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

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 $G$.