An -dimensional space that admits a Poincaré inequality but has no manifold points

Authors:
Bruce Hanson and Juha Heinonen

Journal:
Proc. Amer. Math. Soc. **128** (2000), 3379-3390

MSC (1991):
Primary 43A85; Secondary 28A75

DOI:
https://doi.org/10.1090/S0002-9939-00-05453-8

Published electronically:
May 18, 2000

MathSciNet review:
1690990

Full-text PDF

Abstract | References | Similar Articles | Additional Information

For each integer we construct a compact, geodesic metric space which has topological dimension , is Ahlfors -regular, satisfies the Poincaré inequality, possesses as a unique tangent cone at almost every point, but has no manifold points.

**[BP]**M. Bourdon and H. Pajot, ``Poincaré inequalities and quasiconformal structures on the boundary of some hyperbolic buildings",*Proc. Amer. Math. Soc.***127**(1999), 2315-2324. MR**99j:30024****[C]**J. Cheeger, ``Differentiability of Lipschitz functions on metric measure spaces'',*GAFA, Geom. Funct. Anal*.**9**(1999), 428-517.**[DS]**G. David and S. Semmes, ``Fractured fractals and broken dreams; self-similar geometry through metric and measure", Oxford Lecture Series in Mathematics and its Applications 7, Clarendon Press, Oxford, 1997. MR**99h:28018****[GLP]**M. Gromov, ``Structures Métriques pour les Variétés Riemanniennes", (J. Lafontaine and P. Pansu, eds.), Cedic/Fernand Nathan, Paris, 1981. MR**85e:53051****[HaK 1]**P. Hajlasz and P. Koskela, ``Sobolev meets Poincaré",*C.R. Acad. Sci. Paris***320**(1995), 1211-1215. MR**96f:46062****[HaK 2]**P. Hajlasz and P. Koskela, ``Sobolev met Poincaré",*Mem. Amer. Math. Soc*. (to appear).**[HeK]**J. Heinonen and P. Koskela, ``Quasiconformal maps in metric spaces with controlled geometry",*Acta Math.***181**(1998), 1-61. MR**99j:30025****[L]**T.J. Laakso, ``Ahlfors -regular spaces with arbitrary admitting weak Poincaré inequality",*GAFA, Geom. Funct. Anal*. (to appear).**[S]**E.M. Stein, ``Singular Integrals and Differentiability Properties of Functions", Princeton University Press, Princeton, New Jersey, 1970. MR**44:7280**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
43A85,
28A75

Retrieve articles in all journals with MSC (1991): 43A85, 28A75

Additional Information

**Bruce Hanson**

Affiliation:
Department of Mathematics, St. Olaf College, Northfield, Minnesota 55057

Email:
hansonb@stolaf.edu

**Juha Heinonen**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

DOI:
https://doi.org/10.1090/S0002-9939-00-05453-8

Keywords:
Poincaré inequality,
Ahlfors $n$-regular,
manifold point.

Received by editor(s):
August 14, 1998

Received by editor(s) in revised form:
January 18, 1999

Published electronically:
May 18, 2000

Additional Notes:
The second author was supported by NSF grant DMS 96-22844

Communicated by:
Albert Baernstein II

Article copyright:
© Copyright 2000
American Mathematical Society