Manuscripta Mathematica

, Volume 138, Issue 3–4, pp 371–394 | Cite as

Heat kernel estimates for the \({{\bar{\partial}}}\) -Neumann problem on G-manifolds



We prove heat kernel estimates for the \({\bar{\partial}}\) -Neumann Laplacian \({\square}\) acting in spaces of differential forms over noncompact manifolds with a Lie group symmetry and compact quotient. We also relate our results to those for an associated Laplace-Beltrami operator on functions.

Mathematics Subject Classification (2000)

Primary 32W30 32W05 35H20 


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  1. 1.
    Alexander R., Alexander S.: Geodesics in Riemannian manifolds with boundary. Indiana Univ. Math. J. 30(4), 481–488 (1981)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Beals R., Greiner P.C., Stanton N.K.: The heat equation on a CR manifold. J. Diff. Geom. 20, 343–387 (1984)MathSciNetMATHGoogle Scholar
  3. 3.
    Beals R., Stanton N.K.: The heat equation for the \({{\bar{\partial}}}\) -Neumann problem. I. Comm. Part. Diff. Eq. 12, 351–413 (1987)MathSciNetMATHGoogle Scholar
  4. 4.
    Beals R., Stanton N.K.: The heat equation for the \({{\bar{\partial}}}\) -Neumann problem. II. Canad. J. Math. 40, 502–512 (1988)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bergh J., Löfström J.: Interpolation spaces: an introduction. Springer, New York (1976)MATHCrossRefGoogle Scholar
  6. 6.
    Biroli M., Mosco U.: A Saint-Venant type principle for Dirichlet forms on discontinuous media. Ann. Mat. Pura Appl., IV. Ser. 169, 125–181 (1995)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Boggess, A., Raich, A.: Heat kernels, smoothness estimates and exponential decay, (arXiv:1004.0193) (2010)Google Scholar
  8. 8.
    Boggess A., Raich A.: The \({{\square_b}}\) -heat equation on quadric manifolds. J. Geom. Anal. 21, 256–275 (2011)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Catlin D.: Subelliptic estimates for the \({{\bar{\partial}}}\) -Neumann problem on pseudoconvex domains. Ann. Math. 126, 131–191 (1987)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    D’Angelo J.P.: Real hypersurfaces, orders of contact, and applications. Ann. Math. 115, 615–637 (1982)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Davies, E.B.: One-parameter Semigroups, London Mathematical Society Monographs, 15. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, New York (1980)Google Scholar
  12. 12.
    Davies E.B.: Heat kernels and spectral theory. Cambridge University Press, Cambridge (1990)Google Scholar
  13. 13.
    Dungey, N., ter Elst, A.F.M., Robinson, D.W.: Analysis on lie groups with polynomial growth, Progress in Mathematics, vol. 214. Birkhäuser Boston, Inc., Boston, MA (2003)Google Scholar
  14. 14.
    Della Sala, G., Perez, J.J.: Unitary representations of unimodular Lie groups in Bergman spaces. Math. Z. (in press)Google Scholar
  15. 15.
    Donnelly H., Li P.: Lower bounds for the eigenvalues of Riemannian manifolds. Michigan Math. J. 29, 149–161 (1982)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Driver B.K., Gross L., Saloff-Coste L.: Holomorphic functions and subelliptic heat kernels over Lie groups. J. Eur. Math. Soc. 11, 941–978 (2009)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Engliš M.: Pseudolocal estimates for \({{\bar\partial}}\) on general pseudoconvex domains. Indiana Univ. Math. J. 50, 1593–1607 (2001)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Faris, William G.: Self-adjoint operators, Lecture Notes in Mathematics 433 Springer-Verlag, Berlin, New York (1975)Google Scholar
  19. 19.
    Fefferman C.L., Sanchez-Calle A.: Fundamental solutions for second order subelliptic operators. Ann. Math. 124, 247–272 (1986)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Folland, G.B., Kohn, J.J.: The Neumann problem for the Cauchy–Riemann complex, Ann. Math. Stud. 75 Princeton University Press, Princeton (1972)Google Scholar
  21. 21.
    Folland G.B., Stein E.M.: Estimates for the \({{\bar{\partial}_b}}\) -complex and analysis on the Heisenberg group. Comm. Pure Appl. Math. 27, 429–522 (1974)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Fu S.: Hearing the type of a domain in \({{\mathbb{C}^2}}\) with the \({{\bar{\partial}}}\) -Neumann Laplacian. Dv. Math. 219, 568–603 (2008)MATHGoogle Scholar
  23. 23.
    Fu S.: Hearing pseudoconvexity with the Kohn Laplacian. Math. Ann. 331, 475–485 (2005)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Goldstein J.A.: Semigroups of linear operators and applications, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1985)Google Scholar
  25. 25.
    Gol’dshtei˘n V.M., Kuz’minov V.I., Shvedov I.A.: Dual spaces to spaces of differential forms. Siberian Math. J. 27, 35–44 (1986)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Grauert H.: On Levi’s problem and the imbedding of real-analytic manifolds. Ann. Math. 68, 460–472 (1958)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Greiner, P.C., Stein, E.M.: Estimates for the \({{\bar{\partial}}}\) -Neumann problem, Math. Notes. 19 Princeton Univ. Press, Princeton (1977)Google Scholar
  28. 28.
    Grigor’yan, A.: Heat Kernel and analysis on manifolds, AMS/IP Studies. Adv. Math. 47, p. 498. AMS International Press, New York (2009)Google Scholar
  29. 29.
    Gromov M.: Curvature, diameter, and betti numbers. Comment. Math. Helv. 56, 179–195 (1981)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Gromov, M.: Structures métriques pour les variétés riemanniennes, Rédigés par J. Lafontaine et P. Pansu (ed.), Cedic/F.Nathan, Paris (1981)Google Scholar
  31. 31.
    Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser modern classics. Birkhäuser, Boston (2007)Google Scholar
  32. 32.
    Gromov M., Henkin G., Shubin M.: Holomorphic L 2 functions on coverings of pseudoconvex manifolds. Geom. Funct. Anal. 8, 552– (1998)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Hartogs F.: Zur theorie der analytischen funktionen mehrerer unabhängiger Veränderlichen insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen forschreiten. Math. Ann. 62, 1–88 (1906)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Heinzner, P., Huckleberry, A.T., Kutzschebauch, F.: Abels’ theorem in the real analytic case and applications to complexifications. In: Complex analysis and geometry, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, pp. 229–273 (1995)Google Scholar
  35. 35.
    Jerison D., Sanchez-Calle A.: Estimates for the heat kernel for a sum of squares of vector fields. Indiana Univ. J. Math. 35, 835–854 (1986)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Kato, T.: Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, 132 Springer-Verlag Inc., New York (1966)Google Scholar
  37. 37.
    Kobayashi, S.: Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15 Princeton University Press, Iwanami Shoten, Princeton, Tokyo (1987)Google Scholar
  38. 38.
    Kodaira, K.: Complex manifolds and deformation of complex structures, Grundlehren der Mathematischen Wissenschaften 283, Springer-Verlag, New York (1986)Google Scholar
  39. 39.
    Kohn J.J.: Harmonic Integrals on Strongly Pseudoconvex Manifolds, I. Ann. Math. 78, 112–148 (1963)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Kohn J.J.: Harmonic integrals on strongly pseudoconvex manifolds, II. Ann. Math. 79, 450–472 (1964)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Kohn J.J., Nirenberg L.: Non-coercive boundary value problems. Comm. Pure Appl. Math. 18, 443–492 (1965)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Levi E.E.: Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse. Ann. Mat. Pura Appl. 17, 61–87 (1910)Google Scholar
  43. 43.
    Levi E.E.: Sulle ipersuperficie dello spazio a 4 dimensioni che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse. Ann. Mat. Pura Appl. 18(1), 69–79 (1911)Google Scholar
  44. 44.
    Lions, J.L., Magenes, E.: Non-homgeneous boundary value problems and applications. In: Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, 181 Springer-Verlag, Berlin (1972)Google Scholar
  45. 45.
    McAvity D.M., Osborn H.: Asymptotic expansion of the heat kernel for generalized boundary conditions, Class. Quantum Grav. 8, 1445–1454 (1991)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Margulis G.A.: Discrete subgroups of semisimple lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 17. Springer-Verlag, Berlin (1991)Google Scholar
  47. 47.
    Métivier G.: Spectral asymptotics for the \({{\bar{\partial}}}\) -Neumann problem. Duke Math. J. 48, 779–806 (1981)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Nagel A., Stein E.M., Wainger S.: Balls and metrics defined by vector fields. I: Basic Properties. Acta Math. 155, 103–147 (1985)MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Nash J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Perez J.J.: The G-Fredholm property of the \({{\bar{\partial}}}\) -Neumann Problem. J. Geom. Anal. 19, 87–106 (2009)MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Perez J.J.: The Levi problem on strongly pseudoconvex G-bundles, Ann. Global Anal. Geom. 37, 1–20 (2010)MATHCrossRefGoogle Scholar
  52. 52.
    Perez J.J.: A transversal Fredholm property for the \({{\bar{\partial}}}\) -Neumann problem on G-bundles. Contemp. Math. 535, 187–193 (2011)CrossRefGoogle Scholar
  53. 53.
    Perez, J.J.: Generalized Fredholm properties for invariant pseudodifferential operators. Available at
  54. 54.
    Perez J.J., Stollmann P.: Essential self-adjointness, generalized eigenforms, and spectra for the \({{\bar{\partial}}}\) -Neumann problem on G-manifolds. J. Funct. Anal. 261, 2717–2740 (2011)MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Reed M., Simon B.: Methods of modern mathematical physics. I. Functional analysis. Academic Press Inc., New York (1980)MATHGoogle Scholar
  56. 56.
    Raich A.: Heat equations and the weighted \({{\bar{\partial}}}\) -problem, (arXiv:0704.2768) (2009)Google Scholar
  57. 57.
    Rosenberg S.: Semigroup domination and vanishing theorems. Contemp. Math. 73, 287–302 (1988)CrossRefGoogle Scholar
  58. 58.
    Saloff-Coste L.: Uniformly elliptic operators on Riemannian manifolds. J. Diff. Geom. 36(2), 417–450 (1992)MathSciNetMATHGoogle Scholar
  59. 59.
    Shubin, M.A.: Spectral theory of elliptic operators on noncompact manifolds, Astérisque. 207(5), 35–108 (1992). Méthodes semi-classiques, Vol. 1 (Nantes, 1991)Google Scholar
  60. 60.
    Simon B.: Schrödinger semigroups. Bull. Amer. Math. Soc. 7, 447–526 (1982)MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Siu Y.-T.: Pseudoconvexity and the problem of Levi. Bull. Amer. Math. Soc. 84, 481–512 (1978)MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Stanton N.K.: The heat equation in several complex variables. Bull. Amer. Math. Soc. 11, 65–84 (1984)MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Stanton, N.K.: The heat equation for the \({{\bar{\partial}}}\) -Neumann problem in a strictly pseudoconvex Siegel domain. I, II, J. Analyse Math. 38: 67–112 (1980); 39: 189–202 (1981)Google Scholar
  64. 64.
    Stanton N.K.: The solution of the \({{\bar{\partial}}}\) -Neumann problem in a strictly pseudoconvex Siegel domain. Invent. Math. 65, 137–174 (1981)MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    Stanton N.K., Tartakoff D.S.: The heat equation for the \({{\bar{\partial}_b}}\) -Laplacian. Comm. Part. Diff. Eq. 9, 597–686 (1984)MathSciNetMATHCrossRefGoogle Scholar
  66. 66.
    Stollmann P.: A dual characterization of length spaces with application to Dirichlet metric spaces. Studia Math. 198(3), 221–233 (2010)MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    Straube, E.J.: The L 2-Sobolev theory of the \({{\bar{\partial}}}\) -Neumann problem, ESI Lectures in Mathematics and Physics, EMS (2010)Google Scholar
  68. 68.
    Sturm K.T.: Analysis on local dirichlet spaces. I: Recurrence, conservativeness and L p-Liouville properties. J. Reine Angew. Math. 456, 173–196 (1994)MathSciNetMATHGoogle Scholar
  69. 69.
    Sturm, K.T.: On the geometry defined by dirichlet forms. In: Seminar on stochastic analysis, random fields and applications, Ascona, 1993, (E. Bolthausen et al., eds.), vol 36, pp. 231–242 Progr. Probab., Birkhäuser, Boston (1995)Google Scholar
  70. 70.
    ter Elst, A.F.M., Robinson, D.W., Sikora, A., Zhu, Y.: Dirichlet forms and degenerate elliptic operators. Partial differential equations and functional analysis, 73–95, Oper. Theory Adv. Appl. 168 Birkhäuser, Basel (2006)Google Scholar
  71. 71.
    Wang, H.: L 2-index formula for proper cocompact group actions, preprint.

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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität WienViennaAustria
  2. 2.Fakultät für MathematikTechnische UniversitätChemnitzGermany

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