LAUSR

Let L be a Schrödinger operator of the form L = −Δ+V acting on L2(Rn), n ≥ 3, where the nonnegative potential V belongs to the reverse Hölder class Bq for some q ≥ n: Let BMOL(Rn) denote the BMO space associated to the Schrödinger operator L on Rn. In this article, we show that for every f ∈ BMOL(Rn) with compact support, then there exist g ∈ L∞(Rn) and a finite Carleson measure μ such that $$f\left( x \right) = g\left( x \right) + {S_{\mu ,}}_P\left( x \right)$$f(x)=g(x)+Sμ,P(x) with $${\left\| g \right\|_\infty } + |||\mu ||{|_c} \leqslant C{\left\| f \right\|_{BM{O_L}\left( {{\mathbb{R}^n}} \right)}}$$‖g‖∞+|||μ|||c≤C‖f‖BMOL(ℝn); where $${S_{\mu ,P}} = \int_{\mathbb{R}_ + ^{n + 1}} {{P_t}\left( {x,y} \right)d\mu \left( {y,t} \right)} $$Sμ,P=∫ℝ+n+1Pt(x,y)dμ(y,t), and Pt(x; y) is the kernel of the Poisson semigroup $$\left\{ {{e^{ - t\sqrt L }}} \right\}t > 0$${e−tL}t>0 on L2(Rn). Conversely, if μ is a Carleson measure, then Sμ;P belongs to the space BMOL(Rn). This extends the result for the classical John-Nirenberg BMO space by Carleson (1976) (see also Garnett and Jones (1982), Uchiyama (1980) and Wilson (1988)) to the BMO setting associated to Schrödinger operators.