Carleson measures, BMO spaces and balayages associated to Schrödinger operators

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 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 ƒ ∈ BMO_L(ℝⁿ) with compact support, then there exist g ∈ L^∞(ℝⁿ) and a finite Carleson measure μ such that

f(x)=g(x)+Sμ,P(x)

with ∥g∥_∞+|||μ|||_c⩽C∥f∥_BMOL(ℝⁿ)   ; where

Sμ,P=∫_ℝ+ⁿ⁺¹+Pt(x,y)dμ(y,t),

and P_t(x; y) is the kernel of the Poisson semigroup  {e^−t√L}_t>0 on L ²(ℝ ⁿ ).  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.