Fractional currents and Young geometric integration

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 Hölder continuous maps. In top dimension, fractional currents are the currents represented by functions belonging to a fractional Sobolev space. The space of $\hat{I}\pm$-fractional currents is in duality with a class of cochains, $\hat{I}\pm$-fractional charges, that extend both Whitney's flat cochains and $\hat{I}\pm$-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 $\hat{I}\pm$-fractional $m$-currents as metric currents of the snowflaked metric space $({\mathbb{R}}^d,{\mathrm{d}}_{\mathrm{Eucl}}^{(m+\hat{I}\pm )/(m+1)})$.

to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci