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 - 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 \[ \operatorname*{ess\,inf}_{ x = ( x_1 , \dots , x_m ) } M_p^\lambda \left( \frac{ g_1 ( z_\lambda ( x ) ) }{ f_1 ( x_1 ) }, \dots , \frac{ g_m ( z_\lambda ( x ) ) }{ f_m ( x_m ) } \right) \leq M_{ Q_d ( p ) }^\lambda \left( \frac{ \| g_1 \|_1 }{ \| f_1 \|_1 }, \dots , \frac{ \| g_m \|_1 }{ \| f_m \|_1 } \right) \] 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 Pr\'ekopa--Leindler inequality of Cordero-Erausquin--Maurey.