Subextension of m-Subharmonic Functions

  • Le Mau HaiEmail author
  • Trieu Van Dung


The aim of this note is to establish a result on subextension of m-subharmonic functions in the class \(\mathcal {F}_{m}({\Omega })\) with the precise description of the complex Hessian measure of the subextend function.


m-Subharmonic functions Class \(\mathcal {E}^{0}_{m}({\Omega })\) \(\mathcal {F}_{m}({\Omega })\) Complex Hessian operator Subextension of m-subharmonic functions 

Mathematics Subject Classification (2010)

32U05 32U15 32U40 32W20 



The authors would like to thank the reviewers for valuable suggestions and useful remarks that led to the improvements of the exposition of the paper.


  1. 1.
    Åhag, P., Cegrell, U., CzyŻ, R., Hiep, H.P.: Monge–Ampère measures on pluripolar sets. J. Math. Pures Appl. 92, 613–627 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–40 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Błocki, Z.: Weak solutions to the complex Hessian equation. Ann. Inst. Fourier, Grenoble 55, 1735–1756 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Błocki, Z.: The domain of definition of the complex Monge–Ampère operator. Am. J. Math. 128, 519–530 (2006)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cegrell, U.: The general definition of the complex Monge–Ampère operator. Ann. Inst. Fourier, Grenoble 54, 159–179 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cegrell, U., Kołodziej, S., Zeriahi, A.: Subextension of plurisubharmonic functions with weak singularities. Math. Z. 250, 7–22 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cegrell, U., Zeriahi, A.: Subextension of plurisubharmonic functions with bounded Monge–Ampère operator mass. C.R. Acad. Sci. Paris Ser. I 336, 305–308 (2003)CrossRefzbMATHGoogle Scholar
  8. 8.
    Chinh, H.L.: On Cegrell’s classes of m-subharmonic functions. arXiv:1301.6502v1 (2013)
  9. 9.
    Chinh, H.L.: A variational approach to complex Hessian equations in \(\mathbb {C}^{n}\). J. Math. Anal. Appl. 431, 228–259 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cuong, N.N.: Subsolution theorem for the complex Hessian equation. Univ. Iagell. Acta. Math. 50, 69–88 (2012)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Dinew, S., Kołodziej, S.: A priori estimates for the complex Hessian equations. Anal. PDE 7, 227–244 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hai, M.L., Hong, X.N.: Subextension of plurisubharmonic functions without changing the Monge–Ampère measures and applications. Ann. Pol. Math. 112, 55–66 (2014)CrossRefzbMATHGoogle Scholar
  13. 13.
    Hai, M.L., Hong, X.N., Dung, V.T.: Subextension of plurisubharmonic functions with boundary values in weighted pluricomplex energy classes. Complex Var. Elliptic Equ. 60, 1580–1593 (2015)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hai, M.L., Khiem, V.N., Dung, V.T.: Subextension of plurisubharmonic functions in unbounded hyperconvex domains and applications. Complex Var. Elliptic Equ. 61, 1116–1132 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hiep, H.P.: Pluripolar sets and the subextension in Cegrell’s classes. Complex Var. Elliptic Equ. 53, 675–684 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hung, V.V.: Local property of a class of m-subharmonic functions. Vietnam. J. Math. 44, 603–621 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hung, V.V., Phu, N.V.: Hessian measures on m-polar sets and applications to the complex Hessian equations. Complex Var. Elliptic Equ. 62, 1135–1164 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Klimek, M.: Pluripotential Theory. The Clarendon Press, Oxford University Press, New York (1991)zbMATHGoogle Scholar
  19. 19.
    Sadullaev, A., Abullaev, B.: Potential theory in the class of m-subharmonic functions. Proc. Steklov Inst. Math. 279, 155–180 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of MathematicsHanoi National University of EducationHanoiVietnam
  2. 2.Hung Vuong Gifted High SchoolPhu ThoVietnam

Personalised recommendations