Geodesic
On finitely Lipschitz space mappings
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 8 (2011), pp. 284-295.

Voir la notice de l'article provenant de la source Math-Net.Ru

It is established that a ring $Q$-homeomorphism with respect to $p$-modulus in $\mathbb R^n$, $n\geqslant2$, is finitely Lipschitz if $n-1$ and if the mean integral value of $Q(x)$ over infinitesimal balls $B(x_0,\varepsilon)$ is finite everywhere.
Keywords: $Q$-homeomorphisms, $p$-modulus, $p$-capacity, finite Lipschitz. 
@article{SEMR_2011_8_a23,
     author = {R. R. Salimov},
     title = {On finitely {Lipschitz}  space mappings},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {284--295},
     publisher = {mathdoc},
     volume = {8},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2011_8_a23/}
}
TY  - JOUR
AU  - R. R. Salimov
TI  - On finitely Lipschitz  space mappings
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2011
SP  - 284
EP  - 295
VL  - 8
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2011_8_a23/
LA  - en
ID  - SEMR_2011_8_a23
ER  - 
%0 Journal Article
%A R. R. Salimov
%T On finitely Lipschitz  space mappings
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2011
%P 284-295
%V 8
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2011_8_a23/
%G en
%F SEMR_2011_8_a23
R. R. Salimov. On finitely Lipschitz  space mappings. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 8 (2011), pp. 284-295. http://geodesic.mathdoc.fr/item/SEMR_2011_8_a23/

[1] C.J. Bishop, V.Ya. Gutlyanskii, O. Martio, M. Vuorinen, “On conformal dilatation in space”, Intern. Journ. Math. and Math. Scie., 22 (2003), 1397–1420 | DOI | MR | Zbl

[2] B. Fuglede, “Extremal length and functional completion”, Acta Math., 98 (1957), 171–219 | DOI | MR | Zbl

[3] F.W. Gehring, Quasiconformal mappings in Complex Analysis and its Applications, v. 2, International Atomic Energy Agency, Vienna, 1976 | MR | Zbl

[4] F.W. Gehring, “Lipschitz mappings and the p-capacity of ring in $n$-space”, Advances in the theory of Riemann surfaces (Proc. Conf. Stonybrook, N.Y., 1969), Ann. of Math. Studies, 66, 1971, 175–193 | MR

[5] F.W. Gehring, “Rings and quasiconformal mappings in space”, Trans. Amer. Math. Soc., 103 (1962), 353–393 | DOI | MR | Zbl

[6] A. Golberg, “Differential properties of $(\alpha,Q)$-homeomorphisms”, Further Progress in Analysis, World Scientific Publ., 2009, 218–228 | MR | Zbl

[7] A. Golberg, “Integrally quasiconformal mappings in space”, Intern. Conf. “Analytic methods of mechanics and complex analysis” dedicated to N. A. Kilchevskii and V. A. Zmorovich on the occasion of their birthday centenary (Kiev, June 29 – July 5 2009), 19

[8] A. Golberg , V. Ya. Gutlyanskii, “On Lipschitz continuity of quasiconformal mappings in space”, Journal d'Analyse Mathematique, 109:1 (2009), 233–251 | MR | Zbl

[9] V. Gutlyanskii, T. Sugawa, “On Lipschitz Continuity of Quasiconformal Mappings”, Rep. Univ. Juvaskyla, 83 (2001), 91–108 | MR

[10] J. Hesse, “A $p-$extremal length and $p-$capacity equality”, Ark. Mat., 13 (1975), 131–144 | DOI | MR | Zbl

[11] A. Ignat'ev, V. Ryazanov, “Finite mean oscillation in the mapping theory”, Ukrainian Math. Bull., 2:3 (2005), 403–424 | MR

[12] A. Ignat'ev, V. Ryazanov, “To the theory of the boundary behavior of space mappings”, Ukrainian Math. Bull., 3:2 (2006), 189–201 | MR

[13] D. Kovtonyuk, V. Ryazanov, R. Salimov, E. Sevost'yanov, On mappings in the Orlicz-Sobolev classes, 2011, 69 pp., arXiv: 1012.5010

[14] V.I. Kruglikov, “Capacities of condensors and quasiconformal in the mean mappings in space”, Mat. Sb., 130:2 (1986), 185–206 | MR

[15] O. Lehto, K. Virtanen, Quasiconformal mappings in the Plane, Springer-Verlag, Berlin-Heidelberg-New York, 1973 | MR | Zbl

[16] O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, “Mappings with finite length distortion”, J. Anal. Math., 93 (2004), 215–236 | DOI | MR | Zbl

[17] O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, “$Q$-homeomorphisms”, Contemporary Math., 364 (2004), 193–203 | MR | Zbl

[18] O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, “On $Q$-homeomorphisms”, Ann. Acad. Sci. Fenn., 30 (2005), 49–69 | MR | Zbl

[19] O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in Modern Mapping Theory, Springer Monographs in Mathematies, Springer, New York, 2009 | MR

[20] T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[21] V. Ryazanov, E. Sevost'yanov, “Normal families of space mappings”, Sib. Electron. Math. Reports, 3 (2006), 216–231 | MR | Zbl

[22] V. Ryazanov, E. Sevost'yanov, “Equicontinuous classes of ring $Q$-homeomorphisms”, Sib. Mat. Zh., 48:6 (2007), 1361–1376 | DOI | MR | Zbl

[23] V. Ryazanov, U. Srebro, E. Yakubov, “On ring solutions of Beltrami equation”, J. d'Analyse Math, 96 (2005), 117–150 | DOI | MR | Zbl

[24] S. Saks, Theory of the Integral, Dover Publ. Inc., New York, 1964 | MR | Zbl

[25] Siberian Math. J., 34:6 (1993), 1196–1200 | DOI | MR | Zbl

[26] J. Väisälä, Lectures on $n$-Dimensional Quasiconformal Mappings, Lecture Notes in Math., 229, Springer-Verlag, Berlin, 1971 | MR