Spectral multipliers without semigroup framework and application to random walks

Peng Chen, El Maati Ouhabaz, Adam Sikora*, Lixin Yan

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    1 Citation (Scopus)

    Abstract

    In this paper we prove spectral multiplier theorems for abstract self-adjoint operators on spaces of homogeneous type. We have two main objectives. The first one is to work outside the semigroup context. In contrast to previous works on this subject, we do not make any assumption on the semigroup. The second objective is to consider polynomial off-diagonal decay instead of exponential one. Our approach and results lead to new applications to several operators such as differential operators, pseudo-differential operators as well as Markov chains. In our general context we introduce a restriction type estimates à la Stein-Tomas. This allows us to obtain sharp spectral multiplier theorems and hence sharp Bochner-Riesz summability results in some situation. Finally, we consider the random walk on the integer lattice Zn and prove sharp Bochner-Riesz summability results similar to those known for the standard Laplacian on Rn.

    Original languageEnglish
    Pages (from-to)162-191
    Number of pages30
    JournalJournal des Mathematiques Pures et Appliquees
    Volume143
    DOIs
    Publication statusPublished - Nov 2020

    Keywords

    • Spectral multipliers
    • Polynomial off-diagonal decay kernels
    • Space of homogeneous type
    • Random walk

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