## Relatively and inner uniform domains

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- by Jussi Väisälä PDF
- Conform. Geom. Dyn.
**2**(1998), 56-88 Request permission

## 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.## References

- P. Alestalo, Quasisymmetry in product spaces and uniform domains, Licentiate’s thesis, University of Helsinki, 1991 (Finnish).
- Zoltan Balogh and Alexander Volberg,
*Geometric localization, uniformly John property and separated semihyperbolic dynamics*, Ark. Mat.**34**(1996), no. 1, 21–49. MR**1396621**, DOI 10.1007/BF02559505 - Z. Balogh and A. Volberg,
*Boundary Harnack principle for separated semihyperbolic repellers, harmonic measure applications*, Rev. Mat. Iberoamericana**12**(1996), no. 2, 299–336. MR**1402670**, DOI 10.4171/RMI/200 - M. Bonk, J. Heinonen and P. Koskela, Uniformizing Gromov hyperbolic spaces (in preparation).
- José L. Fernández, Juha Heinonen, and Olli Martio,
*Quasilines and conformal mappings*, J. Analyse Math.**52**(1989), 117–132. MR**981499**, DOI 10.1007/BF02820475 - F. W. Gehring and B. G. Osgood,
*Uniform domains and the quasihyperbolic metric*, J. Analyse Math.**36**(1979), 50–74 (1980). MR**581801**, DOI 10.1007/BF02798768 - F. W. Gehring and B. P. Palka,
*Quasiconformally homogeneous domains*, J. Analyse Math.**30**(1976), 172–199. MR**437753**, DOI 10.1007/BF02786713 - Peter W. Jones,
*Extension theorems for BMO*, Indiana Univ. Math. J.**29**(1980), no. 1, 41–66. MR**554817**, DOI 10.1512/iumj.1980.29.29005 - Peter W. Jones,
*Quasiconformal mappings and extendability of functions in Sobolev spaces*, Acta Math.**147**(1981), no. 1-2, 71–88. MR**631089**, DOI 10.1007/BF02392869 - O. Martio,
*Definitions for uniform domains*, Ann. Acad. Sci. Fenn. Ser. A I Math.**5**(1980), no. 1, 197–205. MR**595191**, DOI 10.5186/aasfm.1980.0517 - O. Martio and J. Sarvas,
*Injectivity theorems in plane and space*, Ann. Acad. Sci. Fenn. Ser. A I Math.**4**(1979), no. 2, 383–401. MR**565886**, DOI 10.5186/aasfm.1978-79.0413 - Vitali D. Milman and Gideon Schechtman,
*Asymptotic theory of finite-dimensional normed spaces*, Lecture Notes in Mathematics, vol. 1200, Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov. MR**856576** - Juan Jorge Schäffer,
*Geometry of spheres in normed spaces*, Lecture Notes in Pure and Applied Mathematics, No. 20, Marcel Dekker, Inc., New York-Basel, 1976. MR**0467256** - W. P. Thurston, The geometry and topology of three-manifolds, Mimeographed notes, Princeton University, 1980.
- Jussi Väisälä,
*Lectures on $n$-dimensional quasiconformal mappings*, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR**0454009**, DOI 10.1007/BFb0061216 - Jussi Väisälä,
*Quasi-Möbius maps*, J. Analyse Math.**44**(1984/85), 218–234. MR**801295**, DOI 10.1007/BF02790198 - Jussi Väisälä,
*Uniform domains*, Tohoku Math. J. (2)**40**(1988), no. 1, 101–118. MR**927080**, DOI 10.2748/tmj/1178228081 - Jussi Väisälä,
*Quasiconformal maps of cylindrical domains*, Acta Math.**162**(1989), no. 3-4, 201–225. MR**989396**, DOI 10.1007/BF02392837 - Jussi Väisälä,
*Free quasiconformality in Banach spaces. I*, Ann. Acad. Sci. Fenn. Ser. A I Math.**15**(1990), no. 2, 355–379. MR**1087342**, DOI 10.5186/aasfm.1990.1527 - Jussi Väisälä,
*Free quasiconformality in Banach spaces. II*, Ann. Acad. Sci. Fenn. Ser. A I Math.**16**(1991), no. 2, 255–310. MR**1139798**, DOI 10.5186/aasfm.1991.1629 - Jussi Väisälä,
*Free quasiconformality in Banach spaces. III*, Ann. Acad. Sci. Fenn. Ser. A I Math.**17**(1992), no. 2, 393–408. MR**1190331**, DOI 10.5186/aasfm.1992.1732 - —, Free quasiconformality in Banach spaces IV, Analysis and Topology, ed. by C. Andreian Cazacu et al., World Scientific (to appear).
- —, The free quasiworld, Proceedings of the fifth Finnish-Polish-Ukranian summer school in complex analysis in Lublin 1996 (to appear).

## Additional Information

**Jussi Väisälä**- Affiliation: Matematiikan laitos, Helsingin yliopisto, PL 4, Yliopistonkatu 5, 00014 Helsinki, Finland
- Email: jvaisala@cc.helsinki.fi
- Received by editor(s): September 18, 1997
- Received by editor(s) in revised form: April 14, 1998
- Published electronically: August 19, 1998
- © Copyright 1998 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**2**(1998), 56-88 - MSC (1991): Primary 30C65
- DOI: https://doi.org/10.1090/S1088-4173-98-00022-8
- MathSciNet review: 1637079