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 Schrödinger operator $L=-\Delta +V$ which is defined by: $g(f)(x)=\left({\int }_0^{\infty }\right|\frac{d}{dt}{e}^{-tL}(f)(x){|}^{2}tdt{)}^{1/2},$ 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 $b\in {\mathrm{CMO}}_{\theta }(\rho )$ and $w\in A_p^{\rho ,\theta }$, where ${\mathrm{CMO}}_{\theta }(\rho )$ is the closure of ${\mathcal{C}}_c^{\infty }({\mathbb{R}}^n)$ in the ${\mathrm{BMO}}_{\theta }(\rho )$ topology which is more larger than the classical $\mathrm{CMO}$ 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 ${\mathcal{T}}^{\ast }(f)(x)={\sup }_{t>0}|e^{-tL}f(x)|.$

16 pages. arXiv admin note: text overlap with arXiv:2102.02105