Abstract
We prove the differentiability of Lipschitz maps X → V, where X denotes a PI space, i.e. a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V denotes a Banach space with the Radon–Nikodym Property (RNP). As a consequence, we obtain a bi-Lipschitz nonembedding theorem for RNP targets. The differentiation theorem depends on a new specification of the differentiable structure for PI spaces involving directional derivatives in the direction of velocity vectors to rectifiable curves. We give two different proofs of this, the second of which relies on a new characterization of the minimal upper gradient. There are strong implications for the infinitesimal structure of PI spaces which will be discussed elsewhere.
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References
J. Bourgain, New Classes of \({\mathcal{L}^p}\) Spaces, Springer Lecture Notes in Math. 889 (1981).
R.D. Bourgin, Geometric aspects of convex sets with the Radon–Nikodym Property, Springer Lecture Notes in Math. 993 (1983).
Bourdon M., Pajot H.: Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings. Proc. Amer. Math. Soc. 127(8), 2315–2324 (1999)
Cheeger J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9(3), 428–517 (1999)
J. Cheeger, B. Kleiner, Embedding Laakso spaces in L 1, preprint, 2006.
J. Cheeger, B. Kleiner, On the differentiability of Lipschitz maps from metric measure spaces into Banach spaces, in “Inspired by S.S. Chern, A Memorial Volume in Honor of a Great Mathematician”, Nankai tracts in Mathematics 11, World Scientific, Singapore (2006), 129–152.
J. Cheeger, B. Kleiner, Characterization of the Radon–Nikodym Property in terms of inverse limits, in “Géométrie différentielle, Physique mathématique, Mathématiques et société pour célébrer les 60 ans de Jean-Pierre Bourguignon”, Séminaires et Congrès, Société Mathématique de France (2008).
Federer H.: Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York (1969)
Heinonen J.: Lectures on Analysis on Metric Spaces. Springer-Verlag, New York (2001)
Heinonen J., Koskela P.: From local to global in quasiconformal structures. Proc. Nat. Acad. Sci. USA 93, 554–556 (1996)
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)
Kirchheim B.: Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Amer. Math. Soc. 121(1), 113–123 (1994)
Laakso T.J.: Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality. Geom. Funct. Anal., 10(1), 111–123 (2000)
Mac Cartney P.W., O’Brien R.C.: Proc. Amer. Math. Soc. 78(1), 40–42 (1980)
Pauls S.D.: The large scale geometry of nilpotent Lie groups. Comm. Anal. Geom. 9(5), 951–982 (2001)
Royden H.L.: Real Analysis third edition. Macmillan Publishing Company, New York (1988)
Shanmugalingam N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16(2), 243–279 (2000)
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The first author was partially supported by NSF Grant DMS 0105128 and the second by NSF Grant DMS 0701515.
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Cheeger, J., Kleiner, B. Differentiability of Lipschitz Maps from Metric Measure Spaces to Banach Spaces with the Radon–Nikodym Property. Geom. Funct. Anal. 19, 1017–1028 (2009). https://doi.org/10.1007/s00039-009-0030-6
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DOI: https://doi.org/10.1007/s00039-009-0030-6
Keywords and phrases
- differentiability
- Lipschitz function
- Banach space
- Radon-Nikodym property
- metric measure space
- doubling measure
- Poincaré inequality
- minimal upper gradient