Journal of Lie Theory

Let $\Gamma$ be a discrete subgroup of a unimodular locally compact group $G$. M. Caspers et al. [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 $\mathfrak{g}$, we develop tools to find explicit bounds on $c(A)$. We show that the problem reduces to:

• The adjoint representation of the semisimple quotient $\mathfrak{s} = \mathfrak{g}/\mathfrak{r}$ of $\mathfrak{g}$ by the radical $\mathfrak{r} \subseteq \mathfrak{g}$ (which was handled in the paper of M. Caspers et al. cited above).
• 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.