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A remark on the boundary behavior of quasiconformal mappings and the classification of Riemann surfaces. (English) Zbl 0568.30040

The author shows that neither the property of being in the Widom class nor the property of being AB-separable (i.e. bounded analytic functions separate points) are preserved by quasiconformal mapping. The required counter examples are constructed as two sheeted covers of the plane and their properties are demonstrated using criteria of C. M. Stanton [Pac. J. Math. 59, 557-565 (1975; Zbl 0328.30040)].
Reviewer: R.Rochberg

MSC:

30F20 Classification theory of Riemann surfaces
30C20 Conformal mappings of special domains

Citations:

Zbl 0328.30040
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References:

[1] Ahlf ors, L. V.: Lectures on Quasiconf ormal Mappings. Van Vostrand, New York (1966). · Zbl 0138.06002
[2] Beurling, A., and Ahlf ors, L.: The boundary correspondence under quasiconf ormal mappings. Acta Math., 96, 125-142 (1956). · Zbl 0072.29602
[3] Sario, L., and Nakai, M.: Classification Theory of Riemann Surfaces. Springer-Verlag, Berlin (1970). · Zbl 0199.40603
[4] Stanton, C. M.: Bounded analytic functions on a class of Riemann surfaces. Pacific J. Math., 59, 557-565 (1975). · Zbl 0328.30040
[5] Widom, H.: Hp sections of vector bundles over Riemann surfaces. Ann. of Math., 94(2), 304-324 (1971). JSTOR: · Zbl 0238.32014
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