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

TL;DR

This paper introduces variable exponent Calderón-Hardy spaces and proves that the iterated Laplacian operator provides a bijective correspondence between these spaces and variable exponent Hardy spaces, solving related PDEs.

Contribution

It defines Calderón-Hardy spaces with variable exponents and establishes the bijective mapping of the Laplacian operator, extending classical harmonic analysis results.

Findings

The operator elta^m is bijective from alderfron-Hardy spaces to Hardy spaces.

The paper characterizes the structure of alderfron-Hardy spaces with variable exponents.

It provides a framework for solving elta^m F = f for f in variable exponent Hardy spaces.

Abstract

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