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( ϕ_1 * ϕ_2 \right) \left( g \right) dg \leq \int_{\mathbb{R}}^{} f \circ \left( ϕ_1^* * ϕ_2^* \right) \left( x \right) dx \end{align*} holds for any measurable functions ${\varphi }_1,{\varphi }_2:G\to {\mathbb{R}}_{\ge 0}$ with $\mu (\mathrm{supp}{\varphi }_1)+\mu (\mathrm{supp}{\varphi }_2)\le m(G)$ and any convex function $f:{\mathbb{R}}_{\ge 0}\to \mathbb{R}$ with $f(0)=0$. Here ${\varphi }^{\ast }$ is the rearrangement of $\varphi$. 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 $\mu (\mathrm{supp}{\varphi }_1)+\mu (\mathrm{supp}{\varphi }_2)\le m(G)$. Then we have $Y_O(P,G)\le Y_O(P,\mathbb{R})$ and $Y_R(P,G)\ge Y_R(P,\mathbb{R})$ as a corollary. Thus, we obtain that $m(G)=\infty$ if and only if $H(p,G)\le H(p,\mathbb{R})$ in the case of $p^{\prime }:=p/(p-1)\in 2\mathbb{Z}$, where $H(p,G)$ is the optimal constant of the Hausdorff--Young inequality.

arXiv comment22 pages, 5 figures