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

Abstract

Let $A$ be an expansive dilation on ${\mathbb{R}}^n$, and $p(\cdot ):{\mathbb{R}}^n\to (0,\infty )$ be a variable exponent function satisfying the globally log-Hölder continuous condition. Let ${\mathcal{H}}_A^{p(\cdot )}({\mathbb{R}}^n)$ 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 $f\in {\mathcal{H}}_A^{p(\cdot )}({\mathbb{R}}^n)$ 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.

15 pages. This is the final version