Interpolation by Lipschitz holomorphic functions (Q1072684)

Let \(B=\{z\in {\mathbb{C}}^ d: | z| <1\}\) be the unit ball in \({\mathbb{C}}^ d\) for \(d>1\), \(S=\partial B\) and \(\sigma\) the Borel measure on S. The main theorem says that for every \(\epsilon >0\) there exists an \(\alpha\in (0,1]\) such that for every real function \(g\in Lip 1\) it is possible to find a nonconstant function \(f\in A(B)\cap Lip \alpha\) such that Re \(f\leq g\) and \(\sigma (\{z\in S:Re f(z)=g(z)\})\geq 1-\epsilon.\) (Here Lip \(\alpha\) is understood with respect to the length of any shortest path on S joining given points.) The following two corollaries are also derived: 1. For every \(\epsilon >0\) there exists an \(\alpha\in (0,1]\) such that for every positive function \(g\in Lip 1\) there exists a nonconstant function \(f\in A(B)\cap Lip \alpha\) such that \(| f| \leq g\) and \(\sigma (\{z\in S:| f(z)| =g(z)\})\geq 1-\epsilon;\) 2. There exists an \(\alpha\in (0,1]\) such that for every \(\epsilon >0\) it is possible to find a nonconstant function \(f\in A(B)\cap Lip \alpha\) such that \(\| f\|_{\infty}\leq 1\) and \(\sigma (\{z\in S: | f(z)| =1\})\geq 1-\epsilon.\)