Properties of removable singularities for Hardy spaces of analytic functions

Removable singularities for Hardy spaces H-p(Omega) = {f is an element of Hol(Omega) : \f\(p) less than or equal to u in Omega for some harmonic u}, 0 < p < infinity are studied. A set E subset of Omega is a weakly removable singularity for H-p(Omega\E) if H-p(Omega\E) subset of Hol(Omega), and a strongly removable singularity for H-p(Omega\E) if H-p(Omega\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 of Omega that is weakly removable for all HP, but not strongly removable for any H-p(Omega\E), 0 < p < infinity, are found. It is easy to show that if E is weakly removable for H-p(Omega\E) and q > p, then E is also weakly removable for H-q(Omega\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.