Modulus of families of sets of finite perimeter and quasiconformal maps between metric spaces of globally $Q$-bounded geometry

Abstract

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 ${\mathcal{H}}^{Q-1}{\lfloor }_{\Sigma E}$ 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 $\Sigma E$, if the image surface $\Sigma f(E)$ 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.

arXiv comment26 pages

Cited by in corpus (1)

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