The polynomial Carleson operator. (English) Zbl 1446.42003

Lie, Victor

Ann. Math. (2) 192, No. 1, 47-163 (2020).

Summary: We prove affirmatively the one-dimensional case of a conjecture of E. M. Stein [J. Fourier Anal. Appl. Spec. Iss., 535–551 (1995; Zbl 0971.42009)] regarding the \(L^p\)-boundedness of the polynomial Carleson operator for \(1<p<\infty\).
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-the strong \(L^2\) bound for the (generalized) Carleson operator, answering a question raised by C. Fefferman [Ann. Math. (2) 98, 551–571 (1973; Zbl 0268.42009)].

MSC:

42A20	Convergence and absolute convergence of Fourier and trigonometric series
42A50	Conjugate functions, conjugate series, singular integrals
35S30	Fourier integral operators applied to PDEs

Keywords:

time-frequency analysis; Carleson’s theorem; polynomial Carleson operator; higher-order wave-packet analysis

Citations:

Zbl 0971.42009; Zbl 0268.42009

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