Riesz Transform Characterizations of Hardy Spaces Associated to Degenerate Elliptic Operators

Let $w$ be a Muckenhoupt $A_2({\mathbb{R}}^n)$ weight and $L_w:=-w^{-1}(A\mathrm{\nabla })$ 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,forsome$r\in[1,,2)$,${tL_w e^{-tL_w}}_{t\geq 0}$satisfiestheweighted$L^r-L^2$ full off-diagonal estimate.