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Sharp Differentiability Results for the Lower Local Lipschitz Constant and Applications to Non-embedding

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We give a sharp condition on the lower local Lipschitz constant of a mapping from a metric space supporting a Poincaré inequality to a Banach space with the Radon–Nikodym property that guarantees differentiability at almost every point. We apply these results to obtain a non-embedding theorem for a corresponding class of mappings.

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  1. Burago, D., Burago, Y., Ivanov, I.: A course in metric geometry. Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence (2001)

    Google Scholar 

  2. Balogh, Z.M., Csörnyei, M.: Scaled-oscillation and regularity. Proc. Am. Math. Soc. 134(9), 2667–2675 (2006) (electronic)

  3. Balogh, Z.M., Rogovin, K., Zürcher, T.: The Stepanov differentiability theorem in metric measure spaces. J. Geom. Anal. 14, 405–422 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bennett, C., Sharpley, R.: Interpolation of operators. Pure Appl. Math., vol. 129. Academic Press, Boston (1988)

    Google Scholar 

  5. Calderón, A.P.: On the differentiability of absolutely continuous functions. Rivista Mat. Univ. Parma 2, 203–213 (1951)

    MATH  MathSciNet  Google Scholar 

  6. Cesari, L.: Sulle funzioni assolutamente continue in due variabili. Ann. Scuola Norm. Super. Pisa 2(10), 91–101 (1941)

    MathSciNet  Google Scholar 

  7. Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9(3), 428–517 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cheeger, J., Kleiner, B.: On the differentiability of Lipschitz maps from metric measure spaces to Banach spaces. Nankai Tracts Math., vol. 11, pp. 129–152. World Sci. Publ, Hackensack (2006)

    Google Scholar 

  9. Cheeger, J., Kleiner, B.: Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon–Nikodým property. Geom. Funct. Anal. 19(4), 1017–1028 (2009)

  10. Cheeger, J., Kleiner, B., Naor, A.: Compression bounds for Lipschitz maps from the Heisenberg group to \(L_1\). Acta Math. 207(2), 291–373 (2011)

  11. David, G., Semmes, S.: Fractured Fractals and Broken Dreams: Self-Similar Geometry Through Metric and Measure. Oxford University Press, Oxford (1997)

  12. Federer, H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag, New York (1969)

    Google Scholar 

  13. Gong, J.: The Lip-lip condition on metric measure spaces. arXiv:1208.2869.

  14. Hanson, B.: Linear dilatation and differentiability of homeomorphisms of \(\mathbb{R}^n\). Proc. Am. Math. Soc. 140(10), 3541–3547 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Heinonen, J.: Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York (2001)

    Book  MATH  Google Scholar 

  16. Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181(1), 1–61 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc 145(688), x+101 pp. (2000)

  18. Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.T.: Sobolev classes of Banach space-valued functions and quasiconformal mappings. J. Anal. Math. 85, 87–139 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Jerison, D.: The Poincaré inequality for vector fields satisfying Hörmander’s condition. Duke Math. J. 53(2), 503–523 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  20. Johnson, W.B., Lindenstrauss, J., Schechtman, G.: Extensions of Lipschitz maps into Banach spaces. Israel J. Math. 54(2), 129–138 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  21. Keith, S.: Modulus and the Poincaré inequality on metric measure spaces. Math. Z. 245(2), 255–292 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Keith, S.: A differentiable structure for metric measure spaces. Adv. Math. 183(2), 271–315 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kauhanen, J., Koskela, P., Malý, J.: On functions with derivatives in a Lorentz space. Manuscripta Math. 100(1), 87–101 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kleiner, B., MacKay, J.: Differentiable structures on metric measure spaces: a primer. arXiv:1108.1324v1.

  25. Koskela, P., MacManus, P.: Quasiconformal mappings and Sobolev spaces. Stud Math. 131(1), 1–17 (1998)

    MATH  MathSciNet  Google Scholar 

  26. Keith, S., Zhong, X.: The Poincaré inequality is an open ended condition. Ann. Math 167(2), 575–599 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Lee, J.R., Naor, A.: Extending Lipschitz functions via random metric partitions. Invent. Math. 160(1), 59–95 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lang, U., Schlichenmaier, T.: Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions. Int. Math. Res. Not. 58, 3625–3655 (2005)

    Article  MathSciNet  Google Scholar 

  29. Malý, J.: Absolutely continuous functions of several variables. J. Math. Anal. Appl. 231(2), 492–508 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  30. Malý, J.: Sufficient conditions for change of variables in integral. In Proceedings on Analysis and Geometry (Russian) (Novosibirsk Akademgorodok, 1999), pages 370–386. Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 2000.

  31. Malý, J., Swanson, D., Ziemer, W.P.: Fine behavior of functions whose gradients are in an Orlicz space. Stud. Math. 190(1), 33–71 (2009)

    Article  MATH  Google Scholar 

  32. Pansu, P.: Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. Math. 129(1), 1–60 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  33. Rademacher, H.: Über partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale. Math. Ann. 79(4), 340–359 (1919)

    Article  MathSciNet  Google Scholar 

  34. Ranjbar-Motlagh, A.: An embedding theorem for Sobolev type functions with gradients in a Lorentz space. Stud. Math. 191(1), 1–9 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  35. Romanov, A.: Absolute continuity of the Sobolev type functions on metric spaces. Siberian. Math. J. 49(5), 911–918 (2008)

    Article  MathSciNet  Google Scholar 

  36. Schioppa, A.: On the relationship between derivations and measurable differentiable structures on metric measure spaces. Ann. Acad. Sci. Fenn. Math. 39, 275–304 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  37. Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana. 16(2), 243–279 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  38. Stepanoff, W.: Über totale Differenzierbarkeit. Math. Ann. 90, 318–320 (1923)

    Article  MATH  MathSciNet  Google Scholar 

  39. Stein, E.M.: Editor’s note: The differentiability of functions in \({ R}^{n}\). Ann. Math. 113(2), 383–385 (1981)

    MATH  Google Scholar 

  40. Wildrick, K., Zürcher, T.: Mappings with an upper gradient in a Lorentz space., 2009. Preprint 382

  41. Wildrick, K., Zürcher, T.: Space filling with metric measure spaces. Math. Z. 270, 103–131 (2012). doi:10.1007/s00209-010-0787-1

    Article  MATH  MathSciNet  Google Scholar 

  42. Zürcher, T.: Local Lipschitz numbers and Sobolev spaces. Mich. Math. J. 55(3), 561–574 (2007)

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K. W. was supported by Academy of Finland Grant 128144, the Swiss National Science Foundation, European Research Council Project CG-DICE, and the European Science Council Project HCAA. T. Z. was supported by the Swiss National Science Foundation Grant PBBEP3_130157 and by the Academy of Finland Grant Number 251650.

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Correspondence to K. Wildrick.

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Communicated by Loukas Grafakos.

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Wildrick, K., Zürcher, T. Sharp Differentiability Results for the Lower Local Lipschitz Constant and Applications to Non-embedding. J Geom Anal 25, 2590–2616 (2015).

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