Hölder Regularity of Distributional Volume Forms

Let $f,g^1,\dots ,g^d:{\mathbb{R}}^d⟶\mathbb{R}$ be Hölder continuous functions. If the Hölder exponents of these functions are less than $1$ but sufficiently large, we use the integral introduced by Züst to construct a distribution, denoted by $f\mathrm{d}{g}^1\wedge \cdots \wedge \mathrm{d}{g}^d$ 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 Hölder regularity. We use this distribution to define the integral ${\int }_{\mathrm{\Omega }}f\mathrm{d}{g}^1\wedge \cdots \wedge \mathrm{d}{g}^d$ by duality, for general domains $\mathrm{\Omega }\subset {\mathbb{R}}^d$. When $\mathrm{\Omega }$ is a rectangle, this integral coincides with Züst’s construction. We then establish a new criterion on the domain $\mathrm{\Omega }$ 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.