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

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 $\Omega$.

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.

THE PLURICOMPLEX GREEN FUNCTION ON PSEUDOCONVEX DOMAINS WITH A SMOOTH BOUNDARY

In this article we deal with the behavior of the pluricomplex Green function GD(·;w), of a pseudoconvex domain D in [Formula: see text], when the pole tends to a boundary point. In [7], it was shown that, given a boundary point w0 of a hyperconvex domain D, then there is a pluripolar set E⊂D, such that lim sup w→w0 GD(z;w)=0 for z∈D\E. Under an additional assumption on D, that can be viewed as natural, one can avoid the pluripolar exceptional set. Our main result is that on a bounded domain [Formula: see text] that admits a Hoelder continuous plurisubharmonic exhaustion function ρ:D→[-1,0), the pluricomplex Green function GD(·,w) tends to zero uniformly on compact subsets of D, if the pole w tends to a boundary point w0 of D.

Hyperconvexity and Bergman completeness

We show that any bounded hyperconvex domain is Bergman complete.

Addendum to “On the Bergman kernel of hyperconvex domains”, Nagoya Math. J. 129 (1993), 43–52

0. In [0-1] it was proved that for any bounded hyperconvex domain D in C2 the Bergman kernel function K(z, w) of D satisfiesIn case n ═ 1, this is due to a behavior of sublevel sets of the Green function. The general case then follows by the extendability of L2 holomorphic functions.