Singular Integrals PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS AMERICAN MATHEMATICAL SOCIETY VOLUME X PROVIDENCE, RHODE ISLAND
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2 VOLUME X PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Singular Integrals AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND 1967
3 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 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
4 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.
5 4- Antoni Zygmund
6 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
7 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
8 Author Index Roman numbers refer to pages on which a reference is made to an author or a work of an author. Italic numbers refer to pages on which a complete reference to a work by the author is given. Boldface numbers indicate the first page c >f the articles in the book. 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 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
9 372 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, 126,127,184,195,196,204,207,209,212 t 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
10 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 x 0 in L Q 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), principalvalued, 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
11 374 Ikehara's theorem, 291 index, 167 of elliptic operator, 312 formula, 292 initial value problem, 213 interpolation theorem of Marcinkiewicz, 318 *th differential in L p, 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, 28 8 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
12 SUBJECT INDEX 375 transform (s; unitary group, Bott's, 78 Hilbert, 2 Riesz, 2,13, 323 vector 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 CDEFGHIJ-AMS
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