Spectral multipliers, Bochner–Riesz means and uniform Sobolev inequalities for elliptic operators

Adam Sikora, Lixin Yan, Xiaohua Yao

    Research output: Contribution to journalArticle

    6 Citations (Scopus)

    Abstract

    This article comprises two parts. In the first, we study Lp to Lq bounds for spectral multipliers and Bochner–Riesz means with negative index in the general setting of abstract self-adjoint operators. In the second, we obtain the uniform Sobolev estimates for constant coefficients higher order elliptic operators P(D)−z and all z∈ℂ∖[0,∞), which give an extension of the second order results of Kenig–Ruiz–Sogge [42]. Next we use perturbation techniques to prove the uniform Sobolev estimates for Schrödingier operators P(D)+V with small integrable potentials V. Finally, we deduce spectral multiplier applications for all these operators, including sharp Bochner–Riesz summability results.
    Original languageEnglish
    Pages (from-to)3070-3121
    Number of pages52
    JournalInternational Mathematics Research Notices
    Volume2018
    Issue number10
    Early online date21 Jan 2017
    DOIs
    Publication statusPublished - 18 May 2018

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