Riesz Transform Characterizations of Hardy Spaces Associated to Degenerate Elliptic Operators

Abstract

Let $w$ be a Muckenhoupt $A_2({\mathbb{R}}^n)$ weight and $L_w:=-w^{-1}\mathrm{div}(A\nabla )$ 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 $n\ge 3$, $q\in [1,2]$ and $p\in (q(\frac{1}{r}+\frac{q-1}{2}+\frac{1}{n}{)}^{-1},1]$ if, for some $r\in [1,2)$, $\{t{L}_w{e}^{-t{L}_w}{\}}_{t\ge 0}$ satisfies the weighted $L^r-L^2$ full off-diagonal estimate.

28 pages; Submitted

Cited by in corpus (1)

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