Smooth fourier multipliers in group algebras via sobolev dimension

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 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 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 L_p estimates for smooth Fourier multipliers in group algebras.