On weighted Compactness of Commutator of semi-group maximal function associated to Schr\"odinger operators

Let $\mathcal{T}^*$ be the semi-group maximal function associated to the Schr\"odinger operator $-\Delta+V(x)$ with $V$ satisfying an appropriate reverse H\"{o}lder inequality. In this paper, we show that the commutator of $\mathcal{T}^*$ 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.