Plurisubharmonic functions

Part 1 of ebook "Holomorphic functions and integral representations in several complex variables" provides readers with contents including: elementary local properties of holomorphic functions; domains of holomorphy and pseudoconvexity; differential forms and hermitian geometry; integral representations in Cn;...

206p lytamnguyet 04-08-2023 3 3   Download

In this paper, we shall construct a new class of weighted energy functions in the class of m-subharmonic functions and study some analysis properties of this class. Specifically, the paper is organized as follows.

7p viharry 15-12-2022 14 3   Download

The aim of the present paper "The Topology on the Space of δ-psh Functions in the Cegrell Classes" is to introduce a metric locally convex topology on the space δE = E−E of δ-psh functions in the Cegrell class E. We prove that with this topology δE is a non-separable and non-reflexive Fr´echet space. At the same time, we extend the Monge–Amp`ere operator from the class E to δE.

14p runordie5 04-07-2022 6 1   Download

The aim of this article "Weak tautness and hyperconvexity in Hilbert spaces" is to introduce the notions of weak tautness and hyperconvexity of a domain in a Banach space and to establish the relation between them. Moreover, some conditions under which a balanced domain in a Hilbert space and a Hartogs domain in a Banach space are hyperconvex are also given.

14p runordie5 04-07-2022 4 1   Download

In the note "Weak Runge pairs in \({C^n}\)", the authors give a characterization for a pair of pseudoconvex domains in \({C^n}\), (D', D), D' \( \subset \) D such that holomorphic functions on D' can be approximated uniformly on compact sets by meromorphic function on D. Explicit examples are also given.

8p runordie5 04-07-2022 8 2   Download

In the paper "Local Polynomial Convexity of Certain Graphs in C²", the authors employ the theory of plurisubharmonic functions and plurisubharmonic hulls to attack the problem. More precisely, the authors construct nonnegative smooth functions vanishing exactly on Lf .

10p runordie5 04-07-2022 7 2   Download

The main aim of this note "A characterization of bounded plurisubharmonic functions" is to give some characterizations for boundedness of functions in the class F. These results are strongly motivated by Theorem 3 in [Xi2] where a characterization for boundedness of psh functions bounded near the boundary is given. The main result of the note is the following

6p runordie5 04-07-2022 6 3   Download

The aim of the paper "Monge–Ampère measures of maximal subextensions of plurisubharmonic functions with given boundary values" is to investigate the Monge–Ampère measures of maximal subextensions of plurisubharmonic functions with given boundary values. As an application, we study the approximation of negative plurisubharmonic function with given boundary values by an increasing sequence of plurisubharmonic functions defined in larger domains.

9p runordie3 27-06-2022 15 3   Download

In this paper "q-subharmonicity and q-convex domains in Cn", the authors study q-subharmonic and q-plurisubharmonic functions in Cn. Next as an application, we give the notion of q-convex domains in Cn which is an extension of weakly q-convex domains introduced and investigated in [10]. In the end of the paper we show that the q-convexity is the local property and give some examples about q-convex domains.

22p runordie3 27-06-2022 15 3   Download

The aim of the paper "Subextension of plurisubharmonic functions without changing the Monge–Ampère measures and applications" is to investigate subextensions with boundary values of certain plurisubharmonic functions without changing the Monge–Ampère measures.

12p runordie3 27-06-2022 19 3   Download

In the paper "The complex Monge–Ampère equation in unbounded hyperconvex domains in Cn", we look for solutions of the complex Monge–Ampère equation with boundary values (ddcu)n = μ in an unbounded hyperconvex domain of Cn.

18p runordie3 27-06-2022 15 3   Download

In the paper, we prove the existence of solutions of the complex Monge–Ampère type equation −χ(u)(ddcu)n = µ in the class Eχ(Ω) if there exist subsolutions in this class. As an application, we prove that the complex Monge–Ampère equation (ddcu)n = µ is solvable in the class E(Ω) if there exist subsolutions locally. Moreover, by an example we show that the conditions in our above result are sharp.

17p runordie3 27-06-2022 18 3   Download

In this paper "The locally F-approximation property of bounded hyperconvex domains", we study the local property of bounded hyperconvex domains Ω which we can approximate each plurisubharmonic function u ∈ F(Ω) by an increasing sequence of plurisubharmonic functions defined on strictly larger domains.

7p runordie3 27-06-2022 16 2   Download

The aim of the paper "Uniqueness properties of m-subharmonic functions in Cegrell classes" is to provide sufficient conditions for unicity of m-subharmonic functions. Since two plurisubharmonic functions may equal on an open subset of a domain without being identical (e.g. u ≡ 0 and v(z) = max(log |z|, 0)), it seems natural to put additional assumptions on Hessian measures of u and v to guarantee that u ≡ v on the whole domain Ω.

15p runordie3 27-06-2022 23 2   Download