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 language||English (US)|
|Number of pages||47|
|Journal||Annales Scientifiques de l'Ecole Normale Superieure|
|State||Published - Jul 1 2017|
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