TY - JOUR
T1 - Local and multilinear noncommutative de Leeuw theorems
AU - Caspers, Martijn
AU - Janssens, Bas
AU - Krishnaswamy-Usha, Amudhan
AU - Miaskiwskyi, Lukas
PY - 2023
Y1 - 2023
N2 - Let Γ<G be a discrete subgroup of a locally compact unimodular group G. Let m∈C
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<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.
AB - Let Γ<G be a discrete subgroup of a locally compact unimodular group G. Let m∈C
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<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.
UR - http://www.scopus.com/inward/record.url?scp=85158084780&partnerID=8YFLogxK
U2 - 10.1007/s00208-023-02611-z
DO - 10.1007/s00208-023-02611-z
M3 - Article
AN - SCOPUS:85158084780
SN - 0025-5831
VL - 388
SP - 4251
EP - 4305
JO - Mathematische Annalen
JF - Mathematische Annalen
IS - 4
ER -