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Sharp Caffarelli–Kohn–Nirenberg inequalities on Riemannian manifolds: the influence of curvature

Part of: Global differential geometry Inequalities Other generalizations

Published online by Cambridge University Press:  22 January 2021

Van Hoang Nguyen Open the ORCID record for Van Hoang Nguyen [Opens in a new window]
Van Hoang Nguyen*
Affiliation:
Department of Mathematics, FPT University, Ha Noi, Viet Nam ([email protected]; [email protected])
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Abstract

We first establish a family of sharp Caffarelli–Kohn–Nirenberg type inequalities (shortly, sharp CKN inequalities) on the Euclidean spaces and then extend them to the setting of Cartan–Hadamard manifolds with the same best constant. The quantitative version of these inequalities also is proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli–Kohn–Nirenberg type inequalities with the same sharp constant as in the Euclidean space of the same dimension. Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály (J. Math. Pures Appl. 119 (2018), 326–346) to a larger class of the sharp CKN inequalities.

Keywords

Caffarelli–Kohn–Nirenberg inequalitiessharp constantRiemannian manifoldcurvature

MSC classification

Primary: 26D10: Inequalities involving derivatives and differential and integral operators
Secondary: 31C12: Potential theory on Riemannian manifolds 53C20: Global Riemannian geometry, including pinching 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 1 , February 2022 , pp. 102 - 127
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Copyright
Copyright © The Author(s) 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Aubin, T.. Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11 (1976), 573–598.Google Scholar
Aubin, T.. Meilleures constantes dans le théorème d'inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire. J. Funct. Anal. 32 (1979), 148–174.10.1016/0022-1236(79)90052-1CrossRefGoogle Scholar
Bakry, D., Concordet, D. and Ledoux, M.. Optimal heat kernel bounds under logarithmic Sobolev inequalities. ESAIM Probab. Statist. 1 (1995/97), 391–407.CrossRefGoogle Scholar
Caffarelli, L., Kohn, R. and Nirenberg, L.. First order interpolation inequalities with weights. Compositio Math. 53 (1984), 259–275.Google Scholar
do Carmo, M. P. and Xia, C.. Complete manifolds with non-negative Ricci curvature and the Caffarelli–Kohn–Nirenberg inequalities. Compositio Math. 140 (2004), 818–826.10.1112/S0010437X03000745CrossRefGoogle Scholar
Catrina, F., Wang, Z. Q. On the Caffarelli–Kohn–Nirenberg inequalities: sharp constants, existence (and non-existence) and symmetry of extremal functions. Commun. Pure Appl. Math. 54 (2001), 229–258.10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I3.0.CO;2-I>CrossRefGoogle Scholar
Chavel, I.. Riemannian geometry. A modern introduction, Cambridge studies in advanced mathematics, vol. 98, 2nd edn (Cambridge: Cambridge University Press, 2006).10.1017/CBO9780511616822CrossRefGoogle Scholar
Cheeger, J. and Colding, T.. On the structure of spaces with Ricci curvature bounded below I. J. Differ. Geom. 46 (1997), 406–480.Google Scholar
Cordero-Erausquin, D., Nazaret, B. and Villani, C.. A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities. Adv. Math. 182 (2004), 307–332.10.1016/S0001-8708(03)00080-XCrossRefGoogle Scholar
Costa, D. G.. Some new and short proofs for a class of Caffarelli–Kohn–Nirenberg type inequalities. J. Math. Anal. Appl. 337 (2008), 311–317.CrossRefGoogle Scholar
Del Pino, M. and Dolbeault, J.. Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81 (2002), 847–875.CrossRefGoogle Scholar
Del Pino, M. and Dolbeault, J.. The optimal Euclidean L p-Sobolev logarithmic inequality. J. Funct. Anal. 197 (2003), 151–161.10.1016/S0022-1236(02)00070-8CrossRefGoogle Scholar
Dolbeault, J. and Esteban, M. J.. Extremal functions for Caffarelli–Kohn–Nirenberg and logarithmic Hardy inequalities. Proc. R. Soc. Edinburgh Sect. A 142 (2012), 745–767.10.1017/S0308210510001101CrossRefGoogle Scholar
Druet, O., Hebey, E. and Vaugon, M.. Optimal Nash's inequalities on Riemannian manifolds the influence of geometry. Int. Math. Res. Not. 14 (1999), 735–779.CrossRefGoogle Scholar
Flynn, J.. Sharp Caffarelli–Kohn–Nirenberg-type inequalities on Carnot groups. Adv. Nonlinear Stud. 20 (2020), 95–111.CrossRefGoogle Scholar
Flynn, J.. Sharp L 2-Caffarelli-Kohn-Nirenberg inequalities for Grushin vector fields. Nonlinear Anal. 199 (2020), 15 pp., 111961.10.1016/j.na.2020.111961CrossRefGoogle Scholar
Gallot, S., Hulin, D. and Lafontaine, J.. Riemannian geometry, 3rd edn (Berlin: Springer-Verlag, 2004).10.1007/978-3-642-18855-8CrossRefGoogle Scholar
Han, Y.. Weighted Caffarelli–Kohn–Nirenberg type inequality on the Heisenberg group. Indian J. Pure Appl. Math. 46 (2015), 147–161.10.1007/s13226-015-0118-7CrossRefGoogle Scholar
Heisenberg, W.. Über den anschaulichen Inhalt de quantentheoretischen Kinematik und Mechenik. Z. Phys. 43 (1927), 175–198.CrossRefGoogle Scholar
Hebey, E.. Nonlinear Analysis on manifolds Sobolev spaces and inequalities. Courant Lecture Notes in Mathematics, vol. 5 (New York; Providence, RI: New York University, Courant Institute of Mathematical Sciences; American Mathematical Society, 1999).Google Scholar
Helgason, S.. Differential geometry, Lie groups and symmetric Spaces, graduate studies in mathematics, Vol. 34 (Providence, RI: Am. Math. Soc., 2001).Google Scholar
Huang, L., Kristály, A. and Zhao, W.. Sharp uncertainty principles on general Finsler manifolds. Trans. Am. Math. Soc. 373 (2020), 8127–8161.10.1090/tran/8178CrossRefGoogle Scholar
Kristály, A.. Metric measure spaces supporting Gagliardo–Nirenberg inequalities volume non-collapsing and rigidities. Calc. Var. Partial Differ. Equ. 55 (2016, 27 pp.).10.1007/s00526-016-1065-9CrossRefGoogle Scholar
Kristály, A.. Sharp uncertainty principles on Riemannian manifolds the influence of curvature. J. Math. Pures Appl. (9) 119 (2018), 326–346.CrossRefGoogle Scholar
Kristály, A. and Ohta, S.. Caffarelli–Kohn–Nirenberg inequality on metric measure spaces with applications. Math. Ann. 357 (2013), 711–726.CrossRefGoogle Scholar
Ledoux, M.. On manifold with nonnegative Ricci curvature and Sobolev inequalities. Commun. Anal. Geom. 7 (1999), 347–353.10.4310/CAG.1999.v7.n2.a7CrossRefGoogle Scholar
Li, P. and Wang, J.. Weighted Poincaré inequality and rigidity of complete manifolds. Ann. Sci. École Norm. Sup. (4), 39 (2006), 921–982.10.1016/j.ansens.2006.11.001CrossRefGoogle Scholar
Lieb, E. H.. Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. 118 (1983), 349–374.CrossRefGoogle Scholar
Minerbe, V.. Weighted Sobolev inequalities and Ricci flat manifolds. Geom. Funct. Anal. 18 (2009), 1696–1749.10.1007/s00039-009-0701-3CrossRefGoogle Scholar
Nguyen, H. M. and Squassina, M.. Fractional Caffarelli–Kohn–Nirenberg inequalities. J. Funct. Anal. 274 (2018), 2661–2672.CrossRefGoogle Scholar
Nguyen, V. H.. New sharp Hardy and Rellich type inequalities on Cartan–Hadamard manifolds and their improvements. Proc. R. Soc. Edinburgh Sect. A, 150 (2020), 2952–2981.CrossRefGoogle Scholar
Ozawa, T., Ruzhansky, M. and Suragan, D.. L p-Caffarelli–Kohn–Nirenberg type inequalities on homogeneous groups. Q. J. Math. 70 (2019), 305–318.CrossRefGoogle Scholar
Ruzhansky, M. and Suragan, D.. Anisotropic L 2-weighted Hardy and L 2-Caffarelli–Kohn–Nirenberg inequalities. Commun. Comtemp. Math. 19 (2007), 12 pp., 1750014.Google Scholar
Ruzhansky, M. and Suragan, D.. On horizontal Hardy, Rellich, Caffarelli–Kohn–Nirenberg and p-sub-Laplacian inequalities on stratified groups. J. Differ. Equ. 262 (2017), 1799–1821.10.1016/j.jde.2016.10.028CrossRefGoogle Scholar
Ruzhansky, M., Suragan, D. and Yessirkegenov, N.. Caffarelli–Kohn–Nirenberg and Sobolev type inequalities on stratified Lie groups. NoDEA Nonlinear Differ. Equ. Appl. 24 (2017), 56.CrossRefGoogle Scholar
Ruzhansky, M., Suragan, D. and Yessirkegenov, N.. Extended Caffarelli–Kohn–Nirenberg inequalities and remainders, stability and superweights for L p-weighted Hardy inequalities. Trans. Am. Math. Soc. Ser. B, 5 (2018), 32–62.CrossRefGoogle Scholar
Ruzhansky, M., Suragan, D. and Yessirkegenov, N.. Extended Caffarelli–Kohn–Nirenberg inequalities and superweights for L p-weighted Hardy inequalities. C. R. Acad. Sci. Paris 355 (2017), 694–698.10.1016/j.crma.2017.04.011CrossRefGoogle Scholar
Talenti, G.. Best constants in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976), 353–372.10.1007/BF02418013CrossRefGoogle Scholar
Weyl, H.. The theory of groups and quantum mechanics (New York: Dover Publications, 1931).Google Scholar
Xia, C.. Complete manifolds with non-negative Ricci curvature and almost best Sobolev constant. Illinois J. Math. 45 (2001), 1253–1259.10.1215/ijm/1258138064CrossRefGoogle Scholar
Xia, C.. The Gagliardo–Nirenberg inequalities and manifolds of non-negative Ricci curvature. J. Funct. Anal. 224 (2005), 230–241.CrossRefGoogle Scholar
Xia, C.. The Caffarelli–Kohn–Nirenberg inequalities on complete manifolds. Math. Res. Lett. 14 (2007), 875–885.CrossRefGoogle Scholar
Yang, Q., Su, D. and Kong, Y.. Hardy inequalities on Riemannian manifolds with negative curvature. Commun. Contemp. Math. 16 (2014), 24 pp.CrossRefGoogle Scholar