Sparse domination results for compactness on weighted spaces

TL;DR

This paper establishes sparse domination techniques to prove compactness of Calderón-Zygmund operators and Haar multipliers on weighted L^p spaces, extending known classes of weights and providing new boundedness results.

Contribution

It introduces sparse bounds to demonstrate compactness of operators on weighted spaces and extends weighted bounds to larger weight classes.

Findings

Calderón-Zygmund operators are compact on weighted L^p spaces under new conditions.

Haar multipliers are bounded and compact on broader weighted spaces.

Weighted bounds are established for weights beyond the classical A_p class.

Abstract

By means of appropriate sparse bounds, we deduce compactness on weighted $L^p(w)$ spaces, $1<p<\infty$, for all Calder\'on-Zygmund operators having compact extensions on $L^2(\mathbb{R}^n)$. Similar methods lead to new results on boundedness and compactness of Haar multipliers on weighted spaces. In particular, we prove weighted bounds for weights in a class strictly larger than the typical $A_p$ class.