Geodesic
On the local behaviour of quasi-regular space mappings
Izvestiya. Mathematics , Tome 62 (1998) no. 6, pp. 1207-1220.

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

We establish the local injectivity of mappings whose dilatation tensor (or matrix dilatation) belongs to the class VMO (vanishing mean oscillation) or to the class BMO (bounded mean oscillation) with sufficiently small BMO-norm, or is at least close to the class VMO in the BMO-norm. 
@article{IM2_1998_62_6_a6,
     author = {M. Vuorinen and O. Martio and V. I. Ryazanov},
     title = {On the local behaviour of quasi-regular space mappings},
     journal = {Izvestiya. Mathematics },
     pages = {1207--1220},
     publisher = {mathdoc},
     volume = {62},
     number = {6},
     year = {1998},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1998_62_6_a6/}
}
TY  - JOUR
AU  - M. Vuorinen
AU  - O. Martio
AU  - V. I. Ryazanov
TI  - On the local behaviour of quasi-regular space mappings
JO  - Izvestiya. Mathematics 
PY  - 1998
SP  - 1207
EP  - 1220
VL  - 62
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1998_62_6_a6/
LA  - en
ID  - IM2_1998_62_6_a6
ER  - 
%0 Journal Article
%A M. Vuorinen
%A O. Martio
%A V. I. Ryazanov
%T On the local behaviour of quasi-regular space mappings
%J Izvestiya. Mathematics 
%D 1998
%P 1207-1220
%V 62
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1998_62_6_a6/
%G en
%F IM2_1998_62_6_a6
M. Vuorinen; O. Martio; V. I. Ryazanov. On the local behaviour of quasi-regular space mappings. Izvestiya. Mathematics , Tome 62 (1998) no. 6, pp. 1207-1220. http://geodesic.mathdoc.fr/item/IM2_1998_62_6_a6/

[1] Goldshtein V. M., “O povedenii otobrazhenii s ogranichennym iskazheniem pri koeffitsiente iskazheniya, blizkom k edinitse”, Sib. matem. zhurn., 12 (1971), 1250–1258 | MR

[2] Martio O., Rickman S., Väisälä J., “Topological and metric properties of quasiregular mappings”, Ann. Acad. Sci. Fenn. Ser. AI, 488 (1971), 1–31 | MR

[3] Reshetnyak Yu. G., Prostranstvennye otobrazheniya s ogranichennym iskazheniem, Nauka, Novosibirsk, 1982 | MR

[4] Ferrand J., “Regularity of conformal mappings of Riemannian manifolds”, Lect. Notes in Math., 743 (1979), 191–203 | MR

[5] Bojarski B., Iwaniec T., Kopiecki R., “Riemannian Manifolds with Non-Smooth Metric Tensors and $QC$-Maps”, Proc. Sem. on Monge-Ampere Equations Related Topics (Firenze 1980), Roma, 1982, 123–165 | MR

[6] Gutlyanskii V. Ya., Martio O., Ryazanov V. I., Vuorinen M., On local injectivity and asymptotic linearity of quasiregular mappings, Preprint 122, Department of Mathematics, University of Helsinki, Helsinki, 1996, p. 29 | MR

[7] John F., Nirenberg L., “On functions of bounded mean oscillation”, Comm. Pure Appl. Math., 14 (1961), 415–426 | DOI | MR | Zbl

[8] Astala K., Gehring F. W., “Injectivity the $BMO$ norm and the universal Teichmüller space”, J. Analyse Math., 46 (1986), 16–57 | DOI | MR | Zbl

[9] Reimann H. M., Rychener Th., Funktionen beschränkter mittlerer Oscillation, Lect. Notes in Math., 487, 1975 | MR | Zbl

[10] Garnett J. B., Bounded analytic functions, Academie Press, New York etc., 1981 | MR | Zbl

[11] Saks S., Theory of the integral, Dover Publ. Inc., New York, 1964 | MR

[12] Bellmann R., Introduction to Matrix Analysis, McGraw-Hill Book Comp., New York etc., 1970 | MR

[13] Ahlfors L., “On a class of quasiconformal mappings”, Österreich. Acad. Wiss. Math. – Naturw. Kl. S. BII, 185 (1976), 5–10 | MR | Zbl

[14] Ahlfors L., “Quasiconformal deformations and mappings in ${{\mathbb R}^n}$”, J. Analyse Math., 30 (1976), 74–97 | DOI | MR | Zbl

[15] Bers L., “On a theorem of Mori and the definition of quasiconformality”, Trans. Amer. Math. Soc., 84 (1957), 78–84 | DOI | MR | Zbl

[16] Boyarskii B. V., “Obobschennye resheniya sitemy differentsialnykh uravnenii pervogo poryadka ellipticheskogo tipa s razryvnymi koeffitsientami”, Matem. sb., 43(85) (1957), 451–503 | MR

[17] Lehto O., Virtanen K., Quasikonforme Abbildungen, Springer-Verlag, Berlin etc., 1965 | MR | Zbl

[18] Iwaniec T., Kopiecki R., “Stability in the differential equations for quasiregular mappings”, Lect. Notes in Math., 798 (1980), 203–214 | DOI | MR | Zbl

[19] Gutlyanskii V., Martio O., Ryazanov V., Vuorinen M., Convergence theorems for quasiregular mappings in the space, Preprint 105, Department of Mathematics, University of Helsinki, Helsinki, 1996

[20] Martio O, Rickman S., Väisälä J., “Definitions for quasiregular mappings”, Ann. Acad. Sci. Fenn. Ser. AI, 448 (1969), 1–40 | MR

[21] Reshetnyak Yu. G., “Lokalnaya struktura otobrazhenii s ogranichennym iskazheniem”, Sib. matem. zhurn., 10:6 (1969), 1319–1333 | MR

[22] Vuorinen M., Conformal Geometry and Quasiregular Mappings, Lect. Notes in Math., 1319, 1988 | MR | Zbl

[23] Zorich V. A., “Teorema Lavrenteva o kvazikonformnykh otobrazheniyakh prostranstva”, Matem. sb., 74(116):3 (1967), 417–433 | MR | Zbl

[24] Martio O., Rickman S., Väisälä J., “Distortion and singularities of quasiregular mappings”, Ann. Acad. Sci. Fenn. Ser. AI, 465 (1970), 1–13 | MR

[25] Bojarski B., Iwaniec T., “Another approach to Liouville theorem”, Math. Nachr., 107 (1982), 253–262 | DOI | MR | Zbl

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

[27] Tukia P., Väisälä J., “A remark on $1$-quasiconformal maps”, Ann. Acad. Sci. Fenn. Ser. AI, 10 (1985), 561–562 | MR | Zbl

[28] Gutlyanskii V., Martio O., Ryazanov V., Vuorinen M., On asymptotical behavior of quasiconformal mappings in the space, Preprint 83, Department of Mathematics, University of Helsinki, Helsinki, 1995 | MR

[29] Dugundji J., Topology, Allyn and Bacon Inc., Boston, 1966 | MR | Zbl