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.
Similar content being viewed by others
References
Bao, D., Chern, S.-S., Shen, Z.: Introduction to Riemann–Finsler geometry. Graduate Texts in Mathematics. Springer, New York (2000)
Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weight. Compos. Math. 53, 259–275 (1984)
Carron, G.: Inégalités de Hardy sur les variétés riemanniennes non-compactes. J. Math. Pures Appl. (9) 76(10), 883–891 (1997)
Kombe, I., Özaydin, M.: Improved Hardy and Rellich inequalities on Riemannian manifolds. Trans. Am. Math. Soc. 361(12), 6191–6203 (2009)
Kristály, A., Ohta, S.: Caffarelli–Kohn–Nirenberg inequality on metric measure spaces with applications. Math. Ann. 357(2), 711–726 (2013)
Maz’ya, V.: Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften, vol. 342. Springer, Heidelberg (2011)
Ohta, S., Sturm, K.-T.: Heat flow on Finsler manifolds. Comm. Pure Appl. Math. 62(10), 1386–1433 (2009)
Shen, Z.: Volume comparison and its applications in Riemann–Finsler geometry. Adv. Math. 128(2), 306–328 (1997)
Van Schaftingen, J.: Anisotropic symmetrization. Ann. Inst. H. Poincaré Anal. Non Linéaire 23, 539–565 (2006)
Xia, C.: The Caffarelli–Kohn–Nirenberg inequalities on complete manifolds. Math. Res. Lett. 14(5), 875–885 (2007)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Marco M. Peloso.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-014-9510-5
Keywords
- Sobolev interpolation inequality
- Sharp constant
- Finsler manifold
- Minkowski space
- Ricci curvature
- Rigidity