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

Abstract

Let ${\mathcal L}\equiv-\Delta+V$ be the Schrödinger operator in ${{\mathbb R}^n}$, 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 $H^1_\rho({{\mathbb R}^n})$ in terms of localized Riesz transforms and establish the boundedness on the BMO-type space ${\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}$ of these operators as well as the boundedness from ${\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}$ to ${\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}$ of their corresponding maximal operators, and as a consequence, the authors obtain the Fefferman–Stein decomposition of ${\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}$ via localized Riesz transforms. When ρ is the known auxiliary function determined by V, ${\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}$ is just the known space $\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})$, and ${\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}$ in this case is correspondingly denoted by $\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})$. As applications, when n ≥ 3, the authors further obtain the boundedness on $\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})$ of Riesz transforms $\nabla{\mathcal L}^{-1/2}$ and their adjoint operators, as well as the boundedness from $\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})$ to $\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})$ of their maximal operators. Also, some endpoint estimates of fractional integrals associated to ${\mathcal L}$ are presented.