An inequality for the convolutions on unimodular locally compact groups and the optimal constant of Young's inequality

Abstract: Let $\mu$ be the Haar measure of a unimodular locally compact group $G$ and $m (G)$ as the infimum of the volumes of all open subgroups of $G$. The main result of this paper is that \begin{align*} \int_{G}^{} f \circ \left( \phi_1 * \phi_2 \right) \left( g \right) dg \leq \int_{\mathbb{R}}^{} f \circ \left( \phi_1^* * \phi_2^* \right) \left( x \right) dx \end{align*} holds for any measurable functions $\phi_1, \phi_2 \colon G \to \mathbb{R}{\geq 0}$ with $\mu ( \mathrm{supp} \; \phi_1 ) + \mu ( \mathrm{supp} \; \phi_2 ) \leq m(G)$ and any convex function $f \colon \mathbb{R}{\geq 0} \to \mathbb{R}$ with $f(0) = 0$. Here $\phi^*$ is the rearrangement of $\phi$. Let $Y_O(P,G)$ and $Y_R(P,G)$ denote the optimal constants of Young's and the reverse Young's inequality, respectively, under the assumption $\mu ( \mathrm{supp} \; \phi_1 ) + \mu ( \mathrm{supp} \; \phi_2 ) \leq m(G)$. Then we have $Y_O(P,G) \leq Y_O(P,\mathbb{R})$ and $Y_R(P,G) \geq Y_R(P,\mathbb{R})$ as a corollary. Thus, we obtain that $m (G) = \infty$ if and only if $H (p,G) \leq H (p, \mathbb{R})$ in the case of $p' := p/(p-1) \in 2 \mathbb{Z}$, where $H (p,G)$ is the optimal constant of the Hausdorff--Young inequality.