VOLUME X

PROCEEDINGS OF

SYMPOSIA IN PURE MATHEMATICS

Singular Integrals

AMERICAN MATHEMATICAL SOCIETY

PROVIDENCE, RHODE ISLAND

1967

http://dx.doi.org/10.1090/pspum/010

Proceedings of the Symposium in Pure Mathematics

of the American Mathematical Society

Held at the University of Chicago

Chicago, Illinois

April 20-22, 1966

Prepared by the American Mathematical Society

under National Science Foundation Grant GP-5500

ALBERTO P. CALDER6N

Editor

Library of Congress Catalog Card Number 67-16553

Copyright (6) 1967 by the American Mathematical Society

Printed in the United States of America

All Rights Reserved Except Those Granted to the United States Government

May not be reproduced in any form without permission of the publishers

This volume of the Proceedings of the Symposia on Pure

Mathematics is dedicated to Professor Antoni Zygmund in

celebration of his sixty-fifth anniversary and in recognition of

his decisive contribution to the field of Singular Integrals.

4-

Antoni Zygmund

Contents

On Singular Integrals 1 B. BAJSANSKI

R. COIFMAN

Algebras of Singular Integral Operators 18 A. P. CALDERON

On the Existence of Singular Integrals 56 A. P. CALDERON

MARY WEISS

A. ZYGMUND

Remarks about -^-Theory of Singular Integral Operators 74 H. O. CORDES

Boundary Value Problems for Second-order Parabolic Equations 82 E. B. FABES

M. JODEIT, JR.

Symbolic Calculus of Kernels with Mixed Homogeneity 106 E. B. FABES

N. M. RIVIERE

On Symmetnzable Differential Operators 128 K. O. FRIEDRICHS

P. D. LAX

Pseudo-differential Operators and Hypoelliptic Equations 138 LARS HORMANDER

Symbols of Parabolic Singular Integrals 184 M. JODEIT, JR.

Singular Integrals and a Boundary Value Problem for the Heat Equation 196 B. FRANK JONES, JR.

A Class of Singular Integrals. Pseudo-differential Operators on Non-quasi-analytic Functions Spaces 208

PAUL KREE

CONTENTS

A Sharp Inequality for Pseudo-differential and Difference Operators 213 P. D. LAX

L. NlRENBERG

Mixed Estimates for Singular Integrals and an Application to Initi'al Value Problems in Parabolic Differential Equations 218

J. E. LEWIS

Singular Integral Operators on Manifolds 232 UMBERTO NERI

Boundary Value Problems for Parabolic Systems of Partial Differential Equations 243

JOHN C. POLKING

On Quasi-homogeneous Bessel Potential Operators 275 CORA SADOSKY

MlSCHA COTLAR

Complex Powers of an Elliptic Operator 288 R. T. SEELEY

Elliptic Singular Integral Equations 308 R. T. SEELEY

Singular Integrals, Harmonic Functions, and Differentiability Properties of Functions of Several Variables 316

E. M. STEIN

Hypersingular Integrals and Summability of Fourier Integrals and Series 336 RICHARD L. WHEEDEN

Author Index 371

Subject Index 373

Author Index Roman numbers refer to pages on which

of an author. Italic numbers refer to pages on which a

is given. Boldface numbers indicate the first page c

Agmon, S., 289, 290, 299, 306, 312, 313 Agronovich, M. S., 308, 312, 313 Arima, R., 101,102,103,105 Arnese, G., 196, 204, 207 Aronson, D. G., 231, 231 Aronszajn, N. , 275, 276,287 Atkinson, F . V., 310,314 Atiyah, M., 143,183, 288, 293, 306, 310,

312,313,313

Bajsanski, B., 1,55 Baouendi,S. ,210 Benedek, A., 57, 73, 201,207, 218, 224,

225,231,287,320,335 Besicovitch, A., 318, 335 Bokobza, J ., 138,139,169,183 Borak, T., 288,307 Borelli, R . ,81 ,81 Bott, R., 78,143,183, 288, 312, 313,313 Boyarskif, B. V., 312,314 Breuer, M., 77, 78,81 Browder, F. E., 288, 291,307 Calderon, A. P., 1 ,2,6,8,10,13,14,17,18,18,

19, 22, 25, 26, 27, 32, 37, 38, 44, 55, 56, 57, 63, 64, 72, 79,106,108,114,116,119,120, 126,127,136,138,139,183, 201, 204,207, 209,210,212,218, 223, 224, 225,231,232, 233, 241, 243, 244, 245, 246, 274, 275,279, 280, 281, 284, 285, 286,287, 289, 310, 311, 312, 313,314, 316, 317, 318, 320, 322, 326, 327, 335,335, 337, 356,369

Coifman, R., 1,55 Cordes, H. O., 74, 77, 78, 79 ,81, 310, 313, 314

Cotlar, M., 2,17, 209, 212, 275

Douglis, A., 312,313 Dunford,N. , 274 Dynin, A. S., 308, 312, 313,313,314

Erdelyi, A., 344,345,346,347,348,349,369 Eskin .G. I., 313,315

a reference is made to an author or a work

complete reference to a work by the author

>f the articles in the book.

Fabes, E. B., 82, 83, 37, 88,104,105, 106, 107, 115,116,117, 118, 119, 120, 121,125, 126,127, 184,195, 196, 204, 207, 209, 212, 218, 219, 224, 226, 227, 228, 231, 231

Federer, H.,294, 307 Feldman, J . , 77 Friedman, A., 105, 229, 231 Friedrichs, K. O., 128,128, 242

Gel'fand, I. M., 312, 314 Giraud,G. , 311,313,314 Gohberg, I., 81, 255, 274, 310, 314 Golovokin, K. K., 287 Grisvard, P., 209, 212

Hardy, G. H., 287 Herman, E., 79,81 Herz ,C .S . , 331, 332, 335 Hilbert, D., 308, 312,314 Hirschmann, 1.1., 320, 323, 335 Hoffman, K., 189,195 Hormander, L., 37, 55, 57, 81 ,81, 138,

138, 139, 162, 164,168, 171, 179, 180,183, 196, 201, 204, 207, 210, 210, 211, 212, 216, 217, 218, 224, 231, 232, 237, 241, 274, 274, 288, 289, 290, 292, 307, 313,314, 316, 320, 335

Horvath, J ., 311, 314, 325, 335

Jessen, B., 199,207 Jodeit, M., Jr. , 82,84,104,105, 184,195 John, R., 321, 335 Jones, B. F. , Jr. , 138,183, 1%, 196, 203, 204, 207, 218, 219, 224, 226

Jones, F., 209, 212

Kahane, J . P., 72 Kato, T., 310,314 Kohn, J . J ., 18,55, 79,81,130,138,158,

183, 216,217, 274, 289, 290, 294, 297, 307, 311,313,314

Koppleman, W., 312, 313, 314

372

Kotake, T., 288, 292,307, 313 Kree, P., 196, 204,207, 208, 208, 209, 210, 211,212, 277, 280,287

Krein, M. G., 255,274, 310,314 Kreiss,H.O., 274,2i 7 Kumano-go,H., 138,183

Labrousse, J . P., 310,314 Lax, P. D., 128,128, 242 Lewis, J . E., 196, 204,207, 218 Littlewood, J . E., 287 Luft, E., 77,81 Luke, Y. I., 358,369 Luxemburg, W. A. J ., 23, 24,55

Magnus, W., 344, 345,346, 347,348, 349, 369

Malgrange, B., 210,212 Marcinkiewicz, J ., 199,207 McKean, H. P., 294, 307 Mihlin, S. G., 37,138,183, 308, 311, 312, 314

Minakshisundaram, S., 288, 292, 294, 306, 307

Muskhelishvili, N. T., 308, 313,314

Narasimhan, M., 288,307 Newberger, S., 80,81 Nirenberg, L., 18,55, 79,81,130,138,158,

164,183, 213, 215, 216,217,274, 288, 289, 290, 294, 297, 307, 312, 313,313, 314, 321,335

Noether, F., 292,307, 310,312,314

Oberhettinger, F., 344, 345, 346, 347, 348, 349,369

O'Neil, R., 213, 215, 216,217, 230, 231,231 Ostrow,E. H.,335

Palais, R. S., 86,105,138,183, 243, 269, 274, 289, 290, 293, 294, 297,307, 312,314

Panejah, B. P., 275,287 Panzone, R., 57, 201,207, 218, 224, 225, 231,320,335

Peetre, J . , 287 Pleijel, A., 288, 292, 294, 306,307 Plemelj, J. ,308, 310,324 Poincare, H., 308,314 Polking, J . C , 243 P61ya,G., 287 Privalof,J.,310,311,314

Richtmyer, R. D., 214,217

AUTHOR INDEX

Riesz, M., 310, 312,315 Riviere, N. M., 87,88,104,105,106,106,

107,115,116,117,118,119,120,121,125, 126,127,184,195,196,204,207,209,212t

224,231

Sadosky, C , 209, 211,212, 275,280,284, 285,287

Schwartz, J . , 274, 320,335 Seeley, R. T., 55,81,85,86,105,138,139,

158,183, 212, 243, 244, 248, 269,274, 288, 288, 289, 230, 293, 302,307,308, 312,313, 315

Shamir, E., 313,315 Shapiro, V. L., 338, 340, 342, 348, 353, 358,

362,365, 366,369 Singer, I. M., 293, 294,306,307, 310, 312, 313,313,315

Smith, K. T., 275,276,287 Stein, E. M., 16, 163, 183, 198, 205, 207,

246,274, 275, 281, 283, 287, 316, 320, 321, 323, 325, 326, 327, 328, 329, 333,335

Taibleson, M. H., 196,198, 201, 204, 207, 275,287,320,331,335

Titchmarsh, E. C , 318,335 Treves, F., 168 Tricomi, F. G., 310, 311,315, 344, 345, 346, 347, 348, 349,369

Unterberger, A., 138,139,169,183

Visik, M. I., 313,315 Volevie, L. R., 275, 287 Volpert, A. I., 312, 313,315

Wainger, S., 138,163,183, 323,335 Watson, G. N., 338, 348, 350, 354, 357, 369 Weiss, G., 325,335 Weiss, M., 56,57, 72, 313,315 Wendroff,B., 215,217 Weyl, H., 294,307 Wheeden, R. L., 325,335, 336 Wiener, N., 291, 307

Zygmund, A., 2, 6, 8,10,13,14,17,18,19, 25, 26, 37, 44, 55, 56, 57, 63, 64, 72, 79, 106, 108, 114,116, 119, 120,126,127,138, 139,183, 199, 201, 204, 207, 209, 210, 212, 218, 223, 224, 231, 233, 284, 285, 286,287, 289, 311, 312, 313, 314, 31$, 316, 317,318, 319, 322,325,326,327,328,329,335, 335, 337,340,356,367,369

Subject Index adjoint, 149 algebra with or-symbols, 77, 79 amplification matrix, 214 area

function, Lusin, 204 integral, 205

Besov spaces, 280 Bessel potentials, quasi-homogeneous, 276 Bochner-Riesz means of order 7, 338 Bott's unitary group, 78 boundary

operator, 101,102 problems, 312

on compact manifolds, 308 oblique, 81

bounded mean oscillation, 321 boundedness, 9

of maximal operator, 1, 2 bundles, vector, 78

C alderdn -Zy gmund kernel, 337 operators, 79, 289

elliptic, 301 Cauchy principal value singular

integral, 309 Cauchy-Riemann equations, 325 class (/, r), operator of, 250 compact

ideal, 77 manifolds, boundary problems on, 308

compactly supported, 148 cone, 201 conjugate function, 59 continuity

in metric L', moduli of, 58 modulus of, 57

convolution singular integral operators, 19 critical index, 340 cube, fundamental, 61

derivative ordinary, 326 at x in harmonic sense, 328 at x0 in LQ sense, 326

difference operator, 213 scheme (s), 213

stability for, 213 Dini condition, 57, 58

elliptic, 232 Calderon-Zygmund operator, 301 differential operators, Garding's

inequality for, 213 operator, 288

index of, 312 partial differential equations, 308

ellipticity condition, 309

fixed points, 142 Fourier transform R (x), principal-valued, 337

Fredholm concept, generalized, 75 equation, 309, 311

function spaces, 22 fundamental cube, 61

^-function, 319 Garding's inequality for elliptic differential

operators, 213 general singular integral equation, 309 generalized Fredholm concept, 75 Grassmann-manifolds, 77

harmonic sense, derivative at x in, 328 Hilbert transform, 2, 309 homogeneous kernel, 56 hypersingular integral, 337 hypoelliptic, 163

374

Ikehara's theorem, 291 index, 167

of elliptic operator, 312 formula, 292

initial value problem, 213 interpolation theorem of Marcinkiewicz, 318

*th differential in Lp, 338 kernel

homogeneous, 56 theorem, Schwartz, 302 truncated, 56

U-topology, 75 Lebesgue

condition, logarithmic, 72 space, 198

Lipschitz spaces, 197,198 Littlewood - Paley

function, 198 g-function, 318

logarithmic Lebesgue condition, 72 Lusin area function, 204

Marcinkiewicz, interpolation theorem of, 318

maximal operator(s), 2, 4,12

boundedne8s of, 1, 2 i?-set(s),75,76

moduli of continuity in metric L', 58 modulus of continuity, 57

^-index, 76 yf*- opera tor, 75 Noether's formula, 293 nonorientable manifolds, 312

oblique boundary problem, 81 operator (s)

of class (J, r), 250 of order zero, 305

ordinary derivative, 326 total differential, 326

oscillation, bounded mean, 321

parabolic, 269 evolution operators, 254 singular integral (s), 184

operator(s),82,86,93 parametrix, 164, 298 partial differential equations, elliptic, 308 Peetre's inequality, 294

SUBJECT INDEX

Poisson integral, 319, 338 potential, 197

of order a of if function, 319 principal value, 56 principal-valued Fourier transform ft(x)t

337 pseudo inverse, 84 pseudo-differential operators, 79, 216,

232 , 288 pseudo-product, 84 pseudo-spectrum, 249

quasi - homogeneous Bessel potentials, 276 functions, 276

#-set, 75 Riesz

operators, 79 transforms(s), 2,13,323

rotation method, 2, 7 rotational, 58

cr-symbol, 77 Schwartz kernel theorem, 302 semielliptic, 233 singular

integral(s), 337 Cauchy principal value, 309 equations, 308

general, 309 operator(s),39,232

convolution, 19 depending on a parameter, 249 parabolic, 82, 86,93 properties of, 243 with variable kernel, 25

parabolic, 184 strongly, 322 truncated, 337

operators with only bounded behaviour atx = oo,80

support, 140 spaces Hk,n 253 space-like, 135 splitting property, 328 stability for difference schemes, 213 subelliptic, 237

properly, 241 symbol, S3,149,185, 214 symmetnzable, 129

Taylor expansion, 297 total differential, ordinary, 326

SUBJECT INDEX 375 transform (s; unitary group, Bott's, 78

Hilbert, 2 Riesz, 2,13, 323 v e c t o r bundles, 78

transversal 142 von-Neumann condition, 214 truncated weak type, 16,17, 57

kernel, 56 weak-type result, 317 singular integral, 337 well-posed, 75

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