Sharp endpoint Lp estimates for Schrödinger groups

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

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    1 Citation (Scopus)

    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. Suppose that the heat operator e- t L satisfies the generalized Gaussian (p0,p′0)-estimates of order m for some 1 ≤ p0< 2. In this paper we prove sharp endpoint Lp-Sobolev bound for the Schrödinger group eitL, that is for every p∈(p0,p0) there exists a constant C= C(n, p) > 0 independent of t such that

    ∥(I+L)-seitLf∥p≤C(1+|t|)s‖f‖p, t∈R, s≥n|1/2-1/p|.

    As a consequence, the above estimate holds for all 1 < p< ∞ when the heat kernel of L satisfies a Gaussian upper bound. This extends classical results due to Feffermann and Stein, and Miyachi for the Laplacian on the Euclidean spaces Rn. We also give an application to obtain an endpoint estimate for Lp-boundedness of the Riesz means of the solutions of the Schrödinger equations.

    Original languageEnglish
    Pages (from-to)667-702
    Number of pages36
    JournalMathematische Annalen
    Volume378
    Issue number1-2
    DOIs
    Publication statusPublished - Oct 2020

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