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Young Integration with respect to Hölder Charges

Philippe Bouafia

TL;DR

This work develops a comprehensive theory of Hölder charges and a multidimensional Young integral that integrates Hölder functions against Hölder charges on BV-sets. The approach combines Haar/Faber–Schauder decompositions, a sewing lemma variant, and Pfeffer integration to define and analyze the indefinite and definite Young integrals, extendable to Hölder differential forms. A key contribution is the intrinsic and isotropic characterization of Hölder charges, their approximation by densities, and a duality with fractional Sobolev spaces, linking generalized differential forms to $W^{1-\alpha,1}$ spaces via a canonical pairing. These results enable pathwise, non-absolute integration in multiple dimensions with potential implications for geometric analysis and rough-path-type theories.

Abstract

We present a multidimensional Young integral that enables to integrate Hölder continuous functions with respect to a Hölder charge. It encompasses the integration of Hölder differential forms introduced by R. Züst: if $f$, $g_1, \dots, g_d$ are merely Hölder continuous functions on the cube $[0, 1]^d$ whose Hölder exponents satisfy a certain condition, it is possible to interpret $\mathrm{d}g_1 \wedge \cdots \wedge \mathrm{d}g_d$ as a Hölder charge and thus to make sense of the integral \[ \int_B f \mathrm{d} g_1 \wedge \cdots \wedge \mathrm{d}g_d \] over a set $B \subset [0, 1]^d$ of finite perimeter.

Young Integration with respect to Hölder Charges

TL;DR

This work develops a comprehensive theory of Hölder charges and a multidimensional Young integral that integrates Hölder functions against Hölder charges on BV-sets. The approach combines Haar/Faber–Schauder decompositions, a sewing lemma variant, and Pfeffer integration to define and analyze the indefinite and definite Young integrals, extendable to Hölder differential forms. A key contribution is the intrinsic and isotropic characterization of Hölder charges, their approximation by densities, and a duality with fractional Sobolev spaces, linking generalized differential forms to spaces via a canonical pairing. These results enable pathwise, non-absolute integration in multiple dimensions with potential implications for geometric analysis and rough-path-type theories.

Abstract

We present a multidimensional Young integral that enables to integrate Hölder continuous functions with respect to a Hölder charge. It encompasses the integration of Hölder differential forms introduced by R. Züst: if , are merely Hölder continuous functions on the cube whose Hölder exponents satisfy a certain condition, it is possible to interpret as a Hölder charge and thus to make sense of the integral over a set of finite perimeter.
Paper Structure (42 sections, 24 theorems, 208 equations)

This paper contains 42 sections, 24 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Charges and Hölder regularity
  3. Young integral
  4. Duality results and characterization of fractional Sobolev regularity
  5. Acknowledgments
  6. Preliminaries and Notations
  7. General notations
  8. $BV$ functions
  9. $BV$-sets
  10. Charges
  11. Restriction of charges
  12. Charges of the form $\boldsymbol{\omega} \circ \Phi$
  13. The ${} \,$ operation
  14. Strong charges
  15. Haar and Faber-Schauder bases in dimension one
  16. ...and 27 more sections

Key Result

Theorem 1

For any $B \in \euscr{BV}([0, 1]^d)$, there is a sequence $(B_n)$ of dyadic figures that $BV$-converges to $B$.

Theorems & Definitions (49)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Lemma 2
  • ...and 39 more