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.
Categories: math.AP