Fractional currents and Young geometric integration
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917390323810304 |
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| author | Bouafia, Philippe |
| author_facet | Bouafia, Philippe |
| contents | We introduce a class of flat currents with fractal properties, called fractional currents, which satisfy a compactness theorem and remain stable under pushforwards by Hölder continuous maps. In top dimension, fractional currents are the currents represented by functions belonging to a fractional Sobolev space.
The space of $α$-fractional currents is in duality with a class of cochains, $α$-fractional charges, that extend both Whitney's flat cochains and $α$-Hölder continuous forms. We construct a partially defined wedge product between fractional charges, enabling a generalization of the Young integral to arbitrary dimensions and codimensions. This helps us identify $α$-fractional $m$-currents as metric currents of the snowflaked metric space $(\mathbb{R}^d, \mathrm{d}_{\mathrm{Eucl}}^{(m+α)/(m+1)})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09298 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractional currents and Young geometric integration Bouafia, Philippe Functional Analysis 49Q15, 28A75, 46E35, 55M25, 26A16 We introduce a class of flat currents with fractal properties, called fractional currents, which satisfy a compactness theorem and remain stable under pushforwards by Hölder continuous maps. In top dimension, fractional currents are the currents represented by functions belonging to a fractional Sobolev space. The space of $α$-fractional currents is in duality with a class of cochains, $α$-fractional charges, that extend both Whitney's flat cochains and $α$-Hölder continuous forms. We construct a partially defined wedge product between fractional charges, enabling a generalization of the Young integral to arbitrary dimensions and codimensions. This helps us identify $α$-fractional $m$-currents as metric currents of the snowflaked metric space $(\mathbb{R}^d, \mathrm{d}_{\mathrm{Eucl}}^{(m+α)/(m+1)})$. |
| title | Fractional currents and Young geometric integration |
| topic | Functional Analysis 49Q15, 28A75, 46E35, 55M25, 26A16 |
| url | https://arxiv.org/abs/2503.09298 |