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

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
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