2022/01/25 by Martijn Caspers, Bas Janssens, Caspers, Martijn +5 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2201.10400
openalex publication_date 2022/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(\widehatG) → Lp(\widehatG) be the corresponding Fourier multiplier. Similarly, let Tm \vertΓ: Lp(\widehatΓ) → Lp(\widehatΓ) be the Fourier multiplier associated to the restriction m|Γ of m to Γ. We show that c( \rm supp( m\vertΓ ) ) \Vert Tm \vertΓ: Lp(\widehatΓ) → Lp(\widehatΓ) \Vert ≤ \Vert Tm : Lp(\widehatG) → Lp(\widehatG) \Vert, 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(BρG) ≥ ρ-d/4 where BρG is the ball of g∈ G with \Vert \rm Adg \Vert < ρ. We further prove several results for multilinear Fourier multipliers. Most significantly, we prove a multilinear De Leeuw restriction theorem for pairs Γ