Structure of measures in Lipschitz differentiability spaces

Bate, David

doi:10.1090/S0894-0347-2014-00810-9

Structure of measures in Lipschitz differentiability spaces
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by David Bate

J. Amer. Math. Soc. 28 (2015), 421-482

DOI: https://doi.org/10.1090/S0894-0347-2014-00810-9

Published electronically: June 4, 2014

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Abstract:

We prove the equivalence of two seemingly very different ways of generalising Rademacher’s theorem to metric measure spaces. One such generalisation is based upon the notion of forming partial derivatives along a very rich structure of Lipschitz curves in a way analogous to the differentiability theory of Euclidean spaces. This approach to differentiability in this generality appears here for the first time and by examining this structure further, we naturally arrive to several descriptions of Lipschitz differentiability spaces.

References

Giovanni Alberti, Marianna Csörnyei, and David Preiss, Structure of null sets, differentiability of Lipschitz functions, and other problems. Preprint.
Giovanni Alberti, Marianna Csörnyei, and David Preiss, Differentiability of Lipschitz functions, structure of null sets, and other problems, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1379–1394. MR 2827846
Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math. 195 (2014), no. 2, 289–391. MR 3152751, DOI 10.1007/s00222-013-0456-1
Luigi Ambrosio and Bernd Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), no. 3, 527–555. MR 1800768, DOI 10.1007/s002080000122
Giovanni Alberti, Rank one property for derivatives of functions with bounded variation, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 2, 239–274. MR 1215412, DOI 10.1017/S030821050002566X
Patrick Bernard, Young measures, superposition and transport, Indiana Univ. Math. J. 57 (2008), no. 1, 247–275. MR 2400257, DOI 10.1512/iumj.2008.57.3163
David Bate and Gareth Speight, Differentiability, porosity and doubling in metric measure spaces, Proc. Amer. Math. Soc. 141 (2013), no. 3, 971–985. MR 3003689, DOI 10.1090/S0002-9939-2012-11457-1
J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), no. 3, 428–517. MR 1708448, DOI 10.1007/s000390050094
Jeff Cheeger and Bruce Kleiner, Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon-Nikodým property, Geom. Funct. Anal. 19 (2009), no. 4, 1017–1028. MR 2570313, DOI 10.1007/s00039-009-0030-6
Jasun Gong, Rigidity of derivations in the plane and in metric measure spaces 33 (2011), available at arXiv:1110.4282.
Jasun Gong, The Lip-lip condition on metric measure spaces 31 (2012), available at arXiv:1208.2869.
Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
Stephen Keith, A differentiable structure for metric measure spaces, Adv. Math. 183 (2004), no. 2, 271–315. MR 2041901, DOI 10.1016/S0001-8708(03)00089-6
Bruce Kleiner and John Mackay, Differentiable structures on metric measure spaces: A primer 23 (2011), available at arXiv:1108.1324.
Stefano Lisini, Characterization of absolutely continuous curves in Wasserstein spaces, Calc. Var. Partial Differential Equations 28 (2007), no. 1, 85–120. MR 2267755, DOI 10.1007/s00526-006-0032-2
M. E. Mera, M. Morán, D. Preiss, and L. Zajíček, Porosity, $\sigma$-porosity and measures, Nonlinearity 16 (2003), no. 1, 247–255. MR 1950786, DOI 10.1088/0951-7715/16/1/315
Emanuele Paolini and Eugene Stepanov, Decomposition of acyclic normal currents in a metric space, J. Funct. Anal. 263 (2012), no. 11, 3358–3390. MR 2984069, DOI 10.1016/j.jfa.2012.08.009
Walter Rudin, Function theory in the unit ball of $\Bbb C^n$, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1980 edition. MR 2446682
S. K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows, Algebra i Analiz 5 (1993), no. 4, 206–238 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 4, 841–867. MR 1246427
L. C. Young, Lectures on the calculus of variations and optimal control theory, W. B. Saunders Co., Philadelphia-London-Toronto, Ont., 1969. Foreword by Wendell H. Fleming. MR 0259704