Carleson's theorem with quadratic phase functions

Michael T. Lacey ¹

¹ School of Mathematics Georgia Institute of Technology Atlanta, GA 30332, U.S.A.

Studia Mathematica, Tome 153 (2002) no. 3, pp. 249-267

Résumé

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].

Détail

DOI : 10.4064/sm153-3-3

Keywords: shown operator below maps itself infty sup hbox int x y right supremum alone gives famous theorem nbsp carleson pointwise convergence fourier series supremum alone observation nbsp nbsp stein nbsp method proof builds steins observation approach carlesons theorem jointly developed author nbsp nbsp thiele nbsp