## 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 BMO_{L}(ℝ^{n}) denote the BMO space associated to the Schrödinger operator L on ℝ^{n}. In this article, we show that for every ƒ ∈ BMO_{L}(ℝ^{n}) with compact support, then there exist g ∈ L^{∞}(ℝ^{n}) 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}(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.

Original language | English |
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Pages (from-to) | 2077–2092 |

Number of pages | 16 |

Journal | Science China Mathematics |

Volume | 60 |

Issue number | 11 |

DOIs | |

Publication status | Published - Nov 2017 |

## Keywords

- BMO space
- Carleson measure
- balayage
- Poisson semigroup
- the reverse Hölder class
- Schrödinger operators