Email this article CONTINUITY OF THE COMPLEX HESSIAN OPERATOR ON CEGRELL’S CLASSES OF MSUBHARMONIC FUNCTIONS AND ITS APPLICATION
About this article
Received: 24/03/20 Revised: 25/05/20 Published: 29/05/20Authors
Nguyen Van Phu
, Electric Power University
Abstract
In 2005, L. H. Chinh proved the existence and continuity of the complex Hessian operator
Hm (u) with u ∈ Fm(Ω) under the sequence of decreasing functions in classes Em0 (Ω). By
using the above result and the integration by parts formula for functions in Fm(Ω), we prove
that if u ∈ Fm(Ω) then operator hHm(u) is continuous under the sequence of decreasing
functions in classes E0 m(Ω) for all functions h ∈ SHm∩ L∞loc(Ω). At the same time, we extend N. V. Khue and P. H. Hiep ’s result from the classes of plurisubharmonic functions to the
class Fm(Ω).
Keywords
Full Text:
PDF (Tiếng Việt)References
[1]. E. Bedford and B. A.Taylor, "A new capacity for plurisubharmonic functions," Acta Math, Vol. 149, pp. 1-40, 1982.
[2]. U. Cegrell, "Pluricomplex energy," Acta Math, Vol. 180, pp. 187-217, 1998.
[3]. U. Cegrell, "The general definition of the complex Monge-Ampere operator," Ann.
Inst. Fourier (Grenoble), Vol. 54, pp.159-179, 2004.
[4]. Z. Blocki, "Weak solutions to the complex Hessian equation," Ann. Inst. Fourier
(Grenoble), Vol. 55, no.5, pp. 1735-1756, 2005.
[5]. L. H. Chinh, "A variational Approach to complex Hessian equations in Cn," J.
Math. Anal. Appl., Vol. 431, no.1, pp. 228-259, 2015.
[6]. N. V. Khue and P. H. Hiep, "A Comparison Principle for the complex MongeAmpere operator in Cegrell’s classes and applications," Trans. Amer. Math. Soc., Vol. 361, pp. 5539-5554, 2010.
[7]. N. Falkner, "Hahn’s Proof of the Hahn Decomposition Theorem, and Related Matters," Amer. Math. Monthly., Vol. 126, no. 3, pp. 264-268, 2019.
Refbacks
- There are currently no refbacks.