The Polynomial Carleson operator

Pages 47-163 from Volume 192 (2020), Issue 1 by Victor Lie

Abstract

We prove affirmatively the one-dimensional case of a conjecture of Stein regarding the $L^p$-boundedness of the Polynomial Carleson operator for $1\lt p\lt \infty $. \par Our proof relies on two new ideas: (i) we develop a framework for \emph higher-order wave-packet analysis that is consistent with the time-frequency analysis of the (generalized) Carleson operator, and (ii) we introduce a \emph local analysis adapted to the concepts of mass and counting function, which yields a new tile discretization of the time-frequency plane that has the major consequence of eliminating the exceptional sets from the analysis of the Carleson operator. As a further consequence, we are able to deliver the full $L^p$-boundedness range and prove directly—without interpolation techniques—the strong $L^2$ bound for the (generalized) Carleson operator, answering a question raised by C. Fefferman.

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Keywords
, , ,
Mathematical Subject Classification
Primary: ,
DOI
https://doi.org/10.4007/annals.2020.192.1.2
MR
4125450
zbMATH
07239272
Milestones
Received: 1 August 2017
Revised: 14 May 2020
Accepted: 12 November 2019
Published online: 17 July 2020

Authors

Victor Lie

Department of Mathematics, Purdue University, West Lafayette, IN 46907, USA
and
Institute of Mathematics of the Romanian Academy, Bucharest, RO 70700, P.O.Box 1-764, Romania