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.