Carleson measures, BMO spaces and balayages associated to Schrödinger operators

Abstract

Let L be a Schrödinger operator of the form L = −Δ+V acting on L ²(Rⁿ), n ≥ 3, where the nonnegative potential V belongs to the reverse Hölder class B ^q for some q ≥ n: Let BMO_L(Rⁿ) denote the BMO space associated to the Schrödinger operator L on Rⁿ. In this article, we show that for every f ∈ BMO_L(Rⁿ) with compact support, then there exist g ∈ L∞(Rⁿ) and a finite Carleson measure μ such that

$$f\left( x \right) = g\left( x \right) + {S_{\mu ,}}_P\left( x \right)$$

with ${\left\| g \right\|_\infty } + |||\mu ||{|_c} \leqslant C{\left\| f \right\|_{BM{O_L}\left( {{\mathbb{R}^n}} \right)}}$; where

$${S_{\mu ,P}} = \int_{\mathbb{R}_ + ^{n + 1}} {{P_t}\left( {x,y} \right)d\mu \left( {y,t} \right)} $$

, and P _t (x; y) is the kernel of the Poisson semigroup $\left\{ {{e^{ - t\sqrt L }}} \right\}t > 0$ on L ²(Rⁿ). Conversely, if μ is a Carleson measure, then S _μ;P belongs to the space BMO_L(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.