Maximal polynomial modulations of singular integrals

TL;DR

This paper proves that maximal operators formed by modulating Calderón–Zygmund kernels with polynomial phases are bounded on L^p spaces, extending classical Carleson theorems to higher dimensions and polynomial phases.

Contribution

It establishes the boundedness of polynomial modulated maximal operators for Calderón–Zygmund kernels, extending previous Carleson theorems to multidimensional and polynomial phase settings.

Findings

Boundedness of polynomial modulated maximal operators on L^p

Extension of Carleson theorem to higher dimensions

Generalization to polynomial phase modulations

Abstract

Let $K$ be a standard H\"older continuous Calder\'on--Zygmund kernel on $\mathbb{R}^{\mathbf{d}}$ whose truncations define $L^2$ bounded operators. We show that the maximal operator obtained by modulating $K$ by polynomial phases of a fixed degree is bounded on $L^p(\mathbb{R}^{\mathbf{d}})$ for $1 < p < \infty$. This extends Sj\"olin's multidimensional Carleson theorem and Lie's polynomial Carleson theorem.