Allocation Problems, and Elliptic

Systems with Affine Invariance

117

Luis A. Caffarelli

8. On a Fourth-Order Partial Differential Equation in

Conformal Geometry 127

Sun-Yung A. Chang

9. Riesz Transforms, Commutators, and Stochastic Integrals

151

A. B. Cruzeiro and P Malliavin

10. An Application of a Formula of Alberto Calder6n to Speaker

Identification

163

Ingrid Daubechies and Stephane Maes

11. Analytic Capacity, Cauchy Kernel, Menger Curvature,

and Rectifiability

183

Guy David

12. Symplectic Subunit Balls and Algebraic Functions

199

Charles Fefferman

13. Multiparameter Calder6n-Zygmund Theory 207

Robert A. Fefferman

Contents

14. Nodal Sets of Sums of Eigenfunctions 223

David Jerison and Gilles Lebeau

15. Large-Time Behavior and Self-Similar Solutions of Some

Semilinear Diffusion Equations

241

Yves Meyer

16. Estimates for Elliptic Equations in Unbounded Domains and

Applications to Symmetry and Monotonicity 263

Louis Nirenberg

17. Asymptotic Expansions for Atiyah-Patodi-Singer Problems 275

Robert Seeley

18. Analysis on Metric Spaces 285

Stephen Semmes

19. Developments in Inverse Problems since Calder6n's

Foundational Paper 295

Gunther Uhlmann

Remarks Addressed to the Conference 347

Alberto P. Calder6n

Contributors 349

Index 353

Preface

Alberto P. Calderon turned seventy-five in September 1995, and an interna-

tional conference in his honor was convened at the University of Chicago in

February 1996, with the participation of more than three hundred mathe-

maticians.

Professor Calderon was among this century's leading analysts, and his

work is characterized by its great originality and depth. Calderon's contri-

butions have been of wide scope, and changed the way researchers approach

and think of a wide variety of areas in mathematics and its applications. His

fundamental influence is felt strongly in harmonic analysis, partial differen-

tial equations, and complex analysis, as well as in such applied areas as signal

processing, geophysics, and tomography.

The Chicago conference was a joyous occasion to celebrate Calderdn's

outstanding achievements. Many of his friends, students, and collaborators

from Europe and North and South America came to join him in the cele-

bration. This volume contains the contributions of nineteen of the speakers,

including a personal account of the history of the Chicago School of analysis

by Elias M. Stein.

The mathematical community was deeply saddened by the loss of Alberto

P. Calderon, who died in Chicago on April 16, 1998, after a short illness. We

have included, in an introduction, a short account of his life and mathematical

career. This volume, originally conceived to mark a joyous milestone, has now

also become a memorial tribute.

vii

Introduction

Michael Christ, Carlos E. Kenig, and Cora Sadosky

Alberto P. Calderon was born in Mendoza, Argentina, on September 14,

1920. His father, a physician, instilled in him the love for mathematics and

music. Impressed by Alberto's early interest in everything mechanical, he

sent him at age twelve to school in Switzerland, in preparation for the ETH,

the leading engineering school. This was not to be, and after two years in

Zurich, Calderon was called back to Mendoza, where he finished high school

and then attended the University of Buenos Aires, graduating as a civil

engineer in 1947. Throughout his studies he had been increasingly interested

in mathematics, but at the time engineering seemed a more viable career.

Upon graduation as an engineer, Calderon took a job in a geophysics

research lab at YPF, the national oil corporation of Argentina. He enjoyed

his work, where he dealt with mathematical problems arising from the design

of tools for oil prospecting. He was always proud of the problems he solved

there and frequently remarked with satisfaction that some problems he could

not solve then are still open. But he joked how fortunate it was that his

YPF supervisor made his life there difficult, since otherwise he would have

remained in Argentina as a state employee until retirement!

This did not come to pass, and Calderon's close relation with the few

active mathematicians then at the University of Buenos Aires was instru-

mental to the breakthrough that led to his mathematical celebrity. In 1948

two concurrent events changed his future: he resigned his position at YPF,

and Antoni Zygmund, one of the world's leading analysts, and a professor

at the University of Chicago, visited the Institute of Mathematics of the

University of Buenos Aires.

At that time the Institute of Mathematics-a single room plus a small

good library-was an appendix of the School of Engineering, with Julio Rey

Pastor, the Goettingen-trained Spanish mathematician who introduced mod-

ern mathematics to Argentina, as its sole professor, and young Dr. Alberto

Gonzalez Dominguez as Rey's only assistant.

Still, its seminar was active,

with the participation of serious young mathematicians, including Spanish