Brascamp-Lieb type inequalities on weighted Riemannian manifolds with boundary

23 Nov 2017  ·  Kolesnikov Alexander V., Milman Emanuel ·

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. read more

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Differential Geometry Functional Analysis Spectral Theory