Local and multilinear noncommutative de Leeuw theorems

Let Gamma< G be a discrete subgroup of a locally compact unimodular group G. Let m is an element of C-b(G) be a p-multiplier on G with 1 <= p < infinity and let T-m: L-p ((G) over cap) -> L-p ((G) over cap) be the corresponding Fourier multiplier. Similarly, let Tm-|Gamma : Lp((Gamma) over cap) -> L-p((Gamma) over cap) be the Fourier multiplier associated to the restriction m|(Gamma) of m to Gamma. We show that c(supp (m)vertical bar Gamma))vertical bar vertical bar T-m vertical bar Gamma : L-p ((Gamma) over cap) -> Lp((Gamma) over cap)vertical bar vertical bar <= vertical bar vertical bar T-m:L-p((G) over cap) -> L-p((G) over cap)parallel to for a specific constant 0 <= c(U) <= 1 that is defined for every U subset of Gamma. The function c quantifies the failure of G to admit small almost Gamma-invariant neighbourhoods and can be determined explicitly in concrete cases. In particular, c(Gamma)=1 when G has small almost Gamma-invariant neighbourhoods. Our result thus extends the de Leeuw restriction theorem from Caspers et al. (Forum Math Sigma 3(e21):59, 2015) as well as de Leeuw's classical theorem (Ann Math 81(2):364-379, 1965). 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(B-rho(G)) >= rho(-d/4). where B-rho(G) is the ball of g is an element of G with parallel to Ad(g) parallel to < rho. We further prove several results for multilinear Fourier multipliers. Most significantly, we prove a multilinear de Leeuw restriction theorem for pairs Gamma. 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.