Sharp Global Bounds for the Hessian on Pseudo-Hermitian Manifolds

  • Sagun ChanilloEmail author
  • Juan J. Manfredi
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


We consider an abstract CR manifold equipped with a strictly positive definite Levi form, which defines a pseudo-Hermitian metric on the manifold. On such a manifold it is possible to define a natural sums of squares sub-Laplacian operator. We use Bochner identities to obtain Cordes–Friedrichs type inequalities on such manifolds where the L 2 norm of the Hessian tensor of a function is controlled by the L 2 norm of the sub-Laplacian of the function with a sharp constant for the inequality. By perturbation we proceed to develop a Cordes–Nirenberg type theory for non-divergence form equations on CR manifolds. Some applications are given to the regularity of p-Laplacians on CR manifolds.


Heisenberg Group Levi Form Paneitz Operator Cordes Condition Hessian Tensor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Capogna, L., Regularity of quasi-linear equations in the Heisenberg group. Comm. Pure Appl. Math. 50 (1997), no. 9, 867–889.CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Chang, S.C., Cheng, J.H., Chiu, H.L., A fourth order Q-curvature flow on a CR 3-manifold, Indiana Math. J., 56 (2007), 1793–1826. Scholar
  3. 3.
    Chern, S.S., Hamilton, R.S., On Riemannian metrics adapted to three-dimensional contact manifolds. With an appendix by Alan Weinstein, Lecture Notes in Math., 1111, Workshop Bonn 1984 (Bonn, 1984), 279–308, Springer, Berlin, 1985.Google Scholar
  4. 4.
    Chiu, H.L., The sharp lower bound for the first positive eigenvalue of the sub-Laplacian on a pseudohermitian 3-manifold, Ann. Global Anal. Geom. 30 (2006), no. 1, 81–96.CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Cordes, H.O., Zero order a-priori estimates for solutions of elliptic differential equations, Proceedings of Symposia in Pure Mathematics IV (1961).Google Scholar
  6. 6.
    Domokos, A., Fanciullo, M.S., On the best constant for the Friedrichs-Knapp-Stein inequality in free nilpotent Lie groups of step two and applications to subelliptic PDE, The Journal of Geometric Analysis, 17(2007), 245–252.CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Domokos, A., Manfredi, J.J., Subelliptic Cordes estimates, Proc. Amer. Math. Soc. 133 (2005), no. 4, 1047–1056.CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Gilbarg, D., Trudinger, N.S., Elliptic partial differential equations of second order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.Google Scholar
  9. 9.
    Greenleaf, A., The first eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold, Comm. Partial Differential Equations 10 (1985), no. 2, 191–217.CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Lee, J.M., The Fefferman metric and pseudo-Hermitian invariants, Trans. Amer. Math. Soc. 296 (1986), no. 1, 411–429.zbMATHMathSciNetGoogle Scholar
  11. 11.
    Li, S.Y., Luk, H.S., The sharp lower bound for the first positive eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold, Proc. Amer. Math. Soc. 132 (2004), no. 3, 789–798.CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Lin, F.H., Second derivative L p-estimates for elliptic equations of nondivergent type, Proc. Amer. Math. Soc. 96 (1986), no. 3, 447–451zbMATHMathSciNetGoogle Scholar
  13. 13.
    Manfredi, J.J., Mingione, G., Regularity results for quasilinear elliptic equations in the Heisenberg group, Mathematische Annalen, 339 (2007), no. 3, 485–544.CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Segovia, C., On the area function of Lusin, Studia Math. 33 (1969) 311–343.zbMATHMathSciNetGoogle Scholar
  15. 15.
    Stein, E., Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, NJ, 1970.zbMATHGoogle Scholar
  16. 16.
    Strichartz, R.S., Harmonic analysis and Radon transforms on the Heisenberg group, J. Funct. Analysis, 96(1991), 350–406..CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Talenti, G., Sopra una classe di equazioni ellittiche a coefficienti misurabili, (Italian) Ann. Mat. Pura Appl. (4) 69, 1965, 285–304.CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Trèves, F., Hypo-analytic structures. Local theory, Princeton Mathematical Series, 40. Princeton University Press, Princeton, NJ, 1992.zbMATHGoogle Scholar
  19. 19.
    Webster, S.M., Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), no. 1, 25–41.zbMATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2010

Authors and Affiliations

  1. 1.Rutgers UniversityPiscatawayUSA
  2. 2.Department of MathematicsUniversity of PittsburghPittsburghUSA

Personalised recommendations