Abstract
We give sufficient conditions for mappings defined on the unit ball of ℝn to have radial limits almost everywhere. In particular, we show that if f:B(0,1)→ℝn is a mapping with exponentially integrable distortion satisfying the growth condition
for some a∈[0,n−1), then 
. Here the set E(f) consists of those points in ∂B(0,1) where f does not have radial limits. We also give an example which shows the difference between the classes of mappings of bounded distortion and certain integrable distortion in terms of radial limits.
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References
Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton Mathematical Series, vol. 48. Princeton University Press, Princeton (2009)
Gehring, F.W.: Rings and quasiconformal mappings in space. Trans. Am. Math. Soc. 103, 353–393 (1962)
Iwaniec, T., Koskela, P., Onninen, J.: Mappings of finite distortion: monotonicity and continuity. Invent. Math. 144(3), 507–531 (2001)
Iwaniec, T., Martin, G.: Geometric Function Theory and Non-linear Analysis. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, New York (2001)
Kauhanen, J.: Failure of the condition N below W 1,n. Ann. Acad. Sci. Fenn. Math. 27(1), 141–150 (2002)
Kauhanen, J., Koskela, P., Malý, J.: Mappings of finite distortion: discreteness and openness. Arch. Ration. Mech. Anal. 160(2), 135–151 (2001)
Koskela, P., Manfredi, J.J., Villamor, E.: Regularity theory and traces of
-harmonic functions. Trans. Am. Math. Soc. 348(2), 755–766 (1996)
Koskela, P., Nieminen, T.: Homeomorphisms of finite distortion: discrete length of radial images. Math. Proc. Camb. Philos. Soc. 144(1), 197–205 (2008)
Kovalev, L., Onninen, J.: Boundary values of mappings of finite distortion. In: Future Trends in Geometric Function Theory. Rep. Univ. Jyväskylä Dep. Math. Stat., vol. 92, pp. 175–182. Univ. Jyväskylä, Jyväskylä (2003)
Li, B.Q., Villamor, E.: Boundary limits for bounded quasiregular mappings. J. Geom. Anal. 19(3), 708–718 (2009)
Martio, O., Rickman, S.: Boundary behavior of quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A I No. 507 (1972), 17 pp.
Martio, O., Srebro, U.: Locally injective automorphic mappings in ℝn. Math. Scand. 85(1), 49–70 (1999)
Noshiro, K.: Cluster Sets. Ergebnisse der Mathematik und ihrer Grenzgebiete. N. F., Heft 28, Springer, Berlin (1960)
Rajala, K.: Radial limits of quasiregular local homeomorphisms. Am. J. Math. 130(1), 269–289 (2008)
Reshetnyak, Y.G.: Space Mappings with Bounded Distortion. Translations of Mathematical Monographs, vol. 73. American Mathematical Society, Providence (1989)
Rickman, S.: Quasiregular Mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 26. Springer, Berlin (1993). Engl. transl.: Results in Mathematics and Related Areas (3)
Vuorinen, M.: Conformal Geometry and Quasiregular Mappings. Lecture Notes in Mathematics, vol. 1319. Springer, Berlin (1988)
-harmonic functions. Trans. Am. Math. Soc. 348(2), 755–766 (1996)
Acknowledgements
The author would like to thank his advisor Kai Rajala for many discussions and comments concerning this manuscript. Part of this research was done while the author was visiting the Mathematics Department at University of Illinois at Urbana-Champaign. The author wishes to thank UIUC for the hospitality. The author was financially supported by the foundation of Vilho, Yrjö and Kalle Väisälä during the preparation of this manuscript.
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Communicated by Steven R. Bell.
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Äkkinen, T. Radial Limits of Mappings of Bounded and Finite Distortion. J Geom Anal 24, 1298–1322 (2014). https://doi.org/10.1007/s12220-012-9373-6
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DOI: https://doi.org/10.1007/s12220-012-9373-6