Mappings with finite length distortion and Riemann surfaces
We prove a series of criteria in terms of dilatations for the continuous and homeomorphic extension of the map pings with finite length distortion between domains on Riemann surfaces to the boundary. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to...
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| Datum: | 2020 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Видавничий дім "Академперіодика" НАН України
2020
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| Schriftenreihe: | Доповіді НАН України |
| Schlagworte: | |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/170618 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Mappings with finite length distortion and Riemann surfaces. / V.I. Ryazanov, S.V. Volkov // Доповіді Національної академії наук України. — 2020. — № 6. — С. 7-14. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Zusammenfassung: | We prove a series of criteria in terms of dilatations for the continuous and homeomorphic extension of the
map pings with finite length distortion between domains on Riemann surfaces to the boundary. The criterion for
the continuous extension of the inverse mapping to the boundary is turned out to be a very simple condition on the
integrability of the dilatations in the first power. Moreover, the domain of the mapping is assumed to be locally
connected on the boundary and its range has a weakly flat boundary. The criteria for the continuous extension of
the direct map pings to the boundary have a much more refined nature. One of such criteria is the existence of a
majorant for the dilation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation
from its mean value over infinitesimal disks centered at the corresponding boundary point. A stronger (but simpler)
one is that the mean value of the dilatation over infinitesimal disks centered at the corresponding boundary point is
finite. The domain is again assumed to be locally connected on the boundary and its range has a strongly accessible
boundary. We give also many other criteria for the continuous extension of the direct mappings to the boundary.
As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion
to the closures of domains are obtained. |
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