Sharp Lp estimates for Schrödinger groups on spaces of homogeneous type

The Anh Bui, Piero D'Ancona, Fabio Nicola

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    4 Citations (Scopus)

    Abstract

    We prove an Lp estimate (Equation Presented) for the Schrödinger group generated by a semibounded, self-adjoint operator L on a metric measure space X of homogeneous type (where n is the doubling dimension of X). The assumptions on L are a mild Lp0 → Lp'0 smoothing estimate and a mild L2 → L2 off-diagonal estimate for the corresponding heat kernel e-tL. The estimate is uniform for φ varying in bounded sets of S(R), or more generally of a suitable weighted Sobolev space. We also prove, under slightly stronger assumptions on L, that the estimate extends to (Equation Presented) with uniformity also for θ varying in bounded subsets of (0,+∞). For nonnegative operators uniformity holds for all θ > 0.

    Original languageEnglish
    Pages (from-to)455-484
    Number of pages30
    JournalRevista Matematica Iberoamericana
    Volume36
    Issue number2
    DOIs
    Publication statusPublished - 2020

    Keywords

    • Schrödinger group
    • metric measure spaces
    • doubling measure
    • spectral multipliers
    • heat kernels

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