×
Huang, M.; Li, Y.

On bilipschitz extensions in real Banach spaces. (English) Zbl 1272.30039

Abstr. Appl. Anal. 2013, Article ID 765685, 9 p. (2013).
Summary: Suppose that \(E\) and \(E'\) denote real Banach spaces with dimension at least 2, that \(D \neq E\) and \(D' \neq E'\) are bounded domains with connected boundaries, that \(f : D \to D'\) is an \(M\)-QH homeomorphism, and that \(D'\) is uniform. The main aim of this paper is to prove that \(f\) extends to a homeomorphism \(\bar{f} : \bar{D} \to \bar{D}'\) and \(\bar{f}|_{\partial D}\) is bilipschitz if and only if \(f\) is bilipschitz in \(\bar{D}\). The answer to some open problems of Väisälä is affirmative under a natural additional condition.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
Cite Review PDF
Full Text: DOI arXiv
OA License

References:

[1] Väisälä, J., The free quasiworld, freely quasiconformal and related maps in Banach spaces, Banach Center Publications, 48, 55-118 (1999) · Zbl 0934.30018
[2] Klén, R.; Rasila, A.; Talponen, J.; Minda, D.; Ponnusamy, S.; Shanmugalingam, N., Quasihyperbolic geometry in euclidean and Banach spaces, Proceedings of the ICM2010 Satellite Conference International Workshop on Harmonic and Quasiconformal Mappings (HMQ ’10) · Zbl 1237.30008
[3] Rasila, A.; Talponen, J., Convexity properties of quasihyperbolic balls on Banach, Annales Academiæ Scientiarum Fennicæ, 37, 1, 215-228 (2012) · Zbl 1316.30023
[4] Väisälä, J., Free quasiconformality in Banach spaces. I, Annales Academiae Scientiarum Fennicae, 15, 2, 355-379 (1990) · Zbl 0696.30022
[5] Väisälä, J., Free quasiconformality in Banach spaces. II, Annales Academiae Scientiarum Fennicae, 16, 2, 255-310 (1991) · Zbl 0761.30014
[6] Tukia, P.; Väisälä, J., Quasisymmetric embeddings of metric spaces, Suomalaisen Tiedeakatemian Toimituksia, 5, 1, 97-114 (1980) · Zbl 0403.54005
[7] Väisälä, J., Quasimöbius maps, Journal d’Analyse Mathematique, 44, 218-234 (1985) · Zbl 0593.30022
[8] Väisälä, J., Uniform domains, The Tohoku Mathematical Journal, 40, 1, 101-118 (1988) · Zbl 0627.30017 · doi:10.2748/tmj/1178228081
[9] Martio, O., Definitions for uniform domains, Suomalaisen Tiedeakatemian Toimituksia, 5, 1, 197-205 (1980) · Zbl 0469.30017
[10] Ahlfors, L. V., Quasiconformal reflections, Acta Mathematica, 109, 291-301 (1963) · Zbl 0121.06403
[11] Gehring, F. W., Injectivity of local quasi-isometries, Commentarii Mathematici Helvetici, 57, 2, 202-220 (1982) · Zbl 0528.30010 · doi:10.1007/BF02565857
[12] Tukia, P.; Väisälä, J., Bilipschitz extensions of maps having quasiconformal extensions, Mathematische Annalen, 269, 651-572 (1984) · Zbl 0533.30021
[13] Gehring, F. W., Extension of quasi-isometric embeddings of Jordan curves, Complex Variables, 5, 2-4, 245-263 (1986) · Zbl 0603.30024
[14] Gehring, F. W.; Palka, B. P., Quasiconformally homogeneous domains, Journal d’Analyse Mathématique, 30, 172-199 (1976) · Zbl 0349.30019
[15] Väisälä, J., Quasihyperbolic geodesics in convex domains, Results in Mathematics, 48, 1-2, 184-195 (2005) · Zbl 1093.30018
[16] Hästö, P.; Ibragimov, Z.; Minda, D.; Ponnusamy, S.; Sahoo, S., Isometries of some hyperbolic-type path metrics, and the hyperbolic medial axis, The Tradition of Ahlfors-Bers. The Tradition of Ahlfors-Bers, Contemporary Mathematics, 432, 63-74 (2007), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 1147.30029 · doi:10.1090/conm/432/08300
[17] Gehring, F. W.; Osgood, B. G., Uniform domains and the quasihyperbolic metric, Journal d’Analyse Mathématique, 36, 50-74 (1979) · Zbl 0449.30012 · doi:10.1007/BF02798768
[18] Vuorinen, M., Conformal invariants and quasiregular mappings, Journal d’Analyse Mathématique, 45, 69-115 (1985) · Zbl 0601.30025 · doi:10.1007/BF02792546
[19] Väisälä, J., Relatively and inner uniform domains, Conformal Geometry and Dynamics, 2, 56-88 (1998) · Zbl 0902.30017 · doi:10.1090/S1088-4173-98-00022-8
[20] Huang, M.; Li, Y.; Vuorinen, M.; Wang, X., On quasimöbius maps in real Banach spaces · Zbl 1285.30007
[21] Li, Y.; Wang, X., Unions of John domains and uniform domains in real normed vector spaces, Annales Academiæ Scientiarum Fennicæ, 35, 2, 627-632 (2010) · Zbl 1208.30026 · doi:10.5186/aasfm.2010.3539
[22] Huang, M.; Li, Y., Decomposition properties of John domains in normed vector spaces, Journal of Mathematical Analysis and Applications, 388, 1, 191-197 (2012) · Zbl 1244.30039 · doi:10.1016/j.jmaa.2011.10.038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.