Positive sparse domination of variational Carleson operators

TL;DR

This paper establishes positive sparse domination for the nonlocal variational Carleson operator, leading to improved weighted inequalities and strengthening existing $L^p$ estimates.

Contribution

It proves the dual form of the $r$-variation norm Carleson operator can be dominated by a positive sparse form, overcoming nonlocality challenges.

Findings

Strengthens $L^p$ estimates for the Carleson operator.

Provides quantitative weighted norm inequalities.

Extends sparse domination techniques to nonlocal operators.

Abstract

Due to its nonlocal nature, the $r$-variation norm Carleson operator $C_r$ does not yield to the sparse domination techniques of Lerner, Di Plinio and Lerner, Lacey. We overcome this difficulty and prove that the dual form to $C_r$ can be dominated by a positive sparse form involving $L^p$ averages. Our result strengthens the $L^p$ estimates by Oberlin et. al. As a corollary, we obtain quantitative weighted norm inequalities improving on previous results by Do and Lacey. Our proof relies on the localized outer $L^p$-embeddings of Di Plinio-Ou and Uraltsev.