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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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