Smooth fourier multipliers in group algebras via sobolev dimension

Adrián Gonzalez-Perez, Marius Junge, Javier Parcet

Research output: Contribution to journalArticle

Abstract

We investigate Fourier multipliers with smooth symbols defined over locally com-pact Hausdorff groups. Our main results in this paper establish new Hormander-Mikhlin criteria for spectral and non-spectral multipliers. The key novelties which shape our approach are three. First, we control a broad class of Fourier multipliers by certain maximal operators in noncommutative Lp spaces. This general principle-exploited in Euclidean harmonic analysis during the last 40 years-is of independent interest and might admit further applications. Second, we replace the formerly used cocycle dimension by the Sobolev dimension. This is based on a noncommutative form of the Sobolev embedding theory for Markov semigroups initiated by Varopoulos, and yields more fiexibility to mea-sure the smoothness of the symbol. Third, we introduce a dual notion of polynomial growth to further exploit our maximal principle for non-spectral Fourier multipliers. The combination of these ingredi-ents yields new Lp estimates for smooth Fourier multipliers in group algebras.

Original languageEnglish (US)
Pages (from-to)879-925
Number of pages47
JournalAnnales Scientifiques de l'Ecole Normale Superieure
Volume50
Issue number4
DOIs
StatePublished - Jul 1 2017

ASJC Scopus subject areas

  • Mathematics(all)

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