Inverse Limit Spaces Satisfying a Poincaré Inequality

Jeff Cheeger; Bruce Kleiner

Analysis and Geometry in Metric Spaces (2015)

  • Volume: 3, Issue: 1, page 15-39, electronic only
  • ISSN: 2299-3274

Abstract

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We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs which imply that the measured Gromov-Hausdorff limit (equivalently, the inverse limit) is a PI space i.e., it satisfies a doubling condition and a Poincaré inequality in the sense of Heinonen-Koskela [12]. The Poincaré inequality is actually of type (1, 1). We also give a systematic construction of examples for which our conditions are satisfied. Included are known examples of PI spaces, such as Laakso spaces, and a large class of new examples. As follows easily from [4], generically our examples have the property that they do not bilipschitz embed in any Banach space with Radon-Nikodym property. For Laakso spaces, thiswas noted in [4]. However according to [7] these spaces admit a bilipschitz embedding in L1. For Laakso spaces, this was announced in [5].

How to cite

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Jeff Cheeger, and Bruce Kleiner. "Inverse Limit Spaces Satisfying a Poincaré Inequality." Analysis and Geometry in Metric Spaces 3.1 (2015): 15-39, electronic only. <http://eudml.org/doc/268843>.

@article{JeffCheeger2015,
abstract = {We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs which imply that the measured Gromov-Hausdorff limit (equivalently, the inverse limit) is a PI space i.e., it satisfies a doubling condition and a Poincaré inequality in the sense of Heinonen-Koskela [12]. The Poincaré inequality is actually of type (1, 1). We also give a systematic construction of examples for which our conditions are satisfied. Included are known examples of PI spaces, such as Laakso spaces, and a large class of new examples. As follows easily from [4], generically our examples have the property that they do not bilipschitz embed in any Banach space with Radon-Nikodym property. For Laakso spaces, thiswas noted in [4]. However according to [7] these spaces admit a bilipschitz embedding in L1. For Laakso spaces, this was announced in [5].},
author = {Jeff Cheeger, Bruce Kleiner},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {Convergent inverse systems; metric measure graphs; PI space; convergent inverse systems; Poincaré inequality; Radon-Nikodým property},
language = {eng},
number = {1},
pages = {15-39, electronic only},
title = {Inverse Limit Spaces Satisfying a Poincaré Inequality},
url = {http://eudml.org/doc/268843},
volume = {3},
year = {2015},
}

TY - JOUR
AU - Jeff Cheeger
AU - Bruce Kleiner
TI - Inverse Limit Spaces Satisfying a Poincaré Inequality
JO - Analysis and Geometry in Metric Spaces
PY - 2015
VL - 3
IS - 1
SP - 15
EP - 39, electronic only
AB - We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs which imply that the measured Gromov-Hausdorff limit (equivalently, the inverse limit) is a PI space i.e., it satisfies a doubling condition and a Poincaré inequality in the sense of Heinonen-Koskela [12]. The Poincaré inequality is actually of type (1, 1). We also give a systematic construction of examples for which our conditions are satisfied. Included are known examples of PI spaces, such as Laakso spaces, and a large class of new examples. As follows easily from [4], generically our examples have the property that they do not bilipschitz embed in any Banach space with Radon-Nikodym property. For Laakso spaces, thiswas noted in [4]. However according to [7] these spaces admit a bilipschitz embedding in L1. For Laakso spaces, this was announced in [5].
LA - eng
KW - Convergent inverse systems; metric measure graphs; PI space; convergent inverse systems; Poincaré inequality; Radon-Nikodým property
UR - http://eudml.org/doc/268843
ER -

References

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