The Schrödinger equation in Lp spaces for operators with heat kernel satisfying Poisson type bounds

Peng Chen, Xuan Thinh Duong, Zhijie Fan, Ji Li, Lixin Yan

    Research output: Contribution to journalArticlepeer-review

    Abstract

    Let L be a non-negative self-adjoint operator acting on L2(X) where X is a space of homogeneous type with a dimension n. In this paper, we study sharp endpoint Lp-Sobolev estimates for the solution of the initial value problem for the Schrödinger equation i∂tu+Lu = 0 and show that for all f ∈ Lp(X), 1 < p < ∞, ∥eitL(I + L)−σnf∥p ≤ C(1 + |t|)σn∥f∥p, t ∈ R, σ ≥ |1/2 − 1/p|, where the semigroup e−tL generated by L satisfies a Poisson type upper bound.

    Original languageEnglish
    Pages (from-to)285-331
    Number of pages47
    JournalJournal of the Mathematical Society of Japan
    Volume74
    Issue number1
    DOIs
    Publication statusPublished - Jan 2022

    Keywords

    • sharp Lp estimate
    • Schrödinger equation
    • elliptic operator
    • heat kernel
    • space of homogeneous type

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