DOI QR코드

DOI QR Code

THE HARDY TYPE INEQUALITY ON METRIC MEASURE SPACES

  • Du, Feng ;
  • Mao, Jing ;
  • Wang, Qiaoling ;
  • Wu, Chuanxi
  • Received : 2017.11.02
  • Accepted : 2018.06.26
  • Published : 2018.11.01

Abstract

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Hardy type inequality with the same exponent n ($n{\geq}3$), then it has exactly the n-dimensional volume growth. Besides, three interesting applications of this fact have also been given. The first one is that we prove that complete noncompact smooth metric measure space with non-negative weighted Ricci curvature on which the Hardy type inequality holds with the best constant are isometric to the Euclidean space with the same dimension. The second one is that we show that if a complete n-dimensional Finsler manifold of nonnegative n-Ricci curvature satisfies the Hardy type inequality with the best constant, then its flag curvature is identically zero. The last one is an interesting rigidity result, that is, we prove that if a complete n-dimensional Berwald space of non-negative n-Ricci curvature satisfies the Hardy type inequality with the best constant, then it is isometric to the Minkowski space of dimension n.

Keywords

Hardy inequality;metric measure spaces;volume doubling condition;Finsler manifolds;smooth metric measure spaces

References

  1. P. Petersen, Comparison geometry problem list, in Riemannian geometry (Waterloo, ON, 1993), 87-115, Fields Inst. Monogr., 4, Amer. Math. Soc., Providence, RI, 1996.
  2. Z. Shen, Volume comparison and its applications in Riemann-Finsler geometry, Adv. Math. 128 (1997), no. 2, 306-328.
  3. J. Van Schaftingen, Anisotropic symmetrization, Ann. Inst. H. Poincare Anal. Non Lineaire 23 (2006), no. 4, 539-565.
  4. G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353-372. https://doi.org/10.1007/BF02418013
  5. G. Wei and W. Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377-405.
  6. C. Xia, Complete manifolds with nonnegative Ricci curvature and almost best Sobolev constant, Illinois J. Math. 45 (2001), no. 4, 1253-1259. https://doi.org/10.1215/ijm/1258138064
  7. A. Gray, Tubes, Addison-Wesley Publishing Company, Advanced Book Program, Red-wood City, CA, 1990.
  8. J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001.
  9. N. N. Katz and K. Kondo, Generalized space forms, Trans. Amer. Math. Soc. 354 (2002), no. 6, 2279-2284.
  10. A. Kristaly, Metric measure spaces supporting Gagliardo-Nirenberg inequalities: volume non-collapsing and rigidities, Calc. Var. Partial Differential Equations 55 (2016), no. 5, Art. 112, 27 pp.
  11. A. Kristaly and S. Ohta, Caffarelli-Kohn-Nirenberg inequality on metric measure spaces with applications, Math. Ann. 357 (2013), no. 2, 711-726.
  12. M. Ledoux, On manifolds with non-negative Ricci curvature and Sobolev inequalities, Comm. Anal. Geom. 7 (1999), no. 2, 347-353.
  13. E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349-374.
  14. F. Madani, Le probleme de Yamabe avec singularites, Bull. Sci. Math. 132 (2008), no. 7, 575-591.
  15. V. G. Maz'ja, Sobolev Spaces, translated from the Russian by T. O. Shaposhnikova, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985.
  16. J. Mao, Eigenvalue inequalities for the p-Laplacian on a Riemannian manifold and estimates for the heat kernel, J. Math. Pures Appl. (9) 101 (2014), no. 3, 372-393.
  17. J. Mao, The Caffarelli-Kohn-Nirenberg inequalities and manifolds with nonnegative weighted Ricci curvature, J. Math. Anal. Appl. 428 (2015), no. 2, 866-881.
  18. J. Mao, The Gagliardo-Nirenberg inequalities and manifolds with non-negative weighted Ricci curvature, Kyushu J. Math. 70 (2016), no. 1, 29-46.
  19. J. Mao, Volume comparison theorems for manifolds with radial curvature bounded, Czechoslovak Math. J. 66(141) (2016), no. 1, 71-86.
  20. J. Mao, Functional inequalities and manifolds with nonnegative weighted Ricci curvature, submitted.
  21. S. Ohta, Finsler interpolation inequalities, Calc. Var. Partial Differential Equations 36 (2009), no. 2, 211-249.
  22. A. Alvino, V. Ferone, P.-L. Lions, and G. Trombetti, Convex symmetrization and applications, Ann. Inst. H. Poincare Anal. Non Lineaire 14 (1997), no. 2, 275-293.
  23. T. Aubin, Problemes isoperimetriques et espaces de Sobolev, J. Differential Geometry 11 (1976), no. 4, 573-598.
  24. T. Aubin, Nonlinear Analysis on Manifolds, Monge-Ampere Equations, Grundlehren der Mathematischen Wissenschaften, 252, Springer-Verlag, New York, 1982.
  25. T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.
  26. D. Bakry and M. Emery, Hypercontractivite de semi-groupes de diffusion, C. R. Acad. Sci. Paris Ser. I Math. 299 (1984), no. 15, 775-778.
  27. D. Bakry and M. Emery, Diffusions hypercontractives, in Seminaire de probabilites, XIX, 1983/84, 177-206, Lecture Notes in Math., 1123, Springer, Berlin, 1985.
  28. D. Bao, S.-S. Chern, and Z. Shen, An introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics, 200, Springer-Verlag, New York, 2000.
  29. L. Caffarelli, R. Kohn, and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math. 53 (1984), no. 3, 259-275.
  30. F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math. 54 (2001), no. 2, 229-258. https://doi.org/10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I
  31. K. S. Chou and C. W. Chu, On the best constant for a weighted Sobolev-Hardy inequality, J. London Math. Soc. (2) 48 (1993), no. 1, 137-151.
  32. M. do Carmo and C. Xia, Ricci curvature and the topology of open manifolds, Math. Ann. 316 (2000), no. 2, 391-400.
  33. M. do Carmo and C. Xia, Complete manifolds with non-negative Ricci curvature and the Caffarelli-Kohn-Nirenberg inequalities, Compos. Math. 140 (2004), no. 3, 818-826.
  34. F. Du, J. Mao, Q.-L. Wang, and C.-X. Wu, The Gagliardo-Nirenberg inequality on metric measure spaces, available online at arXiv:1511.04696.
  35. P. Freitas, J. Mao, and I. Salavessa, Spherical symmetrization and the flrst eigenvalue of geodesic disks on manifolds, Calc. Var. Partial Differential Equations 51 (2014), no. 3-4, 701-724. https://doi.org/10.1007/s00526-013-0692-7
  36. L. Adriano and C. Xia, Hardy type inequalities on complete Riemannian manifolds, Monatsh. Math. 163 (2011), no. 2, 115-129.

Acknowledgement

Supported by : NSF