Local and multilinear noncommutative de Leeuw theorems

Abstract Let $$\Gamma < G$$ Γ < G be a discrete subgroup of locally compact unimodular group G . $$m\in C_b(G)$$ m ∈ C b ( ) p -multiplier on with $$1 \le \infty $$ 1 ≤ p ∞ and let $$T_{m}: L_p({\widehat{G}}) \rightarrow L_p({\widehat{G}})$$ T : L ^ → the corresponding Fourier multiplier. Similarly, $$T_{m \vert _\Gamma }: L_p({\widehat{\Gamma }}) }})$$ | multiplier associated to restriction $$m|_{\Gamma }$$ m We show that $$\begin{aligned} c( {{\,\textrm{supp}\,}}( m|_{\Gamma } ) \Vert T_{m , \end{aligned}$$ c supp ‖ , for specific constant $$0 c(U) 1$$ 0 U is defined every $$U \subseteq \Gamma ⊆ The function c quantifies failure admit small almost -invariant neighbourhoods can determined explicitly in concrete cases. In particular, $$c(\Gamma =1$$ = when has neighbourhoods. Our result thus extends de Leeuw theorem from Caspers et al. (Forum Math Sigma 3(e21):59, 2015) as well Leeuw’s classical (Ann 81(2):364–379, 1965). For real reductive Lie groups we provide an explicit lower bound terms maximal dimension d nilpotent orbit adjoint representation. $$c(B_\rho ^G) \ge \rho ^{-d/4}$$ B ρ ≥ - d / 4 where $$B_\rho ^G$$ ball $$g\in g $$\Vert {{\,\textrm{Ad}\,}}_g Ad further prove several results multilinear multipliers. Most significantly, pairs <G$$ = also obtain versions lattice approximation theorem, compactification periodization theorem. Consequently, are able first examples bilinear multipliers nonabelian groups.