Hölder Regularity of Distributional Volume Forms
Thomas Jaffard
TL;DR
This paper develops a distributional framework for the volume form $f\, \mathrm{d}g^1 \wedge \cdots \wedge \mathrm{d}g^d$ when $f,g^1,\dots,g^d$ are Hölder with exponents in $(0,1)$ and the sum of exponents exceeds the dimension in a suitable sense. It constructs a globally Hölder distribution $\mathcal{M}(f,g^1,\dots,g^d)$ that equals $f\det(\mathrm{d}g)$ when the $g^i$ are Lipschitz and proves a sharp Hölder–Besov regularity bound, placing the object in $\mathcal{C}^{-\gamma}$ with $\gamma=d-\sum\beta_i'$. The authors then extend the integral to general domains $\Omega$ via Besov duality, defining $\int_{\Omega} f\, \mathrm{d}g^1 \wedge \cdots \wedge \mathrm{d}g^d = \langle f\, \mathrm{d}g^1 \wedge \cdots \wedge \mathrm{d}g^d, \mathbb{1}_{\Omega} \rangle$ whenever $\mathbb{1}_{\Omega} \in \mathcal{B}^{\gamma}_{1,1}$, and they provide a geometric criterion on $\Omega$ through the Lebesgue boundary $\partial^*\Omega$ that guarantees this Besov membership. The paper also connects to prior work on higher-dimensional Young-type integrals, Bouafia’s Hölder charges, and upper box-dimension criteria, and extends the theory to homogeneous and local Hölder settings, enabling localization and broader applicability. Overall, it delivers a rigorous, uniquely determined distributional construction, a coherent domain-extension framework, and sharp geometric conditions for well-defined integrals of rough volume forms.
Abstract
Let $f, g^1, \dots, g^d : \mathbb{R}^d \longrightarrow \mathbb{R}$ 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 $f \, \mathrm{d}g^1 \wedge \dots \wedge \, \mathrm{d}g^d$ 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 Hölder regularity. We use this distribution to define the integral $ \int_Ω f \, \mathrm{d}g^1 \wedge \dots \wedge \, \mathrm{d}g^d$ by duality, for general domains $Ω\subset \mathbb{R}^d$. When $Ω$ is a rectangle, this integral coincides with Züst's construction. We then establish a new criterion on the domain $Ω$ 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.