Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Chain rule for pointwise Lipschitz mappings

Jan Kolář, Olga Maleva

TL;DR

This work extends the classical chain rule to compositions where outer and inner maps are only pointwise Lipschitz, by developing Hadamard derivative assignments that exist on large, nonopen domains. A central idea is to use sigma-ideals, notably $\mathcal{L}_1$, to describe sets where differentiability may fail, and to construct complete, Borel measurable derivative assignments (via distTan and regTan) that enable a rigorous almost-everywhere chain rule for compositions. The framework also strengthens Rademacher–Stepanov-type a.e. differentiability results in Banach spaces with the Radon–Nikodým property and extends to infinite dimensions, with an iterated chain-rule formalism for any finite number of mappings. The approach yields robust differentiation tools for Lipschitz and Hadamard-differentiable structures, with potential impact on analysis in nonsmooth contexts and variational problems.

Abstract

The classical Chain Rule formula $(f\circ g)'(x;u)=f'(g(x);g'(x;u))$ gives the (partial, or directional) derivative of the composition of mappings $f$ and $g$. We show how to get rid of the unnecessarily strong assumption of differentiability at all of the relevant points: the mappings do not need to be defined on the whole space and it is enough for them to be pointwise Lipschitz. The price to pay is that the Chain Rule holds almost everywhere. We extend this construction to infinite-dimensional spaces with good properties (Banach, separable, Radon-Nikodým) with an appropriate notion of almost everywhere. Pointwise Lipschitzness is a local condition in contrast to the global Lipschitz property: we do not need the mappings to be defined on the whole space, or even locally in a neighbourhood, nor to know their behaviour far away from the points we consider. This distinguishes our results from recent research on the differentiation of the composition of Lipschitz mappings. The methods we develop for the purpose of proving the Chain Rule also allow us to strengthen the Rademacher-Stepanov type theorem on almost everywhere differentiability of a mapping.

Chain rule for pointwise Lipschitz mappings

TL;DR

This work extends the classical chain rule to compositions where outer and inner maps are only pointwise Lipschitz, by developing Hadamard derivative assignments that exist on large, nonopen domains. A central idea is to use sigma-ideals, notably , to describe sets where differentiability may fail, and to construct complete, Borel measurable derivative assignments (via distTan and regTan) that enable a rigorous almost-everywhere chain rule for compositions. The framework also strengthens Rademacher–Stepanov-type a.e. differentiability results in Banach spaces with the Radon–Nikodým property and extends to infinite dimensions, with an iterated chain-rule formalism for any finite number of mappings. The approach yields robust differentiation tools for Lipschitz and Hadamard-differentiable structures, with potential impact on analysis in nonsmooth contexts and variational problems.

Abstract

The classical Chain Rule formula gives the (partial, or directional) derivative of the composition of mappings and . We show how to get rid of the unnecessarily strong assumption of differentiability at all of the relevant points: the mappings do not need to be defined on the whole space and it is enough for them to be pointwise Lipschitz. The price to pay is that the Chain Rule holds almost everywhere. We extend this construction to infinite-dimensional spaces with good properties (Banach, separable, Radon-Nikodým) with an appropriate notion of almost everywhere. Pointwise Lipschitzness is a local condition in contrast to the global Lipschitz property: we do not need the mappings to be defined on the whole space, or even locally in a neighbourhood, nor to know their behaviour far away from the points we consider. This distinguishes our results from recent research on the differentiation of the composition of Lipschitz mappings. The methods we develop for the purpose of proving the Chain Rule also allow us to strengthen the Rademacher-Stepanov type theorem on almost everywhere differentiability of a mapping.

Paper Structure

This paper contains 11 sections, 56 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. Sigma-ideals $\mathcal{L}_1$ and $\mathcal{L}$
  5. Hadamard derivative assignments. Chain Rule
  6. Differentiation and derived sets: composition with a test mapping
  7. Assignments $\mathop{\mathscr{H}\!_{\Phi}}\nolimits f$ and $\mathop{\mathscr{H\!\!A}\!_{\Phi}}\nolimits f$
  8. Completeness of Hadamard derivative assignment $\mathop{\mathscr{H\!\!A}\!_{\Phi}}\nolimits f$
  9. Borel measurability of $\mathop{\mathscr{H\!\!A}\!_{\Phi}}\nolimits f$
  10. Derivative assignments for pointwise Lipschitz and general functions
  11. Chain Rule for Hadamard derivative assignments

Key Result

Theorem 2.1

Let $f$ be a map of a subset $E$ of a separable Banach space $Y$ to a Banach space $Z$ with the Radon-Nikodým property. Then $f$ is Hadamard differentiable at $\hbox{$$ \scriptsize$\hbox{${\circ}$} {\sim}$} \mathcal{C}$-almost every $y\in E$ at which $\mathop{\mathrm

Theorems & Definitions (185)

  • Theorem 2.1: Preiss2014
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.7
  • Lemma 2.8
  • proof
  • Corollary 2.9
  • ...and 175 more