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Complex Hessian Operator and Lelong Number for Unbounded m-subharmonic Functions

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Abstract

m-subharmonic functions are the right class of admissible solutions to the complex Hessian equation. In this paper, we generalize the definition of the complex Hessian operator to some unbounded m-subharmonic functions, and we prove that the complex Hessian operator is continuous on the monotonically decreasing sequences of m-subharmonic functions. Moreover we establish the Lelong-Jensen type formula and introduce the Lelong number for m-subharmonic functions. A useful inequality for the mixed Hessian operator is showed.

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References

  1. Sadullaev, A., Abdullaev, B.: Potential theory in the class of m-subharmonic functions. Tr. Mat. Inst. Steklova 279, 166–192 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Sadullaev, A., Abdullaev, B.: Capacities and Hessians in the class of m-subharmonic functions. Dokl. Akad. Nauk 448, 515–517 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math. 37, 1–44 (1976)

    MathSciNet  MATH  Google Scholar 

  4. Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–40 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  5. Blocki, Z.: Estimates for the complex Monge-Ampère operator. Bull. Polish Acad. Sci. Math. 41, 151–157 (1993)

    MathSciNet  MATH  Google Scholar 

  6. Blocki, Z.: Weak solutions to the complex Hessian equation. Ann. Inst. Fourier (Grenoble) 55(5), 1735–1756 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian. Acta Math. 155, 261–301 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chou, K.-S., Wang, X.-J.: Variational theory for Hessian equations. Comm. Pure Appl. Math. 54, 1029–1064 (2001)

    MathSciNet  Google Scholar 

  9. Demailly, J.P.: Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines. Mém. Soc. Math. France (N.S.) 19, 1–124 (1985)

    MATH  Google Scholar 

  10. Demailly, J.P.: Mesures de Monge-Ampère et mesures pluriharmoniques. Math. Z. 194, 519–564 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  11. Demailly, J.P.: Potential Theory in Several Complex Variables. Available online at: http://www-fourier.ujf-grenoble.fr/demailly/documents.html

  12. Demailly, J.P.: Monge-Ampère operators, Lelong numbers and intersection theory. Complex analysis and geometry, pp. 115–193. Univ. Ser. Math.,Plenum, NewYork (1993)

  13. Demailly, J.P.: Complex Analytic and Differential Geometry, book available online at: http://www-fourier.ujf-grenoble.fr/demailly/documents.html

  14. Dinew, S., Kolodziej, S.: Liouville Calabi-Yau Type Theorems for Complex Hessian Equations. arXiv:1203.3995v1

  15. Dinew, S., Kolodziej, S.: A priori estimates for complex Hessian equations. Anal. PDE 7, 227–244 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hou, Z.: Complex Hessian equation on Kähler manifold. Int. Math. Res. Not. 16, 3098–3111 (2009)

    MATH  Google Scholar 

  17. Hou, Z., Ma, X.-N., Wu, D.: A second order estimate for complex Hessian equations on a compact Kähler manifold. Math. Res. Lett. 17, 547–561 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ivochkina, N., Trudinger, N.S., Wang, X.-J.: The Dirichlet problem for degenerate Hessian equations. Comm. Partial Diff. Equations 29, 219–235 (2004)

    MathSciNet  MATH  Google Scholar 

  19. Klimek, M.: Pluripotential Theory. Clarendon Press (1991)

  20. Labutin, D.: Potential estimates for a class of fully nonlinear elliptic equations. Duke Math. J. 111, 1–49 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lelong, P.: Fonctionnelles analytiques et fonctions enti`eres (n variables), Séminaire de Mathématiques Supérieures, No. 13 (Été, 1967), Les Presses de \(l^{\prime }\)Université de Montréal. Montreal, Que., 298 (1968)

  22. Lelong, P., Gruman, L.: Entire Functions of Several Complex Variables. Springer-Verlag (1986)

  23. Li, S.-Y.: On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian. Asian J. Math. 8(1), 87–106 (2004)

  24. Lu, H.C.: Viscosity solutions to complex Hessian equations. J. Funct. Anal. 264, 1355–1379 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lu, H.C.: Solutions to degenerate complex Hessian equations. J. Math. Pures Appl. 100, 785–805 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lu, H.C.: A Variational Approach to Complex Hessian Equations in \(\mathbb {C}^{n}\), arXiv: 1301.6502

  27. Nguyen, N.-C.: Subsolution theorem for the complex Hessian equation, Univ. Iagel. Acta Math 50, 69–88 (2013)

    MATH  Google Scholar 

  28. Nguyen, N.-C.: Hölder continuous solutions to complex Hessian equations. Potential Anal. 41, 887–902 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Plis, S.: The Smoothing of m-subharmonic Functions. arXiv: 1312.1906

  30. Trudinger, N.S., Wang, X.J.: Hessian measures I. Topol. Methods Nonlinear Anal. 19, 225–239 (1997)

    MathSciNet  MATH  Google Scholar 

  31. Trudinger, N.S., Wang, X.J.: Hessian measures II. Ann. Math. 150, 579–604 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  32. Trudinger, N.S., Wang, X.J.: Hessian measures III. J. Funct. Anal. 193, 1–23 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wan, D.: Estimates for k-Hessian operator and some applications. Czech. Math. J. 63, 547–564 (2013)

    MATH  MathSciNet  Google Scholar 

  34. Wan, D., Wang, W.: Lelong-Jensen type formula, k-Hessian boundary measure and Lelong number for k-convex functions. J. Math. Pures Appl. 99, 635–654 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, X.J.: The k-Hessian equation, Geometric Analysis and PDEs. Lect. Not. Math 1977, 177–252 (2009)

    Article  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Statistics, Shenzhen University, Room 412 Science and Technology Building, Shenzhen, 518060, People’s Republic of China

    Dongrui Wan

  2. Department of Mathematics, Zhejiang University, Room 103 Ouyang Building, Hangzhou, 310027, People’s Republic of China

    Wei Wang

Authors
  1. Dongrui Wan
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  2. Wei Wang
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Corresponding author

Correspondence to Dongrui Wan.

Additional information

This work is supported by Natural Science Foundation of SZU (grant no. 201424) and National Nature Science Foundation in China (No. 11401390; No. 11171298).

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Wan, D., Wang, W. Complex Hessian Operator and Lelong Number for Unbounded m-subharmonic Functions. Potential Anal 44, 53–69 (2016). https://doi.org/10.1007/s11118-015-9498-x

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  • DOI: https://doi.org/10.1007/s11118-015-9498-x

Keywords

Mathematics Subject Classification (2010)