带有限多个输出函数的Borell–Brascamp–Lieb不等式
Borell--Brascamp--Lieb inequality with finitely many output functions
AI总结：
本文将经典Borell–Brascamp–Lieb不等式扩展为每个输入函数对应不同输出函数的形式，建立了含加权幂平均、幂变换的新不等式，其特例可还原经典不等式或对应多输出Prékopa–Leindler不等式的特例。
AI中文摘要：
经典的针对多个函数的Borell–Brascamp–Lieb不等式是将有限个输入函数与单个输出函数关联起来的积分不等式。在本文中，我们给出了一种扩展形式，该形式允许为每个输入函数对应一个不同的输出函数。更确切地说，我们建立如下不等式：对于权重λ=(λ₁,…,λₘ)，定义z_λ(x):=∑_{i=1}^m λ_i x_i，且将M_p^λ记为幂次为p的加权幂平均。此外，幂变换Q_d定义为p↦p/(1−dp)的连续延拓。我们考虑可积函数f₁,…,fₘ,g₁,…,gₘ:ℝᵈ→ℝ≥0，满足对每个i=1,2,…,m，有0<‖f_i‖₁<∞。那么对任意−∞≤p≤1/d，均有ess inf_{x=(x₁,…,xₘ)} M_p^λ( g₁(z_λ(x))/f₁(x₁), …, gₘ(z_λ(x))/fₘ(xₘ) ) ≤ M_{Q_d(p)}^λ( ‖g₁‖₁/‖f₁‖₁, …, ‖gₘ‖₁/‖fₘ‖₁ )，其中本质下确界是在满足对每个i=1,2,…,m有f_i(x_i)>0的点集上取的。当所有输出函数g₁,…,gₘ均相等时，该式可还原为经典的针对多个函数的Borell–Brascamp–Lieb不等式；当p=0且假设逐点成立时，它与Cordero-Erausquin–Maurey提出的多输出Prékopa–Leindler不等式的一个特例重合。
英文摘要：
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.
