Sparse dominations and weighted variation inequalities for singular integrals and commutators

TL;DR

This paper establishes pointwise sparse domination and weighted inequalities for variation operators of singular integrals and commutators, providing new quantitative bounds and decay estimates.

Contribution

It introduces novel pointwise sparse domination results and weighted bounds for variation operators, extending the theory to commutators and unweighted cases.

Findings

Strong type quantitative weighted bounds for variation operators.

Weak-type quantitative weighted bounds for singular integrals.

Local exponential decay estimates for variation operators.

Abstract

This paper gives the pointwise sparse dominations for variation operators of singular integrals and commutators with kernels satisfying the $L^r$-H\"{o}rmander conditions. As applications, we obtain the strong type quantitative weighted bounds for such variation operators as well as the weak-type quantitative weighted bounds for the variation operators of singular integrals and the quantitative weighted weak-type endpoint estimates for variation operators of commutators, which are completely new even in the unweighted case. In addition, we also obtain the local exponential decay estimates for such variation operators.