Abstract
Let D and D′ be domains in real Banach spaces of dimension at least 2. The main aim of this paper is to study certain arc distortion properties in the quasihyperbolic metric defined in real Banach spaces. In particular, when D′ is a QH inner ψ-uniform domain with ψ being a slow (or a convex domain), we investigate the following: For positive constants c,h,C,M, suppose a homeomorphism f: D → D′ takes each of the 10-neargeodesics in D to (c, h)-solid in D′. Then f is C-coarsely M-Lipschitz in the quasihyperbolic metric. These are generalizations of the corresponding result obtained recently by Väisälä.
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Supported by National Natural Science Foundation of China (Grant No. 11071063), Tianyuan Foundation (Grant No. 10926068) and Scientific Research Fund of Hunan Provincial Education Department (Grant No. 09C635)
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Huang, M.Z., Wang, X.T. The arc distortion in QH inner ψ-uniform (or convex) domains in real Banach spaces. Acta. Math. Sin.-English Ser. 27, 2039–2050 (2011). https://doi.org/10.1007/s10114-011-8672-3
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DOI: https://doi.org/10.1007/s10114-011-8672-3
Keywords
- Uniform domain
- QH ψ-uniform domain
- inner uniform domain
- QH inner ψ-uniform domain
- convex domain
- quasihyperbolic geodesic
- neargeodesic
- quasiconvexity
- real Banach space