# Zygmund, Antoni

# ZYGMUND, ANTONI

(*b*. Warsaw, Poland, 26 December 1900;

*d*. Chicago, Illinois, 30 May 1992), *mathematics, harmonic analysis, trigonometric series, singular integrals*.

The mathematician Zygmund was one of the twentieth century’s leading exponents of Fourier analysis. He was a major figure in taking the subject from one to several variables and in the creation of a theory of singular integrals.

**Early Life and Career.** Zygmund’s schooling was interrupted by World War I when the family moved from Poland to Ukraine. On his return to Warsaw he found there was no opportunity to study astronomy, his first interest, so he switched to mathematics at the University of Warsaw. There he became attached to Aleksander Rajchman and Stanislaw Saks. He learned Riemann’s theory of trigonometric series from Rajchman and Saks and, after taking his doctorate in 1923, wrote several research papers with them. A Rockefeller Fellowship enabled him to spend the academic year 1929–1930 in England with Godfrey H. Hardy at Oxford University and John E. Littlewood at Cambridge University; he also met and worked with Raymond E. A. C. Paley that year. Zygmund found the visit to England was a tremendous stimulus. He went on to write five papers with Paley and a joint paper with Paley and Norbert Wiener on lacunary and random series. On his return to Poland he became a professor of mathematics at Stefan Batory University in Wilno. There he met Jozef Marcinkiewicz, who was then a student but soon became a collaborator with Zygmund.

Even in the early 1930s, Zygmund showed his distaste for manifestations of anti-Semitism, which eventually cost him his job in a politically motivated purge of the university in 1931. Hardy, Littlewood, Henri Lebesgue in France, and other eminent mathematicians wrote in protest, and Zygmund was reinstated. It was during his time at Wilno that Zygmund wrote the first edition of his book *Trigonometric Series*(1935). This volume was so complete in its treatment that it was revised and reprinted by Cambridge University Press in 1959 and was reprinted six times, to become the standard work in its subject. The 1930s were productive for Zygmund in many ways, but they ended in tragedy. Zygmund and Marcinkiewicz joined the Polish army, but the partition of Poland between the Nazis and the Soviets saw Wilno fall in the Russian zone. Many of the Polish Officer Corps were rounded up by the Russians and massacred at Katyn in Poland, and most likely Marcinkiewicz was among them. Saks and Rajchman were murdered by the Nazis, and so in a brief and surely terrifying period of time, Zygmund lost most of his collaborators and close friends.

Given Zygmund’s international reputation and the efforts of the mathematicians Jerzy Neyman and Norbert Wiener, Zygmund was able to get out of Europe, and starting in 1940 he survived the war teaching at Mount Holyoke College in South Hadley, Massachusetts. In 1945 he transferred to the University of Pennsylvania in Philadelphia, and two years later he moved to the University of Chicago, where he stayed for the rest of his career. There he joined a remarkably strong Mathematics Department put together by Marshall Stone, and the result was a golden period for Zygmund, marked most famously by his collaboration with Alberto Calderón. It was in the 1950s that he and Calderón formulated the theory of singular integrals that now carries their names. A few years later, Elias Stein came to Chicago as a doctoral student of Zygmund’s and began another remarkable collaboration. All told, Zygmund had thirty-five students, many of whom went on to have distinguished careers. Robert Fefferman described Zygmund’s own career as being characterized by “his tremendous desire to work with people of the greatest mathematical ability, and his absolute devotion to those people.”

Zygmund received many honors in his lifetime. They included the National Medal of Science in 1986, the highest honor awarded by the U.S. government for work in science, and membership in the U.S. National Academy of Sciences (1961) and other national science academies, including those of Poland, Argentina, and Spain.

**Trigonometric Series.** Zygmund’s early work was on trigonometric series. These had been introduced by Bernhard Riemann in 1854 as a significant generalization of the usual Fourier series. As Zygmund learned from Rajchman, the key questions here concern the uniqueness of the series and its local properties (such as continuity at a point). Uniqueness requires the nontrivial result that a trigonometric series that converges to zero everywhere has all of its coefficients zero (and is therefore the trivial series). This leads to the study of sets *E* such that any trigonometric series that converges to zero outside *E* is necessarily the trivial series. Such sets have measure zero, so their study requires new analytic tools, and Zygmund made a profound investigation of these sets.

The book *Trigonometric Series* owes much to the influences of Saks and Marcinkiewicz. Integrable functions of one variable have an averaging property that is easy to generalize to functions of *n* variables. However, Zygmund showed in 1927, by using a construction of Otton Nikodym, that the generalization is false in dimensions higher than 1, and Saks then showed that even modest generalizations will fail. Zygmund was able to show, however, that the generalization can be made to work for functions of several variables that are in the class *L ^{p}* for some

*p*. In later work with Marcinkiewicz, the class of functions for which the generalization holds was widened considerably.

This work undoubtedly stimulated Zygmund’s interest in extending the results of classical one variable harmonic function theory to several variables, but there was a major obstacle. Single variable harmonic function theory is almost interchangeable with single variable complex function theory; indeed, that was the key insight of Riemann. To create a deep theory of harmonic functions in several variables required building up the theory of functions of several real variables, and this was to be the theme of Zygmund’s work with Calderón.

Calderón and Zygmund embarked on the task of producing a *n*-real variable version of the Hilbert transform. Their paper, “On the Existence of Singular Integrals,” published in *Acta Mathematica* in 1956, was described by Stein in these terms: “There is probably no paper in the last fifty years which has had such widespread influence in analysis” (1998, p. 1133). They used ideas of Marcinkiewicz (as Zygmund was later to acknowledge) and a number of powerful original ideas to establish the existence of the relevant singular integrals. Both the techniques and the results of this paper exerted a considerable influence on the future direction of work in this field.

**Singular Integrals.** The integral operators in which they were interested have their roots in the classical theory of partial differential equations. In pursuing this connection, Calderón and Zygmund were led to a theory of symbols, with implications for the theory of linear partial differential operators. Calderón pushed for a version of the theory of singular integrals that applied to manifolds, and the breakthrough came with his student Robert Seeley’s discovery that the symbol is actually a function on the cotangent space of the manifold. This allowed their theory to merge with the ideas of Russian mathematician Israel M. Gelfand about elliptic operators on manifolds, and Seeley’s calculus was very useful in the first proof of the Atiyah-Singer index theorem, undoubtedly one of the major mathematical events of the century. A profusion of work by many authors in many countries saw the theory of singular integrals become a major part of a much broader theory of what are called pseudo-differential operators. Singular integrals, however, remain central to the study of real functions of several variables, and the work of Zygmund and his collaborators in the Chicago school of analysis decisively deepened that whole branch of mathematics.

*Trigonometric Series* was first published in 1935 and in a third edition in 2002. The book is remarkable for both its thoroughness and its many highlights, among which is the Marcinkiewicz interpolation theorem, which was to play an important part in the creation of the theory of singular integrals. It applies to operators of weak type, and singular integrals on the space *L ^{1}* are of weak type. The Marcinkiewicz integral is another highlight, and here it illuminates the

*L*theory of the Hilbert transform, which is a preview of the Calderón-Zygmund theory. Zygmund also drew on his time with Hardy, Littlewood, and Paley in writing the book, and the Hardy-Littlewood maximal theorem is central to it. Indeed, it is Zygmund, Calderón, and Stein who gave the theorem its central role in analysis. It led Zygmund to deepen the approach to Fourier series of a single variable by using complex variable methods, and it has implications for the study of the Hardy spaces

^{p}*H*, all of which are described in the book. Zygmund himself had a particular liking for the material on the Littelwood-Paley functional, which applied to a function produces a new function with an

^{p}*L*norm comparable in size with the

^{p}*L*norm of the original function. This makes it very useful, and Zygmund used it to study Hardy spaces. Later it was given a simple conceptual proof using the theory of singular integrals, an important moment in the generalization of the single variable theory to the several variables theory that was one of Zygmund’s most profound achievements.

^{p}## BIBLIOGRAPHY

Selected Papers of Antoni Zygmund, *cited below, contains a complete bibliography of Zygmund’s published works*.

### WORKS BY ZYGMUND

With Alberto Calderón. “On the Existence of Certain Singular Integrals.” *Acta Mathematica* 78 (1956): 289–309.

*Selected Papers of Antoni Zygmund*. Edited by Andrzej Hulanicki, Przemyslav Wojtaszczyk, and Wieslaw Zelazko. 3 vols. Dordrecht, Netherlands; Boston: Kluwer Academic, 1989. Includes a complete bibliography of Zygmund’s published works.

*Trigonometric Series*, 3rd ed. Cambridge, U.K.; New York: Cambridge University Press, 2002.

### OTHER SOURCES

Stein, Elias. “Singular Integrals: The Roles of Calderón and Zygmund.” *Notices of the American Mathematical Society* 45 (1998): 1130–1140.

*Jeremy Gray*

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