Endpoint Boundedness of Riesz Transforms on Hardy Spaces Associated with Operators (1107.5097v3)
Abstract: Let $L_1$ be a nonnegative self-adjoint operator in $L2({\mathbb R}n)$ 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 $-\Delta+V$, where $\Delta$ is the Laplace operator on ${\mathbb R}n$ and $0\le V\in L1_{\mathop\mathrm{loc}} ({\mathbb R}n)$. Let $Hp_{L_i}(\mathbb{R}n)$ be the Hardy space associated to $L_i$ for $i\in{1,\,2}$. In this paper, the authors prove that the Riesz transform $D (L_i{-1/2})$ is bounded from $Hp_{L_i}(\mathbb{R}n)$ to the classical weak Hardy space $WHp(\mathbb{R}n)$ in the critical case that $p=n/(n+1)$. Recall that it is known that $D (L_i{-1/2})$ is bounded from $Hp_{L_i}(\mathbb{R}n)$ to the classical Hardy space $Hp(\mathbb{R}n)$ when $p\in(n/(n+1),\,1]$.