文摘
Let L 1 be a nonnegative self-adjoint operator in L 2(?sup class="a-plus-plus"> 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?dinger operator ?οV, where Δ is the Laplace operator on ?sup class="a-plus-plus"> n and $0\le V\in L^{1}_{\mathop{\mathrm{loc}}} ({\mathbb{R}}^{n})$ . Let $H^{p}_{L_{i}}(\mathbb{R}^{n})$ be the Hardy space associated to L i for i∈{1,-}. In this paper, the authors prove that the Riesz transform $D (L_{i}^{-1/2})$ is bounded from $H^{p}_{L_{i}}(\mathbb{R}^{n})$ to the classical weak Hardy space WH p (?sup class="a-plus-plus"> n ) in the critical case that p=n/(n+1). Recall that it is known that $D(L_{i}^{-1/2})$ is bounded from $H^{p}_{L_{i}}(\mathbb{R}^{n})$ to the classical Hardy space H p (?sup class="a-plus-plus"> n ) when p?n/(n+1),-].