Riesz Transform Characterizations of Hardy Spaces Associated to Degenerate Elliptic Operators

TL;DR

This paper characterizes Hardy spaces associated with degenerate elliptic operators using Riesz transforms, extending classical results to weighted and degenerate contexts under specific conditions.

Contribution

It establishes Riesz transform characterizations of Hardy spaces linked to degenerate elliptic operators with weights in certain Muckenhoupt classes, under new off-diagonal estimate conditions.

Findings

Riesz transform characterizations hold for specified weighted Hardy spaces.

Conditions on weights and off-diagonal estimates are crucial for the results.

Extends classical Hardy space theory to degenerate elliptic operators with weights.

Abstract

Let $w$ be a Muckenhoupt $A_2(\mathbb{R}^n)$ weight and $L_w:=-w^{-1}\mathop\mathrm{div}(A\nabla)$ the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$. In this article, the authors establish the Riesz transform characterization of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$, for $w\in A_{q}(\mathbb{R}^n)$ and $w^{-1}\in A_{2-\frac{2}{n}}(\mathbb{R}^n)$ with $n\geq 3$, $q\in[1,2]$ and $p\in(q(\frac{1}{r}+\frac{q-1}{2}+\frac{1}{n})^{-1},1]$ if, for some $r\in[1,\,2)$, $\{tL_w e^{-tL_w}\}_{t\geq 0}$ satisfies the weighted $L^r-L^2$ full off-diagonal estimate.