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

TL;DR

This paper establishes an inequality for convolutions on unimodular locally compact groups, relates it to rearrangements, and determines the optimal constants for Young's inequality, linking group structure to classical analysis results.

Contribution

It introduces a new convolution inequality on unimodular groups, compares optimal constants of Young's inequality with the real line, and characterizes when the group has infinite measure based on Hausdorff--Young constants.

Findings

Proves an inequality relating convolutions on groups and real line.

Shows bounds for optimal Young's inequality constants on groups.

Characterizes infinite measure groups via Hausdorff--Young constants.

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.