On the flows associated to selfadjoint operators on metric measure spaces

The Anh Bui, Piero D'Ancona, Xuan Thinh Duong, Detlef Müller

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    4 Citations (Scopus)

    Abstract

    Let X be a metric space with a doubling measure satisfying μ(B)≳rBn for any ball B with any radius rB> 0. Let L be a non negative selfadjoint operator on L2(X). We assume that e-tL satisfies a Gaussian upper bound and that the flow eitL satisfies a typical L1- L dispersive estimate of the form

    ‖eitLL1→L≲|t|-n/2.

    Then we prove a similar L1- L dispersive estimate for a general class of flows eitϕ(L), with φ(r) of power type near 0 and near ∞. In the case of fractional powers φ(L) = Lν, ν∈ (0 , 1) , we deduce dispersive estimates for eitLν with data in Sobolev, Besov or Hardy spaces HLp with p∈ (0 , 1] , associated to the operator L.

    Original languageEnglish
    Pages (from-to)1393-1426
    Number of pages34
    JournalMathematische Annalen
    Volume375
    Issue number3-4
    Early online date15 Jun 2019
    DOIs
    Publication statusPublished - Dec 2019

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