Endpoint boundedness of Riesz transforms on Hardy spaces associated with operators

摘要

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 (0le Vin 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].