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  • 1. Avelin, Benny
    et al.
    Hed, Lisa
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Persson, Håkan
    A Note on the Hyperconvexity of Pseudoconvex Domains Beyond Lipschitz Regularity2015In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, no 3, p. 531-545Article in journal (Refereed)
    Abstract [en]

    We show that bounded pseudoconvex domains that are Holder continuous for all alpha < 1 are hyperconvex, extending the well-known result by Demailly (Math. Z. 194(4) 519-564, 1987) beyond Lipschitz regularity.

  • 2. Avelin, Benny
    et al.
    Hed, Lisa
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Persson, Håkan
    Approximation of plurisubharmonic functions2016In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 61, no 1, p. 23-28Article in journal (Refereed)
    Abstract [en]

    We extend a result by Fornaaess and Wiegerinck [Ark. Mat. 1989;27:257-272] on plurisubharmonic Mergelyan type approximation to domains with boundaries locally given by graphs of continuous functions.

  • 3.
    Cegrell, Urban
    et al.
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Hed, Lisa
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Subextension and approximation of negative plurisubharmonic functions2008In: The Michigan mathematical journal, ISSN 0026-2285, E-ISSN 1945-2365, Vol. 56, p. 593-601Article in journal (Refereed)
  • 4. Czyż, Rafal
    et al.
    Hed, Lisa
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Persson, Håkan
    Plurisubharmonic functions on compact sets2012In: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 106, p. 133-144Article in journal (Refereed)
    Abstract [en]

    Poletsky has introduced a notion of plurisubharmonicity for functions defined on compact sets in Cn. We show that these functions can be completely characterized in terms of monotone convergence of plurisubharmonic functions defined on neighborhoods of the compact.

  • 5.
    Hed, Lisa
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Approximation of negative plurisubharmonic functions with given boundary values2010In: International Journal of Mathematics, ISSN 0129-167X, Vol. 21, no 9, p. 1135-1145Article in journal (Refereed)
    Abstract [en]

    In this paper, we study the approximation of negative plurisubharmonic functions with given boundary values. We want to approximate a plurisubharmonic function by an increasing sequence of plurisubharmonic functions defined on strictly larger domains.

  • 6.
    Hed, Lisa
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    The plurisubharmonic Mergelyan property2012Doctoral thesis, monograph (Other academic)
    Abstract [en]

    In this thesis, we study two different kinds of approximation of plurisubharmonic functions.

    The first one is a Mergelyan type approximation for plurisubharmonic functions. That is, we study which domains in C^n have the property that every continuous plurisubharmonic function can be uniformly approximated with continuous and plurisubharmonic functions defined on neighborhoods of the domain. We will improve a result by Fornaess and Wiegerinck and show that domains with C^0-boundary have this property. We will also use the notion of plurisubharmonic functions on compact sets when trying to characterize those continuous and plurisubharmonic functions that can be approximated from outside. Here a new kind of convexity of a domain comes in handy, namely those domains in C^n that have a negative exhaustion function that is plurisubharmonic on the closure. For these domains, we prove that it is enough to look at the boundary values of a plurisubharmonic function to know whether it can be approximated from outside.

    The second type of approximation is the following: we want to approximate functions u that are defined on bounded hyperconvex domains Omega in C^n and have essentially boundary values zero and bounded Monge-Ampère mass, with increasing sequences of certain functions u_j that are defined on strictly larger domains. We show that for certain conditions on Omega, this is always possible. We also generalize this to functions with given boundary values. The main tool in the proofs concerning this second approximation is subextension of plurisubharmonic functions.

  • 7.
    Hed, Lisa
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Persson, Håkan
    Plurisubharmonic approximation and boundary values of plurisubharmonic functions2014In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 413, no 2, p. 700-714Article in journal (Refereed)
    Abstract [en]

    We study the problem of approximating plurisubharmonic functions on a bounded domain Omega by continuous plurisubharmonic functions defined on neighborhoods of (Omega) over bar. It turns out that this problem can be linked to the problem of solving a Dirichlet type problem for functions plurisubharmonic on the compact set (Omega) over bar in the sense of Poletsky. A stronger notion of hyperconvexity is introduced to fully utilize this connection, and we show that for this class of domains the duality between the two problems is perfect. In this setting, we give a characterization of plurisubharmonic boundary values, and prove some theorems regarding the approximation of plurisubharmonic functions.

    (C) 2013 Elsevier Inc. All rights reserved.

  • 8.
    Åhag, Per
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Czyz, Rafal
    Hed, Lisa
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    The Geometry of m-Hyperconvex Domains2018In: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 28, no 4, p. 3196-3222Article in journal (Refereed)
    Abstract [en]

    We study the geometry of m-regular domains within the Caffarelli–Nirenberg–Spruck model in terms of barrier functions, envelopes, exhaustion functions, and Jensen measures. We prove among other things that every m-hyperconvex domain admits an exhaustion function that is negative, smooth, strictly m-subharmonic, and has bounded m-Hessian measure.

  • 9.
    Åhag, Per
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Czyż, Rafał
    Hed, Lisa
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Extension and approximation of m-subharmonic functions2018In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 63, no 6, p. 783-801Article in journal (Refereed)
    Abstract [en]

    Let  be a bounded domain, and let f be a real-valued function defined on the whole topological boundary . The aim of this paper is to find a characterization of the functions f which can be extended to the inside to a m-subharmonic function under suitable assumptions on . We shall do so using a function algebraic approach with focus on m-subharmonic functions defined on compact sets. We end this note with some remarks on approximation of m-subharmonic functions.

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