HARMONIC ANALYSIS IN EUCLIDEAN SPACES

Part 2

http://dx.doi.org/10.1090/pspum/035.2

PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS

Volume XXXV, Part 2

HARMONIC ANALYSIS IN EUCLIDEAN SPACES

AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND

1979

PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

HELD AT WILLIAMS COLLEGE WILLIAMSTOWN, MASSACHUSETTS

JULY 1 0 - 2 8 , 1978

EDITED BY GUIDO WEISS

STEPHEN WAINGER

Prepared by the American Mathematical Society with partial support from National Science Foundation grant MCS 77-23480

Library of Congress Cataloging in Publication Data

Symposium in Pure Mathematics, Williams College, 1978. Harmonic analysis in Euclidean spaces. (Proceedings of symposia in pure mathematics; v. 35) Includes bibliographies. 1. Harmonic analysis—Congresses. 2. Spaces, Generalized—Congresses. I. Wainger, Stephen,

1 9 3 6 - II. Weiss, Guido L., 1 9 2 8 - III. American Mathematical Society. IV. Title. V. Series. QA403.S9 1978 515'.2433 79-12726 ISBN 0-8218-1438-9 (v.2)

AMS (MOS) subject classifications (1970). Primary 22Exx, 26A51, 28A70, 30A78, 30A86, 31Bxx, 31C05, 31C99, 32A30, 32C20, 35-XX,

42-XX, 43-XX, 44-XX, 47A35, 47B35 Copyright © 1979 by the American Mathematical Society

Printed in the United States of America All rights reserved except those granted to the United States Government.

This book may not be reproduced in any form without the permission of the publishers.

CONTENTS OF PART 2

Chapter 4. Several complex variables

Singular integrals in complex analysis 3 N. KERZMAN

B.M.O. functions in complex analysis 43 N. TH. VAROPOULOS

Smoothness of harmonic and holomorphic functions 63 STEVEN G. KRANTZ

Remarks on zeros of IP functions in balls 69 NICHOLAS WEYLAND

Inverting Hb on some weakly pseudo-convex domains..... 73 PETER C. GREINER

On some estimates for the d Neumann problem in nonpseudo convex domains 77 M. DERRIDJ

Remarks on isolated degeneracies of the Levi form and the analytic hypoellip-ticity of Db and related partial differential operators 85

DAVID S. TARTAKOFF

Remarks on the Casorati-Weierstrass theorem. 91 NESSIM SIBONY and PIT-MANN WONG

Peak sets for A °°(D) 97 JACQUES CHAUMAT and ANNE-MARIE CHOLLET

Chapter 5. Pseudo differential operators and Partial differential equations

Sharp estimates for pseudodifferential operators 105 YVES MEYER

Fourier integral operators and harmonic analysis on compact manifolds 115 MICHAEL E. TAYLOR

o

Sharp Garding inequalities 137 C. FEFFERMAN and D. PHONG

Subelliptic estimates 143 J. J. KOHN

Lp and Holder estimates for pseudodifferential operators: Necessary conditions 153 RICHARD BEALS

Some new classes of pseudo-differential operators... 159 ALEXANDER NAGEL and E. M. STEIN

v

VI CONTENTS

LP estimates for the wave equation 171 JUAN C. PERAL

Estimates from Lpto its dual for the Klein-Gordon equation 175 BERNARD MARSHAL, WALTER STRAUSS and STEPHEN WAINGER

Dirichlet and Neumann problems for the heat equation in C1-cylinders 179 E. B. FABES and N. M. RIVIERE

Applications of A. P. Calderon's Cauchy integral theorem...... 197 MAX JODEIT, JR

Edge of the wedge theorem for partial differential equations 203 LEON EHRENPREIS

Remarks on decay rates of partial differential equations in infinite domains inRn 213

WALTER LITTMAN

Necessary and sufficient conditions for local solvability on the Heisenberg group 219

DARYL GELLER

Smoothness of solutions of certain partial differential equations constructed from vector fields 227

LINDA PREISS ROTHSCHILD

Chapter 6. Harmonic analysis in other settings: Probability, martingales, local fields, Lie groups and functional analysis

Niki&n-Stein theory and factorization with applications.. 233 JOHN E. GILBERT

Complex interpolation for families of Banach spaces.... 269 R. COIFMAN, M. CWIKEL, R. ROCHBERG, Y. SAGHER and G. WEISS

Martingale theory and harmonic analysis in Euclidean spaces 283 D. L. BURKHOLDER

Applications of the conformal invariance of Brownian motion 303 BURGESS DAVIS

An introduction to Hardy spaces on local fields 311 MITCHELL H. TAIBLESON

Singular integrals on local fields and generalizations to martingales..... 317 SVANTE JANSON

Triadic and dyadic conjugate martingales 321 J.-A. CHAO

i/^-sets for the group of integers of a p-series field....... 325 WILLIAM R. WADE

CONTENTS Vll

Pointwise differentiation on local fields 329 C. W. ONNEWEER

Singular integral multipliers on local fields 333 JAMES E. DALY

Extendibility of convolution operators and convergence of Fourier series on local fields; integral homogeneity of distributions 337

KEITH PHILLIPS

Multipliers on symmetric spaces...... 345 JEAN-LOUIS CLERC

LP multipliers on the Heisenberg group.... 355 LEONEDE DE MICHELE and GIANCARLO MAUCERI

Invariant distrubutions 361 CARL HERZ

Singular integral operators on nilpotent Lie groups..... 375 ROE GOODMAN

On the Fourier transform for the ax 4- b group........ 379 PIERRE EYMARD

Some estimates on symmetric spaces related to the Laplace-Beltrami operator.. 387 N. LOHOUE

Non-isotropic Lipschitz spaces 391 G. B. FOLLAND and H. T. HUNG

Fractional integration on nilpotent groups and representations of semisimple groups 395

MICHAEL COWLING

A tauberian property of the convolution semi-group generated by X2 - \Y\7 on the Heisenberg group 403

ANDRZEJ HULANICKI

A new class of Bessel functions and applications in harmonic analysis.... 407 KENNETH I. GROSS, WAYNE J. HOLMAN, III and RAY A. KUNZE

Multipliers and Parseval type fromulas for Jacobi series.. 417 GEORGE GASPER and WALTER TREBELS

The harmonic machinery for eigenfunction expansions 429 W. C. CONNETT and A. L. SCHWARTZ

The square function for Lx, L^ contractions 435 ALBERTO DE LA TORRE