H\"older Regularity of Distributional Volume Forms

TL;DR

This paper constructs and analyzes a distributional volume form for Hölder continuous functions, extending classical integrals and establishing conditions for their well-definedness in various domains.

Contribution

It introduces a new distributional volume form for Hölder functions, characterizes its regularity, and defines integrals over domains with criteria ensuring their existence.

Findings

The distribution coincides with classical forms for Lipschitz functions.

The paper provides a new regularity estimate for the distribution.

It establishes domain conditions for the integral's well-definedness.

Abstract

Let $f, g^1, \dots, g^d : \mathbb{R}^d \longrightarrow \mathbb{R}$ be H\"older continuous functions. If the H\"older exponents of these functions are less than $1$ but sufficiently large, we use the integral introduced by Z\"ust to construct a distribution, denoted by $f \, \mathrm{d}g^1 \wedge \dots \wedge \, \mathrm{d}g^d$ which depends continuously on the functions $f, g^1, \dots, g^d$ in a sense that we shall specify, and which coincides with the function $f\det(\, \mathrm{d} g)$ when the functions $g^i$ are Lipschitz. We show that this distribution is entirely characterized by these properties and determine its H\"older regularity. We use this distribution to define the integral $ \int_{\Omega} f \, \mathrm{d}g^1 \wedge \dots \wedge \, \mathrm{d}g^d$ by duality, for general domains $\Omega \subset \mathbb{R}^d$. When $\Omega$ is a rectangle, this integral coincides with Z\"ust's construction. We then establish a new criterion on the domain $\Omega$ ensuring that the integral is well defined. This criterion allows to recover a condition of Bouafia on the perimeter of the domain, and in the case when $d = 2$, the condition of Alberti-Stepanov-Trevisan on the upper box dimension of the boundary.