BMO spaces associated to operators with generalised Poisson bounds on non-doubling manifolds with ends

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

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    Abstract

    Consider a non-doubling manifold with ends M=Rn♯Rm where Rn=Rn×Sm−n for m>n≥3. We say that an operator L has a generalised Poisson kernel if L generates a semigroup e−tL whose kernel pt(x,y) has an upper bound similar to the kernel of e−tΔ where Δ is the Laplace-Beltrami operator on M. An example for operators with generalised Gaussian bounds is the Schrödinger operator L=Δ+V where V is an arbitrary non-negative locally integrable potential. In this paper, our aim is to introduce the BMO space BMOL(M) associated to operators with generalised Poisson bounds which serves as an appropriate setting for certain singular integrals with rough kernels to be bounded from L(M) into this new BMOL(M). On our BMOL(M) spaces, we show that the John–Nirenberg inequality holds and we show an interpolation theorem for a holomorphic family of operators which interpolates between Lq(M) and BMOL(M). As an application, we show that the holomorphic functional calculus m(L) is bounded from L(M) into BMOL(M), and bounded on Lp(M) for 1<p<∞.

    Original languageEnglish
    Pages (from-to)114-184
    Number of pages71
    JournalJournal of Differential Equations
    Volume270
    DOIs
    Publication statusPublished - 5 Jan 2021

    Keywords

    • BMO spaces
    • Non-doubling manifold with ends
    • John-Nirenberg inequality
    • Interpolation
    • Singular integrals

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