WEAK CONTINUITY OF THE COMPLEX -HESSIAN OPERATORS WITH RESPECT TO LOCAL UNIFORM CONVERGENCE

In this paper, we study the properties of $k$-plurisubharmonic functions defined on domains in ${ \mathbb{C} }^{n} $. By the monotonicity formula, we give an alternative proof of the weak continuity of complex $k$-Hessian operators with respect to local uniform convergence.

On the class gamma and related classes of functions

The gamma class ??(g) consists of positive and measurable functions that satisfy f(x + yg(x))/f(x) ? exp(?y). In most cases the auxiliary function g is Beurling varying and self-neglecting, i.e., g(x)/x ? 0 and g??0(g). Taking h = log f, we find that h?E??(g, 1), where E??(g, a) is the class of positive and measurable functions that satisfy (f(x + yg(x))? f(x))/a(x) ? ?y. In this paper we discuss local uniform convergence for functions in the classes ??(g) and E??(g, a). From this, we obtain several representation theorems. We also prove some higher order relations for functions in the class ??(g) and related classes. Two applications are given.

LIMITS OF MULTIPOLE PLURICOMPLEX GREEN FUNCTIONS

Let Ω be a bounded hyperconvex domain in ℂn, 0 ∈ Ω, and Sε a family of N poles in Ω, all tending to 0 as ε tends to 0. To each Sε we associate its vanishing ideal [Formula: see text] and pluricomplex Green function [Formula: see text]. Suppose that, as ε tends to 0, [Formula: see text] converges to [Formula: see text] (local uniform convergence), and that (Gε)ε converges to G, locally uniformly away from 0; then [Formula: see text]. If the Hilbert–Samuel multiplicity of [Formula: see text] is strictly larger than its length (codimension, equal to N here), then (Gε)ε cannot converge to [Formula: see text]. Conversely, if [Formula: see text] is a complete intersection ideal, then (Gε)ε converges to [Formula: see text]. We work out the case of three poles.

On the Convergence of the Density of the Power Normalized Maximum1

Necessary and sufficient conditions for local uniform convergence of the density of the power normalized maximum to the corresponding density of a p-max stable law is derived for each of the six p-max stable laws. An unified criterian is also obtained.