An Inequality for the Convolutions on Unimodular Locally Compact Groups and the Optimal Constant of Young’s Inequality

Abstract

AbstractLet $$\mu $$ 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 $$\begin{aligned} \int _{G}^{} f \circ \left( \phi _1 * \phi _2 \right) \left( g \right) dg \le \int _{\mathbb {R}}^{} f \circ \left( \phi _1^* * \phi _2^* \right) \left( x \right) dx \end{aligned}$$ ∫ G f ∘ ϕ 1 ∗ ϕ 2 g d g ≤ ∫ R f ∘ ϕ 1 ∗ ∗ ϕ 2 ∗ x d x holds for any measurable functions $$\phi _1, \phi _2 :G \rightarrow \mathbb {R}_{\ge 0}$$ ϕ 1 , ϕ 2 : G → R ≥ 0 with $$\mu ( \textrm{supp} \; \phi _1 ) + \mu ( \textrm{supp} \; \phi _2 ) \le m(G)$$ μ ( supp ϕ 1 ) + μ ( supp ϕ 2 ) ≤ m ( G ) and any convex function $$f :\mathbb {R}_{\ge 0} \rightarrow \mathbb {R}$$ f : R ≥ 0 → R with $$f(0) = 0$$ f ( 0 ) = 0 . Here $$\phi ^*$$ ϕ ∗ is the rearrangement of $$\phi $$ ϕ . Let $$Y_O(P,G)$$ Y O ( P , G ) and $$Y_R(P,G)$$ Y R ( P , G ) denote the optimal constants of Young’s and the reverse Young’s inequality, respectively, under the assumption $$\mu ( \textrm{supp} \; \phi _1 ) + \mu ( \textrm{supp} \; \phi _2 ) \le m(G)$$ μ ( supp ϕ 1 ) + μ ( supp ϕ 2 ) ≤ m ( G ) . Then we have $$Y_O(P,G) \le Y_O(P,\mathbb {R})$$ Y O ( P , G ) ≤ Y O ( P , R ) and $$Y_R(P,G) \ge Y_R(P,\mathbb {R})$$ Y R ( P , G ) ≥ Y R ( P , R ) as a corollary. Thus, we obtain that $$m (G) = \infty $$ m ( G ) = ∞ if and only if $$H (p,G) \le H (p, \mathbb {R})$$ H ( p , G ) ≤ H ( p , R ) in the case of $$p' := p/(p-1) \in 2 \mathbb {Z}$$ p ′ : = p / ( p - 1 ) ∈ 2 Z , where H(p, G) is the optimal constant of the Hausdorff–Young inequality.