Local noncommutative De Leeuw Theorems beyond reductive Lie groups
Bas Janssens, Benjamin Oudejans
TL;DR
The work extends local noncommutative De Leeuw-type bounds for Fourier multipliers to discrete subgroups Γ within unimodular connected Lie groups G beyond reductive cases. It reduces the problem to linear data from the adjoint action of the semisimple quotient 𝔰 = 𝔤/𝔯 and to the action of 𝔰 on quotients arising from the radical 𝔯, using linearization, the Reduction Lemma, and shifting-characters techniques. A key result is that c(G) = 1 for unimodular connected solvable Lie groups, and for general G the δ-bound splits into a semisimple contribution a_ss, controlled by the Ad-action of S and nilpotent-orbit dimension, and a radical contribution a_rad, governed by the root-space decomposition of 𝔯 and, when needed, central extensions arising from nonreal roots. The findings provide explicit pathways to bound local L_p Fourier multipliers on Γ from those on G in broader classes of groups and outline open problems for achieving nontrivial radical bounds via refined neighborhood bases and representation-theoretic decompositions.
Abstract
Let $Γ$ be a discrete subgroup of a unimodular locally compact group $G$. In Math. Ann. 388, 4251-4305 (2024), it was shown that the $L_p$ norm of a Fourier multiplier $m$ on $Γ$ can be bounded locally by its $L_p$-norm on $G$, modulo a constant $c(A)$ which depends on the support $A$ of $m$. In the context where $G$ is a connected Lie group with Lie algebra $\mathfrak{g}$, we develop tools to find explicit bounds on $c(A)$. We show that the problem reduces to: 1) The adjoint representation of the semisimple quotient $\mathfrak{s} = \mathfrak{g}/\mathfrak{r}$ of $\mathfrak{g}$ by the radical $\mathfrak{r}$ of $\mathfrak{g}$ (which was handled in the paper mentioned above). 2) The action of $\mathfrak{s}$ on a set of real irreducible representations that arise from quotients of the commutator series of $\mathfrak{r}$. In particular, we show that $c(G) = 1$ for unimodular connected solvable Lie groups.