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@article{AA_2013_25_6_a3, author = {D. A. Kovtonyuk and V. I. Ryazanov and R. R. Salimov and E. A. Sevost'yanov}, title = {Toward the theory of the {Orlicz--Sobolev} classes}, journal = {Algebra i analiz}, pages = {50--102}, publisher = {mathdoc}, volume = {25}, number = {6}, year = {2013}, language = {ru}, url = {http://geodesic.mathdoc.fr/item/AA_2013_25_6_a3/} }
TY - JOUR AU - D. A. Kovtonyuk AU - V. I. Ryazanov AU - R. R. Salimov AU - E. A. Sevost'yanov TI - Toward the theory of the Orlicz--Sobolev classes JO - Algebra i analiz PY - 2013 SP - 50 EP - 102 VL - 25 IS - 6 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/AA_2013_25_6_a3/ LA - ru ID - AA_2013_25_6_a3 ER -
D. A. Kovtonyuk; V. I. Ryazanov; R. R. Salimov; E. A. Sevost'yanov. Toward the theory of the Orlicz--Sobolev classes. Algebra i analiz, Tome 25 (2013) no. 6, pp. 50-102. http://geodesic.mathdoc.fr/item/AA_2013_25_6_a3/
[1] Ahlfors L., “On quasiconformal mappings”, J. Analyse Math., 3 (1954), 1–58 | DOI | MR | Zbl
[2] Alberico A., Cianchi A., “Differentiability properties of Orlicz–Sobolev functions”, Ark. Mat., 43:1 (2005), 1–28 | DOI | MR | Zbl
[3] Aleksandrov P. S., “O razmernosti zamknutykh mnozhestv”, Uspekhi mat. nauk, 4:6 (1949), 17–88 | MR | Zbl
[4] Ambrosio L., “Metric space valued functions of bounded variation”, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (4), 17:3 (1990), 439–478 | MR | Zbl
[5] Andreian Cazacu C., “Foundations of quasiconformal mappings”, Handbook of complex analysis: geometric function theory, v. 2, Elsevier, Amsterdam, 2005, 687–753 | DOI | MR | Zbl
[6] Andreian Cazacu C., “On the length-area dilatation”, Complex Var. Theory Appl., 50:7–11 (2005), 765–776 | MR | Zbl
[7] Andreian Cazacu C., “Module inequalities for quasiregular mappings”, Ann. Acad. Sci. Fenn. Math., 2 (1976), 17–28 | DOI | MR | Zbl
[8] Andreian Cazacu C., “Modules and quasiconformality”, Symposia Mathematica, 18, Academic Press, London, 1976, 519–534 | MR
[9] Andreian Cazacu C., “A generalization of the quasiconformality”, Topics in Analysis, Colloq. Math. Anal. (Jyvaskyla, 1970), Lecture Notes in Math., 419, Springer-Verlag, Berlin, 1974, 4–17 | DOI | MR
[10] Aseev V. V., “Moduli semeistv lokalno kvazisimmetricheskikh poverkhnostei”, Sib. mat. zh., 30:3 (1989), 9–15 | MR | Zbl
[11] Astala K., Iwaniec T., Koskela P., Martin G., “Mappings of BMO-bounded distortion”, Math. Ann., 317:4 (2000), 703–726 | DOI | MR | Zbl
[12] Afanaseva E. S., Ryazanov V. I., Salimov R. R., “Ob otobrazheniyakh v klassakh Orlicha–Soboleva na rimanovykh mnogoobraziyakh”, Ukr. mat. vestn., 8:3 (2011), 319–342 | MR
[13] Bakhtin A. K., Bakhtina G. P., Zelinskii Yu. B., Topologo-algebraicheskie struktury i geometricheskie metody v kompleksnom analize, Pratsi In-tu mat. NAN Ukraini, 73, 2008, 308 pp. | MR | Zbl
[14] Balogh Z. M., “Hausdorff dimension distribution of quasiconformal mappings on the Heisenberg group”, J. Anal. Math., 83 (2001), 289–312 | DOI | MR | Zbl
[15] Balogh Z. M., Monti R., Tyson J. T., Frequency of Sobolev and qusiconformal dimension distortion, Research Report 2010-11, 22.07.2010, 36 pp.
[16] Bates S. M., “On the image size of singular maps”, Proc. Amer. Math. Soc., 114:3 (1992), 699–705 | DOI | MR | Zbl
[17] Belinskii P. P., Obschie svoistva kvazikonformnykh otobrazhenii, Nauka, Novosibirsk, 1974, 98 pp. | MR | Zbl
[18] Biluta P. A., “Ekstremalnye problemy dlya otobrazhenii, kvazikonformnykh v srednem”, Sib. mat. zh., 6 (1965), 717–726 | MR | Zbl
[19] Birnbaum Z., Orlicz W., “Über die Verallgemeinerungen des Begriffes der zueinander konjugierten Potenzen”, Studia Math., 3 (1931), 1–67 | Zbl
[20] Bishop C. J., “Quasiconformal mappings which increase dimension”, Ann. Acad. Sci. Fenn. Math., 24:2 (1999), 397–407 | MR | Zbl
[21] Bojarski B., Gutlyanskii V., Martio O., Ryazanov V., Infinitesimal geometry of quasiconformal and bi-Lipschitz mappings in the plane, EMS Tracts in Mathematics, 19, EMS Publ. House, Zürich, 2013, 205 pp. | MR | Zbl
[22] Bojarski B., Hajlasz P., Strzelecki P., “Sard's theorem for mappings in Hölder and Sobolev spaces”, Manuscripta Math., 118:4 (2005), 383–397 | DOI | MR | Zbl
[23] Bojarski B., Iwaniec T., “Analytical foundations of the theory of quasiconformal mappings in $\mathbb R^n$”, Ann. Acad. Sci. Fenn. Math., 8:2 (1983), 257–324 | MR | Zbl
[24] Bourgain J., Korobkov M. V., Kristensen J., “On the Morse–Sard property and level sets of Sobolev and BV functions”, Rev. Mat. Iberoam., 29:1 (2013), 1–23 | DOI | MR | Zbl
[25] Vodopyanov S. K., “O zamknutosti klassov otobrazhenii s ogranichennym iskazheniem”, Mat. trudy, 5:2 (2002), 92–137 | MR | Zbl
[26] Vodopyanov S. K., “Foundations of the theory of mappings with bounded distortion on Carnot groups”, The interaction of analysis and geometry, Contemp. Math., 424, Amer. Math. Soc., Providence, RI, 2007, 303–344 | DOI | MR | Zbl
[27] Vodopyanov S. K., Goldshtein V. M., Prostranstva Soboleva i spetsialnye klassy otobrazhenii, NGU, Novosibirsk, 1981 | MR
[28] Vodopyanov S. K., Goldshtein V. M., Reshetnyak Yu. G., “O geometricheskikh svoistvakh funktsii s pervymi oboschennymi proizvodnymi”, Uspekhi mat. nauk, 34:1 (1979), 17–65 | MR | Zbl
[29] Vodop'yanov S., Ukhlov A., “Mappings associated with weighted Sobolev Spaces”, Complex Anal. Dynam. Sys., v. III, Contemp. Math., 455, 2008, 363–382 | MR
[30] Vodopyanov S. K., Ukhlov A. D., “Prostranstva Coboleva i $(P,Q)$-kvazikonformnye otobrazheniya grupp Karno”, Sib. mat. zh., 39:4 (1998), 776–795 | MR | Zbl
[31] Calderon A. P., “On the differentiability of absolutely continuous functions”, Riv. Math. Univ. Parma, 2 (1951), 203–213 | MR | Zbl
[32] Caraman P., $n$-dimensional quasiconformal mappings, Haessner Publ. Inc., Newfoundland, NJ, 1974 | MR
[33] Cesari L., “Sulle transformazioni continue”, Ann. Mat. Pura Appl. (4), 21 (1942), 157–188 | DOI | MR | Zbl
[34] Cianchi A., “A sharp embedding theorem for Orlicz–Sobolev spaces”, Indiana Univ. Math. J., 45:1 (1996), 39–65 | DOI | MR | Zbl
[35] Chiarenza F., Frasca M., Longo P., “$W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with $VMO$ coefficients”, Trans. Amer. Math. Soc., 336:2 (1993), 841–853 | MR | Zbl
[36] Csörnyei M., Hencl S., Maly J., Homeomorphisms in the Sobolev space $W^{1,n-1}$, Preprint MATH-KMA No 252, Charles Univ., Prague, 2007, 15 pp. | MR
[37] Church P. T., Timourian J. G., “Differentiable maps with small critical set or critical set image”, Indiana Univ. Math. J., 27:6 (1978), 953–971 | DOI | MR | Zbl
[38] Church P. T., Timourian J. G., “Maps having $0$-dimensional critical set image”, Indiana Univ. Math. J., 27:5 (1978), 813–832 | DOI | MR
[39] Cristea M., “Dilatations of homeomorphisms satisfying some modular inequalities”, Rev. Roumaine Math. Pures Appl., 56:4 (2011), 275–282 | MR | Zbl
[40] Cristea M., “Open discrete mapping having local $ACL^n$ inverses”, Complex Var. Elliptic Equ., 55:1–3 (2010), 61–90 | DOI | MR | Zbl
[41] Cristea M., “Local homeomorphisms having local $ACL^n$ inverses”, Complex Var. Elliptic Equ., 53:1 (2008), 77–99 | DOI | MR | Zbl
[42] Cristea M., “Mappings of finite distortion: Zoric's theorem, and equicontinuity results”, Rev. Roumaine Math. Pures Appl., 52:5 (2007), 539–554 | MR | Zbl
[43] Cristea M., “Mappings of finite distortion: boundary extension”, Rev. Roumaine Math. Pures Appl., 51:5–6 (2006), 607–631 | MR | Zbl
[44] Danford N., Shvarts Dzh., Lineinye operatory. Obschaya teoriya, IL, M., 1962, 895 pp.
[45] Donaldson T., “Nonlinear elliptic boundary-value problems in Orlicz–Sobolev spaces”, J. Differential Equations, 10 (1971), 507–528 | DOI | MR | Zbl
[46] Dubinin V. N., Emkosti kondensatorov i simmetrizatsiya v geometricheskoi teorii funktsii kompleksnogo peremennogo, Dalnauka, Vladivostok, 2009, 390 pp.
[47] Dubovitskii A. Ya., “O strukture mnozhestv urovnya differentsiruemykh otobrazhenii $n$-mernogo kuba v $k$-mernyi kub”, Izv. AN SSSR. Cer. mat., 21:3 (1957), 371–408 | MR | Zbl
[48] Fadell A. G., “A note on a theorem of Gehring and Lehto”, Proc. Amer. Math. Soc., 49 (1975), 195–198 | DOI | MR | Zbl
[49] Federer G., Geometricheskaya teoriya mery, Nauka, M., 1987, 760 pp. | MR | Zbl
[50] Fuglede B., “Extremal length and functional completion”, Acta Math., 98 (1957), 171–219 | DOI | MR | Zbl
[51] Gehring F. W., “Rings and quasiconformal mappings in space”, Trans. Amer. Math. Soc., 103 (1962), 353–393 | DOI | MR | Zbl
[52] Gehring F. W., Lehto O., “On the total differentiability of functions of a complex variable”, Ann. Acad. Sci. Fenn. Math., 272 (1959), 3–8 | MR
[53] Gehring F. W., Martio O., “Quasiextremal distance domains and extension of quasiconformal mappings”, J. Anal. Math., 45 (1985), 181–206 | DOI | MR | Zbl
[54] Gehring F. W., Väisälä J., “Hausdorff dimension and quasiconformal mappings”, J. London Math. Soc. (2), 6:2 (1973), 504–512 | DOI | MR | Zbl
[55] Golberg A., “Homeomorphisms with integrally restricted moduli”, Contemp. Math., 553, Amer. Math. Soc., Providence, RI, 2011, 83–98 | DOI | MR | Zbl
[56] Golberg A., “Directional dilatations in space”, Complex Var. Elliptic Equ., 55:1–3 (2010), 13–29 | DOI | MR | Zbl
[57] Golberg A., “Homeomorphisms with finite mean dilatations”, Contemp. Math., 382, Amer. Math. Soc., Providence, RI, 2005, 177–186 | DOI | MR | Zbl
[58] Golberg A., Gutlyanskii V., “On Lipschitz continuity of quasiconformal mappings in space”, J. Anal. Math., 109 (2009), 233–251 | DOI | MR | Zbl
[59] Goldshtein V. M., Reshetnyak Yu. G., Vvedenie v teoriyu funktsii s obobschennymi proizvodnymi i kvazikonformnye otobrazheniya, Nauka, Novosibirsk, 1983 | MR
[60] Gossez J.-P., Mustonen V., “Variational inequalities in Orlicz–Sobolev spaces”, Nonlinear Anal. Theory Meth. Appl., 11 (1987), 379–392 | DOI | MR | Zbl
[61] Grinberg E. L., “On the smoothness hypothesis in Sard's theorem”, Amer. Math. Monthly, 92:10 (1985), 733–734 | DOI | MR | Zbl
[62] Gutlyanskii V. Ya., Ryazanov V. I., Geometricheskaya i topologicheskaya teoriya funktsii i otobrazhenii, Nauk. dumka, Kiev, 2011, 425 pp.
[63] Gutlyanskii V., Ryazanov V., Srebro U., Yakubov E., The Beltrami equation: A geometric approach, Developments of Mathematics, 26, Springer, New York, 2012, 301 pp. | DOI | MR | Zbl
[64] Zorich V. A., “O dopustimom poryadke rosta kharakteristiki kvazikonformnosti v teoreme M. A. Lavrenteva”, Dokl. AN SSSR, 181:3 (1968), 530–533 | Zbl
[65] Zorich V. A., “Kvazikonformnye otobrazheniya i asimptoticheskaya geometriya mnogoobrazii”, Uspekhi mat. nauk, 57:3 (2002), 3–28 | DOI | MR | Zbl
[66] Hajlasz P., “Sobolev spaces on an arbitrary metric space”, Potential Anal., 5:4 (1996), 403–415 | MR | Zbl
[67] Hajlasz P., “Whitney's example by way of Assouad's embedding”, Proc. Amer. Math. Soc., 131:11 (2003), 3463–3467 | DOI | MR | Zbl
[68] Hardy G. H., Littlewood J. E., Polia G., Inequalities, Cambridge Univ. Press, Cambridge, 1934 | Zbl
[69] Heinonen J., Lectures on analysis on metric spaces, Springer, New York etc., 2000 | MR
[70] Heinonen J., Kilpelainen T., Martio O., Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, Clarendon Press, Oxford Univ. Press, New York, 1993 | MR | Zbl
[71] Heinonen J., Koskela P., “Quasiconformal maps in metric spaces with controlled geometry”, Acta Math., 181:1 (1998), 1–41 | DOI | MR
[72] Heinonen J., Koskela P., Shanmugalingam P., Tyson J. T., “Sobolev spaces of Banach space-valued functions and quasiconformal mappings”, J. Anal. Math., 85 (2001), 87–139 | DOI | MR | Zbl
[73] Herron D. A., Koskela P., “Locally uniform domains and quasiconformal mappings”, Ann. Acad. Sci. Fenn. Math., 20:1 (1995), 187–206 | MR | Zbl
[74] Hesse J., “A $p$-extremal length and $p$-capacity equality”, Ark. Mat., 13 (1975), 131–144 | DOI | MR | Zbl
[75] Hsini M., “Existence of solutions to a semilinear elliptic system through generalized Orlicz–Sobolev spaces”, J. Partial Differ. Equ., 23:2 (2010), 168–193 | MR | Zbl
[76] Hurewicz W., Wallman H., Dimension Theory, Princeton Univ. Press, Princeton, 1948 | MR | Zbl
[77] Ignatev A., Ryazanov V., “Konechnoe srednee kolebanie v teorii otobrazhenii”, Ukr. mat. vestn., 2:3 (2005), 395–417 | MR
[78] Iwaniec T., Koskela P., Onninen J., “Mappings of finite distortion: compactness”, Ann. Acad. Sci. Fenn. Math., 27:2 (2002), 391–417 | MR | Zbl
[79] Iwaniec T., Martin G., Geometrical function theory and non-linear analysis, Oxford Mathematical Monographs, Clarendon Press, Oxford Univ. Press, New York, 2001 | MR
[80] Iwaniec T., Sbordone C., “Riesz transforms and elliptic PDEs with VMO coefficients”, J. Anal. Math., 74 (1998), 183–212 | DOI | MR | Zbl
[81] Iwaniec T., Sverák V., “On mappings with integrable dilatation”, Proc. Amer. Math. Soc., 118:1 (1993), 181–188 | DOI | MR | Zbl
[82] John F., Nirenberg L., “On functions of bounded mean oscillation”, Comm. Pure Appl. Math., 14 (1961), 415–426 | DOI | MR | Zbl
[83] Kaufman R., “A singular map of a cube onto a square”, J. Differential Geom., 14:4 (1979), 593–594 | MR | Zbl
[84] Kauhanen J., Koskela P., Maly J., “On functions with derivatives in a Lorentz space”, Manuscripta Math., 10:1 (1999), 87–101 | DOI | MR
[85] Khruslov E. Ya., Pankratov L. S., “Homogenization of the Dirichlet variational problems in Orlicz–Sobolev spaces”, Operator theory and its applications (Winuipeg, MB, 1998), Fields Inst. Commun., 25, Amer. Math. Soc., Providence, RI, 2000, 345–366 | MR | Zbl
[86] Kovalev L. V., “Monotonnost obobschennogo privedennogo modulya”, Zap. nauch. semin. POMI, 276, 2001, 219–236 | MR | Zbl
[87] Kovtonyuk D., Petkov I., Ryazanov V., On homeomorphisms with finite distortion in the plane, 18 Nov., 2010, 16 pp., arXiv: 1011.3310v2[math.CV]
[88] Kovtonyuk D., Ryazanov V., “K teorii granits prostranstvennykh oblastei”, Tr. IPMM NAN Ukrainy, 13, 2006, 110–120 | MR | Zbl
[89] Kovtonyuk D., Ryazanov V., “On the theory of lower $Q$-homeomorphisms”, Ukr. mat. vestn., 5:2 (2008), 157–181 | MR
[90] Kovtonyuk D., Ryazanov V., “On the theory of mappings with finite area distortion”, J. Anal. Math., 104 (2008), 291–306 | DOI | MR
[91] Kovtonyuk D., Ryazanov V., “On boundary behavior of generalized quasi-isometries”, J. Anal. Math., 115 (2001), 103–119 | DOI | MR
[92] Kovtonyuk D., Ryazanov V., Salimov R., Sevost'yanov E., On mappings in the Orlicz–Sobolev classes, 12 Jan., 2011, 42 pp., arXiv: 1012.5010v4[math.CV]
[93] Kovtonyuk D. A., Ryazanov V. I., Salimov R. R., Sevost'yanov E. A., “On mappings in the Orlicz–Sobolev classes”, Ann. Univ. Buchar., Math. Ser., 3(61):1 (2012), 67–78 | MR | Zbl
[94] Koronel J. D., “Continuity and $k$-th order differentiability in Orlicz–Sobolev spaces: $W^kL_A''$”, Israel J. Math., 24:2 (1976), 119–138 | DOI | MR | Zbl
[95] Krasnoselskii M. A., Rutitskii Ya. B., Vypuklye funktsii i prostranstva Orlicha, Fizmatgiz, M., 1958 | MR
[96] Kruglikov V. I., “Emkosti kondensatorov i prostranstvennye otobrazheniya, kvazikonformnye v srednem”, Mat. sb., 130(172):2 (1986), 185–206 | MR | Zbl
[97] Krushkal S. L., “Ob otobrazheniyakh, kvazikonformnykh v srednem”, Dokl. AN SSSR, 157:3 (1964), 517–519 | Zbl
[98] Krushkal S. L., Kyunau R., Kvazikonformnye otobrazheniya – novye metody i prilozheniya, Nauka, Novosibirsk, 1984 | MR | Zbl
[99] Kudyavin V. S., “Otsenki iskazheniya rasstoyaniya pri otobrazheniyakh, kvazikonformnykh v srednem”, Dinamika splosh. sr., 52 (1981), 168–171 | MR
[100] Kudyavin V. S., “Lokalnye i granichnye svoistva otobrazhenii, kvazikonformnykh v srednem”, Sb. nauch. tr. IM SO AN SSSR, Novosibirsk, 1981, 168–171 | MR
[101] Kudyavin V. S., “Povedenie klassa otobrazhenii, kvazikonformnykh v srednem, v izolirovannoi osoboi tochke”, Dokl. AN SSSR, 277:5 (1984), 1056–1058 | MR
[102] Kuratovskii K., Topologiya, v. 1, Mir, M., 1966 | MR
[103] Kuratovskii K., Topologiya, v. 2, Mir, M., 1969 | MR
[104] Kuzmina G. V., “Moduli semeistv krivykh i kvadratichnye differentsialy”, Tr. Mat. in-ta AN SSSR, 139, 1980, 3–241 | MR | Zbl
[105] Kühnau R., “Uber Extremalprobleme bei im Mittel quasiconformen Abbildungen”, Lecture Notes in Math., 1013, Springer-Verlag, Berlin, 1971, 113–124 | DOI | MR
[106] Landes R., Mustonen V., “Pseudo-monotone mappings in Sobolev-Orlicz spaces and nonlinear boundary value problems on unbounded domains”, J. Math. Anal. Appl., 88 (1982), 25–36 | DOI | MR | Zbl
[107] Lappalainen V., Lehtonen A., “Embedding of Orlicz-Sobolev spaces in Hölder spaces”, Ann. Acad. Sci. Fenn. Math., 14:1 (1989), 41–46 | MR | Zbl
[108] Lehto O., Virtanen K., Quasiconformal mappings in the plane, Die Grundlehren Math. Wiss., 126, Springer-Verlag, New York, 1973 | MR | Zbl
[109] Lelong-Ferrand J., Representation conforme et transformations à integrale de Dirichlet bornée, Gauthier-Villars, Paris, 1955 | MR | Zbl
[110] Lomako T. V., “O rasprostranenii nekotorykh obobschenii kvazikonformnykh otobrazhenii na granitsu”, Ukr. mat. zh., 61:10 (2009), 1329–1337 | MR | Zbl
[111] Lomako T., Salimov R., Sevost'yanov E., “On equicontinuity of solutions to the Beltrami equations”, Ann. Univ. Buchar., Math. Ser., 1(59):2 (2010), 263–274 | MR | Zbl
[112] Maly J., “A simple proof of the Stepanov theorem on differentiability alsmost everywhere”, Exposition Math., 17 (1999), 59–61 | MR | Zbl
[113] Maly J., Martio O., “Lusin's condition $(N)$ and mappings of the class $W^{1,n}$”, J. Reine Angew. Math., 485 (1995), 19–36 | MR
[114] Marcus M., Mizel V., “Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems”, Bull. Amer. Math. Soc., 79:4 (1973), 790–795 | DOI | MR | Zbl
[115] Martio O., Modern tools in the theory of quasiconformal maps, Textos de Math. Ser. B, 27, Univ. Coimbra, Dept. Mat., Coimbra, 2000, 43 pp. | MR | Zbl
[116] Martio O., Rickman S., Väisälä J., “Definitions for quasiregular mappings”, Ann. Acad. Sci. Fenn. Math., 448 (1969), 1–40 | MR
[117] Martio O., Ryazanov V., Srebro U., Yakubov E., “Mappings with finite length distortion”, J. Anal. Math., 93 (2004), 215–236 | DOI | MR | Zbl
[118] Martio O., Ryazanov V., Srebro U., Yakubov E., “$Q$-homeomorphisms”, Complex analysis and dynamical systems, Contemp. Math., 364, Amer. Math. Soc., Providence, RI, 2004, 193–203 | DOI | MR | Zbl
[119] Martio O., Ryazanov V., Srebro U., Yakubov E., “On $Q$-homeomorphisms”, Ann. Acad. Sci. Fenn. Math., 30:1 (2005), 49–69 | MR | Zbl
[120] Martio O., Ryazanov V., Srebro U., Yakubov E., Moduli in modern mapping theory, Springer Monographs in Mathematics, Springer, New York etc., 2009, 367 pp. | MR | Zbl
[121] Martio O., Ryazanov V., Vuorinen M., “BMO and injectivity of space quasiregular mappings”, Math. Nachr., 205 (1999), 149–161 | DOI | MR | Zbl
[122] Martio O., Sarvas J., “Injectivity theorems in plane and space”, Ann. Acad. Sci. Fenn. Math., 4:2 (1974), 384–401 | MR
[123] Martio O., Vuorinen M., “Whitney cubes, $p$-capacity and Minkowski content”, Exposition. Math., 5:1 (1987), 17–40 | MR | Zbl
[124] Mattila P., Geometry of sets and measures in Euclidean spaces, Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, 44, Cambridge Univ. Press, Cambridge, 1995 | MR | Zbl
[125] Mazya V. G., Prostranstva S. L. Soboleva, LGU, L., 1985, 416 pp. | MR
[126] Menchoff D., “Sur les differentielles totales des fonctions univalentes”, Math. Ann., 105:1 (1931), 75–85 | DOI | MR | Zbl
[127] Näkki R., “Boundary behavior of quasiconformal mappings in $n$-space”, Ann. Acad. Sci. Fenn. Math., 484 (1970), 1–50 | MR
[128] Norton A., “A critical set with nonnull image has large Hausdorff dimension”, Trans. Amer. Math. Soc., 296:1 (1986), 367–376 | DOI | MR | Zbl
[129] Onninen J., “Differentiability of monotone Sobolev functions”, Real. Anal. Exchange, 26:2 (2000/2001), 761–772 | MR
[130] Orlicz W., “Über eine gewisse Klasse von Räumen vom Typus B.”, Bull. Intern. de l'Acad. Pol. Ser. A (Cracovie), 1932 (1932), 207–220 | Zbl
[131] Orlicz W., “Über Räume $(L^M)$”, Bull. Intern. de l'Acad. Pol. Serie A (Cracovie), 1936 (1936), 93–107 | Zbl
[132] Palagachev D. K., “Quasilinear elliptic equations with VMO coefficients”, Trans. Amer. Math. Soc., 347:7 (1995), 2481–2493 | DOI | MR | Zbl
[133] Perovich M., “Isolated singularity of the mean quasiconformal mappings”, Lecture Notes in Math., 743, Springer-Verlag, Berlin, 1979, 212–214 | DOI | MR
[134] Perovich M., “Globalnaya gomeomorfnost otobrazhenii, kvazikonformnykh v srednem”, Dokl. AN SSSR, 230:4 (1976), 781–784 | MR
[135] Pesin I. N., “Otobrazheniya, kvazikonformnye v srednem”, Dokl. AN SSSR, 187:4 (1969), 740–742 | MR | Zbl
[136] Ponomarëv S. P., “Ob $N$-svoistve gomeomorfizmov klassa $W_p^1$”, Sib. mat. zh., 28:2 (1987), 140–148 | MR | Zbl
[137] Quinn F., Sard A., “Hausdorff conullity of critical images of Fredholm maps”, Amer. J. Math., 94 (1972), 1101–1110 | DOI | MR | Zbl
[138] Rado T., Reichelderfer P. V., Continuous transformations in analysis, Die Grundlehren der Math. Wiss., 75, Springer-Verlag, Berlin, 1955 | MR
[139] Ragusa M. A., “Elliptic boundary value problem in vanishing mean oscillation hypothesis”, Comment. Math. Univ. Carolin., 40:4 (1999), 651–663 | MR | Zbl
[140] Rajala K., Zapadinskaya A., Zürcher T., Generalized Hausdorff dimension distortion in euclidean spaces under Sobolev mappings, 2010, 13 pp., arXiv: 1007.2091v1[math.CA]
[141] Reimann H. M., “On the absolute continuity of surface representation”, Comment. Math. Helv., 46 (1971), 44–47 | DOI | MR | Zbl
[142] Reimann H. M., Rychener T., Funktionen beschränkter mittlerer oscillation, Lecture Notes in Math., 487, Springer-Verlag, Berlin–New York, 1975 | MR | Zbl
[143] Reshetnyak Yu. G., Prostranstvennye otobrazheniya s ogranichennym iskazheniem, Nauka, Novosibirsk, 1982 | MR | Zbl
[144] Reshetnyak Yu. G., “Ob uslovii $N$ dlya prostranstvennykh otobrazhenii klassa $W_{n,loc}^1$”, Sib. mat. zh., 28:5 (1987), 149–153 | MR | Zbl
[145] Reshetnyak Yu. G., “Sobolevskie klassy funktsii so znacheniyami v metricheskom prostranstve”, Sib. mat. zh., 38:3 (1997), 657–675 | MR | Zbl
[146] Reshetnyak Yu. G., “Nekotorye geometricheskie svoistva funktsii i otobrazhenii s obobschennymi proizvodnymi”, Sib. mat. zh., 7:4 (1966), 886–919 | MR | Zbl
[147] Rickman S., Quasiregular mappings, Ergebnisse der Math. und ihrer Grenzgebiete (3), 26, Springer-Verlag, Berlin, 1993 | MR | Zbl
[148] Rudin U., Teoriya funktsii v polikruge, Mir, M., 1974 | MR | Zbl
[149] Ryazanov V. I., “Ob otobrazheniyakh, kvazikonformnykh v srednem”, Sib. mat. zh., 37:2 (1996), 378–388 | MR | Zbl
[150] Ryazanov V. I., Salimov R. R., “Slabo ploskie prostranstva i granitsy v teorii otobrazhenii”, Ukr. mat. vestn., 4:2 (2007), 199–234 | MR
[151] Ryazanov V., Salimov R. R., Sevost'yanov E., “On convergence analysis of space homeomorphisms”, Siberian Adv. in Math., 23:4 (2013), 1–52 | DOI | MR
[152] Ryazanov V. I., Sevostyanov E. A., “Ravnostepenno nepreryvnye klassy koltsevykh $Q$-gomeomorfizmov”, Sib. mat. zh., 48:6 (2007), 1361–1376 | MR | Zbl
[153] Ryazanov V. I., Sevostyanov E. A., “Ravnostepennaya nepreryvnost kvazikonformnykh v srednem otobrazhenii”, Sib. mat. zhurn., 52:3 (2011), 665–679 | MR | Zbl
[154] Ryazanov V., Sevost'yanov E., “On compactness of Orlicz–Sobolev mappings”, Ann. Univ. Buchar. Ser. Math., 3(61):1 (2012), 79–87 | MR | Zbl
[155] Ryazanov V., Srebro U., Yakubov E., “On ring solutions of Beltrami equation”, J. Anal. Math., 96 (2005), 117–150 | DOI | MR | Zbl
[156] Ryazanov V., Srebro U., Yakubov E., “On convergence theory for Beltrami equations”, Ukr. mat. vestn., 5:4 (2008), 524–535 | MR
[157] Ryazanov V., Srebro U., Yakubov E., “To strong ring solutions of the Beltrami equations”, Uzbek. Math. J., 2009, no. 1, 127–137 | MR
[158] Ryazanov V., Srebro U., Yakubov E., “On strong solutions of the Beltrami equations”, Complex Var. Elliptic Equ., 55:1–3 (2010), 219–236 | DOI | MR | Zbl
[159] Ryazanov V., Srebro U., Yakubov E., “Integral conditions in the theory of the Beltrami equations”, Complex Var. Elliptic Equ., 57:12 (2012), 1247–1270 | DOI | MR | Zbl
[160] Saks S., Teoriya integrala, IL, M., 1949, 495 pp.
[161] Sarason D., “Functions of vanishing mean oscillation”, Trans. Amer. Math. Soc., 207 (1975), 391–405 | DOI | MR | Zbl
[162] Sard A., “The measure of the critical values of differentiable maps”, Bull. Amer. Math. Soc., 48 (1942), 883–890 | DOI | MR | Zbl
[163] Sard A., “The equivalence of $n$-measure and Lebesgue measure in $E_n$”, Bull. Amer. Math. Soc., 49 (1943), 758–759 | DOI | MR | Zbl
[164] Sard A., “Images of critical sets”, Ann. of Math., 68:2 (1958), 247–259 | DOI | MR | Zbl
[165] Sard A., “Hausdorff measure of critical images on Banach manifolds”, Amer. J. Math., 87 (1965), 158–174 | DOI | MR | Zbl
[166] Smolovaya E. S., “Granichnoe povedenie koltsevykh $Q$-gomeomorfizmov v metricheskikh prostranstvakh”, Ukr. mat. zh., 62:5 (2010), 682–689 | MR | Zbl
[167] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Nauka, M., 1988, 333 pp. | MR
[168] Solynin A. Yu., “Moduli i ekstremalno-metricheskie problemy”, Algebra i analiz, 11:1 (1999), 3–86 | MR | Zbl
[169] Stein E. M., “Editor's note: The differentiability of functions in $\mathbb R^n$”, Ann. Math. (2), 113:2 (1981), 383–385 | MR | Zbl
[170] Stepanoff W., “Sur la résolution du probléme de Dirichlet á l'aide de l'intégrale de Poisson”, Mat. sb., 32:1 (1924), 111–114 | Zbl
[171] Suvorov G. D., Obobschennyi printsip dliny i ploschadi v teorii otobrazhenii, Nauk. dumka, Kiev, 1985, 280 pp. | MR | Zbl
[172] Suvorov G. D., Metricheskaya teoriya prostykh kontsov i granichnye svoistva ploskikh otobrazhenii s ogranichennymi integralami Dirikhle, Nauk. dumka, Kiev, 1981, 168 pp. | MR
[173] Suvorov G. D., Semeistva ploskikh topologicheskikh otobrazhenii, Izd-vo SO AN SSSR, Novosibirsk, 1965, 164 pp. | MR
[174] Sychev A. V., Moduli i prostranstvennye kvazikonformnye otobrazheniya, Nauka, Novosibirsk, 1983 | Zbl
[175] Tuominen H., “Characterization of Orlicz–Sobolev space”, Ark. Mat., 45:1 (2007), 123–139 | DOI | MR | Zbl
[176] Shlyk V. A., “O ravenstve $p$-emkosti i $p$-modulya”, Sib. mat. zh., 34:6 (1993), 216–221 | MR | Zbl
[177] Väisälä J., Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Math., 229, Springer-Verlag, Berlin, 1971 | MR
[178] Väisälä J., “On the null-sets for extremal distances”, Ann. Acad. Sci. Fenn. Math., 322 (1962), 1–12 | MR
[179] Väisälä J., “Two new characterizations for quasiconformality”, Ann. Acad. Sci. Fenn. Math., 362 (1965), 1–12 | MR
[180] Vasil'ev A., Moduli of families of curves for conformal and quasiconformal mappings, Lecture Notes in Math., 1788, Springer-Verlag, Berlin, 2002 | DOI | MR
[181] Vuillermot P. A., “Hölder-regularity for the solutions of strongly nonlinear eigenvalue problems on Orlicz–Sobolev space”, Houston J. Math., 13:2 (1987), 281–287 | MR | Zbl
[182] Vuorinen M., Conformal geometry and quasiregular mappings, Lecture Notes in Math., 1319, Springer-Verlag, Berlin, 1988, 209 pp. | MR | Zbl
[183] Wilder R. L., Topology of manifolds, Amer. Math. Soc. Colloquium Publications, 32, Amer. Math. Soc., New York, N.Y., 1949 | MR | Zbl
[184] Whitney H., “A function not constant on a connected set of critical points”, Duke Math. J., 1:4 (1935), 514–517 | DOI | MR | Zbl
[185] Ziemer W. P., “Extremal length and conformal capacity”, Trans. Amer. Math. Soc., 126:3 (1967), 460–473 | DOI | MR | Zbl