PROPERTIES OF REMOVABLE SINGULARITIES FOR HARDY SPACES OF ANALYTIC FUNCTIONS

Abstract

Removable singularities for Hardy spaces $H^p(\Omega) = \{f \in \hbox{Hol}(\Omega):\vert f\vert^p \le u \hbox{ in } \Omega \hbox{ for some harmonic }u\}, 0 < p < \infty$ are studied. A set $E \subset \Omega$ is a weakly removable singularity for $H^p(\Omega\backslash E)$ if $H^p(\Omega\backslash E) \subset \hbox{Hol}(\Omega)$ , and a strongly removable singularity for $H^p(\Omega\backslash E)$ if $H^p(\Omega\backslash E) = H^p(\Omega)$ . The two types of singularities coincide for compact $E$ , and weak removability is independent of the domain $\Omega$ .

The paper looks at differences between weak and strong removability, the domain dependence of strong removability, and when removability is preserved under unions. In particular, a domain $\Omega$ and a set $E \subset \Omega$ that is weakly removable for all $H^p$ , but not strongly removable for any $H^p (\Omega\backslash E), 0 < p < \infty$ , are found.

It is easy to show that if $E$ is weakly removable for $H^p(\Omega\backslash E)$ and $q > p$ , then $E$ is also weakly removable for $H^q(\Omega\backslash E)$ . It is shown that the corresponding implication for strong removability holds if and only if $q/p$ is an integer.

Finally, the theory of Hardy space capacities is extended, and a comparison is made with the similar situation for weighted Bergman spaces.