End Point Estimate of Littlewood-Paley Operator Associated to the Generalized Schrödinger Operator

Abstract

Let $\mathcal{L}=-\Delta +\mu$ be the generalized Schrödinger operator on ${ℝ}^d,d\ge 3,$ where $\mu \ne 0$ is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. In this work, we give a new $\text{BMO}$ space associated to the generalized Schrödinger operator $\mathcal{L},\text{BM}{\text{O}}_{\theta ,\mathcal{L}}$ , which is bigger than the $\text{BMO}$ spaces related to the classical Schrödinger operators $\mathcal{A}=-\Delta +V$ , with $V$ a potential satisfying a reverse Hölder inequality introduced by Dziubański et al. in 2005. Besides, the boundedness of the Littlewood-Paley operators associated to $\mathcal{L}$ in $\text{BM}{\text{O}}_{\theta ,\mathcal{L}}$ also be proved.