Integration of Hölder Forms and Currents in Snowflake Spaces

M f dg1 ∧ · · · ∧ dgn in case the functions f, g1, . . . , gn 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 ∈ ( 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). On the other hand, for n ≥ 1 and α ≤ n n+1 the latter space consists of the zero functional only.