  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 $G$.
