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Conformal Geometry and Dynamics

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Uniform spaces and weak slice spaces

Authors: Stephen M. Buckley and David A. Herron
Journal: Conform. Geom. Dyn. 11 (2007), 191-206
MSC (2000): Primary 30C65; Secondary 51F99
Published electronically: September 24, 2007
MathSciNet review: 2346216
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Abstract: We characterize uniform spaces in terms of a slice condition. We also establish the Gehring-Osgood-Väisälä theorem for uniformity in the metric space context.

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  • [BHK01] M. Bonk, J. Heinonen, and P. Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque. 270 (2001), 1-99. MR 1829896 (2003b:30024)
  • [Buc03] S. Buckley, Slice conditions and their applications, Future Trends In Geometric Function Theory (Univ. Jyväskylä), vol. 92, Rep. Univ. Jyväskylä Dept. Math. Stat., 2003, RNC Workshop held in Jyväskylä, June 15-18, 2003, pp. 63-76. MR 2058111 (2005c:30048)
  • [Buc04] S.M. Buckley, Quasiconfomal images of Hölder domains, Ann. Acad. Sci. Fenn. Ser. Math. 29 (2004), 21-42. MR 2041697 (2004m:30030)
  • [BH07] S. Buckley and D.A. Herron, Uniform domains and capacity, Israel J. Math 158 (2007), 129-157.
  • [BO99] S.M. Buckley and J. O'Shea, Weighted Trudinger-type inequalities, Indiana Univ. Math. J. 48 (1999), 85-114. MR 1722194 (2001a:46031)
  • [BS01] S.M. Buckley and A. Stanoyevitch, Weak slice conditions, product domains, and quasiconformal mappings, Rev. Mat. Iberoam. 17 (2001), 1-37. MR 1900897 (2003b:30025)
  • [BS03] -, Distinguishing properties of weak slice conditions, Conform. Geom. Dyn. 7 (2003), 49-75. MR 1992037 (2004f:30014)
  • [CGN00] L. Capogna, N. Garofalo, and D-M. Nhieu, Examples of uniform and NTA domains in Carnot groups, Proceedings on Analysis and Geometry (Novosibirsk), Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., 2000, (Novosibirsk Akad., 1999), pp. 103-121. MR 1847513 (2002k:30037)
  • [CT95] L. Capogna and P. Tang, Uniform domains and quasiconformal mappings on the Heisenberg group, Manuscripta Math. 86 (1995), no. 3, 267-281. MR 1323792 (96f:30019)
  • [Geh87] F.W. Gehring, Uniform domains and the ubiquitous quasidisk, Jahresber. Deutsch. Math.-Verein 89 (1987), 88-103. MR 880189 (88j:30042)
  • [GM85] F.W. Gehring and O. Martio, Quasiextremal distance domains and extension of quasiconformal mappings, J. Analyse Math. 45 (1985), 181-206. MR 833411 (87j:30043)
  • [GO79] F.W. Gehring and B.G. Osgood, Uniform domains and the quasi-hyperbolic metric, J. Analyse Math. 36 (1979), 50-74. MR 581801 (81k:30023)
  • [GP76] F.W. Gehring and B.P. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172-199. MR 0437753 (55:10676)
  • [Gre01] A.V. Greshnov, On uniform and NTA-domains on Carnot groups, Sibirsk. Mat. Zh. 42 (2001), no. 5, 1018-1035. MR 1861631 (2002h:53048)
  • [HK98] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta. Math. 181 (1998), 1-61. MR 1654771 (99j:30025)
  • [Joh61] F. John, Rotation and strain, Comm. Pure Appl. Math. 14 (1961), 391-413. MR 0138225 (25:1672)
  • [Kos99] P. Koskela, Removable sets for Sobolev spaces, Ark. Mat. 37 (1999), 291-304. MR 1714767 (2001g:46077)
  • [Mar80] O. Martio, Definitions for uniform domains, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 197-205. MR 595191 (82c:30028)
  • [MS79] O. Martio and J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1978/79), no. 2, 383-401. MR 565886 (81i:30039)
  • [Väi88] J. Väisälä, Uniform domains, Tôhoku Math. J. 40 (1988), 101-118. MR 927080 (89d:30027)
  • [Väi91] -, Free quasiconformality in Banach spaces II, Ann. Acad. Sci. Fenn. Math. 16 (1991), 255-310. MR 1139798 (94c:30028)
  • [Väi94] -, Exhaustions of John domains, Ann. Acad. Sci. Fenn. Math. 19 (1994), 47-57. MR 1246886 (94i:30024)
  • [Väi98] -, Relatively and inner uniform domains, Conformal Geometry and Dynamics 2 (1998), 56-88. MR 1637079 (99e:30014)
  • [Väi99] -, The free quasiworld. Freely quasiconformal and related maps in Banach spaces., Quasiconformal Geometry and Dynamics (Lublin, 1996) (Warsaw), vol. 48, Inst. Math., Polish of Academy Sciences, Banach Center Publ., 1999, pp. 55-118.

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Additional Information

Stephen M. Buckley
Affiliation: Department of Mathematics, National University of Ireland, Maynooth, Co. Kildare, Ireland

David A. Herron
Affiliation: Department of Mathematics, University of Cincinnati, Cincinnati, Ohio 45221

Keywords: Uniform spaces, quasihyperbolic metric, slice conditions
Received by editor(s): January 9, 2007
Published electronically: September 24, 2007
Additional Notes: The first author was supported in part by Enterprise Ireland and Science Foundation Ireland. Both authors were supported by the Charles Phelps Taft Memorial Fund.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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