Measuring Adjoint-invariance of Neighborhoods in Solvable Lie Groups

Abstract

In this thesis, we derive a lower bound on a quantity appearing in a Fourier multiplier inequality on solvable Lie groups.
In Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi (2022), a classical result by de Leeuw about the restriction of Fourier multipliers on $\mathbb{R}^n$ to a discrete subgroup is extended to a noncommutative setting. It is shown that a compactly supported $p$-multiplier $m$ on a locally compact group $G$ has the following relation to its restriction to a discrete subgroup $\Gamma$:
$$c(\operatorname{supp}(m|_{\Gamma})) \norm{T_{m|_{\Gamma}}}_p \leq \norm{T_m}_p.$$

Here $c(U) = \inf\left\{ \sqrt{\delta_F} \mid F\subseteq U \mathrm{ finite} \right\}$, where $\delta_F$ is a quantity that determines to what extent small neighborhoods of the identity in $G$ are left invariant by conjugation by elements of $F$. In this thesis, we estimate $\delta_F$ for connected solvable Lie groups.\\

Our main result is theorem 9, which states that for a connected solvable Lie group $G$ with Lie algebra $\g$, if $\lambda_1,\ldots,\lambda_n\colon \g_{\mathbb{C}}\to\mathbb{C}$ are the generalized weights of the complexification $\g_{\mathbb{C}}$, there exist unique homomorphisms $\chi_1,\ldots,\chi_n\colon G\to\mathbb{R}_{>0}$ such that $\chi_i = \mathrm{d}\lambda_i$, and $$\delta_F \geq \prod\limits_{i=1}^n \inf\limits_{g\in F}\chi_i(g).$$