Carleson measures, BMO spaces and balayages associated to Schrödinger operators

Peng Chen, Xuan Thinh Duong, Ji Li, Liang Song, LiXin Yan*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    2 Citations (Scopus)

    Abstract

    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 ƒ ∈ BMOL(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 Pt(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 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.

    Original languageEnglish
    Pages (from-to)2077–2092
    Number of pages16
    JournalScience China Mathematics
    Volume60
    Issue number11
    DOIs
    Publication statusPublished - Nov 2017

    Keywords

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

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