Measurable differentiable structures and the Poincare inequality

Scaled-oscillation and regularity

“…These functions were recently used by Keith [Ke1], [Ke2] to give sufficient conditions on X under which the Cheeger-Rademacher theorem of differentiability of Lipschitz functions [C] holds. Let us recall that if f is a Lipschitz function, then there exists a constant L > 0 such that l f ≤ L f ≤ L. If X = Ω ⊂ R n we can apply the classical Rademacher differentiability theorem according to which any Lipschitz function f is differentiable for L n a.e.…”

Modulus and the Poincar� inequality on metric measure spaces

“…Moreover, a surprisingly rich structure can be deduced for a metric measure space, by merely knowing that it admits a Poincaré inequality with a doubling measure. This has been extensively studied, see [HKM93, HK95a, HK96, HK98, HK99, BMS01, HST01, Sha01, HKST01, KST01,Kei02]. In particular, Cheeger [Che99] demonstrated that spaces which admit a Poincaré inequality with a doubling measure, admit a sort of measurable differentiable structure similar to rectifiability.…”

Derivations and Alberti representations

“…In [Che99] Cheeger formulated a generalization of Rademacher's differentiability Theorem for metric measure spaces admitting a Poincaré inequality (this is an analytic condition that has been intoduced in [HK98] and has proven useful to generalize notions of calculus on metric measure spaces; knowing about the Poincaré inequality is not a prerequisite for understanding this paper); a metric measure space satisfying the conclusion of Cheeger's result is often called a (Lipschitz) differentiability space or is said to have a (measurable / strong measurable) differentiable structure. Applications of differentiability spaces include the study of Sobolev and quasiconformal maps in metric measure spaces [BRZ04,Kei04b] and the study of metric embeddings [CK09,Che99]. Recently Bate [Bat15] approached this subject from a different angle by showing that differentiability spaces have a 1 rich 1-rectifiable structure, which can be described in terms of Fubini-like representations of the measure which are called Alberti representations or 1-rectifiable representations [ACP10].…”