Capacity densities and angular limits of quasiregular mappings
HTML articles powered by AMS MathViewer
- by Matti Vuorinen PDF
- Trans. Amer. Math. Soc. 263 (1981), 343-354 Request permission
Abstract:
It is shown that if a bounded quasiregular mapping of the unit ball ${B^n} \subset {R^n}$, $n \geqslant 2$, has a limit at $b \in \partial {B^n}$ through a set $E \subset {B^n}$ with $b \in \bar E$, then it has an angular limit at $b$ provided that $E$ is contained in an open cone $C \subset {B^n}$ with vertex $b$ and that $E$ is thick enough at $b$. The thickness condition is expressed in terms of the $n$-capacity density.References
- F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. Analyse Math. 36 (1979), 50–74 (1980). MR 581801, DOI 10.1007/BF02798768
- F. W. Gehring and B. P. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172–199. MR 437753, DOI 10.1007/BF02786713
- N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027
- O. Martio, Equicontinuity theorem with an application to variational integrals, Duke Math. J. 42 (1975), no. 3, 569–581. MR 380599
- O. Martio, Capacity and measure densities, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), no. 1, 109–118. MR 538093, DOI 10.5186/aasfm.1978-79.0412
- O. Martio and S. Rickman, Boundary behavior of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A. I. 507 (1972), 17. MR 379846
- O. Martio, S. Rickman, and J. Väisälä, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 448 (1969), 40. MR 0259114
- O. Martio, S. Rickman, and J. Väisälä, Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 465 (1970), 13. MR 0267093
- O. Martio and J. Sarvas, Density conditions in the $n$-capacity, Indiana Univ. Math. J. 26 (1977), no. 4, 761–776. MR 477038, DOI 10.1512/iumj.1977.26.26059
- Raimo Näkki, Extension of Loewner’s capacity theorem, Trans. Amer. Math. Soc. 180 (1973), 229–236. MR 328062, DOI 10.1090/S0002-9947-1973-0328062-9 M. Ohtsuka, Dirichlet problem, extremal length, and prime ends, Van Nostrand Reinhold, New York, 1970.
- Seppo Rickman, On the number of omitted values of entire quasiregular mappings, J. Analyse Math. 37 (1980), 100–117. MR 583633, DOI 10.1007/BF02797681
- Seppo Rickman, Asymptotic values and angular limits of quasiregular mappings of a ball, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 185–196. MR 595190, DOI 10.5186/aasfm.1980.0523 J. Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Math., vol. 229, Springer-Verlag, Berlin, New York and Heidelberg, 1971. —, A survey of quasiregular maps in ${R^n}$, Proc. 1978 Internat. Congr. Math. (Helsinki, Finland), Finnish Academy of Science and Letters, Helsinki, 1980.
- Matti Vuorinen, Exceptional sets and boundary behavior of quasiregular mappings in $n$-space, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 11 (1976), 44. MR 437757 —, On the existence of angular limits of $n$-dimensional quasiconformal mappings, Ark. Mat. (to appear).
- Matti Vuorinen, On the boundary behavior of locally $K$-quasiconformal mappings in space, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 79–95. MR 595179, DOI 10.5186/aasfm.1980.0532 —, Lindelöf-type theorems for quasiconformal and quasiregular mappings, Proc. Complex Analysis Semester, Banach Center, Warsaw, Poland, 1979 (to appear).
- William P. Ziemer, Extremal length and $p$-capacity, Michigan Math. J. 16 (1969), 43–51. MR 247077
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 263 (1981), 343-354
- MSC: Primary 30C60
- DOI: https://doi.org/10.1090/S0002-9947-1981-0594412-6
- MathSciNet review: 594412