Endpoint Properties of Localized Riesz Transforms and Fractional Integrals Associated to Schrödinger Operators

摘要

Let L º -D+V{mathcal L}equiv-Delta+V be the Schrödinger operator in mathbb Rn{{mathbb R}^n}, where V is a nonnegative function satisfying the reverse Hölder inequality. Let ρ be an admissible function modeled on the known auxiliary function determined by V. In this paper, the authors characterize the localized Hardy spaces H1_r(mathbb Rn)H^1_rho({{mathbb R}^n}) in terms of localized Riesz transforms and establish the boundedness on the BMO-type space mathopBMO_r(mathbb Rn){mathopmathrm{BMO_rho({mathbb R}^n)}} of these operators as well as the boundedness from mathopBMO_r(mathbb Rn){mathopmathrm{BMO_rho({mathbb R}^n)}} to mathopBLO_r(mathbb Rn){mathopmathrm{BLO_rho({mathbb R}^n)}} of their corresponding maximal operators, and as a consequence, the authors obtain the Fefferman–Stein decomposition of mathopBMO_r(mathbb Rn){mathopmathrm{BMO_rho({mathbb R}^n)}} via localized Riesz transforms. When ρ is the known auxiliary function determined by V, mathopBMO_r(mathbb Rn){mathopmathrm{BMO_rho({mathbb R}^n)}} is just the known space mathopBMO_L(mathbb Rn)mathopmathrm{BMO}_{mathcal L}({{mathbb R}^n}), and mathopBLO_r(mathbb Rn){mathopmathrm{BLO_rho({mathbb R}^n)}} in this case is correspondingly denoted by mathopBLO_L(mathbb Rn)mathopmathrm{BLO}_{mathcal L}({{mathbb R}^n}). As applications, when n ≥ 3, the authors further obtain the boundedness on mathopBMO_L(mathbb Rn)mathopmathrm{BMO}_{mathcal L}({{mathbb R}^n}) of Riesz transforms ÑL-1/2nabla{mathcal L}^{-1/2} and their adjoint operators, as well as the boundedness from mathopBMO_L(mathbb Rn)mathopmathrm{BMO}_{mathcal L}({{mathbb R}^n}) to mathopBLO_L(mathbb Rn)mathopmathrm{BLO}_{mathcal L}({{mathbb R}^n}) of their maximal operators. Also, some endpoint estimates of fractional integrals associated to L{mathcal L} are presented.