Carleson measures, BMO spaces and balayages associated to Schrodinger operators

Abstract

Let $Ł$ be a Schrödinger operator of the form $Ł=-\Delta +V$ acting on $L^2({\mathbb{R}}^n)$, $n\ge 3$, where the nonnegative potential $V$ belongs to the reverse Hölder class $B_q$ for some $q\ge n.$ Let ${\rm BMO}_{\mathcal{L}}(\RR)$ denote the BMO space associated to the Schrödinger operator $Ł$ 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 $\|g\|_{\infty} +\||μ\||_{c}\leq C \|f\|_{{\rm BMO}_{\mathcal{L}}(\RR)},$ 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 $\{e^{-t\sqrt{Ł}}{\}}_{t>0}$ 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 Schrödinger operators.

arXiv comment17 pages