Smooth Fourier multipliers in group algebras via Sobolev dimension

2015/05/20 by González-Pérez, Adrián M., Junge, Marius, Parcet, Javier · 2 citations
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1505.05316

Abstract

We investigate Fourier multipliers with smooth symbols defined over locally compact Hausdorff groups. Our main results in this paper establish new Hörmander-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 L_p 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 flexibility to measure 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 ingredients yields new L_p estimates for smooth Fourier multipliers in group algebras.

Cited by

• Local noncommutative De Leeuw Theorems beyond reductive Lie groups
• The impact of Schur multipliers in harmonic analysis and operator algebras

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