Endpoint Boundedness of Riesz Transforms on Hardy Spaces Associated with Operators

Abstract

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 Schrödinger operator $-\hat{I}”+V$, where $\hat{I}”$ is the Laplace operator on ${\mathbb{R}}^n$ and $0\le V\in L_{\mathrm{loc}}^1({\mathbb{R}}^n)$. Let $H_{L_i}^p({\mathbb{R}}^n)$ 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 $H_{L_i}^p({\mathbb{R}}^n)$ to the classical weak Hardy space $W{H}^p({\mathbb{R}}^n)$ in the critical case that $p=n/(n+1)$. Recall that it is known that $D(L_i^{-1/2})$ is bounded from $H_{L_i}^p({\mathbb{R}}^n)$ to the classical Hardy space $H^p({\mathbb{R}}^n)$ when $p\in (n/(n+1),1]$.

Rev. Mat. Complut. (to appear)