Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Xuan Thinh Duong*, Adam Sikora, Lixin Yan

*Corresponding author for this work

    Research output: Contribution to journalArticle

    17 Citations (Scopus)

    Abstract

    Let L be a non-negative self-adjoint operator acting on L2(X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e-tL whose kernels pt(x,y) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted Lp-norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L2 estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials.

    Original languageEnglish
    Pages (from-to)1106-1131
    Number of pages26
    JournalJournal of Functional Analysis
    Volume260
    Issue number4
    DOIs
    Publication statusPublished - 28 Feb 2011

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