Skip to main content
SpringerLink
Log in

Envelope of plurisubharmonic functions in Cegrell’s classes

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

This paper aims to study the complex Monge–Ampère measure of an envelope of plurisubharmonic functions in Cegrell’s classes. We prove the Hölder continuity of the envelope of Hölder continuous plurisubharmonic functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

Data sharing not applicable to this article as no data sets were generated or analysed during the current study

References

  1. Åhag, P., Cegrell, U., Czyż, R., Hiep, P.H.: Monge–Ampère measures on pluripolar sets. J. Math. Pures Appl. 92, 613–627 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Cegrell, U.: The general definition of the complex Monge–Ampère operator. Ann. Inst. Fourier (Grenoble) 54(1), 159–179 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cegrell, U.: A general Dirichlet problem for the complex Monge–Ampère operator. Ann. Polon. Math. 94, 131–147 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cegrell, U.: Convergence in capacity. Can. Math. Bull. 55(2), 242–248 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cuong, N.N.: On the Hölder continuous subsolution problem for the complex Monge–Ampère equation, II. Anal. PDE 13(2), 435–453 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. Demailly, J.P., Dinew, S., Guedj, V., Hiep, P.H., Kołodziej, S., Zeriahi, A.: Hölder continuous solutions to Monge–Ampère equations. J. Eur. Math. Soc. (JEMS) 16(4), 619–647 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dinh, T.-C., Nguyen, V.-A.: Characterization of Monge–Ampère measures with Hölder continuous potentials. J. Funct. Anal. 266(1), 67–84 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. El Kadiri, M., Fuglede, B., Wiegerinck, J.: Plurisubharmonic and holomorphic functions relative to the plurifine topology. J. Math. Anal. Appl. 381, 107–126 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. El Kadiri, M., Smit, I.M.: Maximal plurifinely plurisubharmonic functions. Potential Anal. 41, 1329–1345 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. El Marzguioui, S., Wiegerinck, J.: Continuity properties of finely plurisubharmonic functions. Indiana Univ. Math. J. 59, 1793–1800 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gogus, N.G.: Continuity of plurisubharmonic envelopes. Ann. Polon. Math. 86(3), 197–217 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gogus, N.G., Sahutoglu, S.: Continuity of plurisubharmonic envelopes in \({\mathbb{C}^{2}}\). Intern. J. Math. 23(12), 1250124 (2012)

    Article  MATH  Google Scholar 

  15. Guedj, V., Kołodziej, S., Zeriahi, A.: Hölder continuous solutions to the complex Monge–Ampère equations. Bull. Lond. Math. Soc. 40(6), 1070–1080 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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)

    Article  MATH  Google Scholar 

  17. Hai, L.M., Trao, N.V., Hong, N.X.: The complex Monge–Ampère equation in unbounded hyperconvex domains in \({\mathbb{C}}^n\). Complex Var. Elliptic Equ. 59(12), 1758–1774 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hiep, P.H.: Hölder continuity of solutions to the Monge–Ampère equations on compact Kähler manifolds. Ann. Inst. Fourier 60(5), 1857–1869 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hong, N.X., Lieu, P.T.: Local Hölder continuity of solutions of the complex Monge–Ampère equation. J. Math. Anal. Appl. 507, 125737 (2022)

    Article  MATH  Google Scholar 

  20. Hong, N.X., Lieu, P.T.: The Dirichlet problem for the complex Monge–Ampère operator on strictly plurifinely pseudoconvex domains. Complex Anal. Oper. Theory 15, 124 (2021)

    Article  MATH  Google Scholar 

  21. Hong, N.X., Thuy, T.V.: Hölder continuous solutions to the complex Monge–Ampère equations in non-smooth pseudoconvex domains. Anal. Math. Phys. 8(3), 465–484 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nilsson, M.: Continuity of envelopes of unbounded plurisubharmonic functions. Math. Z. (2022). https://doi.org/10.1007/s00209-022-03043-2

    Article  MathSciNet  MATH  Google Scholar 

  23. Kiselman, C.O.: Sur la définition de l’opérateur de Monge-Ampère complexe. In: Lee, L. (ed.) Proceedings of the Analyse Complexe, Toulouse, Lecture Notes in Math. 1984, vol. 1094, pp. 139–150. Springer, Berlin (1983)

  24. Lien, N.T., Can, H.V., Viet, H.: The complex Monge–Ampère type equation for \({\cal{F}}\)-plurisubharmonic functions. Acta Math. Vietnam. 46, 737–746 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  25. Xing, Y.: Convergence in capacity. Ann. Inst. Fourier (Grenoble) 58(5), 1839–1861 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Walsh, J.B.: Continuity of envelopes of plurisubharmonic functions. J. Math. Mech. 18, 143–148 (1968)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Street, Cau Giay District, Hanoi, Vietnam

    Pham Nguyen Thu Trang & Nguyen Xuan Hong

  2. Department of Basis Sciences, University of Transport Technology, 54 Trieu Khuc, Thanh Xuan District, Hanoi, Vietnam

    Nguyen Duc Hung

Authors
  1. Pham Nguyen Thu Trang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nguyen Duc Hung
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nguyen Xuan Hong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pham Nguyen Thu Trang.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Additional information

Dedicated to Prof. Do Duc Thai on the occasion of his 60th birthday.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research is funded by the Vietnam Ministry of Education and Training under Grant Number B2022-SPH-03. The authors would like to thank the referees for valuable remarks which led to the improvements of the exposition of the paper.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Trang, P.N.T., Hung, N.D. & Hong, N.X. Envelope of plurisubharmonic functions in Cegrell’s classes. Anal.Math.Phys. 12, 138 (2022). https://doi.org/10.1007/s13324-022-00742-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13324-022-00742-1

Keywords

Search

Navigation