The pointwise convergence of Fourier Series (II). Strong L¹ case for the lacunary Carleson operator

2019/02/10 by Lie, Victor
#42A20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.03630

Abstract

We prove that the lacunary Carleson operator is bounded from L log L to L¹. This result is sharp. The proof is based on two newly introduced concepts: 1) the time-frequency regularization of a measurable set and 2) the set-resolution of the time-frequency plane at 0-frequency. These two concepts will play the central role in providing a special tile decomposition adapted to the interaction between the structure of the lacunary Carleson operator and the corresponding structure of a fix measurable set. Another key insight of our paper is that it provides for the first time a simultaneous treatment of families of tiles with distinct mass parameters. This should be regarded as a fundamental feature/difficulty of the problem of the pointwise convergence of Fourier Series near L¹, context in which, unlike the standard L^p, p>1 case, no decay in the mass parameter is possible.

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• On the Boundedness of the Carleson Operator near L¹