Regularities of semigroups, Carleson measures and the characterizations of BMO-type spaces associated with generalized Schrödinger operators

Abstract

Let $\mathcal{L}=-\Delta +\mu$ be the generalized Schrödinger operator on ${\mathbb{R}}^n,n\ge 3$, where $\Delta$ is the Laplacian and $\mu \not\equiv 0$ is a nonnegative Radon measure on ${\mathbb{R}}^n$. In this article, we introduce two families of Carleson measures $\left\{d{\nu }_{h,k}\right\}$ and $\left\{d{\nu }_{P,k}\right\}$ generated by the heat semigroup $\left\{e^{-t\mathcal{L}}\right\}$ and the Poisson semigroup $\left\{e^{-t\sqrt{\mathcal{L}}}\right\}$, respectively. By the regularities of semigroups, we establish the Carleson measure characterizations of BMO-type spaces ${\mathrm{BMO}}_{\mathcal{L}}\left({\mathbb{R}}^n\right)$ associated with the generalized Schrödinger operators.