Characterizing spaces satisfying Poincaré Inequalities and applications to differentiability

  • Sylvester Eriksson-BiqueEmail author


We characterize complete RNP-differentiability spaces as those spaces which are rectifiable in terms of doubling metric measure spaces satisfying some local (1, p)-Poincaré inequalities. This gives a full characterization of spaces admitting a strong form of a differentiability structure in the sense of Cheeger, and provides a partial converse to his theorem. The proof is based on a new “thickening” construction, which can be used to enlarge subsets into spaces admitting Poincaré inequalities. We also introduce a new notion of quantitative connectivity which characterizes spaces satisfying local Poincaré inequalities. This characterization is of independent interest, and has several applications separate from differentiability spaces. We resolve a question of Tapio Rajala on the existence of Poincaré inequalities for the class of MCP(K, n)-spaces which satisfy a weak Ricci-bound. We show that deforming a geodesic metric measure space by Muckenhoupt weights preserves the property of possessing a Poincaré inequality. Finally, the new condition allows us to show that many classes of weak, Orlicz and non-homogeneous Poincaré inequalities “self-improve” to classical (1, q)-Poincaré inequalities for some \({q \in [1,\infty)}\), which is related to Keith’s and Zhong’s theorem on self-improvement of Poincaré inequalities.


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The author is thankful to professor Bruce Kleiner for suggesting the problem on the local geometry of Lipschitz differentiability spaces, for numerous helpful discussions on the topic and several comments that improved the exposition of this paper. Kleiner was instrumental in restructuring the proofs in the third and fifth sections and thus helped greatly simplify the presentation. The author also thanks a number of people who have given useful comments in the process of writing this paper, such as Sirkka-Liisa Eriksson, Jana Björn, Nagesvari Shanmugalingam, Pekka Koskela, Guy C. David and Ranaan Schul. Some results of the paper were heavily influenced by conversations with Tatiana Toro and Jeff Cheeger. An earlier draft of this paper had a more complicated construction used to resolve Theorem 1.15. This construction is here rephrased in terms of a modified hyperbolic filling which is much clearer than the earlier version. This modification was encouraged by Bruce Kleiner, and suggested to the author by Daniel Meyer. We also thank the anonymous referees for numerous comments and corrections. The research was supported by a NSF graduate student fellowship DGE-1342536 and NSF Grant DMS-1405899.


  1. AGMR15.
    Ambrosio, L., Gigli, N., Mondino, A., Rajala, T.: Riemannian Ricci curvature lower bounds in metric measure spaces with \(\sigma \)-finite measure. Transactions of the American Mathematical Society 367(7), 4661–4701 (2015)MathSciNetzbMATHGoogle Scholar
  2. AGS08.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows: in Metric Spaces and in the Space of Probability Measures. Lectures in math, Springer, Berlin (2008)zbMATHGoogle Scholar
  3. AGS14.
    Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Mathematical Journal 163(7), 1405–1490 (2014)MathSciNetzbMATHGoogle Scholar
  4. And16.
    S. Andrea. An example of a differentiability space which is PI-unrectifiable, arXiv preprint arXiv:1611.01615 (2016)
  5. BS10.
    Bacher, K., Sturm, K.-T.: Localization and tensorization properties of the curvature-dimension condition for metric measure spaces. Journal of Functional Analysis 259(1), 28–56 (2010)MathSciNetzbMATHGoogle Scholar
  6. Bat15.
    Bate, D.: Structure of measures in Lipschitz differentiability spaces. Journal of the American Mathematical Society 28(2), 421–482 (2015)MathSciNetzbMATHGoogle Scholar
  7. BL14.
    D. Bate and S. Li. Characterizations of rectifiable metric measure spaces. Ann. Sci. Éc. Norm. Supér. (4), 50(1) (2017), 1–37Google Scholar
  8. BL18.
    Bate, D., Li, S.: Differentiability and Poincaré-type inequalities in metric measure spaces. Adv. Math. 333, 868–930 (2018)MathSciNetzbMATHGoogle Scholar
  9. BS13.
    D. Bate and G. Speight. Differentiability, porosity and doubling in metric measure spaces. Proceedings of the American Mathematical Society, (3)141 (2013), 971–985Google Scholar
  10. BB18.
    Björn, A., Björn, J.: Local and semilocal Poincaré inequalities on metric spaces. Journal de Mathématiques Pures et Appliquées (2018)Google Scholar
  11. Bjö10.
    Björn, J.: Orlicz-Poincaré inequalities, maximal functions and \(A_\Phi \)-conditions. Proceedings of the Royal Society of Edinburgh: Section A Mathematics 140(1), 3148 (2010)Google Scholar
  12. Bon06.
    M. Bonk. Quasiconformal geometry of fractals, International Congress of Mathematicians, vol. 2, Citeseer, pp. 1349–1373. (2006)Google Scholar
  13. BK02.
    Bonk, M., Kleiner, B.: Quasisymmetric parametrizations of two-dimensional metric spheres. Inventiones mathematicae 150(1), 127–183 (2002)MathSciNetzbMATHGoogle Scholar
  14. BK05.
    Bonk, M., Kleiner, B.: Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary. Geometry & Topology 9(1), 219–246 (2005)MathSciNetzbMATHGoogle Scholar
  15. BS14.
    M. Bonk and E. Saksman. Sobolev spaces and hyperbolic fillings. Journal für die reine und angewandte Mathematik (Crelle's Journal) (2014)Google Scholar
  16. BK13.
    Bourdon, M., Kleiner, B.: Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups. Groups Geom. Dyn. 7, 39–107 (2013)MathSciNetzbMATHGoogle Scholar
  17. BP99.
    Bourdon, M., Pajot, H.: Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings. Proceedings of the American Mathematical Society 127(8), 2315–2324 (1999)MathSciNetzbMATHGoogle Scholar
  18. BP03.
    M. Bourdon and H. Pajot. Cohomologie \(\ell _p\) espaces de Besov. Journal für die reine und angewandte Mathematik (Crelle's Journal), 558 (2003), 85–108Google Scholar
  19. Che99.
    Cheeger, J.: Differentiability of Lipschitz Functions on Metric Measure Spaces. Geometric & Functional Analysis (GAFA) 9(3), 428–517 (1999)MathSciNetzbMATHGoogle Scholar
  20. CC95.
    J. Cheeger and T.H. Colding. Almost rigidity of warped products and the structure of spaces with Ricci curvature bounded below. C. R. Acad. Sci., Paris, Sér. I, 320(3) (1995), 353–357Google Scholar
  21. CC97.
    Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below \(I\). Journal of Differential Geometry 406–480 (1997)Google Scholar
  22. CK08.
    J. Cheeger and B. Kleiner. Characterization of the Radon-Nikodým property in terms of inverse limits. Géométrie différentielle, physique mathématique, mathématiques et société. I, Astérisque, (2008), no. 321, 129–138Google Scholar
  23. CK09.
    Cheeger, J., Kleiner, B.: Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon-Nikodym property. Geometric & Functional Analysis (GAFA) 19(4), 1017–1028 (2009)MathSciNetzbMATHGoogle Scholar
  24. CK13.
    Cheeger, J., Kleiner, B.: Realization of metric spaces as inverse limits, and bilipschitz embedding in \(L^1\). Geometric & Functional Analysis (GAFA) 23(1), 96–133 (2013)zbMATHGoogle Scholar
  25. CKS16.
    Cheeger, J., Kleiner, B., Schioppa, A.: Infinitesimal structure of differentiability spaces, and metric differentiation. Anal. Geom. Metr. Spaces 4, 104–159 (2016)MathSciNetzbMATHGoogle Scholar
  26. CM98.
    Colding, T.H., Minicozzi, W.P.: Liouville theorems for harmonic sections and applications. Communications on pure and applied mathematics 51(2), 113–138 (1998)MathSciNetzbMATHGoogle Scholar
  27. DS89.
    David, G., Semmes, S.: Strong \(A_\infty \) weights, Sobolev inequalities and quasiconformal mappings. Analysis and partial differential equations 122, 101–111 (1989)MathSciNetzbMATHGoogle Scholar
  28. DeJ14.
    N. DeJarnette. Self improving Orlicz-Poincare inequalities, Ph.D. thesis, University of Illinois at Urbana-Champaign, (2014)Google Scholar
  29. DJS12.
    Durand-Cartagena, E., Jaramillo, J.A., Shanmugalingam, N.: The \(\infty \)-Poincaré inequality in metric measure spaces. Michigan Math. J 61(1), 63–85 (2012)MathSciNetzbMATHGoogle Scholar
  30. DJS16.
    Durand-Cartagena, E., Jaramillo, J.A., Shanmugalingam, N.: Geometric Characterizations of \(p\)-Poincaré Inequalities in the Metric Setting. Publicacions matemàtiques 60(1), 81–111 (2016)MathSciNetzbMATHGoogle Scholar
  31. Eri17.
    S. Eriksson-Bique. Quantitative Embeddability and Connectivity in Metric Spaces, Ph.D. thesis, New York University, (2017)Google Scholar
  32. FKS82.
    Fabes, E.B., Kenig, C.E., Serapioni, R.P.: The local regularity of solutions of degenerate elliptic equations. Communications in Partial Differential Equations 7(1), 77–116 (1982)MathSciNetzbMATHGoogle Scholar
  33. Fed69.
    Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften 153, (1969)Google Scholar
  34. FKP91.
    Fefferman, R.A., Kenig, C.E., Pipher, J.: The theory of weights and the dirichlet problem for elliptic equations. Annals of Mathematics 134(1), 65–124 (1991)MathSciNetzbMATHGoogle Scholar
  35. FLW95.
    Franchi, B., Lu, G., Wheeden, R.L.: Representation formulas and weighted Poincaré inequalities for Hörmander vector fields. Annales de l'institut Fourier 45(2), 577–604 (1995)MathSciNetzbMATHGoogle Scholar
  36. Fug57.
    Fuglede, B.: Extremal length and functional completion. Acta Mathematica 98(1), 171–219 (1957)MathSciNetzbMATHGoogle Scholar
  37. GMS15.
    N. Gigli, A. Mondino, and G. Savaré. Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows. Proc. Lond. Math. Soc. (3) 111(5) (2015), 1071–1129Google Scholar
  38. HK95.
    Hajlasz, P., Koskela, P.: Sobolev meets Poincaré. C. R. Acad. Sci. Paris. Sér. I(320), 1211–1215 (1995)zbMATHGoogle Scholar
  39. Hei10.
    Heikkinen, T.: Sharp self-improving properties of generalized Orlicz-Poincaré inequalities in connected metric measure spaces. Indiana Univ. Math. J. 59, 957–986 (2010)MathSciNetzbMATHGoogle Scholar
  40. HT10.
    Heikkinen, T., Tuominen, H.: Orlicz-Sobolev extensions and measure density condition. J. Math. Anal. Appl. 368(2), 508–524 (2010)MathSciNetzbMATHGoogle Scholar
  41. Hei00.
    Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, Berlin (2000)zbMATHGoogle Scholar
  42. Hei07.
    Heinonen, J.: Nonsmooth calculus. Bulletin of the American Mathematical Society 44(2), 163–232 (2007)MathSciNetzbMATHGoogle Scholar
  43. HK98.
    Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Mathematica 181(1), 1–61 (1998)MathSciNetzbMATHGoogle Scholar
  44. HKST15.
    J. Heinonen, P. Koskela, N. Shanmugalingam, and J.T. Tyson. Sobolev spaces on metric measure spaces, New Mathematical Monographs, no. 27, Cambridge University Press, (Feb 2015)Google Scholar
  45. Jer86.
    Jerison, D.: The Poincaré inequality for vector fields satisfying Hörmanders condition. Duke Math. J 53(2), 503–523 (1986)MathSciNetzbMATHGoogle Scholar
  46. Jui09.
    Juillet, N.: Geometric Inequalities and Generalized Ricci Bounds in the Heisenberg Group. International Mathematics Research Notices 2009(13), 2347–2373 (2009)MathSciNetzbMATHGoogle Scholar
  47. KK11.
    Kansanen, O.E., Korte, R.: Strong \(A_\infty \)-weights are \(A_\infty \)-weights on metric spaces. Revista Matemática Iberoamericana 27(1), 335–354 (2011)MathSciNetzbMATHGoogle Scholar
  48. Kei03.
    Keith, S.: Modulus and the Poincaré inequality on metric measure spaces. Mathematische Zeitschrift 245(2), 255–292 (2003)MathSciNetzbMATHGoogle Scholar
  49. Kei04.
    Keith, S.: A differentiable structure for metric measure spaces. Advances in Mathematics 183(2), 271–315 (2004)MathSciNetzbMATHGoogle Scholar
  50. KZ08.
    Keith, S., Zhong, X.: The Poincaré inequality is an open ended condition. Annals of Mathematics 575–599 (2008)Google Scholar
  51. Kle06.
    B. Kleiner. The asymptotic geometry of negatively curved spaces. 25th International Congress of Mathematicians, ICM 2006, (2006), pp. 743–768Google Scholar
  52. Kos99.
    Koskela, P.: Removable sets for Sobolev spaces. Arkiv för Matematik 37(2), 291–304 (1999)MathSciNetzbMATHGoogle Scholar
  53. Laa00.
    Laakso, T.J.: Ahlfors \(Q\)-regular spaces with arbitrary \(Q>1\) admitting weak Poincarè inequality. Geomet. func. anal 10, 111–123 (2000)MathSciNetzbMATHGoogle Scholar
  54. LV09.
    Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Annals of Mathematics 903–991 (2009)Google Scholar
  55. MTW13.
    Mackay, J.M., Tyson, J.T., Wildrick, K.: Modulus and Poincaré inequalities on non-self-similar Sierpiński carpets. Geometric & Functional Analysis (GAFA) 23(3), 985–1034 (2013)zbMATHGoogle Scholar
  56. Mat99.
    Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press (1999)Google Scholar
  57. Oht07.
    S. Ohta. On the measure contraction property of metric measure spaces. Comment. Math. Helv. (82)(2007), 805–828Google Scholar
  58. Pis16.
    Pisier, G.: Martingales in Banach spaces, vol. 155. Cambridge University Press (2016)Google Scholar
  59. Raj12a.
    Rajala, T.: Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm. Journal of Functional Analysis 263(4), 896–924 (2012)MathSciNetzbMATHGoogle Scholar
  60. Raj12b.
    Rajala, T.: Local Poincaré inequalities from stable curvature conditions on metric spaces. Calculus of Variations and Partial Differential Equations 44(3–4), 477–494 (2012)MathSciNetzbMATHGoogle Scholar
  61. Riz16.
    Rizzi, L.: Measure contraction properties of Carnot groups. Calculus of Variations and Partial Differential Equations 55(3), 1–20 (2016)MathSciNetzbMATHGoogle Scholar
  62. Sch14.
    A. Schioppa. On the relationship between derivations and measurable differentiable structures. Ann. Acad. Sci. Fenn. Ser. AI Math., 39 (2014), 275–304Google Scholar
  63. Sch15.
    Schioppa, A.: Poincaré inequalities for mutually singular measures. Anal. Geom. Metr. Spaces 3, 40–45 (2015)MathSciNetzbMATHGoogle Scholar
  64. Sch16.
    Schioppa, A.: The Poincaré inequality does not improve with blow-up. Anal. Geom. Metr. Spaces 4, 363–398 (2016)MathSciNetzbMATHGoogle Scholar
  65. Sem93.
    S. Semmes. Bilipschitz mappings and strong \(A_\infty \) weights. Ann. Acad. Sci. Fenn. Ser. AI Math., (2)18 (1993), 211–248Google Scholar
  66. Sem96a.
    Semmes, S.: Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities. Selecta Mathematica 2(2), 155–295 (1996)MathSciNetzbMATHGoogle Scholar
  67. Sem96b.
    Semmes, S.: On the nonexistence of bilipschitz parametrizations and geometric problems about \(A_\infty \)-weights. Revista Matemática Iberoamericana 12(2), 337–410 (1996)MathSciNetzbMATHGoogle Scholar
  68. Sem03.
    Semmes, S.: An introduction to analysis on metric spaces. Notices of the AMS 50(4), 438–443 (2003)MathSciNetzbMATHGoogle Scholar
  69. Ste16.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43. Princeton University Press (2016)Google Scholar
  70. Stu06a.
    Sturm, K.-T.: On the geometry of metric measure spaces I. Acta Mathematica 196(1), 65–131 (2006)MathSciNetzbMATHGoogle Scholar
  71. Stu06b.
    Sturm, K.-T.: On the geometry of metric measure spaces. II. Acta Mathematica 196(1), 133–177 (2006)MathSciNetzbMATHGoogle Scholar
  72. Tuo04.
    H. Tuominen. Orlicz-Sobolev spaces on metric measure spaces. Ann. Acad. Sci. Fenn. Math. Diss. (2004), no. 135, 86, Dissertation, University of Jyväskylä, Jyväskylä, 2004Google Scholar
  73. Tuo07.
    Tuominen, H.: Characterization of Orlicz-Sobolev space. Ark. Mat. 45(1), 123–139 (2007)MathSciNetzbMATHGoogle Scholar
  74. Väi06.
    Väisälä, J.: Lectures on \(n\)-dimensional quasiconformal mappings, vol. 229. Springer, Berlin (2006)zbMATHGoogle Scholar
  75. Ren08.
    Max-K. von Renesse. On Local Poincaré via Transportation. Math. Z., 259 (2008), 21–31.Google Scholar
  76. Wan08.
    F.-Y. Wang. Orlicz-Poincaré inequalities. Proc. Edinb. Math. Soc. (2), 51(2) (2008), 529–543Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA

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