Potential Analysis

, Volume 34, Issue 1, pp 43–56 | Cite as

Some Weighted Energy Classes of Plurisubharmonic Functions

Article

Abstract

In this paper we study relations between the weighted energy class \(\mathcal{E}_{\chi}\) introduced by S. Benelkourchi, V. Guedj and A. Zeriahi recently with Cegrell’s classes \(\mathcal{E}\) and \(\mathcal{N}\). Next we establish a generalized comparison principle for the operator M χ . As an application, we prove a version of existence of solutions of Monge–Ampère type equations in the class \(\mathcal{E}_{\chi}(H,\Omega)\).

Keywords

Class \(\mathcal{E}_{\chi}\) Class \(\mathcal{E}\) Class \(\mathcal{N}\) Strong comparison principle Operator Mχ Monge–Ampère equation 

Mathematics Subject Classifications (2010)

32U05 32W20 

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References

  1. 1.
    Ahag, P., Cegrell, U., Czyz, R., Hiep, P.H.: Monge-Ampre measures on pluripolar sets. J. Math. Pures Appl. 92, 613–627 (2009)MATHMathSciNetGoogle Scholar
  2. 2.
    Ahag, P., Czyz, R., Hiep, P.H.: Concerning the energy class \(\mathcal{E}_p\) for 0 < p < 1. Ann. Pol. Math. 91, 119–131 (2007)MATHCrossRefGoogle Scholar
  3. 3.
    Bedford, E., Taylor, B.A.: The Dirichlet problem for an equation of complex Monge–Ampère type. In: Byrnes, C. (ed.) Partial Differential Equations and Geometry, pp. 39–50. Dekker (1979)Google Scholar
  4. 4.
    Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149(1–2),1–40 (1982)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Benelkourchi, S.: Weighted Pluricomplex energy. Potential Anal. 31(N1), 1–20 (2009)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Benelkourchi, S., Guedj, V., Zeriahi, A.: Plurisubharmonic functions with weak singularities. In: Proceedings from the Kiselmanfest, Uppsala University, Västra Aros, pp.57–74 (2009)Google Scholar
  7. 7.
    Cegrell, U.: Pluricomplex energy. Acta Math. 180, 187–217 (1998)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cegrell, U.: The general definition of the complex Monge–Ampère operator. Ann. Inst. Fourier (Grenoble) 54, 159–179 (2004)MathSciNetGoogle Scholar
  9. 9.
    Cegrell, U.: A general Dirichlet problem for the complex Monge–Ampère operator. Ann. Pol. Math. 94, 131–147 (2008)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cegrell, U., Kolodziej, S.: The equations of complex Monge–Ampère type and stability of solutions. Math. Ann. 334, 713–729 (2006)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Czyz, R.: On a Monge–Ampère type equation in the Cegrell class \(\mathcal{E}_{\chi}\). arXiv:0805.3246v1 (math.CV)
  12. 12.
    Guedj, V., Zeriahi, A.: The weighted Monge–Ampère energy of quasiplurisubharmonic functions. J. Funct. Anal. 250(2), 442–482 (2007)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hiep, P.H.: A characterization of bounded plurisubharmonic functions. Ann. Pol. Math. 85, 233–238 (2005)MATHCrossRefGoogle Scholar
  14. 14.
    Hiep, P.H.: The comparison principle and Dirichlet problem in the class \(\mathcal{E}_p(f), p>0\). Ibid 88, 247–261 (2006)MATHGoogle Scholar
  15. 15.
    Hiep, P.H.: Pluripolar sets and the subextension in Cegrell’s classes. Complex Var. Elliptic Equations. 53(7), 675–684 (2008)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Khue, N.V., Hiep, P.H.: A comparision principle for the complex Monge–Ampère operator in Cegrell’s classes and applications. Trans. Am. Math. Soc. 361, 5539–5554 (2009)MATHCrossRefGoogle Scholar
  17. 17.
    Klimek, M.: Pluripotential Theory. The Clarendon Press/Oxford University Press, New York (1991)MATHGoogle Scholar
  18. 18.
    Kolodziej, S.: The range of the complex Monge–Ampère operator, II. Indiana Univ. Math. J. 44, 765–782 (1995)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kolodziej, S.: Weak solutions of equations of complex Monge–Ampère type. Ann. Pol. Math. 73, 59–67 (2000)MATHMathSciNetGoogle Scholar
  20. 20.
    Xing, Y.: Continuity of the complex Monge–Ampère operator. Proc. Am. Math. Soc. 124, 457–467 (1996)MATHCrossRefGoogle Scholar
  21. 21.
    Xing, Y.: A strong comparison principle for plurisubharmonic functions with finite Pluricomplex energy. Mich. Math. J. 56, 563–581 (2008)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of MathematicsHanoi National University of EducationHanoiVietnam

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