Let\(\mathcal{L} \equiv - \Delta + V\) be the Schrödinger operator in ℝⁿ, whereV is a nonnegative function satisfying the reverse Hölder inequality. Let ρ be an admissible function modeled on the known auxiliary function determined byV. In this paper, the authors establish several characterizations of the space BMO_ρ(ℝⁿ) in terms of commutators of several different localized operators associated to ρ, 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\(\mathcal{L} \equiv - \Delta + V\) introduced by J. Dziubański, G. Garrigóset al.

中文翻译：

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通过局部Riesz变换的换向子和与Schrödinger算子相关的分数积分来表征局部BMO（ℝn）

让\（\ mathcal {L} \当量- \德尔塔+ V \）是薛定谔操作者在ℝ ^Ñ，其中V是满足反向Hölder不等式非负函数。设ρ为根据V确定的已知辅助函数建模的可允许函数。在本文中，作者建立空间BMO的几个刻画_ρ（ℝ ^ñ）分别对应于ρ的几个不同局部算子的换向器; 这些局部算子包括局部Riesz变换及其伴随算子，局部分数积分及其伴随算子，局部分数最大算子和局部Hardy-Littlewood型最大算子。这些结果甚至对于由J.Dziubański，G。Garrigós等人引入的空间\（\ mathcal {L} \ equiv- \ Delta + V \）都是新的。