A Hilbert space approach to bounded analytic extension in the ball (Q1424083)

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.