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

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 $1\le p_1,p_2,p\le \mathrm{\infty }$ with $1\mathrm{/}{p}_1+1\mathrm{/}{p}_2=1+1\mathrm{/}p$. Here $p^{\mathrm{\prime }}$ is the Hölder conjugate of $p$, $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_p$ is the $L^p$-norm on a left Haar measure, and $\mathrm{\Delta }\text{\hspace{0.17em}⁣}:G\to {\mathbb{R}}_{>0}$ 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 $G$.