Endpoint boundedness of Riesz transforms on Hardy spaces associated with operators

Abstract

Let L 1 be a nonnegative self-adjoint operator in L 2(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 −Δ+V, where Δ is the Laplace operator on ℝ 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, 2}. 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 (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 $H^{p}_{L_{i}}(\mathbb{R}^{n})$ to the classical Hardy space H p (ℝ n ) when p∈(n/(n+1), 1].