Integration of Hölder forms and currents in snowflake spaces

Züst, Roger
January 2011
Calculus of Variations & Partial Differential Equations;Jan2011, Vol. 40 Issue 1/2, p99
Academic Journal
For an oriented n-dimensional Lipschitz manifold M we give meaning to the integral $${\int_M f \, dg_1 \wedge \cdots \wedge dg_n}$$ in case the functions $${f, g_1, \ldots, g_n}$$ 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 $${\alpha \in (\tfrac{n}{n+1},1]}$$ 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 $${n \geq 1}$$ and $${\alpha \leq \tfrac{n}{n+1}}$$ the vector space of n-dimensional currents in ( X, d) is zero.


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