论文标题
对单型局部紧凑型组的卷积的不平等和杨氏不平等的最佳常数
An inequality for the convolutions on unimodular locally compact groups and the optimal constant of Young's inequality
论文作者
论文摘要
令$μ$为单模型的本地紧凑型组$ g $和$ m（g）$的HAAR度量，这是所有开放子组的最小值$ g $。本文的主要结果是\ begin {align*} \ int_ {g}^{} f \ circ \ left（ϕ_1 * ϕ_2 \ right）\ left（g \ firt）dg \ leq \ leq \ leq \ int _ {\ Mathbb {r}}}^{}^{}^{} \ end {align*}具有任何可测量的函数$ ϕ_1，ϕ_2 \ colon g \ to \ mathbb {r} _ {\ geq 0} $，带有$μ（\ mathrm {\ mathrm {supp} \; ϕ_1） +μ（supp）功能$ f \ colon \ mathbb {r} _ {\ geq 0} \ to \ mathbb {r} $带有$ f（0）= 0 $。这里$ ϕ^*$是$ ϕ $的重排。 令$ y_o（p，g）$和$ y_r（p，g）$分别表示杨氏和反向杨的不等式的最佳常数，在假设$μ（\ mathrm {supp} \; ϕ_1） +μ（\ mathrm {supp} \; n;然后，我们有$ y_o（p，g）\ leq y_o（p，\ mathbb {r}）$和$ y_r（p，g）\ geq y_r（p，\ mathbb {r}）$作为推论。因此，在$ h（p，g）\ leq H（p，p，p，\ sathbb {r}）$的情况下，在$ p'：= p/（p/（p/（p/（p/（p/（p/p））中
Let $μ$ 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 $ϕ_1, ϕ_2 \colon G \to \mathbb{R}_{\geq 0}$ with $μ( \mathrm{supp} \; ϕ_1 ) + μ( \mathrm{supp} \; ϕ_2 ) \leq m(G)$ and any convex function $f \colon \mathbb{R}_{\geq 0} \to \mathbb{R}$ with $f(0) = 0$. Here $ϕ^*$ is the rearrangement of $ϕ$. 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 $μ( \mathrm{supp} \; ϕ_1 ) + μ( \mathrm{supp} \; ϕ_2 ) \leq m(G)$. 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.
