Structure of measures in Lipschitz differentiability spaces

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.

61 pages. Fixed a technical error in section 6. Also tied up various parts of section 6

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• Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential
• Quantitative Bi-Lipschitz embeddings of bounded curvature manifolds and orbifolds
• The Rank-One Theorem on $\mathrm{RCD}$ spaces
• Cheeger's differentiation theorem via the multilinear Kakeya inequality
• Rigidity for convex-cocompact actions on rank-one symmetric spaces
• Differentiability of Lipschitz Functions in Lebesgue Null Sets
• Tangents and rectifiability of Ahlfors regular Lipschitz differentiability spaces
• Ultralimits of pointed metric measure spaces
• Sobolev spaces via chains in metric measure spaces
• Thin Loewner carpets and their quasisymmetric embeddings in $S^2$
• On the differentiability of Lipschitz functions with respect to measures in the Euclidean space
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