# Journal of theKorean Mathematical SocietyJKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc.

Published online August 1, 2022

Copyright © The Korean Mathematical Society.

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

Jun Liu, Yaqian Lu, and Mingdong Zhang

China University of Mining and Technology

### 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 $H_A^{\vec{p}}(\mathbb{R}^n)$.

Keywords: dilation, mixed-norm Hardy space, Fourier transform, Hardy-Littlewood inequality

MSC numbers: 42B35, 42B30, 42B10, 46E30