Smooth Fourier multipliers on group von Neumann algebras

Details Smooth Fourier multipliers on group von Neumann algebras Smooth Fourier multipliers on group von Neumann algebras

2010-10-26

arxiv.org/abs/1010.5320v4

Mathematics

Classical Analysis and ODEs

New dimension free estimates added; substantial length reduction

Scientific paper

In this paper, we extend classical methods from harmonic analysis to study smooth Fourier multipliers in the compact dual of arbitrary discrete groups. The main results are a H\"ormander-Mihlin multiplier theorem in finite dimensions, Littlewood-Paley inequalities on group von Neumann algebras and a dimension free Lp estimate for noncommutative Riesz transforms. As a byproduct of our approach, we provide new examples of Lp Fourier multipliers in Rn. The key novelty is to exploit cocycles and cross products in Fourier multiplier theory, in conjunction with quantum probability techniques and a noncommutative form of Calder\'on-Zygmund theory.

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