Abstract
The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings in the Sobolev classes (mappings with generalized derivatives) on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class of FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio–Ryazanov–Srebro–Yakubov at 2004 and then included in the known book of these authors at 2009 on the modern mapping theory. As was shown in the recent papers of Kovtonyuk–Petkov–Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes, because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasiisometries.
We prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extensions to the boundary of the mappings with finite length distortion between domains on Riemann surfaces by Carathéodory prime ends. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to be the very simple condition on the integrability of the dilatations in the first power. The criteria for the continuous extension of the direct mappings to the boundary have a much more refined nature. One of such criteria is the existence of a majorant for the dilatation 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 boundary points. As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion to the closures of domains by Carathéodory prime ends are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. F. Beardon, The Geometry of Discrete Groups, Springer, New York, 1983.
N. Bourbaki, General Topology, Springer, Berlin, 1995.
B. Fuglede, “Extremal length and functional completion,” Acta Math., 98, 171–219 (1957).
V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov, The Beltrami Equation: A Geometric Approach, Springer, New York, 2012.
A. A. Ignat’ev and V. I. Ryazanov, “Finite mean oscillation in mapping theory,” Ukr. Math. Bull., 2(3), 403–424 (2005).
T. Iwaniec and V. Sverak, “On mappings with integrable dilatation,” Proc. Amer. Math. Soc., 118, 181–188 (1993).
T. Iwaniec and G. Martin, Geometric Function Theory and Nonlinear Analysis, Oxford Univ. Press, Oxford, 2001.
D. Kovtonyuk, I. Petkov, and V. Ryazanov, “On the boundary behavior of mappings with finite distortion in the plane,” Lobachevskii J. Math., 38(2), 290–306 (2017).
D. Kovtonyuk, I. Petkov, and V. Ryazanov, “Prime ends in theory of mappings with finite distortion in the plane,” Filomat, 31(5), 1349–1366 (2017).
D. Kovtonyuk and V. Ryazanov, “On the boundary behavior of generalized quasi-isometries,” J. Anal. Math., 115, 103–120 (2011).
D. Kovtonyuk and V. Ryazanov, “On the theory of mappings with finite area distortion,” J. Anal. Math., 104, 291–306 (2008).
S. L. Krushkal’, B. N. Apanasov, and N. A. Gusevskii, Kleinian Groups and Uniformization in Examples and Problems, AMS, Providence, RI, 1986.
K. Kuratowski, Topology, Vol. 1, Academic Press, New York, 1966.
K. Kuratowski, Topology, Vol. 2, Academic Press, New York, 1968.
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “Mappings with finite length distortion,” J. Anal. Math., 93, 215–236 (2004).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “On Q-homeomorphisms,” Ann. Acad. Sci. Fenn. Math., 30(1), 49–69 (2005).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory, Springer, New York, 2009.
O. Martio and J. Väisälä, “Elliptic equations and maps of bounded length distortion,” Math. Ann., 282(3), 423–443 (1988).
V. Ryazanov and R. Salimov, “Weakly flat spaces and boundaries in the theory of mappings,” Ukr. Math. Bull., 4(2), 199–234 (2007).
V. Ryazanov, U. Srebro, and E. Yakubov, “Integral conditions in the mapping theory,” J. Math. Sci., 173(4), 397–407 (2011).
V. Ryazanov, U. Srebro, and E. Yakubov, “Integral conditions in the theory of the Beltrami equations,” Complex Var. Elliptic Equ., 57(12), 1247–1270 (2012).
S. V. Volkov and V. I. Ryazanov, “On the boundary behavior of mappings in the class \( {W}_{\mathrm{loc}}^{1,1} \) on Riemann surfaces,” Trudy Inst. Prikl. Mat. Mekh., 29, 34–53 (2015).
S. V. Volkov and V. I. Ryazanov, “Toward a theory of the boundary behavior of mappings of Sobolev class on Riemann surfaces,” Dopov. NAN Ukr., No. 10, 5–9 (2016).
V. Ryazanov and S. Volkov, “On the boundary behavior of mappings in the class \( {W}_{\mathrm{loc}}^{1,1} \) on Riemann surfaces,” Complex Anal. Oper. Theory, 11(7), 1503–1520 (2017).
V. Ryazanov and S. Volkov, “Prime ends on the Riemann surfaces,” Dopov. NAN Ukr., No. 9, 20–25 (2017).
V. Ryazanov and S. Volkov, “Prime ends in the Sobolev mapping theory on Riemann surfaces,” Mat. Stud., 48(1), 24–36 (2017).
V. Ryazanov and S. Volkov, “Prime ends in the mapping theory on the Riemann surfaces,” J. Math. Sci., 227(1), 81–97 (2017).
P. S. Uryson, “Zum Metrisationsproblem,” Math. Ann., 94, 309–315 (1925).
J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin, 1971.
G. Th. Whyburn, Analytic Topology, AMS, Providence, RI, 1942.
H. Zieschang, E. Vogt, and H.-D. Coldewey, Surfaces and Planar Discontinuous Groups, Springer, Berlin, 1980.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 17, No. 1, pp. 60–76 January–March, 2020.
Rights and permissions
About this article
Cite this article
Ryazanov, V., Volkov, S. Mappings with Finite Length Distortion and Prime Ends on Riemann Surfaces. J Math Sci 248, 190–202 (2020). https://doi.org/10.1007/s10958-020-04869-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-04869-z