Journal of Lie Theory 35 (2025), No. 4, 845--860
Copyright Heldermann Verlag 2025
Local Noncommutative De Leeuw Theorems Beyond Reductive Lie Groups
Bas Janssens
TU Delft, DIAM/EEMCS, Delft, The Netherlands
b.janssens@tudelft.nl
Benjamin Oudejans
TU Delft, DIAM/EEMCS, Delft, The Netherlands
benjamin.oudejans@gmail.com
[Abstract-pdf]
\newcommand{\fs}{\mathfrak{s}} \newcommand{\fg}{\mathfrak{g}} \newcommand{\fr}{\mathfrak{r}} Let $\Gamma$ be a discrete subgroup of a unimodular locally compact group $G$. M.\,Caspers et al. [\emph{Local and multilinear noncommutative de Leeuw theorems}, Math. Ann. 388 (2024) 4251--4305] showed that the $L_p$-norm of a Fourier multi\-plier $m \colon G \rightarrow \mathbb{C}$ on $\Gamma$ can be bounded locally by its $L_p$-norm on $G$, modulo a constant $c(A)$ which depends on the support $A$ of $m|_{\Gamma}$. In the context where $G$ is a connected Lie group with Lie algebra $\fg$, we develop tools to find explicit bounds on $c(A)$. We show that the problem reduces to: \begin{itemize} \item[(1)]\vskip-1mm The adjoint representation of the semisimple quotient $\fs = \fg/\fr$ of $\fg$ by the radical $\fr \subseteq \fg$ (which was handled in the paper of M.\,Caspers et al. cited above). \item[(2)]\vskip-1mm The action of $\fs$ on a set of real irreducible representations that arise from quotients of the commutator series of $\fr$. \end{itemize} In particular, we show that $c(G) = 1$ for unimodular connected solvable Lie groups.
Keywords: Fourier multipliers, almost invariant neighbourhoods.
MSC: 22E15, 43A15, 43A22, 22D25, 46L51.
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