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Isometrically removable sets for functions in the Hardy space are polar. (English) Zbl 0625.32004

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 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 \(H^ p\).
Reviewer: J.Burbea

MSC:

32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions

Citations:

Zbl 0532.32004
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