Non-isotropic Hausdorff capacity of exceptional sets for pluri-Green potentials in the unit ball of ${\Bbb C}^n$

Kuzman Adzievski¹

¹ Department of Mathematics and Computer Science South Carolina State University Orangeburg, SC 29117, U.S.A.

Annales Polonici Mathematici, Tome 88 (2006) no. 1, pp. 59-82.

Résumé

We study questions related to exceptional sets of pluri-Green potentials $V_{\mu }$ in the unit ball $B$ of ${\mathbb C}^n$ in terms of non-isotropic Hausdorff capacity. For suitable measures $\mu $ on the ball $B$, the pluri-Green potentials $V_{\mu }$ are defined by $$ V_{\mu }(z)=\int _B\mathop {\rm log}\nolimits {1\over |\phi _z(w)|}\, d\mu (w), $$ where for a fixed $z\in B$, $\phi _z$ denotes the holomorphic automorphism of $B$ satisfying $\phi _z(0)=z$, $\phi _z(z)=0$ and $(\phi _z\circ \phi _z)(w)=w$ for every $w \in B$. If $d\mu (w) =f(w)d\lambda (w)$, where $f$ is a non-negative measurable function of $B$, and $\lambda $ is the measure on $B$, invariant under all holomorphic automorphisms of $B$, then $V_{\mu }$ is denoted by $V_f$. The main result of this paper is as follows: Let $f$ be a non-negative measurable function on $B$ satisfying $$ \int _B(1-|z|^2)f^p(z)\, d\lambda (z)\infty $$ for some $p$ with $1 p {n/(n-1)}$ and some $\alpha $ with $0 \alpha n+p-np$. Then for each $\tau $ with $1\le \tau \le {n/\alpha }$, there exists a set $E_{\tau }\subseteq S$ with $H_{\alpha \tau }(E_{\tau })=0$ such that $$ \mathop {\rm lim}_{\textstyle { z\to \zeta \atop z\in {\mathcal T}_{\tau ,c}(\zeta )}} V_f(z)=0 $$ for all points $\zeta \in S\setminus E_{\tau }$. In the above, for $\alpha > 0$, $H_{\alpha }$ denotes the non-isotropic Hausdorff capacity on $S$, and for $\zeta \in S =\partial B$, $\tau \ge 1$, and $c> 0$, ${\mathcal T}_{\tau ,c}(\zeta )$ are the regions defined by $$ {\mathcal T}_{\tau ,c}(\zeta )= \{ z\in B:|1-\langle z,\zeta \rangle |^{\tau } c(1-{|z|}^2) \} . $$

Détail

DOI : 10.4064/ap88-1-5

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