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A Hilbert space approach to bounded analytic extension in the ball. (English) Zbl 1045.32015

We say that a \(C^1\)-map \(\varphi:\overline D\to\overline B\) of the closed unit disk into a unit ball in \(\mathbb{C}^n\) is an analytic disk transversally attached to the unit sphere if \(\varphi\) is holomorphic on \(D\), injective on \(\overline D\), \(\|\varphi(u)\|= 1\Leftrightarrow| u|= 1\) and \(\langle\varphi'(u),\varphi(u)\rangle\neq 0\) for \(| u|= 1\).
The main result of the paper is the following: Theorem 2.3. Let \(A\) be an analytic disk in the unit ball of \(\mathbb{C}^n\), transversally attached to the unit sphere. Then any bounded analytic function on \(A\) admits an analytic extension to the Schur class of the ball.

MSC:

32D15 Continuation of analytic objects in several complex variables
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32A70 Functional analysis techniques applied to functions of several complex variables
47A48 Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
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