Calderón-Hardy Spaces with variable exponents and the Solution of the Equation ${\Delta }^{m}F=f$ for $f\in H^{p(\cdot )}({\mathbb{R}}^n)$

Abstract

In this article we define the Calderón-Hardy spaces with variable exponents on ${\mathbb{R}}^n$, ${\mathcal{H}}_{q,\gamma }^{p(.)}({\mathbb{R}}^n)$, and we show that for $m\in \mathbb{N}$ the operator ${\Delta }^m$ is a bijective mapping from ${\mathcal{H}}_{q,2m}^{p(.)}({\mathbb{R}}^n)$ onto $H^{p(.)}({\mathbb{R}}^n)$.