Authors: Jun Cao, School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China(caojun1860@mail.bnu.edu.cn), Yu Liu, Department of Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, People's Republic of China (pkyuliu@yahoo.com.cn) and Dachun Yang (Corresponding author), School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China (dcyang@bnu.edu.cn).
Hardy spaces H_L¹(Rⁿ) associated to Schrödinger Type Operators (-Δ)²+V², pp. 1067-1095.
ABSTRACT. Let L=(-Δ)²+V² be the Schrödinger Type Operator in Rⁿ with n≥ 5, where the nonnegative potential V belongs to the reverse Hölder class B_q0(Rⁿ) with q₀ ∈ (n/2, ∞). In this paper, the authors introduce the Hardy spaces H_L¹(Rⁿ), which are defined in terms of radial maximal functions associated with the heat semigroup {e^-tL}_t>0, and establish their atomic decomposition characterizations. As applications, the authors show that the Hardy space H_L¹(Rⁿ) coincides with the Hardy space H¹_-Δ+V(Rⁿ) with equivalent norms. Moreover, the authors also prove that the higher order Riesz transform ∇²(L^-1/2) is bounded from H_L¹(Rⁿ) to the classical Hardy space H¹(Rⁿ) and the corresponding fractional integral L^-α/4 for α ∈ (0, n) maps H_L¹(Rⁿ) continuously into the Lebesgue space L^n/(n-α)(Rⁿ).