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

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

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