Relatively and inner uniform domains

Author:
Jussi Väisälä

Journal:
Conform. Geom. Dyn. **2** (1998), 56-88

MSC (1991):
Primary 30C65

DOI:
https://doi.org/10.1090/S1088-4173-98-00022-8

Published electronically:
August 19, 1998

MathSciNet review:
1637079

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We generalize the concept of a uniform domain in Banach spaces into two directions. (1) The ordinary metric of a domain is replaced by a metric , 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 , where is a Banach space and is a ball.

**[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 -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).

Retrieve articles in *Conformal Geometry and Dynamics of the American Mathematical Society*
with MSC (1991):
30C65

Retrieve articles in all journals with MSC (1991): 30C65

Additional Information

**Jussi Väisälä**

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

Email:
jvaisala@cc.helsinki.fi

DOI:
https://doi.org/10.1090/S1088-4173-98-00022-8

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