Characterizations of Localized BMO(${\mathbb R}^n$) via Commutators of Localized Riesz Transforms and Fractional Integrals Associated to Schr\"odinger Operators

Let ${\mathcal L}\equiv-\Delta+V$ be the Schr\"odinger operator in ${\mathbb R^n}$, where $V$ is a nonnegative function satisfying the reverse H\"older inequality. Let $\rho$ be an admissible function modeled on the known auxiliary function determined by $V$. In this paper, the authors establish several characterizations of the space ${\mathop\mathrm{BMO_\rho(\rn)}}$ in terms of commutators of several different localized operators associated to $\rho$, respectively; these localized operators include localized Riesz transforms and their adjoint operators, the localized fractional integral and its adjoint operator, the localized fractional maximal operator and the localized Hardy-Littlewood-type maximal operator.
These results are new even for the space ${\mathop\mathrm{BMO_{\mathcal L}(\rn)}}$ introduced by J. Dziuba\'nski, G. Garrig\'os et al.