Carleson measures, BMO spaces and balayages associated to Schrödinger operators

Abstract

Let $\L$ be a Schr\"odinger operator of the form $\L=-\Delta+V$ acting on $L^2(\mathbb R^n)$, $n\geq3$, where the nonnegative potential $V$ belongs to the reverse H\"older class $B_q$ for some $q\geq n.$ Let ${\rm BMO}_{{\mathcal{L}}}(\RR)$ denote the BMO space associated to the Schr\"odinger operator $\L$ on $\RR$. In this article we show that for every $f\in {\rm BMO}_{\mathcal{L}}(\RR)$ with compact support, then there exist $g\in L^{\infty}(\RR)$ and a finite Carleson measure $\mu$ such that $$ f(x)=g(x) + S_{\mu, {\mathcal P}}(x) $$ with $\|g\|_{\infty} +\||\mu\||_{c}\leq C \|f\|_{{\rm BMO}_{\mathcal{L}}(\RR)},$ where $$ S_{\mu, {\mathcal P}}=\int_{{\mathbb R}^{n+1}_+} {\mathcal P}_t(x,y) d\mu(y, t), $$ and ${\mathcal P}_t(x,y)$ is the kernel of the Poisson semigroup $\{e^{-t\sqrt{\L}}\}_{t> 0} $ on $L^2(\mathbb R^n)$. Conversely, if $\mu$ is a Carleson measure, then $S_{\mu, {\mathcal P}}$ belongs to the space ${\rm BMO}_{{\mathcal{L}}}(\RR)$. This extends the result for the classical John--Nirenberg BMO space by Carleson \cite{C} (see also \cite{U,GJ,W}) to the BMO setting associated to Schr\"odinger operators.