Integration of Hölder forms and currents in snowflake spaces

Abstract

For an oriented $n$-dimensional Lipschitz manifold $M$ we give meaning to the integral ${\int }_{M}fd{g}_1\wedge ...\wedge d{g}_n$ 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 $\alpha \in (\frac{n}{n+1},1]$ the $n$-dimensional locally normal currents in a locally compact metric space $(X,d)$ represent a subspace of the $n$-dimensional currents in $(X,d^{\alpha })$. On the other hand, for $n\ge 1$ and $\alpha \le \frac{n}{n+1}$ the latter space consists of the zero functional only.

arXiv comment21 pages, polished and slightly extended