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