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Let $A$ be an expansive dilation on $\mathbb{R}^n$, and$p(\cdot):\mathbb{R}^n\rightarrow(0,\,\infty)$ be a variable exponent functionsatisfying the globally log-H\"{o}lder continuous condition. Let$\mathcal{H}^{p(\cdot)}_A({\mathbb {R}}^n)$ be the variable anisotropic Hardyspace defined via the non-tangential grand maximal function. In this paper, theauthors 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 continuousfunction $F$ at the origin and then give a variant of the Hardy-Littlewoodinequality in the setting of anisotropic Hardy spaces with variable exponents.

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