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

Abstract

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 $\lambda =({\lambda }_1,\dots ,{\lambda }_m)$, we set $z_{\lambda }(x):={\sum }_{i=1}^m{\lambda }_i{x}_i$ and denote by $M_p^{\lambda }$ the weighted power mean of power $p$. In addition, the power transform $Q_d$ is defined as the continuous extension of $p\mapsto p/(1-dp)$. We consider integrable functions $f_1,\dots ,f_m,g_1,\dots ,g_m:{\mathbb{R}}^d\to {\mathbb{R}}_{\ge 0}$ satisfying $0<\Vert f_i{\Vert }_1<\infty$ 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 $-\infty \le p\le 1/d$, 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.

arXiv comment30 pages, 1 Table