Carleson's theorem with quadratic phase functions

Volume 153 / 2002

Michael T. Lacey Studia Mathematica 153 (2002), 249-267 MSC: 42B20, 42A20. DOI: 10.4064/sm153-3-3


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].


  • Michael T. LaceySchool of Mathematics
    Georgia Institute of Technology
    Atlanta, GA 30332, U.S.A.

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