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arXiv · 2503.09298

Fractional currents and Young geometric integration

Philippe Bouafia
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Abstract

We introduce a class of flat currents with fractal properties, called fractional currents, which satisfy a compactness theorem and remain stable under pushforwards by H\"older continuous maps. In top dimension, fractional currents are the currents represented by functions belonging to a fractional Sobolev space. The space of $\alpha$-fractional currents is in duality with a class of cochains, $\alpha$-fractional charges, that extend both Whitney's flat cochains and $\alpha$-H\"older 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 $\alpha$-fractional $m$-currents as metric currents of the snowflaked metric space $(\mathbb{R}^d, \mathrm{d}_{\mathrm{Eucl}}^{(m+\alpha)/(m+1)})$.

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Philippe Bouafia. 2025-03-12. Fractional currents and Young geometric integration. https://doi.org/10.2422/2036-2145.202503_012

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