On the lower order of mappings with finite length distortion

E. A. Sevostyanov

Matematičeskie trudy, Tome 18 (2015) no. 1, pp. 98-117.

Résumé

We study the problem of the so-called lower order for one kind of mappings with finite distortion, actively investigated in the recent 15–20 years. We prove that mappings with finite length distortion $f:D\rightarrow \mathbb{R}^n$, $n\ge 2$, whose outer dilatation is integrable to the power $\alpha>n-1$ with finite asymptotic limit have lower order bounded from below.

Bibliographie

[1] Reshetnyak Yu. G., Prostranstvennye otobrazheniya s ogranichennym iskazheniem, Nauka, Novosibirsk, 1982

[2] Saks S., Teoriya integrala, Izd-vo inostr. lit., M., 1949

[3] Sevostyanov E. A., “Obobschenie odnoi lemmy E. A. Poletskogo na klassy prostranstvennykh otobrazhenii”, Ukr. matem. zhurn., 61:7 (2009), 969–975

[4] Sevostyanov E. A., “K teorii ustraneniya osobennostei otobrazhenii s neogranichennoi kharakteristikoi kvazikonformnosti”, Izv. RAN. Ser. matem., 74:1 (2010), 159–174 | DOI

[5] Federer G., Geometricheskaya teoriya mery, Nauka, M., 1987

[6] Bishop C. J., Gutlyanskii V. Ya., Martio O., Vuorinen M., “On conformal dilatation in space”, Internat. J. Math. Sci., 22 (2003), 1397–1420 | DOI

[7] Caraman P., “Relations between $p$-capacity and $p$-module, I”, Rev. Roumaine Math. Pures Appl., 39:6 (1994), 509–553

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

[9] Golberg A., Salimov R., “Topological mappings of integrally bounded $p$-moduli”, Ann. Univ. Buchar. Math. Ser., 3(LXI):1 (2012), 49–66

[10] Gutlyanskii V. Ya., Ryazanov V. I., Srebro U., Yakubov E., The Beltrami Equation: A Geometric Approach, Developments in Mathematics, 26, Springer, New York, etc., 2012

[11] Iwaniec T., Martin G., Geometric Function Theory and Nonlinear Analysis, Oxford Mathematical Monographs, Clarendon Press, Oxford, 2001

[12] Manfredi J. J., Villamor E., “Mappings with integrable dilatation in higher dimensions”, Bull. Amer. Math. Soc. (N.S.), 32:2 (1995), 235–240 | DOI

[13] Manfredi J. J., Villamor E., “An extension of Reshetnyak's theorem”, Indiana Univ. Math. J., 47:3 (1998), 1131–1145

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

[15] Martio O., Ryazanov V., Srebro U., Yakubov E., Moduli in Modern Mapping Theory, Springer Monographs in Mathematics, Springer, New York, 2009

[16] Rado T., Reichelderfer P. V., Continuous Transformations in Analysis, Springer-Verlag, Berlin etc., 1955

[17] Rajala K., “Mappings of finite distortion: the Rickman–Picard theorem for mappings of finite lower order”, J. Anal. Math., 94 (2004), 235–248 | DOI

[18] Rickman S., Vuorinen M., “On the order of quasiregular mappings”, Ann. Acad. Sci. Fenn. Ser. A I Math., 7:2 (1982), 221–231 | DOI

[19] Rickman S., Quasiregular Mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 26, Springer-Verlag, Berlin, 1993

[20] Sevost'yanov E. A., “The Väisälä inequality for mappings with finite length distortion”, Complex Variables Elliptic Equ., 55:1–3 (2010), 91–101 | DOI

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