Robin functions and extremal functions
Tom 80 / 2003
Streszczenie
Given a compact set $K\subset {\mathbb C}^N$, for each positive integer $n$, let $$\eqalign{ V^{(n)}(z)={}&V^{(n)}_K(z)\cr :={}&\sup\left\{ {1\over \hbox{deg}\,p}\,V_{p(K)}(p(z)): p \ \hbox{holomorphic polynomial}{,} \, 1\leq \hbox{deg}\,p \leq n\right\}.\cr} $$ These “extremal-like” functions $V^{(n)}_K$ are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, $$ V_K(z):=\max\left [0,\sup \left\{{1\over \hbox{deg}\,p}\log {|p(z)|}: p \ \hbox{holomorphic polynomial}{,} \, \|p\|_K\leq 1\right\}\right]. $$ Our main result is that if $K$ is regular, then all of the functions $V^{(n)}_K$ are continuous; and their associated Robin functions $$ \varrho_{V^{(n)}_K}(z):=\limsup_{|\lambda|\to \infty} [{V^{(n)}_K}(\lambda z)-\log(|\lambda|)] $$ increase to $\varrho_K:=\varrho_{V_K}$ for all $z$ outside a pluripolar set.
