|1Ryazanov, VI |
1Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk
|Dopov. Nac. akad. nauk Ukr. 2020, 6:7-14|
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.
|Keywords: boundary behavior, mappings with finite length distortion, Riemann surfaces, strongly accessible and weakly flat boundaries|
1. Volkov, S. V. & Ryazanov, V. I. (2015). On the boundary behavior of mappings in the class 1, 1 loc W on Riemannian surfaces. Trudy Instituta Prikladnoi Matematiki i Mehaniki NAN Ukrainy, 29, pp. 34-53 (in Russian).
2. Volkov, S. V. & Ryazanov, V. I. (2016). Toward a theory of the boundary behavior of mappings of Sobolev class on Riemann surfaces. Dopov. Nac. akad. nauk Ukr., No. 10, pp. 5-9. https://doi.org/10.15407/dopovidi2016.10.005
3. Ryazanov, V. & Volkov, S. (2017). On the boundary behavior of mappings in the class 1, 1 loc W on Riemann surfaces. Complex Anal. Oper. Theory, 11, No. 7, pp. 1503-1520. https://doi.org/10.1007/s11785-016-0618-4
4. Martio, O., Ryazanov, V., Srebro, U. & Yakubov, E. (2004). Mappings with finite length distortion. J. Anal. Math., 93, pp. 215-236. https://doi.org/10.1007/BF02789308
5. Martio, O., Ryazanov, V., Srebro, U. & Yakubov, E. (2009). Moduli in modern mapping theory. Springer Monographs in Mathematics. New York: Springer. https://doi.org/10.1007/978-0-387-85588-2
6. Kovtonyuk, D., Petkov, I. & Ryazanov, V. (2017). Prime ends in theory of mappings with finite distortion in the plane. Filomat, 31, No. 5, pp. 1349-1366. https://doi.org/10.2298/FIL1705349K
7. Martio, O., Ryazanov, V., Srebro, U. & Yakubov, E. (2005). On Q-homeomorphisms. Ann. Acad. Sci. Fenn. Math., 30, No. 1, pp. 49-69.
8. Väisälä, J. (1971). Lectures on n-dimensional quasiconformal mappings. Lecture Notes in Mathematics, Vol. 229. Berlin, New York: Springer.
9. Krushkal’, S. L., Apanasov, B. N. & Gusevskii, N. A. (1986). Kleinian groups and uniformization in examples and problems. Translations of Mathematical Monographs, Vol. 62. Providence, RI: AMS.
10. Beardon, A. F. (1983). The geometry of discrete groups. Graduate Texts in Matheamatics, Vol. 91. New York: Springer.
11. Ryazanov, V. & Salimov, R. (2007). Weakly flat spaces and boundaries in the theory of mappings. Ukrainian Math. Bull., 4, No. 2, pp. 199-234.
12. Volkov, S. V. & Ryazanov, V. I. (2019). On mappings of finite length distortion on Riemannian surfaces. Trudy Instituta Prikladnoi Matematiki i Mehaniki NAN Ukrainy, 33, pp. 1-16 (in Ukrainian).
13. Ryazanov, V., Srebro, U. & Yakubov, E. (2010). On integral conditions in the mapping theory. Math. Sci. J., 173, No. 4, pp. 397-407. https://doi.org/10.1007/s10958-011-0257-2
14. Kovtonyuk, D. & Ryazanov, V. (2008). On the theory of mappings with finite area distortion. J. Anal. Math., 104, pp. 291-306. https://doi.org/10.1007/s11854-008-0025-5