Abstract
We study the covering properties of mappings of bounded and exponentially integrable distortion on the unit ball. We extend the results of Eremenko (Proc Am Math Soc 128:557–560, 2000) by proving Bloch-type theorems for mappings of exponentially integrable distortion. In the case of mappings of bounded distortion, we formulate and prove Bloch’s theorem in its most natural form.
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Chen H. and Gauthier P.M. (2001). Bloch constants in several variables. Trans. Am. Math. Soc. 353(4): 1371–1386
Eremenko A. (2000). Bloch radius, normal families and quasiregular mappings. Proc. Am. Math. Soc. 128(2): 557–560
Herron D. and Koskela P. (2003). Mappings of finite distortion: gauge dimension of generalized quasicircles. Ill. J. Math. 47(4): 1243–1259
Iwaniec T., Koskela P. and Onninen J. (2001). Mappings of finite distortion: monotonicity and continuity. Invent. Math. 44(3): 507–531
Iwaniec, T., Martin, G.: Geometric function theory and nonlinear analysis. Oxford Mathematical Monographs (2001)
Kauhanen J., Koskela P. and Malý J. (2001). Mappings of finite distortion: discreteness and openness. Arch. Ration. Mech. Anal. 160: 135–151
Kauhanen J., Koskela P. and Malý J. (2001). Mappings of finite distortion: condition N. Mich. Math. J. 9(1): 169–181
Kauhanen J., Koskela P., Malý J., Onninen J. and Zhong X. (2003). Mappings of finite distortion: sharp Orlicz-conditions. Rev. Mat. Iberoam. 19(3): 857–872
Koskela P. and Onninen J. (2006). Mappings of finite distortion: capacity and modulus inequalities. J. Reine Angew. Math. 599: 1–26
Koskela, P., Onninen, J., Rajala, K.: Mappings of finite distortion: injectivity radius of a local homeomorphism. Future trends in Geometric Function Theory, pp. 169–174, Rep. Univ. Jyväskylä Dep. Math. Stat., 92, Univ. of Jyväskylä, 2003
Manfredi J. and Villamor E. (1998). An extension of Reshetnyak’s theorem. Indiana Univ. Math. J. 47(3): 1131–1145
Manfredi J. and Villamor E. (2001). Traces of monotone functions in weighted Sobolev spaces. Ill. J. Math. 45(2): 403–422
McAuley L.F. and Robinson E.E. (1983). On Newman’s theorem for finite-to-one open mappings on manifolds. Proc. Am. Math. Soc. 87(3): 561–566
Rajala K. (2004). A lower bound for the Bloch radius of K-quasiregular mappings. Proc. Am. Math. Soc. 132(9): 2593–2601
Rajala K. (2004). Mappings of finite distortion: the Rickman-Picard theorem for mappings of finite lower order. J. Anal. Math. 94: 235–248
Reshetnyak, Yu.G.: Space mappings with bounded distortion, Transactions of Mathematical Monographs. Am. Math. Soc., vol. 73 (1989)
Rickman S. (1985). The analogue of Picard’s theorem for quasiregular mappings in dimension three. Acta Math. 154(3–4): 195–242
Rickman S. (1993). Quasiregular mappings. Springer, Heidelberg
Väisälä J. (1971). Lectures on n-dimensional quasiconformal mappings. Springer, Heidelberg
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Research supported by the Academy of Finland, and by NSF grant DMS 0244421. Part of this research was done when the author was visiting at the University of Michigan. He wishes to thank the department for hospitality.
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Rajala, K. Bloch’s theorem for mappings of bounded and finite distortion. Math. Ann. 339, 445–460 (2007). https://doi.org/10.1007/s00208-007-0124-0
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DOI: https://doi.org/10.1007/s00208-007-0124-0