Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 905-917

QUASICONFORMAL MAPS WITH BILIPSCHITZ OR IDENTITY BOUNDARY VALUES IN BANACH SPACES

Yaxiang Li, Matti Vuorinen and Xiantao Wang

Central South University of Forestry and Technology, College of Science
Changsha, Hunan 410004, P.R. China; yaxiangli 'at' 163.com

University of Turku, Department of Mathematics and Statistics
FIN-20014 Turku, Finland; vuorinen 'at' utu.fi

Hunan Normal University, Department of Mathematics
Changsha, Hunan 410081, P.R. China; xtwang 'at' hunnu.edu.cn

Abstract. Suppose that E and E' denote real Banach spaces with dimension at least 2 and that D \varsubsetneq E and D' \varsubsetneq E' are uniform domains with homogeneously dense boundaries. We consider the class of all φ-FQC (freely φ-quasiconformal) maps of D onto D' with bilipschitz boundary values. We show that the maps of this class are η-quasisymmetric. As an application, we show that if D is bounded, then maps of this class satisfy a two sided Hölder condition. Moreover, replacing the class φ-FQC by the smaller class of M-QH maps, we show that M-QH maps with bilipschitz boundary values are bilipschitz. Finally, we show that if f is a φ-FQC map which maps D onto itself with identity boundary values, then there is a constant C, depending only on the function φ, such that for all x ∈ D, the quasihyperbolic distance satisfies k_D(x,f(x)) ≤ C.

2010 Mathematics Subject Classification: Primary 30C65, 30L10, 30F45; Secondary 30C20.

Key words: Uniform domain, FQC map, quasisymmetric, bilipschitz boundary values, Hölder condition.

Reference to this article: Y. Li, M. Vuorinen and X. Wang: Quasiconformal maps with bilipschitz or identity boundary values in Banach spaces. Ann. Acad. Sci. Fenn. Math. 39 (2014), 905-917.

Full document as PDF file

doi:10.5186/aasfm.2014.3954