Resumen de On weighted compactness of commutators of square function and semi-group maximal function associated to Schrödinger operators
Shifen Wang, Chunmei Zhang, Qingying Xue
Let \Delta be the Laplacian operator on {\mathbb{R}}^n and V be a nonnegative potential satisfying an appropriate reverse Hölder inequality. The Littlewood–Paley square function g associated with the Schrödinger operator L=-\Delta +V is defined by:
\begin{aligned} g(f)(x)=\Big (\int _{0}^{\infty }\Big |\frac{d}{dt}e^{-tL}(f)(x)\Big |^2tdt\Big )^{1/2}. \end{aligned} In this paper, we show that the commutators of g are compact operators on L^p(w) for 1 p \infty if b \in \text{CMO}_{\theta}(p) and w \in A_{p}^{p, \theta}, where CMO_{\theta, \rho}(\mathbb{R}^n) denotes the closure of \mathcal{C}_c^\infty ({\mathbb{R}}^n) in the \text{BMO}_{\theta}(\rho) topology and A_{p}^{p, \theta} is a weighted class which is more larger than Muckenhoupt A_p weight class. An extra weight condition in a previous 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)|.
