# Celebratio Mathematica

## Antoni Zygmund

### The contributions of Antoni Zygmundto real analysis

#### by Daniel Waterman

At the be­gin­ning of Zyg­mund’s ca­reer he wrote a few pa­pers on real ana­lys­is which were not dir­ectly re­lated to tri­go­no­met­ric series. It is in­ter­est­ing to note, however, that these pa­pers and the few such that he wrote later in life were al­ways joint, the prob­lems pre­sum­ably arising from a con­flu­ence of in­terests of the parties.

Typ­ic­al of these are the pa­per of 1923 with Si­er­piński show­ing that there ex­ist func­tions dis­con­tinu­ous on every set hav­ing the power of the con­tinuum and an­oth­er with Saks in 1924 on the sheaf of tan­gent lines from a point to a planar curve.1 An­oth­er such res­ult ap­pears in Saks’s The­ory of the In­teg­ral, noted (in the ori­gin­al French ver­sion) as Zyg­mund’s Lemma, and used in the proof of the Fun­da­ment­al The­or­em of the Cal­cu­lus for the Per­ron in­teg­ral.

Most of Zyg­mund’s con­tri­bu­tions are sub­sumed un­der the rub­ric of “dif­fer­en­ti­ation the­ory”, which is a nat­ur­al con­sequence of his primary in­terests in tri­go­no­met­ric series. The se­quence of par­tial sums ${s_n(x)}$ of the tri­go­no­met­ric Four­i­er series of a func­tion $f$ is trans­formed by a sum­mab­il­ity meth­od in­to a se­quence $\{\sigma_n(x)\}$ and the dif­fer­ence $\sigma_n (x)-f(x)$ is rep­res­en­ted by the con­vo­lu­tion of a sym­met­ric dif­fer­ence and a sum­mab­il­ity ker­nel. The simplest ex­ample is the case where the meth­od is con­ver­gence it­self and $\sigma_n(x)-f(x)$ is es­sen­tially the con­vo­lu­tion of $f(x+t)+f(x-t)-2f(x) \qquad\text{and}\qquad (\sin nt)/t .$ Sim­il­arly, his work in 1934 and his joint pa­per in 1935 (with Jessen and Mar­cinkiewicz) on the dif­fer­en­ti­ab­il­ity of mul­tiple in­teg­rals arose from prob­lems in the sum­mab­il­ity of double Four­i­er series.

In the study of tri­go­no­met­ric series, the con­jug­ate series is of great im­port­ance; in the case of the Four­i­er series of a func­tion, the as­so­ci­ated con­jug­ate func­tion was the cen­ter of much of Zyg­mund’s work. In 1929 he ex­ten­ded the res­ult of M. Riesz on the in­teg­rabil­ity of the con­jug­ate func­tion from the class $L^p$, $p > 1$, to the class $L\log L$, which proved to be an es­sen­tial in­gredi­ent of much of his later work. In the same pa­per he also in­tro­duced the spaces $L_{\varphi}$, which are now called Or­licz spaces al­though the pa­per of Or­licz ap­peared in 1931. He ex­plored the re­la­tion­ships between many de­riv­at­ives; the Peano, Schwarz, Borel, Rieman­ni­an and ap­prox­im­ate de­riv­at­ives were all con­sidered. In a par­tic­u­larly deep pa­per of 1936 with Mar­cinkiewicz, al­though most of the pa­per dealt with dif­fer­en­ti­ation, he defined the T in­teg­ral of Per­ron type. With the T in­teg­ral he solved a prob­lem of Den­joy by show­ing that if a tri­go­no­met­ric series con­verges every­where to a func­tion $f$, then the series is the T-Four­i­er series of $f$.

The no­tion of smooth­ness which ap­pears in 1945 is a gen­er­al­iz­a­tion of sym­met­ric con­tinu­ity and $\Lambda_{\ast}$ is a gen­er­al­iz­a­tion of the class of in­teg­rals of $\mathit{Lip}(1)$ func­tions. In this pa­per he also stud­ies frac­tion­al in­teg­rals and de­riv­at­ives. Many of these res­ults were gen­er­al­ized to $n$-di­men­sions with Calder­ón in 1954, and the no­tion of smooth­ness is ap­plied to ex­plain a res­ult of Salem in a pa­per with Mary Weiss in 1959. He re­turned to the ba­sic ques­tions in the 1945 pa­per in a series of deep pa­pers with E. M. Stein in 1961, 1964 and 1965.

In the 1952 pa­per with Calder­ón on the ex­ist­ence of sin­gu­lar in­teg­rals, a cov­er­ing lemma is of crit­ic­al im­port­ance. This lemma provides the basis for the de­com­pos­i­tion of a func­tion in­to its “good” and “bad” parts. These tech­niques are fea­tured in sev­er­al pa­pers pub­lished with Calder­ón in the 50’s and early 60’s. The most not­able may be the 1961 pa­per on loc­al prop­er­ties of solu­tions of el­lipt­ic PDE’s.