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Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary

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Abstract

It is known that by dualizing the Bochner–Lichnerowicz–Weitzenböck formula, one obtains Poincaré-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry–Émery 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.

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Acknowledgments

We thank Franck Barthe, Bo Berndtsson, Andrea Colesanti, Dario Cordero-Erausquin, Bo’az Klartag, Michel Ledoux, Frank Morgan, Van Hoang Nguyen, Shin-ichi Ohta, Yehuda Pinchover and Steve Zelditch for their comments and interest. We also thank the anonymous referees for carefully reading the paper. Alexander V. Kolesnikov was supported by RFBR project 14-01-00237 and the DFG Project RO 1195/12-1. This study (research Grant No. 14-01-0056) was supported by The National Research University—Higher School of Economics’ Academic Fund Program in 2014/2015. Emanuel Milman was supported by ISF (Grant No. 900/10), BSF (Grant No. 2010288) and Marie-Curie Actions (Grant No. PCIG10-GA-2011-304066). The research leading to these results is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 637851).

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  1. Faculty of Mathematics, Higher School of Economics, Moscow, Russia

    Alexander V. Kolesnikov

  2. Department of Mathematics, Technion - Israel Institute of Technology, 32000, Haifa, Israel

    Emanuel Milman

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Kolesnikov, A.V., Milman, E. Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary. J Geom Anal 27, 1680–1702 (2017). https://doi.org/10.1007/s12220-016-9736-5

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