Riesz Transform Characterization of Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

TL;DR

This paper develops a Riesz transform characterization for Hardy spaces associated with ball quasi-Banach function spaces, extending classical results to a broad class of function spaces.

Contribution

It introduces new Hardy spaces of harmonic functions and establishes isomorphisms, enabling the first and higher order Riesz transform characterizations for these spaces.

Findings

First order Riesz transform characterization established

Higher order Riesz transform characterization obtained

Applicable to various classical and generalized Hardy spaces

Abstract

Let $X$ be a ball quasi-Banach function space satisfying some mild assumptions and $H_X(\mathbb{R}^n)$ the Hardy space associated with $X$. In this article, the authors introduce both the Hardy space $H_X(\mathbb{R}^{n+1}_+)$ of harmonic functions and the Hardy space $\mathbb{H}_X(\mathbb{R}^{n+1}_+)$ of harmonic vectors, associated with $X$, and then establish the isomorphisms among $H_X(\mathbb{R}^n)$, $H_{X,2}(\mathbb{R}^{n+1}_+)$, and $\mathbb{H}_{X,2}(\mathbb{R}^{n+1}_+)$, where $H_{X,2}(\mathbb{R}^{n+1}_+)$ and $\mathbb{H}_{X,2}(\mathbb{R}^{n+1}_+)$ are, respectively, certain subspaces of $H_X(\mathbb{R}^{n+1}_+)$ and $\mathbb{H}_X(\mathbb{R}^{n+1}_+)$. Using these isomorphisms, the authors establish the first order Riesz transform characterization of $H_X(\mathbb{R}^n)$. The higher order Riesz transform characterization of $H_X(\mathbb{R}^n)$ is also obtained. The results obtained in this article have a wide range of generality and can be applied to the classical Hardy space, the weighted Hardy space, the Herz-Hardy space, the Lorentz-Hardy space, the variable Hardy space, the mixed-norm Hardy space, the local generalized Herz-Hardy space, and the mixed-norm Herz-Hardy space.