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Quantified compactness in Lipschitz-free spaces of $[-1,1]^n$

Thierry De Pauw

TL;DR

The paper identifies the Lipschitz-free space $\mathscr{F}(\mathbf{C}^n)$ with a space of $0$-dimensional flat cycles $\mathscr{G}(\mathbf{C}^n)$ having vanishing augmentation, via a precise isometric isomorphism. It develops the $\mathbf{G}$-norm, a boundary-minimization framework, and a representation theory that connects $0$-dimensional flat cycles to $\mathscr{L}^n \wedge \eta$ with $\eta\in L^1$, tying the Banach space structure to standard currents. The work then introduces a qualitative and a quantitative notion of $G$-compactness through a moduli function $\boldsymbol{\kappa}$ and the Federer-Fleming deformation theorem, yielding explicit entropy bounds for compact subsets and a constructive density result via polyhedral currents. Collectively, these results provide a detailed, quantitative understanding of compactness in Lipschitz-free spaces over $[-1,1]^n$ and furnish practical, combinatorial tools for entropy estimates and approximations in this geometric-analytic setting.

Abstract

We show that the members of the Lipschitz-free space of $[-1,1]^n$ are exactly the 0-dimensional flat currents whose "boundary" vanishes. The connection with normal and flat currents allows to use the Federer-Fleming compactness and deformation theorems in this context. We characterize the compact subsets of this Lipschitz-free space and we quantify their $ε$-entropy.

Quantified compactness in Lipschitz-free spaces of $[-1,1]^n$

TL;DR

The paper identifies the Lipschitz-free space with a space of -dimensional flat cycles having vanishing augmentation, via a precise isometric isomorphism. It develops the -norm, a boundary-minimization framework, and a representation theory that connects -dimensional flat cycles to with , tying the Banach space structure to standard currents. The work then introduces a qualitative and a quantitative notion of -compactness through a moduli function and the Federer-Fleming deformation theorem, yielding explicit entropy bounds for compact subsets and a constructive density result via polyhedral currents. Collectively, these results provide a detailed, quantitative understanding of compactness in Lipschitz-free spaces over and furnish practical, combinatorial tools for entropy estimates and approximations in this geometric-analytic setting.

Abstract

We show that the members of the Lipschitz-free space of are exactly the 0-dimensional flat currents whose "boundary" vanishes. The connection with normal and flat currents allows to use the Federer-Fleming compactness and deformation theorems in this context. We characterize the compact subsets of this Lipschitz-free space and we quantify their -entropy.
Paper Structure (10 sections, 20 theorems, 73 equations)

This paper contains 10 sections, 20 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The norm $\mathbf{G}$
  4. Representation of $0$-dimensional flat cycles
  5. The Banach space $\mathscr{G}(\mathbf{C}^n)$
  6. A dense subspace of $\mathscr{G}(\mathbf{C}^n)$
  7. $\mathscr{F}(\mathbf{C}^n)$ and $\mathscr{G}(\mathbf{C}^n)$ are isometrically isomorphic
  8. Qualitative $\mathbf{G}$-compactness by means of a function $\boldsymbol{\kappa}$
  9. Quantitative $\mathbf{G}$-compactness by means of the deformation theorem
  10. Upper bound for the cardinality of $\mathscr{P}^n_{k,\varepsilon}$

Key Result

Theorem 2.12

Let $T \in \mathbf{F}_m(\mathbb{R}^n)$ be such that $\mathop{\mathrm{\mathrm{spt}}}\nolimits T \subseteq \mathbf{C}^n$ and $\varepsilon > 0$. Then there exist $R \in \mathscr{D}_m(\mathbb{R}^n)$ and $S \in \mathscr{D}_{m+1}(\mathbb{R}^n)$ such that

Theorems & Definitions (44)

  • Theorem 2.12
  • proof
  • Theorem 2.13
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Theorem 4.2
  • ...and 34 more