Complex Hessian Operator and Lelong Number for Unbounded m-subharmonic Functions

2015 ◽  
Vol 44 (1) ◽  
pp. 53-69 ◽  
Author(s):  
Dongrui Wan ◽  
Wei Wang
Keyword(s):  
Subharmonic Functions ◽  
Lelong Number ◽  
Hessian Operator ◽  
Complex Hessian Operator

Hessian boundary measures

2019 ◽  
Vol 30 (03) ◽  
pp. 1950016
Author(s):  
Van Thien Nguyen
Keyword(s):  
Subharmonic Functions ◽  
Plurisubharmonic Functions ◽  
Lelong Number ◽  
Boundary Measure ◽  
Hessian Operator ◽  
Complex Hessian Operator ◽  
Monge Ampere

We will study certain boundary measures related to [Formula: see text]-subharmonic functions on [Formula: see text]-hyperconvex domains. These measures generalize the boundary measures studied by Wan and Wang (see [Complex Hessian operator and Lelong number for unbounded [Formula: see text]-subharmonic functions, Potential. Anal. 44(1) (2016) 53–69]). For the case of plurisubharmonic functions ([Formula: see text]) the boundary measure has been studied by Cegrell and Kemppe (see [Monge–Ampère boundary measures, Ann. Polon. Math. 96 (2009) 175–196]).

The Dirichlet problem for the complex Hessian operator in the class $\mathcal{N}_m(\Omega,f)$

2021 ◽  
Vol 127 (2) ◽  
pp. 287-316
Author(s):  
Ayoub El Gasmi
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Borel Measure ◽  
Positive Borel Measure ◽  
Hessian Equation ◽  
Hessian Operator ◽  
Hyperconvex Domain ◽  
Complex Hessian Operator

Let $\Omega\subset \mathbb{C}^{n}$ be a bounded $m$-hyperconvex domain, where $m$ is an integer such that $1\leq m\leq n$. Let $\mu$ be a positive Borel measure on $\Omega$. We show that if the complex Hessian equation $H_m (u) = \mu$ admits a (weak) subsolution in $\Omega$, then it admits a (weak) solution with a prescribed least maximal $m$-subharmonic majorant in $\Omega$.

scholarly journals On the mean value property of superharmonic and subharmonic functions

2006 ◽  
Vol 2006 ◽  
pp. 1-3
Author(s):  
Robert Dalmasso
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Subharmonic Functions ◽  
Mean Value Property ◽  
The Mean

We prove a converse of the mean value property for superharmonic and subharmonic functions. The case of harmonic functions was treated by Epstein and Schiffer.

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