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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2510.20427 |
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| _version_ | 1866914109126082560 |
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| author | Jaffard, Thomas |
| author_facet | Jaffard, Thomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20427 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hölder Regularity of Distributional Volume Forms Jaffard, Thomas Functional Analysis Differential Geometry 46F10 (Primary), 49Q15 (Secondary), 42C40 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. |
| title | Hölder Regularity of Distributional Volume Forms |
| topic | Functional Analysis Differential Geometry 46F10 (Primary), 49Q15 (Secondary), 42C40 |
| url | https://arxiv.org/abs/2510.20427 |