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Integration of Hölder forms and currents in snowflake spaces

For an oriented $n$-dimensional Lipschitz manifold $M$ we give meaning to the integral $\int_M f dg_1 \wedge ... \wedge dg_n$ in case the functions $f, g_1, >..., g_n$ are merely Hölder continuous of a certain order by extending the construction of the Riemann-Stieltjes integral to higher dimensions. More generally, we show that for $α\in (\frac{n}{n+1},1]$ the $n$-dimensional locally normal currents in a locally compact metric space $(X,d)$ represent a subspace of the $n$-dimensional currents in $(X,d^α)$. On the other hand, for $n \geq 1$ and $α\leq \frac{n}{n+1}$ the latter space consists of the zero functional only.