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Carleson measures, BMO spaces and balayages associated to Schrödinger operators

Research paper by Peng Chen, Xuan Thinh Duong; Ji Li; Liang Song; LiXin Yan

Indexed on: 22 Dec '17Published on: 01 Nov '17Published in: Science China Mathematics



Abstract

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