A construction of outer functions along curves on the sphere in \(\mathbb{C}^ n\) (Q1923697)

Let \(B\) be the unit ball in \(\mathbb{C}^n\), \(\gamma\) be a simple smooth curve on the unit sphere \(S=\partial B\), transverse at all its points, and let \(\varphi\) be a nonnegative continuous function on \(\gamma\) satisfying \(\int_\gamma \log\varphi ds> -\infty\). Then with a finite number of necessary compatibility conditions there is a holomorphic function \(f\) in \(B\) with no zeros in \(B\), such that \(|f|\in C(\overline B)\) and \(|f|= \varphi\) on \(\gamma\). If \(\gamma\) satisfies a Lipschitz condition of order \(\beta>0\), then \(f\) can be chosen in the ball algebra \(A(B)\). Also several comments are given.