Infinitesimal Structure of Differentiability Spaces, and Metric Differentiation

Abstract

Abstract:We prove metric di erentiation for di erentiability spaces in the sense of Cheeger [10,14,27]. As corollaries we give a new proof of one of the main results of [14], a proof that the Lip-lip constant of any Lip-lip space in the sense of Keith [27] is equal to , and new nonembeddability results.

The Lip-lip equality is stable under blow-up

“…We then move on (subsection 3.2) to explain how rescalings affect the modules of derivations and forms. We conclude this section with a generalization of a result [CKS15,Sec. 7] to the category of Weaver derivations.…”

“…The proof can be reconstructed from the argument in [CKS15,Sec. 7] where the result is stated in a less general context: (X, µ) is a differentiability space and Φ is a finite set of Lipschitz functions.…”

“…The only item that requires further justification is (Bw-up3). We highlight the additional arguments; we will refer to the notation and setting in [CKS15,Sec. 7]; in particular, r n = λ −1 n and for γ ∈ Frag(X) we define:…”

“…However note that we are using a convention to normalize the measures different from that in [CKS15], ). There are straightforward modifications for the regularity requirements to handle a countable collection Φ; here ε and S play the rôle of parameters selecting (S) how long γ is, and (ε) how close γ is to a constant speed segment on which the first ⌊S⌋ elements of Φ are close to affine maps; (PS3): We have chosen an l 1 -sequence {ε m } ⊂ (0, ∞) and used (Lemma 7.35 in [CKS15, Sec.…”

“…While finishing this paper, we have learned from Bate and Li [BL15] that they have studied the class of differentiability spaces for which differentiation of RNP-valued functions (in addition to that of real-valued ones) holds, and have characterized them in terms of an "infinitesimal accessibility" condition. One possibility is that differentiability implies RNP-differentiability, and then extending the results of [CK09] to general differentiability spaces might give a route to answer some questions raised in [CKS15]. Another possibility is that differentiability spaces are organized in a sort of "hierarchy" depending on which Banach-valued Lipschitz maps are differentiable.…”