Spectral multipliers via resolvent type estimates on non-homogeneous metric measure spaces

Peng Chen, Adam Sikora*, Lixin Yan

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    Abstract

    We describe a simple but surprisingly effective technique of obtaining spectral multiplier results for abstract operators which satisfy the finite propagation speed property for the corresponding wave equation propagator. We show that, in this setting, spectral multipliers follow from resolvent or semigroup type estimates. The most notable point of the paper is that our approach is very flexible and can be applied even if the corresponding ambient space does not satisfy the doubling condition or if the semigroup generated by an operator is not uniformly bounded. As a corollary we obtain Lp spectrum independence for several second order differential operators and recover some known results. Our examples include the Laplace–Belltrami operator on manifolds with ends and Schrödinger operators with strongly subcritical potentials.

    Original languageEnglish
    Pages (from-to)555-570
    Number of pages16
    JournalMathematische Zeitschrift
    Volume294
    Issue number1-2
    Early online date8 Apr 2019
    DOIs
    Publication statusPublished - Feb 2020

    Keywords

    • Spectral multipliers
    • Non-homogeneous type spaces
    • Finite propagation speed property
    • Lᴾ spectrum independence
    • Manifolds with ends
    • Schrödinger operators

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