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].
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Cao, J., Yang, D. & Yang, S. Endpoint boundedness of Riesz transforms on Hardy spaces associated with operators. Rev Mat Complut 26, 99–114 (2013). https://doi.org/10.1007/s13163-011-0092-5
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DOI: https://doi.org/10.1007/s13163-011-0092-5
Keywords
- Riesz transform
- Davies-Gaffney estimate
- Schrödinger operator
- Second order elliptic operator
- Hardy space
- Weak Hardy space