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On the exterior product of Hölder differential forms

Philippe Bouafia

Abstract

We introduce a complex of cochains, $α$-fractional charges ($0 < α\leq 1$), whose regularity is between that of De Pauw-Moonens-Pfeffer's charges and that of Whitney's flat cochains. We show that $α$-Hölder differential forms and their exterior derivative can be realized as $α$-fractional charges, and that it is possible to define the exterior product between an $α$-fractional and a $β$-fractional charge, under the condition that $α+ β> 1$. This construction extends the Young integral in arbitrary dimension and codimension.

On the exterior product of Hölder differential forms

Abstract

We introduce a complex of cochains, -fractional charges (), whose regularity is between that of De Pauw-Moonens-Pfeffer's charges and that of Whitney's flat cochains. We show that -Hölder differential forms and their exterior derivative can be realized as -fractional charges, and that it is possible to define the exterior product between an -fractional and a -fractional charge, under the condition that . This construction extends the Young integral in arbitrary dimension and codimension.
Paper Structure (6 sections, 16 theorems, 105 equations)

This paper contains 6 sections, 16 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Charges in middle dimension
  4. $\alpha$-Fractionality
  5. A Littlewood-Paley type decomposition of fractional charges
  6. Main result

Key Result

Theorem 2.2

Let $K \subseteq \mathbb{R}^d$ be compact. For all $c \geqslant 0$, the ball $\{ T \in \mathbf{N}_m(K) : \mathbf{N}(T) \leqslant c\}$ is $\mathbf{F}$-compact.

Theorems & Definitions (28)

  • Theorem 2.2: Compactness
  • Proposition 2.4
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.6
  • proof
  • Proposition 3.9
  • proof
  • Proposition 4.4
  • ...and 18 more