Toward the theory of generalized quasi-isometries

D.A.Kovtonyuk; V.I.Ryazanov

This note is devoted to the study of the so-called finitely bi-Lipschitz mappings that are {far-reaching generalizations} of isometries as well as quasi-isometries. It is establish a {number} of criteria for homeomorphic extension to the boundary of the finitely bi-Lipschitz homeomorphisms $f$ between domains in $R^n$, $n\geqslant2$, whose outer dilatations $K_O(x,f)$ satisfy integral constraints of the type $\int\Phi(K_O^{n-1}(x,f))\,dm(x)<\infty$ with a convex increasing function $\Phi\colon [0,\infty]\to[0,\infty]$. Note that integral conditions on the function $\Phi$ found by us are not only sufficient but also necessary for a continuous extension of $f$ to the boundary.

1. L. Ahlfors, On quasiconformal mappings, J. Analyse Math. 3 (1953, 1954), 1--58.

2. P.A. Biluta, Extremal problems for mappings which are quasiconformal in the mean, Sib. Mat. Zh. 6 (1965), 717--726.

3. F.W. Gehring, O. Martio, Quasiextremal distance domains and extension of quasiconformal mappings, J. Anal. Math. 45 (1985), 181--206.

4. A. Golberg, Homeomorphisms with finite mean dilatations, Contemporary Math. 382 (2005), 177--186.

5. T. Iwaniec, G. Martin, Geometric Function Theory and Nonlinear Analysis, Clarendon Press, Oxford, 2001.

6. D. Kovtonyuk, V. Ryazanov, On the theory of mappings with finite area distortion, J. d'Analyse Math. 104 (2008), 291--306.

7. D. Kovtonyuk, V. Ryazanov, On the theory of lower $Q$-homeomorphisms, Ukr. Mat. Visn. 5 (2008), №2, 159--184.

8. V.I. Kruglikov, Capacities of condensors and quasiconformal in the mean mappings in space, Mat. Sb. 130 (1986), №2, 185--206.

9. S.L. Krushkal', On mappings that are quasiconformal in the mean, Dokl. Akad. Nauk SSSR 157 (1964), №3, 517--519.

10. S.L. Krushkal', R. K\"uhnau, Quasiconformal mappings: new methods and applications, Novosibirsk, Nauka, 1984. (in Russian)

11. R. K"uhnau, "Uber Extremalprobleme bei im Mittel quasiconformen Abbildungen, Lecture Notes in Math. 1013 (1983), 113--124.

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

13. O. Martio, J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A1 Math. 4 (1978, 1979), 384--401.

14. I.N. Pesin, Mappings quasiconformal in the mean, Dokl. Akad. Nauk SSSR 187 (1969), №4, 740--742.

15. V.I. Ryazanov, On mappings that are quasiconformal in the mean, Sibirsk. Mat. Zh. 37 (1996), №2, 378--388.

16. V. Ryazanov, U. Srebro, E. Yakubov, On integral conditions in the mapping theory, Ukrainian Math. Bull. 7 (2010), 73--87.

17. A. Ukhlov, S.K. Vodopyanov, Mappings associated with weighted Sobolev spaces, Complex Anal. Dynam. Syst. III, Contemp. Math. 455 (2008), 369--382.

	Toward the theory of generalized quasi-isometries
Author	D.A.Kovtonyuk, V.I.Ryazanov denis_kovtonyuk@bk.ru, vl_ryazanov@mail.ru, vlryazanov1@rambler.ru Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, 74 Roze Luxemburg str., 83114 Donetsk, Ukraine;
Abstract	This note is devoted to the study of the so-called finitely bi-Lipschitz mappings that are {far-reaching generalizations} of isometries as well as quasi-isometries. It is establish a {number} of criteria for homeomorphic extension to the boundary of the finitely bi-Lipschitz homeomorphisms $f$ between domains in $R^n$, $n\geqslant2$, whose outer dilatations $K_O(x,f)$ satisfy integral constraints of the type $\int\Phi(K_O^{n-1}(x,f))\,dm(x)<\infty$ with a convex increasing function $\Phi\colon [0,\infty]\to[0,\infty]$. Note that integral conditions on the function $\Phi$ found by us are not only sufficient but also necessary for a continuous extension of $f$ to the boundary.
Keywords	generalized quasi-isometry, bi-Lipschitz mapping, outer dilatation
DOI	doi:10.30970/ms.34.2.129-135
Reference	1. L. Ahlfors, On quasiconformal mappings, J. Analyse Math. 3 (1953, 1954), 1--58. 2. P.A. Biluta, Extremal problems for mappings which are quasiconformal in the mean, Sib. Mat. Zh. 6 (1965), 717--726. 3. F.W. Gehring, O. Martio, Quasiextremal distance domains and extension of quasiconformal mappings, J. Anal. Math. 45 (1985), 181--206. 4. A. Golberg, Homeomorphisms with finite mean dilatations, Contemporary Math. 382 (2005), 177--186. 5. T. Iwaniec, G. Martin, Geometric Function Theory and Nonlinear Analysis, Clarendon Press, Oxford, 2001. 6. D. Kovtonyuk, V. Ryazanov, On the theory of mappings with finite area distortion, J. d'Analyse Math. 104 (2008), 291--306. 7. D. Kovtonyuk, V. Ryazanov, On the theory of lower $Q$-homeomorphisms, Ukr. Mat. Visn. 5 (2008), №2, 159--184. 8. V.I. Kruglikov, Capacities of condensors and quasiconformal in the mean mappings in space, Mat. Sb. 130 (1986), №2, 185--206. 9. S.L. Krushkal', On mappings that are quasiconformal in the mean, Dokl. Akad. Nauk SSSR 157 (1964), №3, 517--519. 10. S.L. Krushkal', R. K\"uhnau, Quasiconformal mappings: new methods and applications, Novosibirsk, Nauka, 1984. (in Russian) 11. R. K"uhnau, "Uber Extremalprobleme bei im Mittel quasiconformen Abbildungen, Lecture Notes in Math. 1013 (1983), 113--124. 12. O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in Modern Mapping Theory, Springer, New York, 2009. 13. O. Martio, J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A1 Math. 4 (1978, 1979), 384--401. 14. I.N. Pesin, Mappings quasiconformal in the mean, Dokl. Akad. Nauk SSSR 187 (1969), №4, 740--742. 15. V.I. Ryazanov, On mappings that are quasiconformal in the mean, Sibirsk. Mat. Zh. 37 (1996), №2, 378--388. 16. V. Ryazanov, U. Srebro, E. Yakubov, On integral conditions in the mapping theory, Ukrainian Math. Bull. 7 (2010), 73--87. 17. A. Ukhlov, S.K. Vodopyanov, Mappings associated with weighted Sobolev spaces, Complex Anal. Dynam. Syst. III, Contemp. Math. 455 (2008), 369--382.
Pages	129-135
Volume	34
Issue	2
Year	2010
Journal	Matematychni Studii
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Table of content of issue	html