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Shortcut Laakso spaces, pure PI unrectifiability and differentiability of Lipschitz functions

David Bate, Pietro Wald

Abstract

We construct a family of purely PI unrectifiable Lipschitz differentiability spaces and investigate the possible of Banach spaces targets for which Lipschitz differentiability holds. We provide a general investigation into the geometry of \emph{shortcut} metric spaces and characterise when such spaces are PI rectifiable, and when they are $Y$-LDS, for a given $Y$. The family of spaces arises as an example of our characterisations. Indeed, we show that Laakso spaces satisfy the required hypotheses.

Shortcut Laakso spaces, pure PI unrectifiability and differentiability of Lipschitz functions

Abstract

We construct a family of purely PI unrectifiable Lipschitz differentiability spaces and investigate the possible of Banach spaces targets for which Lipschitz differentiability holds. We provide a general investigation into the geometry of \emph{shortcut} metric spaces and characterise when such spaces are PI rectifiable, and when they are -LDS, for a given . The family of spaces arises as an example of our characterisations. Indeed, we show that Laakso spaces satisfy the required hypotheses.

Paper Structure

This paper contains 32 sections, 106 theorems, 302 equations.

Table of Contents

  1. Introduction
  2. Metric spaces with shortcuts
  3. Quantitative differentiation on Laakso spaces
  4. Shortcut Laakso spaces
  5. Acknowledgements
  6. Preliminaries
  7. General notation
  8. Lipschitz differentiability spaces
  9. Three useful lemmas
  10. PI spaces
  11. PI rectifiability
  12. Ahlfors-David regular spaces and Borel-Cantelli-type lemmas
  13. Shortcut metric spaces
  14. Shortcut metric spaces and PI (un)rectifiability
  15. Pure PI unrectifiability
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $2<s<\infty$. There is a compact $s$-Ahlfors-David regular metric measure space $\mathcal{L}=(\mathcal{L},d,\mu)$ with the following properties:

Theorems & Definitions (247)

  • Theorem 1.1
  • Definition 2.2
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • Definition 2.8
  • ...and 237 more