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Integration of Hölder forms and currents in snowflake spaces. (English) Zbl 1219.49036
Summary: 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 $$f, g_1, \dots, g_n$$ 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 $${n \geq 1}$$ and $$\alpha \leq \frac{n}{n+1}$$ the vector space of $$n$$-dimensional currents in $$(X, d^\alpha)$$ is zero.

##### MSC:
 49Q15 Geometric measure and integration theory, integral and normal currents in optimization 26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
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