H\"older Regularity of Distributional Volume Forms

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.