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JOURNALS // Algebra i Analiz

Algebra i Analiz, 2015, Volume 27, Issue 5, Pages 81–116 (Mi aa1456)

Prime ends and the Orlicz–Sobolev classes
D. A. Kovtonyuk, V. I. Ryazanov

References

1. Afanaseva E. S., Ryazanov V. I., Salimov R. R., “Ob otobrazheniyakh klassa Orlicha–Soboleva na rimanovykh mnogoobraziyakh”, Ukr. mat. vestn., 8:3 (2011), 319–342  mathscinet
2. Vodopyanov S. K., “Otobrazheniya s ogranichennym iskazheniem i s konechnym iskazheniem na gruppakh Karno”, Sib. mat. zh., 40:4 (1999), 764–804  mathnet  mathscinet  zmath
3. Vodopyanov S. K., Goldshtein V. M., Reshetnyak Yu. G., “O geometricheskikh svoistvakh funktsii s pervymi obobschennymi proizvodnymi”, Uspekhi mat. nauk, 34:1 (1979), 17–65  mathnet  mathscinet  zmath
4. Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966  mathscinet
5. Zorich V. A., “Sootvetstvie granits pri $Q$-kvazikonformnykh otobrazheniyakh shara”, Dokl. AN SSSR, 145:6 (1962), 1209–1212  mathnet  zmath
6. Zorich V. A., “Granichnye svoistva odnogo klassa otobrazhenii v rostranstve”, Dokl. AN SSSR, 153:1 (1963), 23–26  mathnet  zmath
7. Zorich V. A., “Opredelenie granichnykh elementov posredstvom sechenii”, Dokl. AN SSSR, 164:4 (1965), 736–739  mathnet  zmath
8. Ignatev A. A., Ryazanov V. I., “Konechnoe srednee kolebanie v teorii otobrazhenii”, Ukr. mat. vestn., 2:3 (2005), 395–417  mathscinet  zmath
9. Kovtonyuk D. A., Petkov I. V., Ryazanov V. I., Salimov R. R., “Granichnoe povedenie i zadacha Dirikhle dlya uravnenii Beltrami”, Algebra i analiz, 25:4 (2013), 101–124  mathnet  mathscinet  zmath  elib
10. Kovtonyuk D. A., Ryazanov V. I., “K teorii granits prostranstvennykh oblastei”, Tr. IPMM NAN Ukrainy, 13, 2006, 110–120  mathscinet  zmath
11. Kovtonyuk D. A., Ryazanov V. I., “K teorii nizhnikh $Q$-gomeomorfizmov”, Ukr. mat. vestn., 5:2 (2008), 159–184  mathscinet
12. Kovtonyuk D. A., Ryazanov V. I., Salimov R. R., Sevostyanov E. A., “K teorii klassov Orlicha–Soboleva”, Algebra i analiz, 25:6 (2013), 50–102  mathnet  mathscinet  zmath
13. Kovtonyuk D. A., Salimov R. R., Sevostyanov E. A., K teorii otobrazhenii klassov Soboleva i Orlicha–Soboleva, ed. Ryazanov V. I., Naukova dumka, Kiev, 2013
14. Krasnoselskii M. A., Rutitskii Ya. B., Vypuklye funktsii i prostranstva Orlicha, Fizmatgiz, 1958
15. Lomako T. V., “O rasprostranenii nekotorykh obobschenii kvazikonformnykh otobrazhenii na granitsu”, Ukr. mat. zh., 61:10 (2009), 1329–1337  mathscinet  zmath
16. Mazya V. G., Prostranstva S. L. Soboleva, LGU, L., 1985  mathscinet
17. Reshetnyak Yu. G., Prostranstvennye otobrazheniya s ogranichennym iskazheniem, Nauka, Novosibirsk, 1982  mathscinet
18. Ryazanov V. I., Salimov R. R., “Slabo ploskie prostranstva i granitsy v teorii otobrazhenii”, Ukr. mat. vestn., 4:2 (2007), 199–234  mathscinet
19. Ryazanov V. I., Sevostyanov E. A., “Ravnostepenno nepreryvnye klassy koltsevykh $Q$-gomeomorfizmov”, Sib. mat. zh., 48:6 (2007), 1361–1376  mathnet  mathscinet  zmath  elib
20. Saks S., Teoriya integrala, IL, M., 1949
21. Sobolev S. L., Prilozheniya funktsionalnogo analiza v matematicheskoi fizike, LGU, L., 1950
22. Stepanoff W., “Sur les conditions de l'existence de la differentielle totale”, Mat. sb., 32:3 (1925), 511–527  mathnet  zmath
23. Suvorov G. D., Obobschennyi printsip “dliny i ploschadi” v teorii otobrazhenii, Naukova dumka, Kiev, 1985  mathscinet
24. Federer G., Geometricheskaya teoriya mery, Nauka, M., 1987  mathscinet
25. Shlyk V. A., “O ravenstve $p$-emkosti i $p$-modulya”, Sib. mat. zh., 34:6 (1993), 216–221  mathnet  mathscinet  zmath
26. Adamowicz T., Björn A., Björn J., Shanmugalingam N., “Prime ends for domains in metric spaces”, Adv. Math., 238 (2013), 459–505  crossref  mathscinet  zmath  isi
27. Birnbaum Z., Orlicz W., “Über die Verallgemeinerungen des Begriffes der zueinauder konjugierten Potenzen”, Studia Math., 3 (1931), 1–67  zmath
28. Calderon A. P., “On the differentiability of absolutely continuous functions”, Riv. Math. Univ. Parma, 2 (1951), 203–213  mathscinet  zmath
29. Caratheodory C., “Über die Begrenzung der einfachzusammenhängender Gebiete”, Math. Ann., 73:3 (1913), 323–370  crossref  mathscinet  zmath
30. 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  mathscinet  zmath  isi
31. Collingwood E. F., Lohwator A. J., The theory of cluster sets, Cambridge Tracts in Math., 56, Cambridge Univ. Press, Cambridge, 1966  mathscinet  zmath
32. Freudenthal H., “Enden and Primenden”, Fund. Math., 39 (1952), 189–210  mathscinet
33. Gehring F. W., “Rings and quasiconformal mappings in space”, Trans. Amer. Math. Soc., 103:3 (1962), 353–393  crossref  mathscinet  zmath
34. Gehring F. W., Martio O., “Quasiextremal distance domains and extension of quasiconformal mappings”, J. Anal. Math., 45 (1985), 181–206  crossref  mathscinet  zmath  isi
35. Gutlyanskii V., Ryazanov V., Srebro U., Yakubov E., The Beltrami equation. A geometric approach, Developments in Math., 26, Springer, New York, 2012  crossref  mathscinet  zmath
36. Heinonen J., Kilpelainen T., Martio O., Nonlinear potential theory of degenerate elliptic equations, Oxford Math. Monogr., Clarendon Press, New York, 1993  mathscinet  zmath
37. Hencl S., Koskela P., Lectures on mappings of finite distortion, Lecture Notes in Math., 2096, Springer, Cham, 2014  crossref  mathscinet  zmath
38. Iwaniec T., Martin G., Geometric function theory and non-linear analysis, Oxford Math. Monogr., Oxford Univ. Press, New York, 2001  mathscinet
39. Iwaniec T., Sbordone C., “Riesz transforms and elliptic PDEs with VMO coefficients”, J. Anal. Math., 74 (1998), 183–212  crossref  mathscinet  zmath  isi
40. Iwaniec T., Sverák V., “On mappings with integrable dilatation”, Proc. Amer. Math. Soc., 118:1 (1993), 181–188  crossref  mathscinet  zmath  isi
41. Kaufmann B., “Über die Berandung ebener und räumlicher Gebiete”, Math. Ann., 103:1 (1930), 70–144  crossref  mathscinet  zmath
42. Kovtonyuk D., Petkov I., Ryazanov V., Salimov R. R., “On the Dirichlet problem for the Beltrami equation”, J. Anal. Math., 122:4 (2014), 113–141  crossref  mathscinet  zmath  elib
43. Kovtonyuk D., Petkov I., Ryazanov V., “On the boundary behaviour of solutions to the Beltrami equations”, Complex Var. Elliptic Equ., 58:5 (2013), 647–663  crossref  mathscinet  zmath  isi  elib
44. Kovtonyuk D., Ryazanov V., “On the theory of mappings with finite area distortion”, J. Anal. Math., 104 (2008), 291–306  crossref  mathscinet  isi
45. Kovtonyuk D., Ryazanov V., “On the boundary behavior of generalized quasi-isometries”, J. Anal. Math., 115 (2011), 103–119  crossref  mathscinet  zmath  isi
46. 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(LXI):1 (2012), 67–78  mathscinet  zmath
47. Lindelöf E., “Sur un principe general de l'analyse et ses applications á la theorie de la representation conforme”, Acta Soc. Sci. Fenn., 46:4 (1915), 1–35
48. Maly J., “A simple proof of the Stepanov theorem on differentiability almost everywhere”, Expo. Math., 17 (1999), 59–61  mathscinet  zmath
49. Martio O., Rickman S., Väisälä J., “Definitions for quasiregular mappings”, Ann. Acad. Sci. Fenn. Math., 448 (1969), 1–40  mathscinet
50. Martio O., Ryazanov V., Srebro U., Yakubov E., Moduli in modern mapping theory, Springer Monogr. Math., Springer, New York, 2009  mathscinet  zmath
51. Martio O., Ryazanov V., Vuorinen M., “BMO and injectivity of space quasiregular mappings”, Math. Nachr., 205 (1999), 149–161  crossref  mathscinet  zmath  isi
52. Martio O., Vuorinen M., “Whitney cubes, $p$-capacity and Minkowski content”, Expo. Math., 5:1 (1987), 17–40  mathscinet  zmath
53. Mazurkiewicz S., “Über die Definition der Primenden”, Fund. Math., 26 (1936), 272–279  zmath
54. Mazurkiewicz S., “Recherches sur la theorie des bouts premiers”, Fund. Math., 33 (1945), 177–228  mathscinet  zmath
55. Näkki R., “Prime ends and quasiconformal mappings”, J. Anal. Math., 35 (1979), 13–40  crossref  mathscinet  zmath
56. Ohtsuka M., Extremal length and precise functions, Gakkotosho Co., Tokyo, 2003  mathscinet  zmath
57. Orlicz W., “Über eine gewisse Klasse von Räumen vom Typus B”, Bull. Intern. de l'Acad. Pol. Ser. A, 1932 (1932), 207–220, Cracovie  zmath
58. Orlicz W., “Über Räume $(L^M)$”, Bull. Intern. de l'Acad. Pol. Ser. A, 1936 (1936), 93–107  zmath
59. Palagachev D. K., “Quasilinear elliptic equations with VMO coefficients”, Trans. Amer. Math. Soc., 347:7 (1995), 2481–2493  crossref  mathscinet  zmath  isi
60. Rado T., Reichelderfer P. V., Continuous transformations in analysis, Grundlehren Math. Wiss., 75, Springer, Berlin, 1955  mathscinet
61. Ragusa M. A., “Elliptic boundary value problem in vanishing mean oscillation hypothesis”, Comment. Math. Univ. Carolin., 40:4 (1999), 651–663  mathscinet  zmath
62. Reimann H. M., Rychener T., Funktionen beschränkter mittlerer oscillation, Lecture Notes in Math., 487, Springer-Verlag, Berlin, 1975  mathscinet
63. Rickman S., Quasiregular mappings, Ergeb. Math. Grenzgeb. (3), 26, Springer-Verlag, Berlin, 1993  mathscinet  zmath
64. Ryazanov V., Salimov R., Srebro U., Yakubov E., “On boundary value problems for the Beltrami equations”, Contemp. Math., 591, Amer. Math. Soc., Providence, RI, 2013, 211–242  crossref  mathscinet  zmath
65. Ryazanov V., Sevost'yanov E., “Equicontinuity of mappings quasiconformal in the mean”, Ann. Acad. Sci. Fenn. Math., 36 (2011), 231–244  crossref  mathscinet  zmath  isi
66. Ryazanov V., Srebro U., Yakubov E., “On ring solutions of Beltrami equation”, J. Anal. Math., 96 (2005), 117–150  crossref  mathscinet  zmath  isi
67. Ryazanov V., Srebro U., Yakubov E., “To strong ring solutions of the Beltrami equations”, Uzbek. Math. J., 2009, no. 1, 127–137  mathscinet
68. Ryazanov V., Srebro U., Yakubov E., “On strong solutions of the Beltrami equations”, Complex Var. Elliptic Equ., 55:1–3 (2010), 219–236  crossref  mathscinet  zmath  isi  elib
69. Ryazanov V., Srebro U., Yakubov E., “Integral conditions in the theory of the Beltrami equations”, Complex Var. Elliptic Equ., 57:12 (2012), 1247–1270  crossref  mathscinet  zmath  isi  elib
70. Ryazanov V., Srebro U., Yakubov E., “Integral conditions in the mapping theory”, Ukr. mat. vestn., 7:1 (2010), 73–87  mathscinet
71. Sarason D., “Functions of vanishing mean oscillation”, Trans. Amer. Math. Soc., 207 (1975), 391–405  crossref  mathscinet  zmath
72. Ursell H. D., Young L. C., Remarks on the theory of prime ends, Mem. Amer. Mat. Soc., 1951, no. 3, 1951, 29 pp.  mathscinet  zmath
73. Vasil'ev A., Moduli of families of curves for conformal and quasiconformal mappings, Lecture Notes in Math., 1788, Springer-Verlag, Berlin, 2002  crossref  mathscinet  zmath
74. Väisälä J., Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Math., 229, Springer-Verlag, Berlin, 1971  mathscinet  zmath
75. Vuorinen M., Conformal geometry and quasiregular mappings, Lecture Notes in Math., 1319, Springer-Verlag, Berlin, 1988  mathscinet  zmath
76. Whyburn G. Th., Analytic topology, Amer. Math. Soc. Collog. Publ., 28, Amer. Math. Soc., New York, 1942  mathscinet  zmath
77. Wilder R. L., Topology of manifolds, Amer. Math. Soc. Collog. Publ., 3, Amer. Math. Soc., New York, 1949  mathscinet
78. Zaanen A. C., Linear analysis. Measure and integral, Banach and Hilbert space, linear integral equations, Noordhoff N. V., Groningen, 1953  mathscinet
79. Ziemer W. P., “Extremal length and conformal capacity”, Trans. Amer. Math. Soc., 126:3 (1967), 460–473  crossref  mathscinet  zmath


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