The Polynomial Carleson operator

• 成果类型: Article
• 署名作者: Lie, Victor
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2020.192.1.2
• 发表日期: 2020
• 页码: 47-163
• 关键词: bilinear hilbert-transforms; singular-integrals; pointwise convergence; harmonic-analysis; fourier-series; maximal functions; ergodic averages; norm convergence; valued inequalities; radon transforms

摘要

We prove affirmatively the one-dimensional case of a conjecture of Stein regarding the L-p-boundedness of the Polynomial Carleson operator for 1 < p < infinity. 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.