Structure of measures in Lipschitz differentiability spaces

By David Bate. 2012

93 citations recorded by the source corpus.

We prove the equivalence of two seemingly very di erent ways of generalising Rademacher's theorem to metric measure spaces. The rst was introduced by Cheeger and is based upon di erentiation with respect to another, xed, chart func- tion. The second approach is new for this generality and originates in some ideas of Alberti. It is based upon forming partial derivatives along a very rich structure of Lipschitz curves, analogous to the di erentiability theory of Euclidean spaces. By examining this structure further, we naturally arrive to several descriptions of Lipschitz di erentiability spaces.

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Citations in the corpus

On a conjecture of Cheeger (2016, background): “One of the main results of [Bat15] asserts that a Lipschitz differentiability space necessarily admits many independent Alberti representations, also cf.”
On the volume measure of non-smooth spaces with Ricci curvature bounded below (2016, methodology): “We decided to give a proof of Theorem 1.2 as self-contained as possibile but we wish to mention that by using more heavily the work of Bate [8] and David [18], it is possibile to give an alter- native argument which avoids the rectifiability Theorem 3.7.”
On the differentiability of Lipschitz functions with respect to measures in the Euclidean space (2014, background): “The function g is clearly of class C1.”
Well posedness of Lagrangian flows and continuity equations in metric measure spaces (2014, background): “2, the (extended) distance is hidden in the choice of the family A, we call the next result “superposition in metric measure spaces”, because in most cases A consists precisely of distance functions from a countable dense set (see also the recent papers [Bate 2013; Schioppa 2014] for related results”
Behaviour of the reference measure on $\sf RCD$ spaces under charts (2016, background): “1, without the need of using also the deep results in [5], [4] and [1].”