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

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

17 pages. arXiv admin note: text overlap with arXiv:2112.08320