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An inequality for mixed Monge–Ampère measures

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Abstract

We generalize an inequality for mixed Monge–Ampère measures from Kołodziej (Indiana Univ. Math. J. 43, 1321–1338, 1994). We also give an example that shows that our assumptions are sharp. The corresponding result in the setting of compact Kähler manifold is also discussed.

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References

  1. Alexandrov, A.D.: Die innere Geometrie der konvexen Flächen. Akademie Verlag, Berlin (1955)

    Google Scholar 

  2. Błocki, Z.: The complex Monge–Ampère operator in hypeconvex domains. Ann. Scuola Norm. Sup. Pisa 23, 721–747 (1996)

    MATH  Google Scholar 

  3. Błocki, Z.: On the definition of the complex Monge–Ampère operator in \({\mathbb C^2}\) . Math. Ann. 328, 415–423 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Błocki, Z.: The domain of definition of the complex Monge–Ampère operator. Am. J. Math. 128(2), 519–530 (2006)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Cegrell, U.: Pluricomplex energy. Acta Math. 180, 187–217 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  9. Cegrell, U., Kołodziej, S.: The Dirichlet problem for the complex Monge–Ampère operator: Perron classes and rotation invariant measures. Mich. Math. J. 41, 563–569 (1994)

    Article  MATH  Google Scholar 

  10. Cegrell, U., Kołodziej, S.: Equation of complex Monge–Ampère type and stability of solutions. Math. Ann. 334, 713–729 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Demailly, J.-P.: Regularization of closed positive currents and intersection theory. J. Alg. Geom. 1, 361–409 (1992)

    MATH  MathSciNet  Google Scholar 

  12. Dinew, S., Zhang, Z.: Stability of bounded solutions for degenerate complex Monge–Ampère equations (in preparation)

  13. Garding, L.: An inequality for hyperbolic polynomials. J. Math. Mech. 8, 957–965 (1959)

    MATH  MathSciNet  Google Scholar 

  14. Guedj, V., Zeriahi, A.: Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal. 15(4), 607–639 (2005)

    MATH  MathSciNet  Google Scholar 

  15. Guedj, V., Zeriahi, A.: The weighted Monge–Ampère energy of quasiplurisubharmonic functions, preprint, /ar**v/math.CV/0612630, J. Funct. Anal. (to appear)

  16. Horn, R.A., Johnson, C.R.: Matrix analysis. Cambridge University Press, London (1985)

    MATH  Google Scholar 

  17. Kołodziej, S.: The range of the complex Monge–Ampère operator. Indiana Univ. Math. J. 43, 1321–1338 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Kołodziej, S.: The Monge–Ampère equation on compact Kähler manifolds. Indiana Univ. Math. J. 52(3), 667–686 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kołodziej, S.: The complex Monge–Ampère equation and pluripotential theory. Mem AMS 178/840, 1–64 (2005)

    Google Scholar 

  21. Khue, N.V., Hiep, P.H.: Some properties of the complex Monge–Ampère operator in Cegrell’s classes and applications (preprint)

  22. Rashkovskii, A.: Newton numbers and residual measures of plurisubharmonic functions. Ann. Polon. Math. 75, 213–231 (2000)

    MATH  MathSciNet  Google Scholar 

  23. Rashkovskii, A.: Lelong numbers with respect to regular plurisubharmonic weights. Results Math. 39, 320–332 (2001)

    MATH  MathSciNet  Google Scholar 

  24. Wiklund, J.: Matrix inequalities and the complex Monge–Ampère operator. Ann. Polon. Math. 83, 211–220 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Wiklund, J.: Pluricomplex charge at weak singularities, preprint ar**v:math/0510671

  26. **ng, Y.: Continuity of the complex Monge–Ampère operator. Proc. AMS 124, 457–467 (1996)

    Article  MATH  Google Scholar 

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  1. Institute of Mathematics, Jagiellonian University, ul. Reymonta 4, 30-059, Kraków, Poland

    Sławomir Dinew

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  1. Sławomir Dinew
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Correspondence to Sławomir Dinew.

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Dinew, S. An inequality for mixed Monge–Ampère measures. Math. Z. 262, 1–15 (2009). https://doi.org/10.1007/s00209-008-0356-z

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  • DOI: https://doi.org/10.1007/s00209-008-0356-z

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