Prime ends in the Sobolev mapping theory on Riemann surfaces

V. Ryazanov; S. Volkov

doi:10.15330/ms.48.1.24-36

We prove criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends of Caratheodory.

1. E.S. Afanaseva, V.I. Ryazanov, R.R. Salimov, On mappings in Orlicz-Sobolev classes on Riemannian manifolds, Ukr. Mat. Visn., 8 (2011), №3, 319.342, 461 (in Russian); transl. in J. Math. Sci. (N.Y.), 181 (2012), №1, 1-17.

2. B. Bojarski, V. Gutlyanskii, O. Martio, V. Ryazanov, Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane, EMS Tracts in Mathematics, V.19, EMS Publishing House, Zurich, 2013.

3. Bourbaki N., General topology. The main structures, Nauka, Moscow, 1968. (in Russian)

4. C. Caratheodory Uber die Begrenzung der einfachzusammenhangender Gebiete, Math. Ann., 73 (1913), 323-370.

5. E.F. Collingwood, A.J. Lohwator, The Theory of Cluster Sets, Cambridge Tracts in Math. and Math. Physics, V.56, Cambridge Univ. Press, Cambridge, 1966.

6. Fuglede B., Extremal length and functional completion, Acta Math., 98 (1957), 171-219.

7. V. Gutlyanskii, V. Ryazanov, On recent advances in boundary value problems in the plane, Ukr. Mat. Visn., 13 (2016), №2, 167.212; transl. in J. Math. Sci. (N.Y.), 221 (2017), №5, 638-670.

8. V. Gutlyanskii, V. Ryazanov, U. Srebro, E. Yakubov, The Beltrami Equation: A Geometric Approach, Developments in Mathematics, V.26, Springer, New York etc., 2012.

9. V. Gutlyanskii, V. Ryazanov, E. Yakubov, The Beltrami equations and prime ends, Ukr. Mat. Visn., 12 (2015), №1, 27.66; transl. in J. Math. Sci. (N.Y.), 210 (2015), №1, 22-51.

10. V. Gutlyanskii, V. Ryazanov, A. Yefimushkin, On the boundary value problems for quasiconformal mappings in the plane, Ukr. Mat. Visn., 12 (2015), №3, 363.389; transl. in J. Math. Sci. (N.Y.), 214 (2016), №2, 200-219.

11. D. Kovtonyuk, I. Petkov, V. Ryazanov, On the boundary behavior of mappings with finite distortion in the plane, Lobachevskii J. Math. (USA), 38 (2017), №2, 290-306.

12. D.A. Kovtonyuk, I.V. Petkov, V.I. Ryazanov, R.R. Salimov, Boundary behaviour and the Dirichlet problem for the Beltrami equations, Algebra and Analysis., 25 (2013), №4, 101.124 (in Russian); transl. in St. Petersburg Math. J., 25 (2014), №4, 587-603.

13. D. Kovtonyuk, I. Petkov, V. Ryazanov, R. Salimov, On the Dirichlet problem for the Beltrami equation, J. Anal. Math., 122 (2014), №4, 113-141.

14. D.A. Kovtonyuk, V.I. Ryazanov, Prime ends and the Orlicz-Sobolev classes, Algebra and Analysis, 27 (2015), №5, 81.116 (in Russian); transl. in St. Petersburg Math. J., 27 (2016), №5, 765-788.

15. S.L. Krushkal, B.N. Apanasov, N.A. Gusevskii, Kleinian groups and uniformization in examples and problems, Transl. of Math. Mon., 62, AMS, Providence, RI, 1986.

16. K. Kuratowski, Topology, V.1, Academic Press, New York, 1968.

17. „K. Kuratowski, Topology, V2, Academic Press, New York.London, 1968.

18. O. Lehto, K. Virtanen, Quasiconformal Mappings in the Plane, Springer-Verlag, New York, 1973.

19. O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in Modern Mapping Theory, Springer, New York etc., 2009.

20. R. Nakki, Prime ends and quasiconformal mappings, J. Anal. Math., 35 (1979), 13-40.

21. M.H.A. Newman, Elements of the topology of plane sets of points, 2nd ed., Cambridge University Press, Cambridge, 1951.

22. I.V. Petkov, The boundary behavior of homeomorphisms of the class W1;1 loc on the plane by prime ends, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki (in Russian), 2015, №6, 19-23.

23. V. Ryazanov, R. Salimov, Weakly flat spaces and bondaries in the mapping theory, Ukr. Mat. Visn., 4 (2007), №2, 199.234 (in Russian); translation in Ukrain. Math. Bull. (N.Y.), 4 (2007), №2, 199-233.

24. V. Ryazanov, R. Salimov, U. Srebro, E. Yakubov, On boundary value problems for the Beltrami equations, Complex analysis and dynamical systems V, Contemp. Math., 591 (2013), 211.242, Amer. Math. Soc., Providence, RI, 2013.

25. V.I. Ryazanov, S.V. Volkov, On the boundary behavior of mappings in the class $W^{1,1}_{\rm loc}$ on Riemann surfaces, Proceedings of Inst. Appl. Math. Mech. of the NAS of Ukraine, 29 (2015), 34.53. (in Russian)

26. V.I. Ryazanov, S.V. Volkov, On the boundary behavior of mappings in the class $W^{1,1}_{\rm loc}$ on Riemann surfaces, Complex Anal. Oper. Theory, 11 (2017), №7, 1503.1520; see also ArXiv: 1604.00280v5 [math.CV], 2016, 24 p.

27. V.I. Ryazanov, S.V. Volkov, On the theory of the boundary behavior of mappings in the Sobolev class on Riemann surfaces, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, 2016, №10, 5-9. (in Russian)

28. E.S. Smolovaya, Boundary behavior of ring Q-homeomorphisms in metric spaces, Ukrain. Mat. Zh., 62 (2010), №5, 682.689 (in Russian); transl. in Ukrainian Math. J., 62 (2010), №5, 785-793.

29. S. Stoilow, Lecons sur les principes topologiques de la theorie des fonctions analytiques, Gauthier-Villars, Paris, 1956.

30. J. Vaisala, Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Math., 229, Springer-Verlag, Berlin, 1971.

31. G.Th. Whyburn, Analytic Topology, AMS, Providence, 1942.

	Prime ends in the Sobolev mapping theory on Riemann surfaces
Author	V. Ryazanov, S. Volkov vl.ryazanov1@gmail.com, serhii.volkov@donntu.edu.ua Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Slovyansk, Ukraine
Abstract	We prove criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends of Caratheodory.
Keywords	Riemann surfaces; Sobolev mappings; prime ends; continuous and homeomorphic extension; boundary behavior
DOI	doi:10.15330/ms.48.1.24-36
Reference	1. E.S. Afanaseva, V.I. Ryazanov, R.R. Salimov, On mappings in Orlicz-Sobolev classes on Riemannian manifolds, Ukr. Mat. Visn., 8 (2011), №3, 319.342, 461 (in Russian); transl. in J. Math. Sci. (N.Y.), 181 (2012), №1, 1-17. 2. B. Bojarski, V. Gutlyanskii, O. Martio, V. Ryazanov, Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane, EMS Tracts in Mathematics, V.19, EMS Publishing House, Zurich, 2013. 3. Bourbaki N., General topology. The main structures, Nauka, Moscow, 1968. (in Russian) 4. C. Caratheodory Uber die Begrenzung der einfachzusammenhangender Gebiete, Math. Ann., 73 (1913), 323-370. 5. E.F. Collingwood, A.J. Lohwator, The Theory of Cluster Sets, Cambridge Tracts in Math. and Math. Physics, V.56, Cambridge Univ. Press, Cambridge, 1966. 6. Fuglede B., Extremal length and functional completion, Acta Math., 98 (1957), 171-219. 7. V. Gutlyanskii, V. Ryazanov, On recent advances in boundary value problems in the plane, Ukr. Mat. Visn., 13 (2016), №2, 167.212; transl. in J. Math. Sci. (N.Y.), 221 (2017), №5, 638-670. 8. V. Gutlyanskii, V. Ryazanov, U. Srebro, E. Yakubov, The Beltrami Equation: A Geometric Approach, Developments in Mathematics, V.26, Springer, New York etc., 2012. 9. V. Gutlyanskii, V. Ryazanov, E. Yakubov, The Beltrami equations and prime ends, Ukr. Mat. Visn., 12 (2015), №1, 27.66; transl. in J. Math. Sci. (N.Y.), 210 (2015), №1, 22-51. 10. V. Gutlyanskii, V. Ryazanov, A. Yefimushkin, On the boundary value problems for quasiconformal mappings in the plane, Ukr. Mat. Visn., 12 (2015), №3, 363.389; transl. in J. Math. Sci. (N.Y.), 214 (2016), №2, 200-219. 11. D. Kovtonyuk, I. Petkov, V. Ryazanov, On the boundary behavior of mappings with finite distortion in the plane, Lobachevskii J. Math. (USA), 38 (2017), №2, 290-306. 12. D.A. Kovtonyuk, I.V. Petkov, V.I. Ryazanov, R.R. Salimov, Boundary behaviour and the Dirichlet problem for the Beltrami equations, Algebra and Analysis., 25 (2013), №4, 101.124 (in Russian); transl. in St. Petersburg Math. J., 25 (2014), №4, 587-603. 13. D. Kovtonyuk, I. Petkov, V. Ryazanov, R. Salimov, On the Dirichlet problem for the Beltrami equation, J. Anal. Math., 122 (2014), №4, 113-141. 14. D.A. Kovtonyuk, V.I. Ryazanov, Prime ends and the Orlicz-Sobolev classes, Algebra and Analysis, 27 (2015), №5, 81.116 (in Russian); transl. in St. Petersburg Math. J., 27 (2016), №5, 765-788. 15. S.L. Krushkal, B.N. Apanasov, N.A. Gusevskii, Kleinian groups and uniformization in examples and problems, Transl. of Math. Mon., 62, AMS, Providence, RI, 1986. 16. K. Kuratowski, Topology, V.1, Academic Press, New York, 1968. 17. „K. Kuratowski, Topology, V2, Academic Press, New York.London, 1968. 18. O. Lehto, K. Virtanen, Quasiconformal Mappings in the Plane, Springer-Verlag, New York, 1973. 19. O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in Modern Mapping Theory, Springer, New York etc., 2009. 20. R. Nakki, Prime ends and quasiconformal mappings, J. Anal. Math., 35 (1979), 13-40. 21. M.H.A. Newman, Elements of the topology of plane sets of points, 2nd ed., Cambridge University Press, Cambridge, 1951. 22. I.V. Petkov, The boundary behavior of homeomorphisms of the class W1;1 loc on the plane by prime ends, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki (in Russian), 2015, №6, 19-23. 23. V. Ryazanov, R. Salimov, Weakly flat spaces and bondaries in the mapping theory, Ukr. Mat. Visn., 4 (2007), №2, 199.234 (in Russian); translation in Ukrain. Math. Bull. (N.Y.), 4 (2007), №2, 199-233. 24. V. Ryazanov, R. Salimov, U. Srebro, E. Yakubov, On boundary value problems for the Beltrami equations, Complex analysis and dynamical systems V, Contemp. Math., 591 (2013), 211.242, Amer. Math. Soc., Providence, RI, 2013. 25. V.I. Ryazanov, S.V. Volkov, On the boundary behavior of mappings in the class $W^{1,1}_{\rm loc}$ on Riemann surfaces, Proceedings of Inst. Appl. Math. Mech. of the NAS of Ukraine, 29 (2015), 34.53. (in Russian) 26. V.I. Ryazanov, S.V. Volkov, On the boundary behavior of mappings in the class $W^{1,1}_{\rm loc}$ on Riemann surfaces, Complex Anal. Oper. Theory, 11 (2017), №7, 1503.1520; see also ArXiv: 1604.00280v5 [math.CV], 2016, 24 p. 27. V.I. Ryazanov, S.V. Volkov, On the theory of the boundary behavior of mappings in the Sobolev class on Riemann surfaces, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, 2016, №10, 5-9. (in Russian) 28. E.S. Smolovaya, Boundary behavior of ring Q-homeomorphisms in metric spaces, Ukrain. Mat. Zh., 62 (2010), №5, 682.689 (in Russian); transl. in Ukrainian Math. J., 62 (2010), №5, 785-793. 29. S. Stoilow, Lecons sur les principes topologiques de la theorie des fonctions analytiques, Gauthier-Villars, Paris, 1956. 30. J. Vaisala, Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Math., 229, Springer-Verlag, Berlin, 1971. 31. G.Th. Whyburn, Analytic Topology, AMS, Providence, 1942.
Pages	24-36
Volume	48
Issue	1
Year	2017
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html