arXiv subjects

Takashi Satomi

Publications and source records attributed to Takashi Satomi.

6 recordsLinked to original sources

arXiv2026

Borell--Brascamp--Lieb inequality with finitely many output functions

The classical Borell--Brascamp--Lieb inequality for multiple functions is an integral inequality relating finitely many input functions to a single output function. In this paper, we give an extension that allows a distinct output function for each input function. More precisely, we establish the following inequality. For a weight $ λ= ( λ_1 , \dots , λ_m ) $, we set $ z_λ( x ) := \sum_{ i = 1 }^m λ_i x_i $ and denote by $ M_p^λ$ the weighted power mean of power $ p $. In addition, the power transform $ Q_d $ is defined as the continuous extension of $ p \mapsto p / ( 1 - d p ) $. We consider integrable functions $ f_1 , \dots , f_m , g_1 , \dots , g_m \colon \mathbb{ R }^d \to \mathbb{ R }_{ \geq 0 } $ satisfying $ 0 < \| f_i \|_1 < \infty $ for every $ i = 1 , 2 , \dots , m $. Then one has \[ \operatorname*{ess\,inf}_{ x = ( x_1 , \dots , x_m ) } M_p^λ\left( \frac{ g_1 ( z_λ( x ) ) }{ f_1 ( x_1 ) }, \dots , \frac{ g_m ( z_λ( x ) ) }{ f_m ( x_m ) } \right) \leq M_{ Q_d ( p ) }^λ\left( \frac{ \| g_1 \|_1 }{ \| f_1 \|_1 }, \dots , \frac{ \| g_m \|_1 }{ \| f_m \|_1 } \right) \] for any $ - \infty \leq p \leq 1 / d $, where the essential infimum is taken over the set of points satisfying $ f_i ( x_i ) > 0 $ for every $ i = 1 , 2 , \dots , m $. When all output functions $ g_1 , \dots , g_m $ are equal, this recovers the classical Borell--Brascamp--Lieb inequality for multiple functions. When $ p = 0 $ and the hypothesis is imposed pointwise, it coincides with a special case of the multi-output Prékopa--Leindler inequality of Cordero-Erausquin--Maurey.

math.CA↗

arXiv2026

Continuous solutions of the complex Kac--Bernstein functional equation on the integers and the real numbers

In this paper, the continuous solutions of the complex Kac--Bernstein functional equation \[ f_1 ( u + v ) f_2 ( u - v ) f_1 ( u' + v' ) f_2 ( u' - v' ) = f_1 ( u + v' ) f_2 ( u - v' ) f_1 ( u' + v ) f_2 ( u' - v ) \] are classified for $ \mathbb{ Z } $ and $ \mathbb{ R } $. By using this classification for $ \mathbb{ Z } $, we consider a generalization of the Kac--Bernstein theorem on the one-dimensional torus $ \mathbb{ T } $ by Baryshnikov--Eisenberg--Stadje from probability Borel measures to complex Borel measures. Similarly, the original Kac--Bernstein theorem on $ \mathbb{ R } $ is generalized from probability Borel measures to complex Borel measures.

math.CA↗

arXiv2024

Pushforward measures on homogeneous spaces of non-unimodular groups and properties of modular functions

This paper shows that a formula for the pushforward measures on the fiber bundle $ G \to G / H $ of homogeneous space of locally compact groups $ G $ and $ H $ can be written down by using the modular functions. As a result, we obtain the equality \begin{align} \frac{ Δ_G ( h ) }{ Δ_{ G / N } ( h N ) } = \frac{ Δ_H ( h ) }{ Δ_{ H / N } ( h N ) } \end{align} on the modular functions for any closed subgroups $ N $ and $ H $ of a locally compact group $ G $ with $ N \lhd G $ and $ N \subset H \subset G $ and any $ h \in H $.

math.GR↗

arXiv2023

Inequality on the optimal constant of Young's convolution inequality for locally compact groups and their closed subgroups

We define the optimal constant $Y ( p_1 , p_2 ; G )$ of Young's convolution inequality as \begin{align} Y ( p_1 , p_2 ; G ) := \sup \{ \| ϕ_1 * ( ϕ_2 Δ^{1 / p_1'} ) \|_p \mid ϕ_1 , ϕ_2 \colon G \to \mathbb{C} , \; \| ϕ_1 \|_{p_1} = \| ϕ_2 \|_{p_2} = 1 \} \end{align} for a locally compact group $G$ and $1 \leq p_1 , p_2 , p \leq \infty$ with $1 / p_1 + 1 / p_2 = 1 + 1 / p$. Here $p'$ is the Hölder conjugate of $p$, $\| \cdot \|_{ p }$ is the $L^p$-norm on a left Haar measure, and $Δ\colon G \to \mathbb{R}_{> 0}$ is the modular function. The main result of this paper is that $Y ( p_1 , p_2 ; G ) \leq Y ( p_1 , p_2 ; H )$ for any closed subgroup $H \subset G$. It follows from this inequality that $Y ( p_1 , p_2 ; G ) \leq Y ( p_1 , p_2 ; \mathbb{R} )^{ \dim G - r ( G ) }$ for any connected Lie group $G$ such that the center of the semisimple part is a finite group such as connected linear Lie groups and connected solvable Lie groups, where $r ( G )$ is the dimension of the maximal compact subgroups of $G$.

math.FA↗

arXiv2023

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

Let $m(G)$ be the infimum of the volumes of all open subgroups of a unimodular locally compact group $G$. Suppose integrable functions $ϕ_1 , ϕ_2 \colon G \to [0,1]$ satisfy $\| ϕ_1 \| \leq \| ϕ_2 \|$ and $\| ϕ_1 \| + \| ϕ_2 \| \leq m (G)$, where $\| \cdot \|$ denotes the $L^1$-norm with respect to a Haar measure $dg$ on $G$. We have the following inequality for any convex function $f \colon [0, \| ϕ_1 \| ] \to \mathbb{R}$ with $f(0) = 0$: \begin{align*} \int_{G}^{} f \circ ( ϕ_1 * ϕ_2 ) (g) dg \leq 2 \int_{0}^{\| ϕ_1 \|} f(y) dy + ( \| ϕ_2 \| - \| ϕ_1 \| ) f( \| ϕ_1 \| ). \end{align*} As a corollary, we have a slightly stronger version of Brunn-Minkowski-Kemperman inequality. That is, we have \begin{align*} \mathrm{vol}_* ( B_1 B_2 ) \geq \mathrm{vol} ( \{ g \in G \mid 1_{B_1} * 1_{B_2} (g) > 0 \} ) \geq \mathrm{vol} (B_1) + \mathrm{vol} (B_2) \end{align*} for any non-null measurable sets $B_1 , B_2 \subset G$ with $\mathrm{vol} (B_1) + \mathrm{vol} (B_2) \leq m(G)$, where $\mathrm{vol}_*$ denotes the inner measure and $1_B$ the characteristic function of $B$.

math.MG↗

arXiv2023

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

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.

math.GR↗