An inequality for the compositions of convex functions with convolutions and an alternative proof of the Brunn-Minkowski-Kemperman inequality

Satomi, Takashi

doi:10.48550/arxiv.2111.15349

Cited by 2 publications

(4 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When φ 1 and φ 2 are characteristic functions on a unimodular locally compact group G, Theorem 1.2 was previously obtained by the author [41,Theorem 1.1]. By using this result and the layer cake representation (Sect.…”

Section: Introductionmentioning

confidence: 82%

“…Theorem 1.2 was proved in the case of G = R or some cases of f as Table 1. For the Brunn-Minkowski inequality and Kemperman's result, see the previous paper [41,Corollary 1.3] of the author. Now, we consider the case of G = R. For f (y) := −y p with 0 < p ≤ 1, Theorem 1.2 was proved by Proposition 9] to improve the reverse Young's inequality (Fact 2.6 (2).)…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Satomi

2023

J Fourier Anal Appl

Self Cite

View full text Add to dashboard Cite

Let $$\mu $$ 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.

show abstract

Section: Introductionmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

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

Satomi

2023

J Fourier Anal Appl

Self Cite

View full text Add to dashboard Cite

show abstract

“…In this case, one can see that Wang-Madiman's result is sharper than Theorem 1.2 by replacing φ 1 and φ 2 in Theorem 1.2 with φ ⋆ 1 and φ ⋆ 2 , respectively. When φ 1 and φ 2 are characteristic functions on a unimodular locally compact group G, Theorem 1.2 was obtained by the author in a previous paper [Sat21,Theorem 1.1]. By using this result and the layer cake representation (Section 5), this result is generalized to any measurable functions φ 1 and φ 2 as in Theorem 1.2.…”

Section: Introductionmentioning

confidence: 88%

“…Theorem 1.2 was proved in the case of G = R or some cases of f as Table 1.1. For the Brunn-Minkowski inequality and Kemperman's result, see our previous paper [Sat21,Corollary 1.3]. Now, we consider the case of G = R. For f (y) := −y p with 0 < p ≤ 1, Theorem 1.2 was proved by Brascamp-Lieb [BL76, Proposition 9] to improve the reverse Young's inequality (Fact 2.6 (2)) and the Prékopa-Leindler inequality.…”

Section: Introductionmentioning

confidence: 99%

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

Satomi¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

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 thatholds for any measurable functions φ1, φ2 : G → R ≥0 with µ(supp φ1) + µ(supp φ2) ≤ m(G) and any convex function f : R ≥0 → R with f (0) = 0. Here φ * is the rearrangement of φ.Let YO(P, G) and YR(P, G) denote the optimal constants of Young's and the reverse Young's inequality, respectively, under the assumption of µ(supp φ1) + µ(supp φ2) ≤ m(G). Then we have YO(P, G) ≤ YO(P, R) and YR(P, G) ≥ YR(P, R) as a corollary. Thus, we obtain that m(G) = ∞ if and only if H(p, G) ≤ H(p, R) in the case of p ′ := p/(p − 1) ∈ 2Z, where H(p, G) is the optimal constant of the Hausdorff-Young inequality.

show abstract

An inequality for the compositions of convex functions with convolutions and an alternative proof of the Brunn-Minkowski-Kemperman inequality

Cited by 2 publications

References 16 publications

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

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

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

Contact Info

Product

Resources

About