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
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
The notion of smoothness which appears in 1945 is a generalization of
symmetric continuity and
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.