Riesz transform characterizations for multidimensional Hardy spaces

TL;DR

This paper characterizes multidimensional Hardy spaces associated with a self-adjoint operator using Riesz transforms, extending the theory to Bessel and Laguerre operators under specific heat semigroup assumptions.

Contribution

It provides a new characterization of Hardy spaces via Riesz transforms for a class of operators, including Bessel and Laguerre, under certain heat semigroup conditions.

Findings

Hardy space $H^1_L(X)$ characterized by Riesz transforms

Extension of Riesz transform characterization to Bessel and Laguerre operators

Applicable under specific assumptions on the heat semigroup $ ext{exp}(-tL)$

Abstract

We study Hardy space $H^1_L(X)$ related to a self-adjoint operator $L$ defined on Euclidean domain $X \subseteq \mathbb{R}^d$. Under certain assumptions on the heat semigroup $\exp(-tL)$ we prove characterization of $H^1_L(X)$ by the Riesz transforms related to $L$. As an application, we prove the Riesz transform characterization for multidimensional Bessel and Laguerre operators.