Brascamp-Lieb type inequalities on weighted Riemannian manifolds with boundary
Abstract
It is known that by dualizing the Bochner-Lichnerowicz-Weitzenb\"{o}ck formula, one obtains Poincar\'e-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry-\'Emery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). When the manifold has a boundary, an appropriate generalization of the Reilly formula may be used instead. By systematically dualizing this formula for various combinations of boundary conditions of the domain (convex, mean-convex) and the function (Neumann, Dirichlet), we obtain new Brascamp-Lieb type inequalities on the manifold. All previously known inequalities of Lichnerowicz, Brascamp-Lieb, Bobkov-Ledoux and Veysseire are recovered, extended to the Riemannian setting and generalized into a single unified formulation, and their appropriate versions in the presence of a boundary are obtained. Our framework allows to encompass the entire class of Borell's convex measures, including heavy-tailed measures, and extends the latter class to weighted-manifolds having negative generalized dimension.
Keywords
Cite
@article{arxiv.1310.2526,
title = {Brascamp-Lieb type inequalities on weighted Riemannian manifolds with boundary},
author = {Alexander V. Kolesnikov and Emanuel Milman},
journal= {arXiv preprint arXiv:1310.2526},
year = {2017}
}
Comments
24 pages. The original submission has been split into two parts for publication. The first part, corresponding to the present arXiv version, was published in J. Geom. Anal. The second part, corresponding to a new arXiv version submitted on Nov 23, 2017, will appear in the Amer. J. of Math