On Witten Laplacians and Brascamp-Lieb's inequality on manifolds with boundary

TL;DR

This paper derives Brascamp-Lieb type inequalities on manifolds with boundary using supersymmetry of the Witten Laplacian, extending classical inequalities to more general geometric contexts.

Contribution

It introduces a novel approach leveraging supersymmetry and quadratic form decompositions to establish inequalities on manifolds with boundary.

Findings

Brascamp-Lieb inequalities derived for manifolds with boundary

Extension of inequalities to manifolds without boundary

Method based on Witten Laplacian supersymmetry

Abstract

In this paper, we derive from the supersymmetry of the Witten Laplacian Brascamp-Lieb's type inequalities for general differential forms on compact Riemannian manifolds with boundary. In addition to the supersymmetry, our results essentially follow from suitable decompositions of the quadratic forms associated with the Neumann and Dirichlet self-adjoint realizations of the Witten Laplacian. They moreover imply the usual Brascamp-Lieb's inequality and its generalization to compact Riemannian manifolds without boundary.