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Non-absolute integration and application to Young geometric integration

Philippe Bouafia

TL;DR

This work surveys non-absolute integration theories, centering on the Henstock–Kurzweil and Pfeffer integrals, and develops a functional-analytic framework for multidimensional Young integration via spaces of strong charges $CH([0,1]^d)$. It builds a comprehensive coupling between one- and higher-dimensional gauge integrals, charges, and geometric integration theories, culminating in a fractional calculus of currents and charges that extends to Hölder regimes and to $m$-codimensional settings. The paper introduces a multidimensional Faber–Schauder basis for $CH([0,1]^d)$, studies stochastic increments on $BV$-sets, and establishes a Young–Pfeffer correspondence under regularity conditions, thereby unifying several non-absolute integration approaches. Through Whitney flat chains, normal currents in middle dimension, and fractional currents, it provides a cohesive geometric integration framework that supports wedge products, pushforwards, and a robust duality between chains and cochains, with implications for analysis on irregular domains and stochastic geometry.

Abstract

We survey several non-absolutely convergent integrals, including the Henstock-Kurzweil and Pfeffer integrals, and use ideas from these theories to investigate the problem of multidimensional Young integration. We further present results on Young geometric integration, namely the integration of certain generalized differential forms over $m$-dimensional subsets of $\mathbb{R}^d$. This is achieved by introducing appropriate notions of chains and cochains, in the spirit of Whitney's geometric integration theory.

Non-absolute integration and application to Young geometric integration

TL;DR

This work surveys non-absolute integration theories, centering on the Henstock–Kurzweil and Pfeffer integrals, and develops a functional-analytic framework for multidimensional Young integration via spaces of strong charges . It builds a comprehensive coupling between one- and higher-dimensional gauge integrals, charges, and geometric integration theories, culminating in a fractional calculus of currents and charges that extends to Hölder regimes and to -codimensional settings. The paper introduces a multidimensional Faber–Schauder basis for , studies stochastic increments on -sets, and establishes a Young–Pfeffer correspondence under regularity conditions, thereby unifying several non-absolute integration approaches. Through Whitney flat chains, normal currents in middle dimension, and fractional currents, it provides a cohesive geometric integration framework that supports wedge products, pushforwards, and a robust duality between chains and cochains, with implications for analysis on irregular domains and stochastic geometry.

Abstract

We survey several non-absolutely convergent integrals, including the Henstock-Kurzweil and Pfeffer integrals, and use ideas from these theories to investigate the problem of multidimensional Young integration. We further present results on Young geometric integration, namely the integration of certain generalized differential forms over -dimensional subsets of . This is achieved by introducing appropriate notions of chains and cochains, in the spirit of Whitney's geometric integration theory.
Paper Structure (19 sections, 24 theorems, 171 equations)

This paper contains 19 sections, 24 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Non-absolute integrals and the space of (strong) charges
  3. Henstock-Kurzweil integration
  4. A topology on $HK[0, 1]$
  5. Henstock integral in higher dimension
  6. Charges
  7. A differentiation result
  8. Pfeffer integral
  9. Analysis in the space of strong charges
  10. An alternative construction of the space of strong charges
  11. One-dimensional Haar and Faber-Schauder bases
  12. A multidimensional Faber-Schauder basis
  13. Increments of stochastic processes over $BV$-sets
  14. Young integral with respect to a fractional charge
  15. Geometric integration theories
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $f \in C^\gamma([0,1]^d)$ and $g_i \in C^{\beta_i}([0,1]^d)$ for $i \in \{1,\dots,d\}$. If then the integral is well defined as a certain limit of Riemann-Stieltjes sums.

Theorems & Definitions (64)

  • Theorem 1.1: Züst, 2011 Zust
  • Definition 2.1
  • Example 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6: Yeong
  • Theorem 2.7: Alexiewicz, Alex
  • Definition 2.9
  • ...and 54 more