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 \{ \| ϕ_1 * ( ϕ_2 Δ^{1 / p_1'} ) \|_p \mid ϕ_1 , ϕ_2 \colon G \to \mathbb{C} , \; \| ϕ_1 \|_{p_1} = \| ϕ_2 \|_{p_2} = 1 \} \end{align} for a locally compact group $G$ and $1\le p_1,p_2,p\le \infty$ with $1/p_1+1/p_2=1+1/p$. Here $p^{\prime }$ is the Hölder conjugate of $p$, $\Vert \cdot {\Vert }_p$ is the $L^p$-norm on a left Haar measure, and $\Delta :G\to {\mathbb{R}}_{>0}$ is the modular function. The main result of this paper is that $Y(p_1,p_2;G)\le Y(p_1,p_2;H)$ for any closed subgroup $H\subset G$. It follows from this inequality that $Y(p_1,p_2;G)\le Y(p_1,p_2;\mathbb{R}{)}^{\dim G-r(G)}$ 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 $G$.

20 pages