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Số xuất bản: Tập 5 Số 2 (2026): JOURNAL OF MATHEMATICS AND MATHEMATICAL SCIENCES
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Nguyễn, V. P., & Lê, M. H. (2027). Weak convergence and solvability of the non-polar part NP (ddc•)n of Monge-Ampère measures in the class D(Ω). Tạp chí Khoa học Thăng Long: Toán và các Khoa học liên quan, 5(2). https://science.thanglong.edu.vn/index.php/volc/article/view/401

Weak convergence and solvability of the non-polar part NP (ddc•)n of Monge-Ampère measures in the class D(Ω)

Nguyễn Văn Phú1,   , Le Mau Hai2,
1 Electric Power University
2 Department of Mathematics, Hanoi National University of Education

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Tóm tắt

The aim of this note is to prove weak convergence in the class D(Ω) and of the corresponding non-polar part (N P )(ddc•)n of Monge-Ampère measures in the relationship with the convergence in Cn-capacity on an open subset Ω of Cn. In the end of the note we investigate the solvability of the Monge–Ampère equation for the non-polar part in the class D(Ω).
 

Từ khóa

class D(Ω), convergence in capacity, weak convergence of the complex Monge-Ampère measures, weak convergence of the non-polar part, solvability of the non-polar part of the complex Monge-Ampère measure

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Tài liệu tham khảo

[1] P. ˚Ahag, U. Cegrell, R. Czy˙z and Hiep, P. H., Monge-Ampère measures on pluripolar sets, J. Math. Pures Appl., 92 (2009), 613–
627.
[2] E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math., 149(1982), no 1–2, 1–40.
[3] E. Bedford and B.A.Taylor, Fine topology, ˆSilov boundary and (ddc), J. Funct. Anal, 72(1987), no. 2, 225–251.
[4] Z. Blocki, The domain definition of the complex Monge-Ampère operator, Amer. J. Math, 128 (2006), no. 2, 519–530.
[5] Z. Blocki, The complex Monge–Ampère operator in pluripotential theory. Lectures notes. 1998. Forthcoming. Available from:
http://www:/gamma.im.uj.edu.pl/∼Blocki.
[6] U. Cegrell, Discontinuité de l’opératur de Monge-Ampère complexe, C. R. Acad.Sci. Paris Sér.I Math, 296(1983), no. 21, 869–871.
[7] U. Cegrell, Convergence in Capacity, Isaac Newton Institute for Math. Science P., 2001, reprint Series NI01046-NPD, also available at arxiv.org: math. CV/0505218.
[8] U. Cegrell, The general definition of the complex Monge-Ampère operator, Ann. Inst. Fourier. 54(2004),159–179.
[9] U. Cegrell, Convergence in Capacity, Canad. Math. Bull. Vol. 55(2) (2012), 242–248.
[10] R. Czy ˙z, The complex Monge-Ampère operator in the Cegrell classes, Diss. Math. 466 (2009) 83pp.
[11] Thai Duong Do, Hoang Son Do, Van Tu Le, Ngoc Thanh Cong Pham, A Dirichlet Type Problem for Non-Pluripolar Complex
Monge-Ampère Equations, Acta Math. Vietnam. 50 (2025), no. 3, 305–328.
[12] Thai Duong Do, Ngoc Thanh Cong Pham, An existence theorem for non-pluripolar complex Monge-Ampère type equations on hyperconvex domains, J. Math. Anal. Appl. 559 (2026), no. 1, Paper No. 130403.
[13] M. El Kadiri, Fonctions plurisousharmoniques et topologie plurifine, Rend. Acad. Naz. Sci. XL Mem. Mat. Appl, (5) 27(2003),77–88.
[14] M. El Kadiri, B. Fuglede, J. Wiegerink, Plurifinely plurisubhar monic functions and holomorphic functions relative to the plurifine topology, J. Math. Anal. Appl, 381(2011), no.2, 706-723.
[15] M. El Kadiri, An equality on Monge-Ampère measures, J. Math.Anal. Appl, 519(2023), no.2, Paper No. 126826.
[16] M. El Kadiri, Remarks on weak convergence of complex Monge Ampère measures, Indag. Math. (N.S) 35(2024), no.1, 28–36.
[17] S. Kolodziej, Existence and regularity of global solutions to the complex Monge-Ampère equation, Preprint.
[18] S. El Marzguioui and J. Wiegerinck, The plurifine topology is locally connected, Potential Anal, 25(2006), 283–288.
[19] Y. Xing, Continuity of the complex Monge- Ampère operator, Proc. Amer. Math. Soc. 124(1996), 457–467.