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Some functional inequalities on non-reversible Finsler manifolds

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Abstract

We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by Ohta and Sturm. Following the approach of the \(\Gamma \)-calculus of Bakry et al (2014), we show the dimensional versions of the Poincaré–Lichnerowicz inequality, the logarithmic Sobolev inequality, and the Sobolev inequality. In the reversible case, these inequalities were obtained by Cavalletti and Mondino (2015) in the framework of curvature-dimension condition by means of the localization method. We show that the same (sharp) estimates hold also for non-reversible metrics.

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Acknowledgements

This work is supported in part by JSPS Grant-in-Aid for Scientific Research (KAKENHI) 15K04844.

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  1. Department of Mathematics, Osaka University, Osaka, 560-0043, Japan

    Shin-Ichi Ohta

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  1. Shin-Ichi Ohta
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Correspondence to Shin-Ichi Ohta.

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Communicating Editor: Parameswaran Sankaran

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Ohta, SI. Some functional inequalities on non-reversible Finsler manifolds. Proc Math Sci 127, 833–855 (2017). https://doi.org/10.1007/s12044-017-0357-0

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  • DOI: https://doi.org/10.1007/s12044-017-0357-0

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