Research article

A note on the space of delta m-subharmonic functions

  • Received: 29 October 2019 Accepted: 25 February 2020 Published: 04 March 2020
  • MSC : 32U15, 32U20

  • In this note, we present some properties of a certain space of delta m-subharmonic functions. We prove that the convergence in this space implies the convergence in m-capacity.

    Citation: Van Thien Nguyen, Samsul Ariffin Abdul Karim, Dinh Dat Truong. A note on the space of delta m-subharmonic functions[J]. AIMS Mathematics, 2020, 5(3): 2369-2375. doi: 10.3934/math.2020156

    Related Papers:

  • In this note, we present some properties of a certain space of delta m-subharmonic functions. We prove that the convergence in this space implies the convergence in m-capacity.


    加载中


    [1] P. Åhag, R. Czyż, On a characterization of m-subharmonic functions with weak singularities, Annales Polonici Mathematici, 123 (2019), 21-29.
    [2] P. Åhag, U. Cergell, R. Czyż, Vector spaces of delta-plurisubharmonic functions and extensions of the complex Monge-Ampère operator, J. Math. Anal. Appl., 422 (2015), 960-980. doi: 10.1016/j.jmaa.2014.09.022
    [3] P. Åhag, R. Czyż, L. Hed, The geometry of m-hyperconvex domains, J. Geo. Anal., 28 (2018), 3196-3222. doi: 10.1007/s12220-017-9957-2
    [4] P. Åhag, R. Czyż, L. Hed, Extension and approximation of m-subharmonic functions, Complex. Var. Elliptic. Equ., 63 (2018), 783-801. doi: 10.1080/17476933.2017.1345888
    [5] P. Åhag, R. Czyż, An inequality for the Beta function with Application to Pluripotential Theory, J. Inequal Appl., 2009 (2009), 1-8.
    [6] P. Åhag, R. Czyż, Modulability and duality of certain cones in pluripotential theory, J. Math. Anal. Appl., 361 (2010), 302-321. doi: 10.1016/j.jmaa.2009.07.013
    [7] A. Benali, N. Ghiloufi, Lelong number of m-subharmonic functions, J. Math. Anal. Appl., 466 (2018), 1373-1392. doi: 10.1016/j.jmaa.2018.06.055
    [8] Z. Błocki, Estimates for the complex Monge-Amp'ere operator, Bull. Polon. Acad. Sci. Math., 41 (1993), 151-157.
    [9] Z. Błocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble), 55 (2005), 1735-1756. doi: 10.5802/aif.2137
    [10] U. Cegrell, J. Wiklund, A Monge-Ampère norm for delta-plurisubharmonic functions, Math. Scand., 97 (2005), 201-216. doi: 10.7146/math.scand.a-14972
    [11] R. Czyż, A note on Le-Pham's paper, Acta. Math. Vietnamica., 34 (2009), 401-410.
    [12] R. Czyż, V.T. Nguyen, On a constant in the energy estimate, Comptes Rendus Math., 355 (2017), 1050-1054. doi: 10.1016/j.crma.2017.09.019
    [13] T. Darvas, The Mabuchi Completion of the Space of Kähler Potentials, Amer. J. Math., 139 (2017), 1275-1313. doi: 10.1353/ajm.2017.0032
    [14] S. Dinew, S. Kołodziej, A priori estimates for the complex Hessian equations, Anal. PDE., 1 (2014), 227-244.
    [15] S. Dinew, S. Kołodziej, Non standard properties of m-subharmonic functions, Dolomites Research Notes on Approximation, 11 (2018), 35-50.
    [16] V. Guedj, A. Zeriahi, The weighted Monge-Amprère energy of quasiplurisubharmonic functions, J. Funct. Anal., 250 (2007), 442-482. doi: 10.1016/j.jfa.2007.04.018
    [17] H. Hawari, M. Zaway, On the space of delta m-subharmonic functions, Analysis Math., 42 (2016), 353-369. doi: 10.1007/s10476-016-0404-6
    [18] L.M. Hai, P.H. Hiep, The topology on the space of δ-psh functions in the Cegrell classes, Results Math., 49 (2006), 127-140. doi: 10.1007/s00025-006-0212-6
    [19] V.V. Hung, N.V. Phu, Hessian measures on m-polar sets and applications to the complex Hessian equations, Complex Var. Elliptic Equ., 62 (2017), 1135-1164. doi: 10.1080/17476933.2016.1273907
    [20] S.Y. Li, On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian, Asian J. Math., 8 (2004), 87-106. doi: 10.4310/AJM.2004.v8.n1.a8
    [21] H.C. Lu, Complex Hessian equations, Doctoral thesis, University of Toulouse III Paul Sabatier, 2012.
    [22] H.C. Lu, A variational approach to complex Hessian equations in $\mathbb{C}^n$, J. Math. Anal. Appl., 431 (2015), 228-259.
    [23] N.C. Nguyen, Subsolution theorem for the complex Hessian equation, Univ. Iagel. Acta Math., 50 (2013), 69-88.
    [24] N.C. Nguyen, Hölder continuous solutions to complex Hessian equations, Potential Anal., 41 (2014), 887-902. doi: 10.1007/s11118-014-9398-5
    [25] V.T. Nguyen, On delta m-subharmonic functions, Ann. Polon. Math., 118 (2016), 25-49.
    [26] V.T. Nguyen, Maximal m-subharmonic functions and the Cegrell class $\mathcal{N}_m$, Indagationes Mathematicae 30 (2019), 717-739.
    [27] V.T. Nguyen, A characterization of Cegrell's classes and generalized m-capacities, Ann. Polon. Math., 121 (2018), 33-43. doi: 10.4064/ap170728-26-1
    [28] V.T. Nguyen, The convexity of radially symmetric m-subharmonic functions, Complex. Var. Elliptic. Equ., 63 (2018), 1396-1407. doi: 10.1080/17476933.2017.1373347
    [29] A. Rashkovskii, Local geodesics for plurisubharmonic functions, Math. Z., 287 (2017), 73-83.
    [30] A. Sadullaev, B. Abdullaev, Potential theory in the class of m-subharmonic functions, Trudy Matematicheskogo Instituta imeni V.A. Steklova, 279 (2012), 166-192.
    [31] D. Wan, W. Wang, Complex Hessian operator and Lelong number for unbounded m-subharmonic functions, Potential. Anal., 44 (2016), 53-69. doi: 10.1007/s11118-015-9498-x
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(434) PDF downloads(285) Cited by(0)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog