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Stepanov theorem for mappings between metric spaces

Iván Caamaño

TL;DR

This work extends Rademacher-type differentiability to mappings between metric spaces in a Stepanov sense within metric measure spaces that admit rectifiable charts. By merging Kirchheim's metric differentiability with Cheeger-type differentiable structures, it shows that for any f:X→Y, a.e. x in a rectifiable chart U with porous sets null, there exists a metric differential md_x f ∈ sn^n such that d_Y(f(y),f(x)) ≈ md_x f(φ(y)−φ(x)) as y→x. The main contribution is a Stepanov-type generalization of metric differentiability a.e. on U∩S(f), encompassing non-Lipschitz maps, and clarifying the role of rectifiability and porosity in enabling differentiability in this broad setting. The results deepen the understanding of differentiability in metric measure spaces and provide a robust framework for analyzing rectifiable structures and metric differentials in non-Euclidean contexts.

Abstract

For Lipschitz maps between a metric measure space and a metric space, combining the ideas of Kirchheim's metric differentiability and Cheeger's differentiable structures leads to a Rademacher-type theorem for a notion of metric differentiability with respect to a rectifiable chart, and in this paper we prove the validity of a Stepanov-type generalization of such result under the same assumptions.

Stepanov theorem for mappings between metric spaces

TL;DR

This work extends Rademacher-type differentiability to mappings between metric spaces in a Stepanov sense within metric measure spaces that admit rectifiable charts. By merging Kirchheim's metric differentiability with Cheeger-type differentiable structures, it shows that for any f:X→Y, a.e. x in a rectifiable chart U with porous sets null, there exists a metric differential md_x f ∈ sn^n such that d_Y(f(y),f(x)) ≈ md_x f(φ(y)−φ(x)) as y→x. The main contribution is a Stepanov-type generalization of metric differentiability a.e. on U∩S(f), encompassing non-Lipschitz maps, and clarifying the role of rectifiability and porosity in enabling differentiability in this broad setting. The results deepen the understanding of differentiability in metric measure spaces and provide a robust framework for analyzing rectifiable structures and metric differentials in non-Euclidean contexts.

Abstract

For Lipschitz maps between a metric measure space and a metric space, combining the ideas of Kirchheim's metric differentiability and Cheeger's differentiable structures leads to a Rademacher-type theorem for a notion of metric differentiability with respect to a rectifiable chart, and in this paper we prove the validity of a Stepanov-type generalization of such result under the same assumptions.

Paper Structure

This paper contains 5 sections, 15 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Lipschitz differentiability spaces.
  3. Rectifiability.
  4. The metric differential.
  5. Stepanov Theorem.

Key Result

Theorem 1.2

Let $(X,d,\mu )$ be a metric measure space whose porous sets are null and $(U,\varphi )$ a rectifiable chart in $X$. Then any mapping $f:X\rightarrow Y$, where $(Y,d_Y)$ is a metric space, is metrically differentiable with respect to $(U,\varphi )$ almost everywhere on $U\cap S(f)$.

Theorems & Definitions (33)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1: Kuratowski embedding Theorem, Kuratowski
  • Lemma 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: Lipschitz charts
  • Definition 2.7: Weak Lipschitz charts
  • Definition 2.8
  • ...and 23 more