Local and multilinear noncommutative de Leeuw theorems

Abstract

Let Γ b(G) be a p-multiplier on G with 1≤p<∞ and let T
m:L
p(G^)→L
p(G^) be the corresponding Fourier multiplier. Similarly, let Tm|
Γ:L
p(Γ^)→L
p(Γ^) be the Fourier multiplier associated to the restriction m|
Γ of m to Γ. We show that (Formula presented.) 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 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
ρ
G)≥ρ
-d/4 where B
ρ
G is the ball of g∈G with ‖Ad
g‖<ρ. We further prove several results for multilinear Fourier multipliers. Most significantly, we prove a multilinear de Leeuw restriction theorem for pairs Γ