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Fragment-wise differentiable structures

David Bate, Sylvester Eriksson-Bique, Elefterios Soultanis

Abstract

The $p$-modulus of curves, test plans, upper gradients, charts, differentials, approximations in energy and density of directions are all concepts associated to the theory of Sobolev functions in metric measure spaces. The purpose of this paper is to give an analogous geometric and ``fragment-wise'' theory for Lipschitz functions and Weaver derivations, where $\infty$-modulus of curve fragments, $\ast$-upper gradients and Alberti representations play a central role. We give a new definition of fragment-wise charts and prove that they exists for spaces with finite Hausdorff dimension. We give a replacement for $p$-duality in terms of Alberti representations and $\infty$-modulus and present the theory of $\ast$-upper gradients. Further, we give new and sharper results for approximations of Lipschitz functions, which yields the density of directions. Our results are applicable to all complete and separable metric measure spaces. In the process, we show that there are strong parallels between the Sobolev and Lipschitz worlds.

Fragment-wise differentiable structures

Abstract

The -modulus of curves, test plans, upper gradients, charts, differentials, approximations in energy and density of directions are all concepts associated to the theory of Sobolev functions in metric measure spaces. The purpose of this paper is to give an analogous geometric and ``fragment-wise'' theory for Lipschitz functions and Weaver derivations, where -modulus of curve fragments, -upper gradients and Alberti representations play a central role. We give a new definition of fragment-wise charts and prove that they exists for spaces with finite Hausdorff dimension. We give a replacement for -duality in terms of Alberti representations and -modulus and present the theory of -upper gradients. Further, we give new and sharper results for approximations of Lipschitz functions, which yields the density of directions. Our results are applicable to all complete and separable metric measure spaces. In the process, we show that there are strong parallels between the Sobolev and Lipschitz worlds.
Paper Structure (27 sections, 39 theorems, 156 equations)

This paper contains 27 sections, 39 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Fragment-wise differentials and density of directions
  4. Lipschitz differentiability spaces and Weaver derivations
  5. Approximation
  6. Preliminaries
  7. Notation and conventions
  8. Lipschitz free space
  9. Curve fragments and $\ast$-upper gradients
  10. $\infty$-Modulus and minimal $\ast$-upper gradients
  11. Plans and Alberti representations
  12. Weaver derivations
  13. Approximation
  14. Curve fragments, line integrals, and lower semicontinuity
  15. Construction of an approximating sequence
  16. ...and 12 more sections

Key Result

Theorem 1.2

Suppose $(U,\varphi)$ is a fragment-wise chart of dimension $n$. Then every $f\in \operatorname{LIP}(X)$ admits a unique fragment-wise differential $\mathrm {d} f:U\to(\mathbb{R}^n)^*$. Moreover for $\mu$-a.e. $x\in U$ there exists a norm $|\cdot|_x$ on $(\mathbb{R}^n)^*$ such that $x\mapsto |\xi|_x for every $f\in \operatorname{LIP}(X)$.

Theorems & Definitions (89)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • ...and 79 more