Endpoint boundedness of Riesz transforms on Hardy spaces associated with operators

Abstract

Let L ₁ be a nonnegative self-adjoint operator in L ²(ℝⁿ) 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 ℝⁿ 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(ℝⁿ) 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(ℝⁿ) when p∈(n/(n+1), 1].