整体$Q$ -有界几何度量空间上有限周长集族的模与拟共形映射
Modulus of families of sets of finite perimeter and quasiconformal maps between metric spaces of globally $Q$-bounded geometry
摘要：
推广了J.C. Kelly推广到了Ahlfors $Q$ -正则度量测度空间支持$1$ -Poincaré不等式.本文证明了：若$X$和$Y$是两个支持$1$ -Poincaré不等式的Ahlfors $Q$ -正则空间，且$f:X\to Y$是拟共形映射，则$f$拟保持对应于任意有限周长集合$E\subset X$的测度集合$\mathcal{H}^{Q-1}\lfloor_{ΣE}$的$Q/(Q-1)$ -模.我们还表明，对于$Q/(Q-1)$ -模几乎每一个$ΣE$，如果图像表面$Σf(E)$没有看到的奇异集的$f$作为一个大的集合，那么$f(E)$也是有限的周长。即使在标准的欧几里德设置我们的结果是更普遍的比凯利，因此是新的，即使在那里。
英文摘要：
We generalize a result of J. C. Kelly to the setting of Ahlfors $Q$-regular metric measure spaces supporting a $1$-Poincaré inequality. It is shown that if $X$ and $Y$ are two Ahlfors $Q$-regular spaces supporting a $1$-Poincaré inequality and $f:X\to Y$ is a quasiconformal mapping, then the $Q/(Q-1)$-modulus of the collection of measures $\mathcal{H}^{Q-1}\lfloor_{ΣE}$ corresponding to any collection of sets $E\subset X$ of finite perimeter is quasi-preserved by $f$. We also show that for $Q/(Q-1)$-modulus almost every $ΣE$, if the image surface $Σf(E)$ does not see the singular set of $f$ as a large set, then $f(E)$ is also of finite perimeter. Even in the standard Euclidean setting our results are more general than that of Kelly, and hence are new even in there.
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