Advertisement

Weakly Solutions to the Complex Monge–Ampère Equation on Bounded Plurifinely Hyperconvex Domains

  • Nguyen Xuan Hong
  • Hoang Van Can
Article
  • 18 Downloads

Abstract

Let \(\mu \) be a non-negative measure defined on bounded \({\mathcal {F}}\)-hyperconvex domain \(\Omega \). We are interested in giving sufficient conditions on \(\mu \) such that we can find a plurifinely plurisubharmonic function satisfying NP\((dd^c u)^n =\mu \) in QB\((\Omega )\).

Keywords

Plurifinely pluripotential theory Plurifinely plurisubharmonic functions Complex Monge–Ampère equations 

Mathematics Subject Classification

32U05 32U15 

Notes

Acknowledgements

This article has been partially completed during a stay of the first author at the Vietnam Institute for Advanced Study in Mathematics. He wishes to thank the institution for their kind hospitality and support. This research is funded by the Vietnam Ministry of Education and Training under Grant Number B2018-SPH-57. The authors would like to thank the referees for valuable remarks which led to the improvements of the exposition of the paper.

References

  1. 1.
    Åhag, P., Cegrell, U., Czyż, R., Hiep, P.H.: Monge–Ampère measures on pluripolar sets. J. Math. Pures Appl. 92, 613–627 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Åhag, P., Cegrell, U., Hiep, P.H.: A product property for the pluricomplex energy. Osaka J. Math. 47, 637–650 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Åhag, P., Cegrell, U., Hiep, P.H.: Monge–Ampère measures on subvarieties. J. Math. Anal. Appl. 423(1), 94–105 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bedford, E., Taylor, B.A.: Fine topology, Silov boundary and \((dd^c)^n\). J. Funct. Anal. 72, 225–251 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Błocki, Z.: On the definition of the Monge–Ampère operator in \({\mathbb{C}}^2\). Math. Ann. 328(3), 415–423 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cegrell, U.: Pluricomplex energy. Acta Math. 180, 187–217 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cegrell, U.: The general definition of the complex Monge–Ampère operator. Ann. Inst. Fourier 54(1), 159–179 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fuglede, B.: Finely Harmonic Functions, vol. 289. Springer, Berlin (1972). Lecture Notes in MathzbMATHGoogle Scholar
  9. 9.
    Hai, L.M., Hiep, P.H.: An equality on the complex Monge–Ampère measures. J. Math. Anal. Appl. 444(1), 503–511 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hai, L.M., Hiep, P.H., Hong, N.X., Phu, N.V.: The Monge-Ampère type equation in the weighted pluricomplex energy class. Int. J. Math. 25(5), 1450042 (2014)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hai, L.M., Trao, N.V., Hong, N.X.: The complex Monge–Ampère equation in unbounded hyperconvex domainsin \({\mathbb{C}}^n\). Complex Var. Elliptic Equ. 59(12), 1758–1774 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hai, L.M., Thuy, T.V., Hong, N.X.: A note on maximal subextensions of plurisubharmonic functions. Acta Math. Vietnam. 43, 137–146 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hong, N.X.: Range of the complex Monge–Ampère operator on plurifinely domain. Complex Var. Elliptic Equ. 63, 532–546 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hong, N.X., Can, H.V.: On the approximation of weakly plurifinely plurisubharmonic functions. Indag. Math. (2018).  https://doi.org/10.1016/j.indag.2018.05.015 Google Scholar
  15. 15.
    Hong, N.X., Hai, L.M., Viet, H.: Local maximality for bounded plurifinely plurisubharmonic functions. Potential Anal. 48, 115–123 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hong, N.X., Thuy, T.V.: Hölder continuity for solutions to the complex Monge-Ampère equations in non-smooth pseudoconvex domains. Anal. Math. Phys. (2017).  https://doi.org/10.1007/s13324-017-0175-7 Google Scholar
  17. 17.
    Hong, N.X., Viet, H.: Local property of maximal plurifinely plurisubharmonic functions. J. Math. Anal. Appl. 441, 586–592 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lien, N.T.: Local property of maximal unbounded plurifinely plurisubharmonic functions. Complex Var. Elliptic Equ. (2018).  https://doi.org/10.1080/17476933.2018.1427082 Google Scholar
  19. 19.
    El Kadiri, M.: Fonctions finement plurisousharmoniques et topologie plurifine. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 27, 77–88 (2003)MathSciNetGoogle Scholar
  20. 20.
    El Kadiri, M., Fuglede, B., Wiegerinck, J.: Plurisubharmonic and holomorphic functions relative to the plurifine topology. J. Math. Anal. Appl. 381, 107–126 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    El Kadiri, M., Smit, I.M.: Maximal plurifinely plurisubharmonic functions. Potential Anal. 41, 1329–1345 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    El Kadiri, M., Wiegerinck, J.: Plurifinely plurisubharmonic functions and the Monge–Ampère operator. Potential Anal. 41, 469–485 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    El Marzguioui, S., Wiegerinck, J.: Continuity properties of finely plurisubharmonic functions. Indiana Univ. Math. J. 59, 1793–1800 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Trao, N.V., Viet, H., Hong, N.X.: Approximation of plurifinely plurisubharmonic functions. J. Math. Anal. Appl. 450, 1062–1075 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wiegerinck, J.: Plurifine potential theory. Ann. Polon. Math. 106, 275–292 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsHanoi National University of EducationHanoiVietnam
  2. 2.Department of Basis SciencesUniversity of Transport TechnologyHanoiVietnam

Personalised recommendations