The Corona Theorem on the Complements of Certain Square Cantor Sets

Details The Corona Theorem on the Complements of Certain Square Cantor Sets The Corona Theorem on the Complements of Certain Square Cantor Sets

2007-12-06

arxiv.org/abs/0712.1039v1

Mathematics

Complex Variables

16 pages, 3 figures. (submitted: Journal d'Analyse Mathematique)

Scientific paper

Let $K$ be a square Cantor set, i.e. the Cartesian product $K=E\times E$ of
two linear Cantor sets. Let $\delta_n$ denote the proportion of the intervals
removed in the $n$th stage of the construction of $E$. It is shown that if
$\delta_n=o(\frac1{\log\log n})$ then the corona theorem holds on the domain
$\Omega=\mathbb C^\ast\setminus K$.

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