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Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 1, Pages 159–174 (Mi izv2791)  

This article is cited in 14 scientific papers (total in 14 papers)

Towards a theory of removable singularities for maps with unbounded characteristic of quasi-conformity

E. A. Sevost'yanov

Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences

Abstract: We prove that sets of zero modulus with weight $Q$ (in particular, isolated singularities) are removable for discrete open $Q$-maps $f\colon D\to\overline{\mathbb R} ^n$ if the function $Q(x)$ has finite mean oscillation or a logarithmic singularity of order not exceeding $n-1$ on the corresponding set. We obtain analogues of the well-known Sokhotskii–Weierstrass theorem and also of Picard's theorem. In particular, we show that in the neighbourhood of an essential singularity, every discrete open $Q$-map takes any value infinitely many times, except possibly for a set of values of zero capacity.

Keywords: maps with bounded distortion and their generalizations, discrete open maps, removing singularities of maps, essential singularities, Picard's theorem, Sokhotskii's theorem, Liouville's theorem.

DOI: https://doi.org/10.4213/im2791

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English version:
Izvestiya: Mathematics, 2010, 74:1, 151–165

Bibliographic databases:

UDC: 517.5
MSC: Primary 30C65; Secondary 57R45
Received: 14.04.2008

Citation: E. A. Sevost'yanov, “Towards a theory of removable singularities for maps with unbounded characteristic of quasi-conformity”, Izv. RAN. Ser. Mat., 74:1 (2010), 159–174; Izv. Math., 74:1 (2010), 151–165

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. E. A. Sevost'yanov, “On the branch points of mappings with the unbounded coefficient of quasiconformality”, Siberian Math. J., 51:5 (2010), 899–912  mathnet  crossref  mathscinet  isi  elib
    2. E. A. Sevost'yanov, “On the local behavior of mappings with unbounded quasiconformality coefficient”, Siberian Math. J., 53:3 (2012), 520–531  mathnet  crossref  mathscinet  isi
    3. Cristea M., “On generalized quasiregular mappings”, Complex Variables and Elliptic Equations, 58:12 (2013), 1745–1764  crossref  mathscinet  isi  scopus
    4. Cristea M., “Some metric relations of the homeomorphisms satisfying generalized modular inequalities (I)”, Math. Rep. (Bucur.), 15:4 (2013), 387–396  mathscinet  zmath  isi  scopus
    5. M. Cristea, “Local homeomorphisms satisfying generalized modular inequalities”, Complex Variables and Elliptic Equations, 59:10 (2014), 1363–1387  crossref  mathscinet  zmath  isi  scopus
    6. R. R. Salimov, “Lower estimates of $p$-modulus and mappings of Sobolev's class”, St. Petersburg Math. J., 26:6 (2015), 965–984  mathnet  crossref  mathscinet  isi  elib  elib
    7. M. Cristea, “Boundary behaviour of the mappings satisfying generalized inverse modular inequalities”, Complex Variables and Elliptic Equations, 60:4 (2015), 437–469  crossref  mathscinet  zmath  isi  scopus
    8. E. A. Sevost'yanov, “On Removable Singularities of Maps with Growth Bounded by a Function”, Math. Notes, 97:3 (2015), 438–449  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. R. R. Salimov, “O konechnoi lipshitsevosti klassov Orlicha–Soboleva”, Vladikavk. matem. zhurn., 17:1 (2015), 64–77  mathnet
    10. E. A. Sevostyanov, “On the lower order of mappings with finite length distortion”, Siberian Adv. Math., 26:2 (2016), 126–138  mathnet  crossref  crossref  mathscinet  elib
    11. E. A. Sevostyanov, “On local and boundary behaviour of mappings in metric spaces”, St. Petersburg Math. J., 28:6 (2017), 807–824  mathnet  crossref  isi  elib
    12. Cristea M., “The limit mapping of generalized ring homeomorphisms”, Complex Var. Elliptic Equ., 61:5 (2016), 608–622  crossref  mathscinet  zmath  isi  elib  scopus
    13. Cristea M., “Some properties of open, discrete, generalized ring mappings”, Complex Var. Elliptic Equ., 61:5 (2016), 623–643  crossref  mathscinet  zmath  isi  elib  scopus
    14. D. P. Ilyutko, E. A. Sevostyanov, “Ustranenie izolirovannykh osobennostei obobschennykh kvaziizometrii na rimanovykh mnogoobraziyakh”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 63, no. 2, Rossiiskii universitet druzhby narodov, M., 2017, 266–277  mathnet  crossref  mathscinet
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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