# The Calder\'on operator and the Stieltjes transform on variable Lebesgue spaces with weights

## TL;DR

This paper characterizes the weights for the boundedness of the Stieltjes transform and Calderón operator on weighted variable Lebesgue spaces with log-Hölder continuous exponents, extending classical results and providing applications to inequalities.

## Contribution

It introduces a unified Muckenhoupt-type condition for these operators on variable Lebesgue spaces with weights, extending previous constant exponent results.

## Key findings

- Established a single Muckenhoupt-type condition for boundedness.
- Extended classical results to variable exponent spaces with weights.
- Provided applications to Hilbert's inequality and integral operators.

## Abstract

We characterize the weights for the Stieltjes transform and the Calder\'on operator to be bounded on the weighted variable Lebesgue spaces $L_w^{p(\cdot)}(0,\infty)$, assuming that the exponent function $p(\cdot)$ is log-H\"older continuous at the origin and at infinity. We obtain a single Muckenhoupt-type condition by means of a maximal operator defined with respect to the basis of intervals $\{ (0,b) : b>0\}$ on $(0,\infty)$. Our results extend those in \cite{DMRO1} for the constant exponent $L^p$ spaces with weights. We also give two applications: the first is a weighted version of Hilbert's inequality on variable Lebesgue spaces, and the second generalizes the results in \cite{SW} for integral operators to the variable exponent setting.