Regularity for a Class of Singular Complex Hessian Equations


In this paper, we study the regularity of the complex Hessian equation when the right hand has pole singularity. We show the Hölder continuity of the solution to the Dirichlet problem. In particular, for the complex Monge-Ampère equation, we improve a result of [7].

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  1. [1]

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

    MathSciNet  Article  Google Scholar 

  2. [2]

    Blocki, Z.: Weak solutions to the complex Hessian equation. Annales de l’Institut Fourier, 55(5), 1735–1756 (2005)

    MathSciNet  Article  Google Scholar 

  3. [3]

    Blocki, Z., Dinew, S.: A local regularity of the complex Monge—Ampère equation. Math. Ann., 351(2), 411–416 (2011)

    MathSciNet  Article  Google Scholar 

  4. [4]

    Caffarelli, L., Kohn, J., Nirenberg, L., et al.: The Dirichlet problem for nonlinear second-order elliptic equations, II. Complex Monge—Ampere, and uniformaly elliptic equations. Comm. Pure Appl. Math., 38, 209–252 (1985)

    MathSciNet  Article  Google Scholar 

  5. [5]

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

    MathSciNet  Article  Google Scholar 

  6. [6]

    Dinew, S., Kolodziej, S.: A priori estimates for complex Hessian equations. Anal. PDE, 7(1), 227–244 (2014)

    MathSciNet  Article  Google Scholar 

  7. [7]

    Feng, K., Shi, Y. L., Xu, Y. Y.: On the Dirichlet problem for a class of singular complex Monge-Ampere equations. Acta Math. Sinica, 34, 209–220 (2018)

    MathSciNet  Article  Google Scholar 

  8. [8]

    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983

    Book  Google Scholar 

  9. [9]

    Guedj, V., Kolodziej, S., Zeriahi, A.: Holder continuous solutions to Monge-Ampere equations. Bull. London Math. Soc., 40, 1070–1080 (2008)

    MathSciNet  Article  Google Scholar 

  10. [10]

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

    MathSciNet  Article  Google Scholar 

  11. [11]

    Kolodziej, S.: The complex Monge-Ampère equation. Acta Math., 180, 69–117 (1998)

    MathSciNet  Article  Google Scholar 

  12. [12]

    Kolodziej, S.: Equicontinuity of families of pluri-subharmonic functions with bounds on their Monge—Ampère masses. Math. Z., 240, 835–847 (2002)

    MathSciNet  Article  Google Scholar 

  13. [13]

    La Nave, G., Tian, G.: A continuity method to construct canonical metrics. Math. Ann., 365, 911–921 (2016)

    MathSciNet  Article  Google Scholar 

  14. [14]

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

    MathSciNet  Article  Google Scholar 

  15. [15]

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

    MathSciNet  Article  Google Scholar 

  16. [16]

    Wang, X. J.: The k-Hessian Equation. Lecture Notes in Math., Vol. 1977, Springer, 2009

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We thank the referees for their time and comments.

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Corresponding author

Correspondence to Bin Zhou.

Additional information

The first author is supported by the CSC (Grant No. 201906320165); the second author is supported by NSFC (Grant Nos. 11571018 and 11822101)

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Wang, J.X., Zhou, B. Regularity for a Class of Singular Complex Hessian Equations. Acta. Math. Sin.-English Ser. (2021).

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  • Complex Hessian equation
  • singularity
  • regularity

MR(2010) Subject Classification

  • 32W20