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Analytic discs with rectifiable simple closed curves as ends. (English) Zbl 0644.32003
The following two theorems are proved: a) Let \(\Gamma\) be a rectifiable simple closed curve in \({\mathbb{C}}^ N\), \(\Delta\) the open unit disc in \({\mathbb{C}}\). If \(f: \Delta\to {\mathbb{C}}^ N-\Gamma\) is a bounded proper holomorphic map, then f’ belongs to the Hardy space \(H^ 1\). Moreover f carries \(b\Delta\) onto \(\Gamma\) as a covering map of order equal to the multiplicity of the map f.
b) Let \(\Gamma\) be a compact set of finite length in \({\mathbb{C}}^ N\) such that, outside a closed subset of zero length, \(\Gamma\) has a structure of an arc. Suppose that \(f: \Delta\to {\mathbb{C}}^ N-\Gamma\) is a bounded proper holomorphic map. Then f’ belongs to \(H^ 1.\)
The proofs depend on an extension result concerning bounded holomorphic functions in \(\Delta\) recently obtained (independently) by Pommerenke and Alexander.
Reviewer: N.Mihalache

32B15 Analytic subsets of affine space
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32A40 Boundary behavior of holomorphic functions of several complex variables
30E25 Boundary value problems in the complex plane
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