Inequalities for weighted spaces with variable exponents

TL;DR

This paper extends classical inequalities to weighted variable exponent spaces and demonstrates the boundedness of Riesz potentials between these spaces, advancing harmonic analysis in non-uniform settings.

Contribution

It develops an off-diagonal Fefferman-Stein inequality and proves Riesz potential boundedness on weighted variable exponent Hardy spaces.

Findings

Established off-diagonal Fefferman-Stein inequality for weighted variable exponent spaces.

Proved boundedness of Riesz potential operators between weighted variable Hardy and Lebesgue spaces.

Extended classical harmonic analysis results to more general variable exponent and weighted contexts.

Abstract

In this article we obtain an "off-diagonal" version of the Fefferman-Stein vector-valued maximal inequality on weighted Lebesgue spaces with variable exponents. As an application of this result and the atomic decomposition developed in [12] we prove, for certain exponents $q(\cdot)$ in $\mathcal{P}^{\log}(\mathbb{R}^{n})$ and certain weights $\omega$, that the Riesz potential $I_{\alpha}$, with $0 < \alpha < n$, can be extended to a bounded operator from $H^{p(\cdot)}_{\omega}(\mathbb{R}^{n})$ into $L^{q(\cdot)}_{\omega}(\mathbb{R}^{n})$, for $\frac{1}{p(\cdot)} := \frac{1}{q(\cdot)} + \frac{\alpha}{n}$.