Free Fourier Multipliers associated with the firstSegment

Abstract

We study Fourier multipliers on free group ${\mathbb{F}}_{\infty }$ associated with the first segment of the reduced words, and prove that they are completely bounded on the noncommutative $L^p$ spaces $L^p({\hat{\mathbb{F}}}_{\infty })$ iff their restriction on $L^p({\hat{\mathbb{F}}}_1)=L^p(\mathbb{T})$ are completely bounded. As a consequence, every classical Mikhlin multiplier extends to a $L^p$ Fourier multiplier on free groups for all $1<p<\infty$.

References in corpus (1)

• Free Hilbert Transforms

Cited by in corpus (1)

• A Hörmander-Mikhlin multiplier theory for free groups and amalgamated free products of von Neumann algebras