Convergence in capacity on compact Kähler manifolds
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 4, pp. 903-919.

The aim of this note is to study the convergence in capacity for functions in the class (X,ø). We study the problem under several restrictions on the Monge-Ampère measures of the functions considered, such as common domination by a fixed measure or control on the variation.

Publié le :
Classification : 32W20, 32Q15
Dinew, Sławomir 1 ; Hiệp, Phạm Hoàng 2

1 Institute of Mathematics Jagiellonian University ul. Łojasiewicza 6 30-348 Kraków, Poland
2 Department of Mathematics University of Education (Dai hoc Su Pham Ha Noi) CauGiay, Hanoi, Vietnam
@article{ASNSP_2012_5_11_4_903_0,
     author = {Dinew, S{\l}awomir and Hiệp, Phạm Ho\`ang},
     title = {Convergence in capacity on compact {K\"ahler} manifolds},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {903--919},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 11},
     number = {4},
     year = {2012},
     zbl = {1268.32008},
     mrnumber = {3060705},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2012_5_11_4_903_0/}
}
TY  - JOUR
AU  - Dinew, Sławomir
AU  - Hiệp, Phạm Hoàng
TI  - Convergence in capacity on compact Kähler manifolds
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2012
SP  - 903
EP  - 919
VL  - 11
IS  - 4
PB  - Scuola Normale Superiore, Pisa
UR  - http://www.numdam.org/item/ASNSP_2012_5_11_4_903_0/
LA  - en
ID  - ASNSP_2012_5_11_4_903_0
ER  - 
%0 Journal Article
%A Dinew, Sławomir
%A Hiệp, Phạm Hoàng
%T Convergence in capacity on compact Kähler manifolds
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2012
%P 903-919
%V 11
%N 4
%I Scuola Normale Superiore, Pisa
%U http://www.numdam.org/item/ASNSP_2012_5_11_4_903_0/
%G en
%F ASNSP_2012_5_11_4_903_0
Dinew, Sławomir; Hiệp, Phạm Hoàng. Convergence in capacity on compact Kähler manifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 4, pp. 903-919. http://www.numdam.org/item/ASNSP_2012_5_11_4_903_0/

[1] S. Benelkourchi, V. Guedj and A. Zeriahi, A priori estimates for weak solutions of complex Monge-Ampère equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (2008), 81–96. | EuDML | Numdam | MR | Zbl

[2] E. Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-Ampère operator, Invent. Math. 37 (1976), 1–44. | EuDML | MR | Zbl

[3] E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1–40. | MR | Zbl

[4] T. Bloom and N. Levenberg, Capacity convergence results and applications to a Bernstein-Markov inequality, Trans. Amer. Math. Soc. 351 (1999), 4753–4767. | MR | Zbl

[5] U. Cegrell, Pluricomplex energy, Acta Math. 180 (1998), 187–217. | MR | Zbl

[6] U. Cegrell, The general definition of the complex Monge-Ampère operator, Ann. Inst. Fourier Grenoble 54 (2004), 159–179. | EuDML | Numdam | MR | Zbl

[7] U. Cegrell, Convergence in capacity, Technical report, Isaac Newton Institute for Mathematical Sciences, 2001 (arXiv: math/0505218). | Zbl

[8] U. Cegrell and S. Kołodziej, The Dirichlet problem for the complex Monge-Ampère operator: Perron classes and rotation invariant measures, Michigan Math. J. 41 (1994), 563–569. | MR | Zbl

[9] U. Cegrell and S. Kołodziej, Equation of complex Monge-Ampère type and stability of solutions, Math. Ann. 334 (2006), 713–729. | MR | Zbl

[10] S. Dinew, An inequality for mixed Monge-Ampère measures, Math. Z. 262 (2009), 1–15. | MR | Zbl

[11] S. Dinew, Uniqueness in (X,ø), J. Funct. Anal. 256 (2009), 2113–2122. | MR | Zbl

[12] V. Guedj and A. Zeriahi, Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal. 15 (2005), 607–639. | MR | Zbl

[13] V. Guedj and A. Zeriahi, The weighted Monge-Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal. 250 (2007), 442–482. | MR | Zbl

[14] P. H. Hiệp, On the convergence in capacity on compact Kähler manifolds and its applications, Proc. Amer. Math. Soc. 136 (2008), 2007–2018. | MR | Zbl

[15] P. H. Hiệp, Convergence in capacity, Ann. Polon. Math. 93 (2008), 91–99. | EuDML | MR | Zbl

[16] L. Hörmander, “Notions of Convexity”, Birkhäuser, Boston, 1994. | MR | Zbl

[17] N. V. Khue and P. H. Hiệp, A comparison principle for the complex Monge-Ampère operator in Cegrell’s classes and applications, Trans. Amer. Math. Soc. 361 (2009), 5539–5554. | MR | Zbl

[18] S. Kołodziej, The Monge-Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J. 52 (2003), 667–686. | MR | Zbl

[19] S. Kołodziej, The set of measures given by bounded solutions of the complex Monge-Ampère equation on compact Kähler manifolds, J. London Math. Soc. 124 (2005), 225–238. | MR | Zbl

[20] S. Kołodziej, The complex Monge-Ampère equation and pluripotential theory, Memoirs Amer. Math. Soc. 178/840 (2005), 1–64. | MR | Zbl

[21] Y. Xing, Continuity of the complex Monge-Ampère operator, Proc. Amer. Math. Soc. 124 (1996), 457–467. | MR | Zbl

[22] Y. Xing, Continuity of the complex Monge-Ampère operator on compact Kähler manifolds, Math. Z. 263 (2009), 331–344. | MR | Zbl