Abstract
Let $\Omega$ be a domain in ${\bf C}$. A point $\lambda$ of the boundary of $\Omega$ is said to be essential if, for every neighborhood $V$ of $\lambda$, there is an $f\in H^\infty(\Omega)$ such that $f$ does not extend analytically to $V$. It is known that there is a smallest domain $\Omega^*$ containing $\Omega$ such that $\Omega^*$ has no nonessential boundary points. The main result here is the following: Suppose $H^\infty(\Omega)$ is nontrivial. Let $\varphi\colon\Omega\to\Omega$ be analytic and let $C_\varphi$ be the bounded linear operator on $H^\infty(\Omega)$, $0<p<\infty$ given by $C_\varphi f=f\circ\varphi$. Then the range of $C_\varphi$ has uncountable codimension unless $\varphi$ extends to a conformal mapping of $\Omega^*$ onto itself.
Original language | English |
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Pages (from-to) | 323-326 |
Journal | Rendiconti del Seminario Matematico |
Volume | 46 |
Issue number | 3 |
Publication status | Published - 1988 |
Externally published | Yes |
Subject classification (UKÄ)
- Mathematical Sciences