Local and multilinear noncommutative de Leeuw theorems

This paper is a collaboration with Martijn Caspers, Bas Janssens, and Amudhan Krishnaswamy-Usha, and is available on arXiv: Click!

Abstract: Let Γ<G be a discrete subgroup of a locally compact unimodular group G. Let m∈Cb(G) be a p-multiplier on G with 1≤p<∞ and let Tm:Lp(Gˆ)→Lp(Gˆ) be the corresponding Fourier multiplier. Similarly, let Tm|Γ:Lp(Γˆ)→Lp(Γˆ) be the Fourier multiplier associated to the restriction m|Γ of m to Γ. We show that c(supp(m|Γ))∥Tm|Γ:Lp(Γˆ)→Lp(Γˆ)∥≤∥Tm:Lp(Gˆ)→Lp(Gˆ)∥, for a specific constant 0≤c(U)≤1 that is defined for every U⊆Γ. The function c quantifies the failure of G to admit small almost Γ-invariant neighbourhoods and can be determined explicitly in concrete cases. In particular, c(Γ)=1 when G has small almost Γ-invariant neighbourhoods. Our result thus extends the De Leeuw restriction theorem from [CPPR15] as well as De Leeuw’s classical theorem [Lee65]. For real reductive Lie groups G we provide an explicit lower bound for c in terms of the maximal dimension d of a nilpotent orbit in the adjoint representation. We show that c(BGρ)≥ρ−d/2 where BGρ is the ball of g∈G with ∥Adg∥<ρ. We further prove several results for multilinear Fourier multipliers. Most significantly, we prove a multilinear De Leeuw restriction theorem for pairs Γ<G with c(Γ)=1. We also obtain multilinear versions of the lattice approximation theorem, the compactification theorem and the periodization theorem. Consequently, we are able to provide the first examples of bilinear multipliers on nonabelian groups.