Selected Papers of Antoni Zygmund

479 Pages·1989·24.68 MB·English

by  A. P. Calderón

Selected Papers of Antoni Zygmund
Volume 3
Mathematics and Its Applications (East European Series)
Managing Editor:
M.HAZEWINKEL
CenJre/or Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. BIALY NICKI-BIRULA. Institute 0/ Mathematics, Warsaw University, Poland
H. KURKE. Humboldt University, Berlin, CD.R.
J. KURZWEIL. Mathematics Institute, Academy 0/ Sciences, Prague, Czechoslovakia
L. LEINDLER. Bolyai Institute, Szeged, Hungary
L. LOVA sz. Bolyai Institute, Szeged, Hungary
D. S. MITRINOVIC. University 0/ Belgrade, Yugoslavia
S. ROLEWICZ. Polish Academy 0/ Sciences, Warsaw, Poland
BL. H. SENDOV. Bulgarian Academy o/Sciences, Sofia, Bulgaria
1. T. TODOROV. Bulgarian Academy o/Sciences, Sofia, Bulgaria
H. TRIEBEL. University 0/ lena, CD.R.
Volume 41/3
Selected Papers of
Antoni Zygmund
Volume 3
edited by
A. Hulanicki
Unil'ersity of Wroclaw, Po/and
and
P. Wojtaszczyk and W. Zelazko
Institute o/Mathematics,
Po/ish Academy of Sciences,
Warsaw, Po/and
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
Library of Congress Cataloging in Publication Data
Zygmund, Antoni, 1900-
[Selections, 19891
Selected papers of Antoni Zygmund I edited by A, Hulanicki and p,
WOjtaszczyk and W, Zelazko.
p, cm. -- (Mathematics and its applications. East European
series)
<U.S. set)
1. Harmonic analysis. 2. Zygmund, Antoni, 1900-
I. Hulanicki, Andrzej, 1933- II. WOjtaszczyk, Przemys{aw, 1940-
III. Zelazko, Wies{aw. IV. Title. V. Series; Mathematics and
its applications (Kluwer Academic Publishers). East European
series.
OA403.Z9425 1989
515' .2433--dc20 89-20033
ISBN-I3: 978-94-010-6962-5 e-ISBN-13: 978-94-009-1045-4
DOl: 10.1007/978-94-009-1045-4
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SERIFS EDITOR'S PREFACE
'Et moi ..... si j'avait su comment en revenir, One service mathematics bas rendered the
je n'y serais point aile: human race. It bas put common sense: back
Jules Verne where it belongs, on the topmost shelf next
The series is divergent; thCldorc we may be 10 tile dusty canister labelled 'discarded 1lOII-
ICDIC'.
able 10 do something with it.
Eric T. Bell
O. Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non
linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for
other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And
all statements obtainable this way form part of the raison d'!tre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred
volumes have appeared it seems opportune to reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However, the 'tree' of knowledge of
mathematics and related fields does not grow only by putting forth new branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related. Further, the kind and level of sophistication
of mathematics applied in various sciences has changed drastically in recent years:
measure theory is used (non-trivially) in regional and theoretical economics; algebraic
geometry interacts with physics; the Minkowsky lemma, coding theory and the structure
of water meet one another in packing and covering theory; quantum fields, crystal
defects and mathematical programming profit from homotopy theory; Lie algebras are
relevant to filtering; and prediction and electrical engineering can use Stein spaces. And
in addition to this there are such new emerging subdisciplines as 'experimental
mathematics', 'CFO', 'completely integrable systems', 'chaos, synergetics and large-scale
order', which are almost impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics."
By and large, all this still applies today. It is still true that at first sight mathematics seems rather
fragmented and that to find, see, and exploit the deeper underlying interreIations more effort is
needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will
continue to try to make such books available.
If anything. the description I gave in 1977 is now an understatement. To the examples of
interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu
lar functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more)
all come together. And to the examples of things which can be usefully applied let me add the topic
'finite geometry'; a combination of words which sounds like it might not even exist, let alone be
applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via
finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to
be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And,
accordingly, the applied mathematician needs to be aware of much more. Besides analysis and
numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
extra mathematical sophistication that this requires. For that is where the rewards are. Linear
models are honest and a bit sad and depressing: proportional efforts and results. It is in the non
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci-
vi SERIES EDITOR'S PREFACE
ate what I am hinting at: if electronics were linear we would have no fun with transistors and com
puters; we would have no TV; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace
and anticommuting integration, p-adic and ultrametric space. All three have applications in both
electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre
quently proved the shortest path between 'rea!' results. Similarly, the first two topics named have
already provided a number of 'wormhole' paths. There is no telling where all this is leading -
fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five sub
series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything
else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis
cipline which are used in others. Thus the series still aims at books dealing with:
- a central concept which plays an important role in several different mathematical and/or
scientific specialization areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
- influences which the results, problems and concepts of one field of enquiry have, and have had,
on the development of another.
These three volumes of selected papers by Antoni Zygmund -of which this is the third one -are the
first of their kind in this series. More are scheduled to appear fairly soon. Often, collected or
selected works are treated as separate projects on their own and are not placed in any series what
ever (or are placed in a series of collected works); so possibly a few words of explanation, or at
least motivation, are in their place.
The interconnectedness of all things is something in which I (like Douglas Adams' creation
Svlad Cjelli) firmly believe, especially in mathematics and science (though for vastly different rea
sons). In fact this trust in the importance of interrelations is a main driving force behind the whole
Mathematics and Its Applications series.
Interconnectedness between things mathematical can be found in monographs, especially when
written with that point in view, and in selected proceedings of conferences of an interspecialistic
character, but it can also be found in the work of a single scientist, especially a great one, a point
that needs no discussion in the present case; and, in fact, in such a case the refusal to restrict con
siderations to the domain of one labelled, classified, and recognized subdiscipline may take on
dramatic forms.
Abel recommended to read (only) the works of the original masters. I have my own relatively
modest collection of collected and selected works, and, with increasing awareness of just how valu
able Abel's remark is, I find myself dipping into them more and more frequently.
In addition, the great ones often have the invigorating habit of scattering their papers far and
wide; add ups and downs in the popularity and status of journals, an increasing tendency to publish
important material only in proceeding volumes, and finding all the relevant stuff of a particular
author on a particular topic may well become something of a serious chore in itself.
Moreover, if present trends continue, very few libraries will be able to afford (more or less) com
plete collections of even only the more important journals, proceedings, and other edited volumes.
All this makes judiciously chosen selected/collected works (and other ways of grouping interest
ing material) particularly inviting and attractive.
The shortest path between two truths in the Never lend books, for no one ever returns
real domain passes through the complex them: the only books I have in my library
domain. are books that other folk have lent me.
J. Hadamard Anatole France
La physique ne nous donne pas seulement The function of an expert is not to be more
l'occasion de ri:soudre des problemes ... eIle right than other people. but to be wrong for
nous fait prcssentir la solution. more sophisticated reasons.
H. Poincare David Butler
Bussum, September 1989 Michiel Hazewinkel
TABLE OF CONTENTS
BIBLIOGRAPHY OF ANTONI ZYGMUND ix
PAPER
[141] (with A. P. Calderon) 'A note on the interpolation of linear
operations', SM 12 (1951), 194-204.
[142] 'Polish mathematics between the two wars (1919-39)', in: Proceed
ings of tke Second Canadian Mathematical Congress, Vancouver,
1949, pp. 3-9, Univ. of Toronto Press, 1951. 12
[143] (with A. P. Calderon) 'On the existence of certain singular in-
tegrals', Acta Math. 88 (1952), 85-139. 19
[148] (with R. Salem) 'Some properties of trigonometric series whose
terms have random signs', Acta Matk. 91 (1954), 245-301. 74
[150] (with A. P. Calderon) 'Singular integrals and periodic func
tions', SM 14 (1954), 249-271. 131
[151] (with R. Salem) 'Sur un theoreme de Piatetc;ki-Shapiro', CRAS
240 (1955), 2040-2042. 154
[153] (with A. P. Calderon) 'On a problem of Mihlin', TAMS 78
(1955), 209-224. 157
[164] (with A. P. Calderon) 'Addenda to the paper "On a problem of
Mihlin"', ibid., 84 (1957), 559-560. 173
[155] (with A. P. Calderon) 'A note on the interpolation of sublinear
operations', Amer. J. Math. 78 (1956), 282-288. 175
[156] (with A. P. Calderon) 'On singular integrals', ibid., 289-309. 182
[157] (with A. P. Calderon) 'Algebras of certain singular operators',
ibid., 310-320. 203
[158] 'On a theorem of Marcinkiewicz concerning interpolation of oper-
ations', J. Matk. Pures Appl. (9) 35 (1956), 223-248. 214
[159] 'Hilbert transforms in En', in: Proceedings of tke International
Congress of Mathematicians, 1954, Amsterdam, Vol. 3, pp. 140-
151, Erven P. Noordhoff N.V., Groningen, and North-Holland
Publishing Co., Amsterdam, 1956. 240
[160] 'On the Littlewood-Paley function g*(O)', Proc. Nat. Acad. Sci.
U.S.A. 42 (1956), 208-212. 252
[162] (with A. P. Calderon) 'Singular integral operators and differen-
tial equations', Amer. J. Matk. 79 (1957), 901-921. 257
The papers appearing in these volumes are numbered according to the comprehensive list of publications of Antoni Zygmtmd which
are presented in chronological order beginning on p. ix.
viii TABLE OF CONTENTS
[166J (with M. Weiss) 'A note on smooth functions', Nederl. Akad.
Wetensch. Proc. Ser. A 62 = Indag. Math. 21 (1959), 52-58. 278
[174J (with A. P. Calderon) 'Local properties of solutions of elliptic
partial differential equations', SM 20 (1961), 171-225. 285
[179J (with E. M. Stein) 'On the differentiability offunctions', ibid.,
23 (1963/1964), 247-283. 340
[180J (with A. P. Calderon) 'On the higher gradients of harmonic
functions', ibid., 24 (1964), 211-226. 377
[182J (with E. M. Stein) 'On the fractional differentiability of func-
tions', Proc. London Math. Soc. (3) 14A (1965), 249-264. 393
[187J (with E. M. Stein) 'Boundedness of translation invariant opera-
tors on Holder spaces and LP-spaces', Ann. of Math. (2) 85 (1967),
337-349. 409
[192J (with M. Weiss) 'On multipliers preserving convergence of tri-
gonometric series almost everywhere', SM 30 (1968), 111-120. 422
[196J 'On certain lemmas of Marcinkiewicz and Carleson', J. Approx.
Theory 2 (1969), 249-257. 432
[202J 'A Cantor-Lebesgue theorem for double trigonometric series', SM
43 (1972),173-178. 441
[212J (with A. P. Calderon) 'A note on singular integrals', ibid., 65
(1979),77-87. 447
BIBLIOGRAPHY OF ANTONI ZYGMUND
Abbreviations
BIAP Bulletin International de l'Academie Polonaise des Sciences et des Let
tres, Classe des Sciences Mathematiques et Naturelles, Serie A: Sciences
M athematiques
CRAS Comptes Rendus hebdomadaires des seances de l'Academie des Sciences
(Paris)
FM Fundamenta Mathematicae
MR Mathematical Reviews
SM Studia M athematica
TAMS Transactions of the American Mathematical Society
Zbl. Zentralblatt fUr Mathematik
[M) J6zef Marcinkiewicz, Collected Papers, PWN-Polish Scientific Publishers,
Warszawa, 1964.
[S] Raphael Salem, (Euvres mathematiques, Hermann, Paris, 1967.
Other abbreviations follow those of Mathematical Reviews.
1923
[1] 'Sur la theorie riemannienne des series trigonometriques', CRAS 177,521-
523; Errata, ibid., 804.
[2] 'Sur les series trigonometriques', ibid., 576-579; Errata, ibid., 804.
[3] (with W. Sierpinski) 'Sur une fonction qui est discontinue sur tout en
semble de puissance du continu', FM 4, 316-318; reprinted in: Waclaw
Sierpinski, (Euvres choisies, Tome II, pp. 497--499, PWN-Polish Scientific
Publishers, Warszawa, 1975.
1924
[4] (with A. Rajchman) 'Sur les principes et les problemes de la theorie rie
mannienne des series trigonometriques', Ann. Soc. Polon. Math. 3, 147-148.
[5] 'Sur un theoreme de M. Marcel Riesz', ibid., 153.
[6] 'Sur les series de Fourier restreintes', CRAS 178, 181-182.
ix
The pape. . reprinted in this three-volume work are marked with •.
Those appearing in this particular volume are marked with ••.
x ANTONI ZYGMUND
[7] 'Sur une generalisation de la methode de Cesaro', ibid., 179, 870-872.
[8] (with S. Saks) 'Sur les faisceaux des tangentes it une courbe', FM 6, 117-
121.
[9] '0 module cia,glosci sumy szeregu sprzt;Zonego z szeregiem Fouriera' (On the
modulus of continuity of the sum of a series conjugate to a Fourier series),
Prace Mat.-Fiz. 33, 125-132 [in Polish, French summary].
1925
[10] (with S. Saks) 'Un teorema suIle curve continue', Boll. Un. Mat. Ital. 4,
7-10.
*[ 11] 'Sur la derivation des series de Fourier', BIAP, Annee 1924, 243-249.
*[12] 'Sur la sommation des series trigonometriques conjuguees aux series de
Fourier', ibid., 251-258.
[13] 'Sur Ia sommation des series trigonometriques et celles de puissances par les
moyennes typiques', CRAS 181, 1122-1123.
1926
[14] 'Sur Ia sommabilite des series de Fourier des fonctions verifiant Ia condition
de Lipschitz', BIAP, Annee 1925, 1-9.
*[15] (with A. Rajchman) 'Sur la relation du procede de sommation de Cesaro
et celui de Riemann', ibid., 69-80.
*[16] 'Sur un theoreme de M. Gronwall', ibid., 207-217.
[17] 'Sur la sommation des series par Ie procede des moyennes typiques', ibid.,
265-287.
*[18] 'Remarque sur Ia sommabilite des series de fonctions orthogonales', ibid.,
Annee 1926, 185-191.
*[19] 'Contribution it l'unicite du developpement trigonometrique', Math. Z. 24,
40-46.
*[20] 'Sur la theorie riemannienne des series trigonometriques', ibid., 47-104.
[21] 'Sur un theoreme de Ia theorie de la sommabilite', ibid., 25, 291-296.
*[22] (with A. Rajchman) 'Sur la possibilite d'appliquer Ia methode de Riemann
aux series trigonometriques sommables par Ie procede de Poisson', ibid.,
261-273.
*[23] 'Sur les series trigonometriques sommables par Ie procede de Poisson', ibid.,
274-290.
*[24] 'Une remarque sur un theoreme de M. Kaczmarz', ibid., 297-298.
[25] '0 teorji srednich arytmetycznych" (On the theory of arithmetic means),
Mathesis Polska 1,75-85 and 119-129 [in Polish].
1927
*[26J 'Sur l'application de la premiere moyenne arithmetique dans la thoorie des
series de fonctions orthogonales', FM 10,356-362.