Inequality on the optimal constant of Young's convolution inequality for locally compact groups and their closed subgroups

TL;DR

This paper investigates the optimal constant in Young's convolution inequality for locally compact groups, establishing inequalities that relate these constants across subgroups and specific classes of Lie groups.

Contribution

It proves that the optimal constant for a group is bounded above by that of its subgroups and derives explicit bounds for certain Lie groups.

Findings

Optimal constant decreases when passing to subgroups.

Bound for connected Lie groups involving their maximal compact subgroups.

Inequality holds for various classes of Lie groups, including solvable and linear Lie groups.

Abstract

We define the optimal constant $Y ( p_1 , p_2 ; G )$ of Young's convolution inequality as \begin{align} Y ( p_1 , p_2 ; G ) := \sup \{ \| \phi_1 * ( \phi_2 \Delta^{1 / p_1'} ) \|_p \mid \phi_1 , \phi_2 \colon G \to \mathbb{C} , \; \| \phi_1 \|_{p_1} = \| \phi_2 \|_{p_2} = 1 \} \end{align} for a locally compact group $G$ and $1 \leq p_1 , p_2 , p \leq \infty$ with $1 / p_1 + 1 / p_2 = 1 + 1 / p$. Here $p'$ is the H\"{o}lder conjugate of $p$, $\| \cdot \|_{ p }$ is the $L^p$-norm on a left Haar measure, and $\Delta \colon G \to \mathbb{R}_{> 0}$ is the modular function. The main result of this paper is that $Y ( p_1 , p_2 ; G ) \leq Y ( p_1 , p_2 ; H )$ for any closed subgroup $H \subset G$. It follows from this inequality that $Y ( p_1 , p_2 ; G ) \leq Y ( p_1 , p_2 ; \mathbb{R} )^{ \dim G - r ( G ) }$ for any connected Lie group $G$ such that the center of the semisimple part is a finite group such as connected linear Lie groups and connected solvable Lie groups, where $r ( G )$ is the dimension of the maximal compact subgroups of $G$.