Local noncommutative De Leeuw Theorems beyond reductive Lie groups

TL;DR

This paper extends noncommutative De Leeuw theorems to a broader class of groups, providing explicit bounds on Fourier multipliers' norms by analyzing Lie algebra structures and group actions.

Contribution

It develops tools to explicitly bound constants in noncommutative De Leeuw theorems for connected Lie groups beyond reductive cases.

Findings

Bounded the $L_p$ norm of Fourier multipliers on discrete subgroups by the ambient group norms.

Reduced the problem to analyzing the adjoint representation of the semisimple quotient.

Established that the constant $c(G)$ equals 1 for unimodular connected solvable Lie groups.

Abstract

Let $\Gamma$ 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 $\Gamma$ can be bounded locally by its $L_p$-norm on $G$, modulo a constant $c(A)$ which depends on the support $A$ of $m$. 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.