On weighted Compactness of Commutator of semi-group maximal function associated to Schrödinger operators

Abstract

Let ${\mathcal{T}}^{\ast }$ be the semi-group maximal function associated to the Schrödinger operator $-\Delta +V(x)$ with $V$ satisfying an appropriate reverse Hölder inequality. In this paper, we show that the commutator of ${\mathcal{T}}^{\ast }$ is a compact operator on $L^p(w)$ for $1<p<\infty$ if $b\in {\text{CMO}}_{\theta }(\rho )({\mathbb{R}}^n)$ and $w\in A_p^{\rho ,\theta }({\mathbb{R}}^n)$. Here ${\text{ CMO}}_{\theta }(\rho )({\mathbb{R}}^n)$ denotes the closure of ${\mathcal{C}}_c^{\infty }({\mathbb{R}}^n)$ in the ${\text{BMO}}_{\theta }(\rho )({\mathbb{R}}^n)$ (which is larger than the classical $\text{BMO}({\mathbb{R}}^n)$ space) topology. The space where $b$ belongs and the weighs class $w$ belongs are more larger than the usual $\text{CMO}({\mathbb{R}}^n)$ space and the Muckenhoupt $A_p$ weights class, respectively.

arXiv admin note: text overlap with arXiv:2012.12747