CONTINUITY OF THE COMPLEX HESSIAN OPERATOR ON CEGRELL’S CLASSES OF MSUBHARMONIC FUNCTIONS AND ITS APPLICATION

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).