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July 2020 The Polynomial Carleson operator
Victor Lie
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Ann. of Math. (2) 192(1): 47-163 (July 2020). DOI: 10.4007/annals.2020.192.1.2

Abstract

We prove affirmatively the one-dimensional case of a conjecture of Stein regarding the $L^p$-boundedness of the Polynomial Carleson operator for $1 \lt p\lt \infty$.

Our proof relies on two new ideas: (i) we develop a framework for higher-order wave-packet analysis that is consistent with the time-frequency analysis of the (generalized) Carleson operator, and (ii) we introduce a 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---strong $L^2$ bound for the (generalized) Carleson operator, answering a question raised by C. Fefferman.

Citation

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Victor Lie. "The Polynomial Carleson operator." Ann. of Math. (2) 192 (1) 47 - 163, July 2020. https://doi.org/10.4007/annals.2020.192.1.2

Information

Published: July 2020
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2020.192.1.2

Subjects:
Primary: 42A20 , 42A50

Keywords: Carleson's theorem , higher-order wave-packet analysis , Polynomial Carleson operator , time-frequency analysis

Rights: Copyright © 2020 Department of Mathematics, Princeton University

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Vol.192 • No. 1 • July 2020
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