Carleson's theorem with quadratic phase functions
Volume 153 / 2002
Studia Mathematica 153 (2002), 249-267
MSC: 42B20, 42A20.
DOI: 10.4064/sm153-3-3
Abstract
It is shown that the operator below maps $L^p$ into itself for $1< p< \infty$. $$ Cf(x):=\sup_{a,b}\left| \hbox{p.v.}\int f(x-y)e^{i(ay^2+by)}{dy\over y}\right|. $$ The supremum over $b$ alone gives the famous theorem of L. Carleson [2] on the pointwise convergence of Fourier series. The supremum over $a$ alone is an observation of E. M. Stein [12]. The method of proof builds upon Stein's observation and an approach to Carleson's theorem jointly developed by the author and C. M. Thiele [7].
