Positive Gaussian kernels also have Gaussian minimizers

TL;DR

This paper investigates conditions under which optimal constants for multilinear operators with Gaussian kernels can be determined by centered Gaussian functions, extending and unifying previous inequalities and results.

Contribution

It introduces criteria for computing optimal constants using Gaussian functions and provides necessary and sufficient conditions for positivity, extending Lieb's results on Gaussian kernel maximizers.

Findings

Criteria for optimal constants via Gaussian functions

Necessary and sufficient conditions for positivity

Extension of inverse inequalities and Lieb's results

Abstract

We study lower bounds on multilinear operators with Gaussian kernels acting on Lebesgue spaces, with exponents below one. We put forward natural conditions when the optimal constant can be computed by inspecting centered Gaussian functions only, and we give necessary and sufficient conditions for this constant to be positive. Our work provides a counterpart to Lieb's results on maximizers of multilinear operators with real Gaussian kernels, also known as the multidimensional Brascamp-Lieb inequality. It unifies and extends several inverse inequalities.