On weighted Compactness of Commutators of square function and semi-group maximal function associated to Schrodinger operator

Abstract: In this paper, the object of our investigation is the following Littlewood-Paley square function $g$ associated with the Schr\"odinger operator $L=-\Delta +V$ which is defined by: $g(f)(x)=\Big(\int_{0}^{\infty}\Big|\frac{d}{dt}e^{-tL}(f)(x)\Big|^2tdt\Big)^{1/2},$ where $\Delta$ is the laplacian operator on $\mathbb{R}^n$ and $V$ is a nonnegative potential. We show that the commutators of $g$ are compact operators from $L^p(w)$ to $L^p(w)$ for $1<p<\infty$ if $b\in {\rm CMO}_\theta(\rho)$ and $w\in A_p^{\rho,\theta}$, where ${\rm CMO}_\theta(\rho)$ is the closure of $\mathcal{C}_c^\infty(\mathbb{R}^n)$ in the ${\rm BMO}_\theta(\rho)$ topology which is more larger than the classical ${\rm CMO}$ space and $A_p^{\rho,\theta}$ is a weights class which is more larger than Muckenhoupt $A_p$ weight class. An extra weight condition in a privious weighted compactness result is removed for the commutators of the semi-group maximal function defined by $\mathcal{T}^*(f)(x)=\sup_{t\>0}|e^{-tL}f(x)|.$