Fourier Transform of Variable Anisotropic Hardy Spaces with Applications to Hardy-Littlewood Inequalities

Abstract

Let $p(\cdot ):{\mathbb{R}}^n\to (0,1]$ be a variable exponent function satisfying the globally log-Hölder continuous condition and $A$ a general expansive matrix on ${\mathbb{R}}^n$. Let $H_A^{p(\cdot )}({\mathbb{R}}^n)$ be the variable anisotropic Hardy space associated with $A$ defined via the radial maximal function. In this article, via the known atomic characterization of $H_A^{p(\cdot )}({\mathbb{R}}^n)$ and establishing two useful estimates on anisotropic variable atoms, the author shows that the Fourier transform $\hat{f}$ of $f\in H_A^{p(\cdot )}({\mathbb{R}}^n)$ coincides with a continuous function $F$ in the sense of tempered distributions, and $F$ satisfies a pointwise inequality which contains a step function with respect to $A$ as well as the Hardy space norm of $f$. As applications, the author also obtains a higher order convergence of the continuous function $F$ at the origin. Finally, an analogue of the Hardy--Littlewood inequality in the variable anisotropic Hardy space setting is also presented. All these results are new even in the classical isotropic setting.

18 pages. arXiv admin note: text overlap with arXiv:2006.11509