The Polynomial Carleson Operator

We prove affirmatively the one dimensional case of a conjecture of Stein regarding the $L^p$-boundedness of the Polynomial Carleson operator, for $1<p<\infty$. The proof is based on two new ideas: i) developing a framework for \emph{higher-order wave-packet analysis} that is consistent with the time-frequency analysis of the (generalized) Carleson operator, and ii) a new tile discretization of the time-frequency plane that has the major consequence of \emph{eliminating the exceptional sets} from the analysis of the Carleson operator. As a further consequence, we are able to provide 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.