Inverse Limit Spaces Satisfying a Poincaré Inequality
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 . 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 , 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 . However according to  these spaces admit a bilipschitz embedding in L1. For Laakso spaces, this was announced in .
-  J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), no. 3, 428–517.
-  J. Cheeger and B. Kleiner, Generalized differentiation and bi-Lipschitz nonembedding in L1, C.R.A.S Paris (2006), no. 5, 297–301.
-  J. Cheeger and B Kleiner, On the differentiability of Lipschtz maps from metric measure spaces into Banach spaces, Inspired by S.S. Chern, A Memorial volume in honor of a greatmathematician, Nankai tracts inMathematics, vol. 11, World Scientific, Singapore, 2006, pp. 129–152.
-  J. Cheeger and B. Kleiner, Differentiation of Lipschitz maps from metric measure spaces to Banach spaces with the Radon- Nikodym Property, Geom. Funct. Anal. 19 (2009), no. 4, 1017–1028. [WoS]
-  , Differentiating maps to L1 and the geometry of BV functions, Ann. of Math. 171 (2010), no. 2, 1347–1385.
-  , Metric differentiation, monotonicity and maps into L1, Invent. Math. 182 (2010), no. 2, 355–370. [WoS]
-  , Realization of metric spaces as inverse limits, and bilipschitz embedding in L1, Geom. Funct. Anal. 23 (2013), no. 1, 1017–1028. [WoS]
-  J. Cheeger, B. Kleiner, and A. Naor, A (log n) (1) integrality gap for the Sparcest Cut SPD, Proceedings of 50th Annual IEEE on Foundations of Computer Science (FOCS 2009), 2009, pp. 555–564.
-  , Compression bounds for Lipschitz maps from the Heisenberg group to L1, Acta Math. 207 (2011), no. 2, 291–373. [WoS]
-  K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent.Math. 87 (1987), no. 3, 517–547.
-  P. Hajlasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. Providence (2000), no. 145.
-  J. Heinonen and P. Koskela, From local to global in quasiconformal structures, Proc. Nat. Acad. Sci. USA 93 (1996), 554–556.
-  S. Keith, Modulus and the Poincaré inequality on metric measure spaces, Math. Z. 245 (2003), no. 2, 255–292. [WoS]
-  S Keith and X. Zhong, The Poincaré inequality is an open ended condition, Ann. of Math. (2) 167 (2008), no. 2, 575–599.
-  T. Laakso, Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality, Geom. Funct. Anal. 10 (2000), no. 1, 111–123.
-  A. Schioppa, Poincaré inequalities for mutually singular measures, Anal. Geom. Metr. Spaces (3):40–45, 2015.
-  S. Semmes, Finding curves on general spaces through quantitative topology with applications for Sobolev and Poincaré inequalities, Selecta Math. (N.S.) 2 (1996) 155–295.[Crossref]