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Conformal Geometry and Dynamics

ISSN 1088-4173



Relatively and inner uniform domains

Author: Jussi Väisälä
Journal: Conform. Geom. Dyn. 2 (1998), 56-88
MSC (1991): Primary 30C65
Published electronically: August 19, 1998
MathSciNet review: 1637079
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Abstract: We generalize the concept of a uniform domain in Banach spaces into two directions. (1) The ordinary metric $d$ of a domain is replaced by a metric $e\ge d$, in particular, by the inner metric of the domain. (2) The uniformity condition is supposed to hold only for certain pairs of points of the domain. We consider neargeodesics and solid arcs in these domains. Applications to the boundary behavior of quasiconformal maps are given. In particular, we study maps between domains of the form $E\times B$, where $E$ is a Banach space and $B$ is a ball.

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  • [Al] P. Alestalo, Quasisymmetry in product spaces and uniform domains, Licentiate's thesis, University of Helsinki, 1991 (Finnish).
  • [BV1] Z. Balogh and A. Volberg, Geometric localization, uniformly John property and separated semihyperbolic dynamics, Ark. Mat. 34, 1996, 21-49. MR 97i:30033
  • [BV2] -, Boundary Harnack principle for separated semihyperbolic repellers, harmonic measure applications, Rev. Mat. Iberoamericana 12, 1996, 299-336. MR 97m:31001
  • [BHK] M. Bonk, J. Heinonen and P. Koskela, Uniformizing Gromov hyperbolic spaces (in preparation).
  • [FHM] J.L. Fernández, J. Heinonen and O. Martio, Quasilines and conformal mappings, J. Analyse Math. 52, 1989, 117-132. MR 90a:30017
  • [GO] F.W. Gehring and B.G. Osgood, Uniform domains and the quasihyperbolic metric, J. Analyse Math. 36, 1979, 50-74. MR 81k:30023
  • [GP] F.W. Gehring and B.P. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30, 1976, 172-199. MR 55:10676
  • [Jo1] P.W. Jones, Extension theorems for BMO, Indiana Univ.Math. J. 29, 1980, 41-66. MR 81b:42047
  • [Jo2] -, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147, 1981, 71-88. MR 83i:30014
  • [Ma] O. Martio, Definitions for uniform domains, Ann. Acad.Sci. Fenn. Math. 5, 1980, 197-205. MR 82c:30028
  • [MaS] O. Martio and J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Math. 4, 1979, 383-401. MR 81i:30039
  • [MiS] V. D. Milman and G. Schechtman, Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics 1200, Springer-Verlag, 1986. MR 87m:46038
  • [Sc] J.J. Schäffer, Geometry of spheres in normed spaces, Marcel Dekker, 1976. MR 57:7120
  • [Th] W. P. Thurston, The geometry and topology of three-manifolds, Mimeographed notes, Princeton University, 1980.
  • [Vä1] J. Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics 229, Springer-Verlag, 1971. MR 56:12260
  • [Vä2] -, Quasimöbius maps, J. Analyse Math. 44, 1984/85, 218-234. MR 87f:30059
  • [Vä3] -, Uniform domains, Tôhoku Math. J. 40, 1988, 101-118. MR 89d:30027
  • [Vä4] -, Quasiconformal maps of cylindrical domains, Acta Math. 162, 1989, 201-225. MR 90f:30034
  • [Vä5] -, Free quasiconformality in Banach spaces. I, Ann. Acad. Sci. Fenn. Math. 15, 1990, 355-379. MR 92d:30012
  • [Vä6] -, Free quasiconformality in Banach spaces. II, Ann. Acad. Sci. Fenn. Math. 16, 1991, 255-310. MR 94c:30028
  • [Vä7] -, Free quasiconformality in Banach spaces. III, Ann. Acad. Sci. Fenn. Math. 17, 1992, 393-408. MR 94c:30029
  • [Vä8] -, Free quasiconformality in Banach spaces IV, Analysis and Topology, ed. by C. Andreian Cazacu et al., World Scientific (to appear).
  • [Vä9] -, The free quasiworld, Proceedings of the fifth Finnish-Polish-Ukranian summer school in complex analysis in Lublin 1996 (to appear).

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Additional Information

Jussi Väisälä
Affiliation: Matematiikan laitos, Helsingin yliopisto, PL 4, Yliopistonkatu 5, 00014 Helsinki, Finland

Received by editor(s): September 18, 1997
Received by editor(s) in revised form: April 14, 1998
Published electronically: August 19, 1998
Article copyright: © Copyright 1998 American Mathematical Society

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