Inequalities for weighted spaces with variable exponents

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 $H_{\omega }^{p(\cdot )}({\mathbb{R}}^n)$ into $L_{\omega }^{q(\cdot )}({\mathbb{R}}^n)$, for $\frac{1}{p(\cdot )}:=\frac{1}{q(\cdot )}+\frac{\alpha }{n}$.