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

TL;DR

This paper proves the weighted compactness of commutators of Littlewood-Paley square functions and semi-group maximal functions related to Schrödinger operators, extending previous results to larger function and weight classes.

Contribution

It introduces broader function spaces and weight classes for the compactness of commutators of these operators associated with Schrödinger operators.

Findings

Commutators are compact on weighted L^p spaces for functions in CMO_θ(ρ).

Extends compactness results to larger weight classes A_p^{ρ,θ}.

Removes extra weight conditions from previous compactness theorems.

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 $\mathcal{T}^*(f)(x)=\sup_{t>0}|e^{-tL}f(x)|.$