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A Sharp Sobolev Interpolation Inequality on Finsler Manifolds

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Abstract

In this paper we study a sharp Sobolev interpolation inequality on Finsler manifolds. We show that Minkowski spaces represent the optimal framework for the Sobolev interpolation inequality on a large class of Finsler manifolds: (1) Minkowski spaces support the sharp Sobolev interpolation inequality; (2) any complete Berwald space with non-negative Ricci curvature which supports the sharp Sobolev interpolation inequality is isometric to a Minkowski space. The proofs are based on properties of the Finsler–Laplace operator and on the Finslerian Bishop–Gromov volume comparison theorem.

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Acknowledgments

The author would like to thank the referee for her/his useful remarks. Research supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project no. PN-II-ID-PCE-2011-3-0241, and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. The present work was initiated during the author’s visit at the Institut des Hautes Études Scientifiques (IHÉS), Bures-sur-Yvette, France.

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Authors and Affiliations

  1. Department of Economics, Babeş-Bolyai University, 400591 , Cluj-Napoca, Romania

    Alexandru Kristály

  2. Institute of Applied Mathematics, Óbuda University (formerly Budapest Tech Polytechnical Institution), Budapest , 1034, Hungary

    Alexandru Kristály

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  1. Alexandru Kristály
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Correspondence to Alexandru Kristály.

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Communicated by Marco M. Peloso.

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Kristály, A. A Sharp Sobolev Interpolation Inequality on Finsler Manifolds. J Geom Anal 25, 2226–2240 (2015). https://doi.org/10.1007/s12220-014-9510-5

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  • DOI: https://doi.org/10.1007/s12220-014-9510-5

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