Endpoint Boundedness of Riesz Transforms on Hardy Spaces Associated with Operators

Let $L_1$ be a nonnegative self-adjoint operator in $L^2({\mathbb{R}}^n)$ satisfying the Davies-Gaffney estimates and $L_2$ a second order divergence form elliptic operator with complex bounded measurable coefficients. A typical example of $L_1$ is the Schr"odinger operator $-\mathrm{\Delta }+V$, where $\mathrm{\Delta }$ is the Laplace operator on ${\mathbb{R}}^n$ and $0\le V\in L^1_{\mathop$\mathrm{l}\mathrm{o}\mathrm{c}$} ({\mathbb R}^n)$\mathrm{.}Let$H^p_{L_i}($\mathbb{R}$^n)$ be the Hardy space associated to $L_i$ for $i\in \{1,2\}$. In this paper, the authors prove that the Riesz transform $D(L_i^{-1\mathrm{/}2})$ is bounded from $H_{L_i}^p({\mathbb{R}}^n)$ to the classical weak Hardy space $W{H}^p({\mathbb{R}}^n)$ in the critical case that $p=n\mathrm{/}(n+1)$. Recall that it is known that $D(L_i^{-1\mathrm{/}2})$ is bounded from $H_{L_i}^p({\mathbb{R}}^n)$ to the classical Hardy space $H^p({\mathbb{R}}^n)$ when $p\in (n\mathrm{/}(n+1),1]$.