On removable singularities of mappings in
uniform spaces

E. A. Sevost’yanov; S. A. Skvortsov; N. S. Ilkevych

doi:10.30970/ms.52.1.24-31

The paper is devoted to the study of mappings of two metric spaces that distort the modulus of families of paths by analogy with the Poletskii inequality.We deal with the situation when the mapping acts in a space that admits weak sphericalization, while the corresponding extended metric space is uniform. For such mappings, the possibility of continuous extension to an isolated boundary point is proved and, as a consequence, an analogue of the Sokhotski-Casorati-Weierstrass theorem is obtained.

1. E. Afanaseva, A. Golberg, R. Salimov, Finite mean oscillation in upper ular metric spaces, Lobachevskii J. Math., 38 (2017), №2, 206-212.

2. E. Afanaseva, R. Salimov, Boundary behavior of mappings in $\lambda(\varepsilon)$-regular metric spaces, J. Math. Sci., 211 (2015), №5, 617-623.

3. A. Ignatev, V. Ryazanov, Finite mean oscillation in the mapping theory, Ukr. Mat. Visn., 2 (2005), №3, 395.417 (in Russian); translation Finite mean oscillation in the mapping theory, Ukr. Math. Bull., 2 (2005), №3, 403-424.

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

5. R. Nakki and B. Palka, Uniform equicontinuity of quasiconformal mappings, Proc. Amer. Math. Soc., 37 (1973), №2, 427-433.

6. S. Rickman, Quasiregular mappings, Berlin etc., Springer-Verlag, 1993.

7. V. Ryazanov, R. Salimov,Weakly flat spaces and boundaries in the mapping theory, Ukr. Math. Visnyk, 4 (2007), №2, 199-233 (in Russian); translation Weakly flat spaces and boundaries in the mapping theory, Ukr. Math. Bull., 4 (2007), №2, 199-233.

8. E.A. Sevostyanov, A.A. Markysh, On Sokhotski-Casorati-Weierstrass theorem on metric spaces, Complex Variables and Elliptic Equations, published online https://www.tandfonline.com/doi/full/ 10.1080/17476933.2018.1557155.

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

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

11. J. Heinonen, Lectures on Analysis on metric spaces, Springer Science+Business Media, New York, 2001.

12. K. Kuratowski, Topology, V.2, Academic Press, New York-London, 1968.

	On removable singularities of mappings in uniform spaces
Author	E. A. Sevost’yanov¹, S. A. Skvortsov², N. S. Ilkevych³ esevostyanov2009@gmail.com¹, serezha.skv@gmail.com², ilkevych@list.ru² 1) Zhytomyr Ivan Franko State University Zhytomyr, Ukraine Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Slovyansk, Ukraine; 2, 3) Zhytomyr Ivan Franko State University Zhytomyr, Ukraine
Abstract	The paper is devoted to the study of mappings of two metric spaces that distort the modulus of families of paths by analogy with the Poletskii inequality.We deal with the situation when the mapping acts in a space that admits weak sphericalization, while the corresponding extended metric space is uniform. For such mappings, the possibility of continuous extension to an isolated boundary point is proved and, as a consequence, an analogue of the Sokhotski-Casorati-Weierstrass theorem is obtained.
Keywords	metric spaces; quasiconformal mappings; mappings with bounded and finite distortion; singularities; moduli of families of paths
DOI	doi:10.30970/ms.52.1.24-31
Reference	1. E. Afanaseva, A. Golberg, R. Salimov, Finite mean oscillation in upper ular metric spaces, Lobachevskii J. Math., 38 (2017), №2, 206-212. 2. E. Afanaseva, R. Salimov, Boundary behavior of mappings in $\lambda(\varepsilon)$-regular metric spaces, J. Math. Sci., 211 (2015), №5, 617-623. 3. A. Ignatev, V. Ryazanov, Finite mean oscillation in the mapping theory, Ukr. Mat. Visn., 2 (2005), №3, 395.417 (in Russian); translation Finite mean oscillation in the mapping theory, Ukr. Math. Bull., 2 (2005), №3, 403-424. 4. O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in Modern Mapping Theory, Springer Monographs in Mathematics, Springer, New York etc., 2009. 5. R. Nakki and B. Palka, Uniform equicontinuity of quasiconformal mappings, Proc. Amer. Math. Soc., 37 (1973), №2, 427-433. 6. S. Rickman, Quasiregular mappings, Berlin etc., Springer-Verlag, 1993. 7. V. Ryazanov, R. Salimov,Weakly flat spaces and boundaries in the mapping theory, Ukr. Math. Visnyk, 4 (2007), №2, 199-233 (in Russian); translation Weakly flat spaces and boundaries in the mapping theory, Ukr. Math. Bull., 4 (2007), №2, 199-233. 8. E.A. Sevostyanov, A.A. Markysh, On Sokhotski-Casorati-Weierstrass theorem on metric spaces, Complex Variables and Elliptic Equations, published online https://www.tandfonline.com/doi/full/ 10.1080/17476933.2018.1557155. 9. J. Vaisala, Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Math., V.229, Springer.Verlag, Berlin etc., 1971. 10. B. Fuglede, Extremal length and functional completion, Acta Math., 98 (1957), 171-219. 11. J. Heinonen, Lectures on Analysis on metric spaces, Springer Science+Business Media, New York, 2001. 12. K. Kuratowski, Topology, V.2, Academic Press, New York-London, 1968.
Pages	24-31
Volume	52
Issue	1
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On removable singularities of mappings in uniform spaces