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

TL;DR

This paper studies the Fourier transform of anisotropic mixed-norm Hardy spaces, proving it coincides with a continuous function and establishing Hardy-Littlewood inequalities in this setting.

Contribution

It introduces a new analysis of Fourier transforms in anisotropic mixed-norm Hardy spaces using atomic decomposition and establishes related inequalities.

Findings

Fourier transform of functions in $H_A^{\vec{p}}(\mathbb{R}^n)$ is continuous.

The Fourier transform can be controlled by the Hardy space norm and a step function.

Higher order convergence of the Fourier transform at the origin is achieved.

Abstract

Let $\vec{p}\in(0,1]^n$ be a $n$-dimensional vector and $A$ a dilation. Let $H_A^{\vec{p}}(\mathbb{R}^n)$ denote the anisotropic mixed-norm Hardy space defined via the radial maximal function. Using the known atomic characterization of $H_{A}^{\vec{p}}(\mathbb{R}^n)$ and establishing a uniform estimate for corresponding atoms, the authors prove that the Fourier transform of $f\in H_A^{\vec{p}}(\mathbb{R}^n)$ coincides with a continuous function $F$ on $\mathbb{R}^n$ in the sense of tempered distributions. Moreover, the function $F$ can be controlled pointwisely by the product of the Hardy space norm of $f$ and a step function with respect to the transpose matrix of $A$. As applications, the authors obtain a higher order of convergence for the function $F$ at the origin, and an analogue of Hardy--Littlewood inequalities in the present setting of $H_A^{\vec{p}}(\mathbb{R}^n)$.