Characterization of weights for the variable fractional maximal operator and weighted inequalities for variable fractional rough operators

TL;DR

This paper extends the characterization of weights for variable fractional maximal operators and introduces a new variable H"ormander-type condition for kernels, establishing weighted inequalities in variable Lebesgue spaces.

Contribution

It generalizes known results to variable fractional operators and introduces a novel variable H"ormander condition for kernels, leading to new weighted inequality results.

Findings

Characterized weights for variable fractional maximal operators.

Introduced a new variable H"ormander-type condition for kernels.

Proved weighted inequalities for operators with kernels satisfying this condition.

Abstract

We characterize the class of weights related to the boundedness of variable fractional maximal operator $M_{\beta(\cdot),r(\cdot)}$ on variable Lebesgue spaces. This extend previously known results, including those corresponding to the fractional operator $M_{\beta(\cdot),1}$. In addition, we introduce a class of kernels $K$ satisfying a new variable H\"ormander-type condition $H_{\beta(\cdot),r(\cdot)}$. For the fractional operator $T_{\beta(\cdot)}$ given by a kernel in $H_{\beta(\cdot),r(\cdot)}$, we prove a Coifman-Fefferman inequality and weighted inequalities in variable Lebesgue space. Finally, we provide examples of kernels in this variable H\"ormander class.