The Fourier Transform of Anisotropic Hardy Spaces with Variable Exponents and Their Applications

TL;DR

This paper studies the Fourier transform of functions in anisotropic Hardy spaces with variable exponents, proving it is a continuous function and exploring convergence and inequalities relevant to these spaces.

Contribution

It establishes the continuity of the Fourier transform for functions in variable anisotropic Hardy spaces and derives related convergence and inequality results.

Findings

Fourier transform of Hardy space functions is a continuous tempered distribution.

Higher order convergence of the Fourier transform at the origin.

A variant of Hardy-Littlewood inequality for variable anisotropic Hardy spaces.

Abstract

Let $A$ be an expansive dilation on $\mathbb{R}^n$, and $p(\cdot):\mathbb{R}^n\rightarrow(0,\,\infty)$ be a variable exponent function satisfying the globally log-H\"{o}lder continuous condition. Let $\mathcal{H}^{p(\cdot)}_A({\mathbb {R}}^n)$ be the variable anisotropic Hardy space defined via the non-tangential grand maximal function. In this paper, the authors obtain that the Fourier transform of $f\in \mathcal{H}^{p(\cdot)}_A({\mathbb {R}}^n)$ coincides with a continuous function $F$ on $\mathbb{R}^n$ in the sense of tempered distributions. As applications, the authors further conclude a higher order convergence of the continuous function $F$ at the origin and then give a variant of the Hardy-Littlewood inequality in the setting of anisotropic Hardy spaces with variable exponents.