Isometrically removable sets for functions in the Hardy space are polar (Q579473)

For a domain D in \({\mathbb{C}}^ n\), \(H^ p=H^ p(D)\), \(0<p<\infty\), denotes the Hardy space of holomorphic functions f on D for which \(| f|^ p\) has a harmonic majorant. For a relatively closed subset E of D, E is said to be removable for \(H^ p(D\setminus E)\) if \(D\setminus E\) is connected and each f in \(H^ p(D\setminus E)\) has an holomorphic extension to a function on \(H^ p\). (In other words the restriction map \(H^ p(D)\to H^ p(D\setminus E)\) is surjective). The set E is said to be isometrically removable for \(H^ p(D\setminus E)\) if the restriction map \(H^ p(D)\to H^ p(D\setminus E)\) is also a surjective isometry. The paper provides a characterization of isometrically removable sets E (provided \(D\setminus E\) supports a nonconstant function in \(H^ p)\). More specifically, it is shown that a set E, which is a relatively closed subset of D such that \(D\setminus E\) is connected and such that D is biholomorphically equivalent to a bounded domain, is isometrically removable for \(H^ p(D\setminus E)\) if and only if E is polar. The fact that the polarity of E implies isometrical removability was already proved by \textit{P. Järvi} [Proc. Am. Math. Soc. 86, 596-598 (1982; Zbl 0532.32004)]. In particular, this characterization shows that isometrical removability is independent of p \((0<p<\infty)\). The proof is based on a lemma in potential theory of \({\mathbb{R}}^ n\); if E is a relatively closed subset of D such that \(D\setminus E\) is connected, if there is a subharmonic function u on \(D\setminus E\) that is not harmonic but which has a least harmonic majorant h, and if u admits a subharmonic continuation to D which is dominated by a superharmonic continuation of h to D, then E is polar. The paper provides a proof of this lemma as well as its applications to Hardy spaces \(h^ p\) of harmonic functions on D. The results in \(h^ p\) do not exactly parallel those for the spaces \(H^ p\).