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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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