On Stein's Conjecture on the Polynomial Carleson Operator

TL;DR

This paper proves that the polynomial Carleson operator $C_d$ is of strong type $(p,r)$ for certain $p$ and $r$, confirming Stein's conjecture for the case $1<p<2$ using advanced time-frequency analysis.

Contribution

It extends the relational time-frequency perspective to polynomial phases, establishing strong type bounds for $C_d$ and confirming Stein's conjecture in the $1<p<2$ range.

Findings

Proves $C_d$ is of strong type $(p,r)$ for $1<r<p< finite$.

Confirms Stein's conjecture for $1<p<2$.

Extends time-frequency analysis methods to polynomial phases.

Abstract

We prove that the generalized Carleson operator $C_d$ with polynomial phase function is of strong type $(p,r)$, $1<r<p<\infty$; this yields a positive answer in the $1<p<2$ case to a conjecture of Stein which asserts that for $1<p<\infty$ we have that $C_d$ is of strong type $(p,p)$. A key ingredient in this proof is the further extension of the {\it relational} time-frequency perspective (introduced in \cite{q}) to the general polynomial phase.