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

TL;DR

This paper studies the Fourier transform of variable anisotropic Hardy spaces with applications to Hardy-Littlewood inequalities, establishing new results on the transform's properties and convergence in this generalized setting.

Contribution

It introduces new Fourier transform properties for variable anisotropic Hardy spaces and extends Hardy-Littlewood inequalities to this context, even in classical cases.

Findings

Fourier transform of functions in variable anisotropic Hardy spaces coincides with a continuous tempered distribution.

Established pointwise inequalities involving the Fourier transform and Hardy space norms.

Proved higher order convergence of the Fourier transform at the origin.

Abstract

Let $p(\cdot):\ \mathbb{R}^n\to(0,1]$ be a variable exponent function satisfying the globally log-H\"{o}lder 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 $\widehat{f}$ 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.