Musielak Orlicz bumps and Bloom type estimates for commutators of Calder\'on Zygmund and fractional integral operators on variable Lebesgue spaces via sparse operators

TL;DR

This paper establishes new weighted boundedness conditions for Calderón Zygmund operators and their commutators on variable Lebesgue spaces, extending Bloom type estimates using sparse domination techniques.

Contribution

It introduces Musielak Orlicz bump conditions for weights and broadens Bloom type estimates to variable Lebesgue spaces with sparse operator methods.

Findings

Weighted boundedness of Calderón Zygmund operators established

Bloom type estimates extended to fractional integral operators

Sparse domination techniques are effectively applied

Abstract

We obtain Musielak Orlicz bumps conditions on a pair of weights for the boundedness of Calder\'on Zygmund operators and their commutators between variable Lebesgue spaces with different weights. The symbols of the commutators belong to a wider class of functions. We also give Bloom type estimates for commutators of Calder\'on Zygmund and fractional integral operators in the variable Lebesgue context. The techniques involved in both type of results are related with the theory of sparse domination.