雪花空间中Hölder形式和流的积分
Integration of Hölder forms and currents in snowflake spaces
摘要：
对于有向的$n$维Lipschitz流形$M$，我们通过将黎曼-斯蒂尔切斯积分的构造推广到高维，在函数$f, g_1, >..., g_n$仅仅是某一阶的Hölder连续的情况下，给出积分$\int_M f dg_1 \wedge ... \wedge dg_n$的意义。更一般地说，我们表明，对于$α\in (\frac{n}{n+1},1]$的$n$维局部正常电流在局部紧度量空间$(X,d)$表示的子空间的$n$维电流在$(X,d^α)$。另一方面，对于$n \geq 1$和$α\leq \frac{n}{n+1}$，后者的空间仅由零泛函组成。
英文摘要：
For an oriented $n$-dimensional Lipschitz manifold $M$ we give meaning to the integral $\int_M f dg_1 \wedge ... \wedge dg_n$ 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 $α\in (\frac{n}{n+1},1]$ the $n$-dimensional locally normal currents in a locally compact metric space $(X,d)$ represent a subspace of the $n$-dimensional currents in $(X,d^α)$. On the other hand, for $n \geq 1$ and $α\leq \frac{n}{n+1}$ the latter space consists of the zero functional only.
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