On the problem of existence of analytic functions with boundary values of given modulus. (English. Russian original) Zbl 0871.30036

Samokhin, M. V.

Sb. Math. 187, No. 1, 111-117 (1996); translation from Mat. Sb. 187, No. 1, 113-120 (1996).

Let \(\pi\) be the universal covering map from the unit disk \(\Delta\) onto a domain \(D\subset \overline\mathbb{C}\), and let \(G\) be the group of all linear-fractional maps \(\gamma: \Delta\to \Delta\) such that \(\pi\circ \gamma= \pi\). The author finds conditions for the domain \(D\) to possess the follwoing property: for any measurable function \(u\geq 0\) on \(\partial \Delta\), automorphic with respect to the group \(G\) with \(|\log u|\in L^1\), there exist an automorphic function \(H(z)\) in the Smirnov class \(S'(\Delta)\) such that \(|H(e^{i\theta}) |= u(\theta)\) a.e. on \(\partial\Delta\).

Reviewer: A.Yu.Rashkovsky (Khar’kov)

MSC:

30F35	Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
30E25	Boundary value problems in the complex plane
31C35	Martin boundary theory
46J20	Ideals, maximal ideals, boundaries

Keywords:

Smirnov class; boundary value; Martin compactification; Wiener compactification; automorphic function