On the Cegrell classes

Abstract

The Cegrell classes with zero boundary data are defined by certain decreasing approximating sequences of functions with different properties depending on the class in question. It is different for Cegrell classes which are given by a continuous function f , these classes are defined by an inequality. It is proved in this article that it is possible to define the Cegrell classes which are given by f in a similar manner as those classes with zero boundary data. An existence result for the Dirichlet problem for certain singular measures is proved. The article ends with three applications. Results connected to convergence in capacity, subextension of plurisubharmonic functions and integrability are proved.

Convergence in Capacity

“…This theorem is a generalization of Theorem 1.1 in [5], where the assumption was that {u s } converges to u in C n -capacity as s tends to +∞. The theorem also generalizes [1,Theorem 5.3], [12,Theorem 1], and [13,Theorem 5] and is quite sharp, as shown in [12,Theorem 2(ii)].…”

“…It is well known that the complex Monge-Ampère operator is continuous under monotone limits, but not continuous in the L 1 loc -topology [3]. Therefore it is important to find conditions on sequences of plurisubharmonic functions so that the sequence converges to a function having Monge-Ampère measure equal to the weak limit of the Monge-Ampère measures of the functions in the sequence.…”

Subextension of plurisubharmonic functions with boundary values in weighted pluricomplex energy classes

“…For systematic and complete study of classes of plurisubharmonic functions with generalized boundary values in other classes, we refer readers to the paper of [21]. Note that functions in K( , f ) not necessarily have finite total Monge-Ampère mass (see [22]).…”

Pluripolar sets and the subextension in Cegrell's classes

“…The aim of the present note is to study pluripolar sets and the subextension in Cegrell's classes. The subextension was studied in [5][6][7][8]. First, in Section 3, we prove that for every complete pluripolar set E in there exists a function ' 2 F 1 () such that E ¼ {' ¼ À1} .…”