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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Quasiconformal mappings and sets of finite perimeter


Author: James C. Kelly
Journal: Trans. Amer. Math. Soc. 180 (1973), 367-387
MSC: Primary 30A60; Secondary 28A75
DOI: https://doi.org/10.1090/S0002-9947-1973-0357783-7
MathSciNet review: 0357783
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Abstract: Let $D$ be a domain in ${R^n},n \geqslant 2,f$ a quasiconformal mapping on $D$. We give a definition of bounding surface of codimension one lying in $D$, and show that, given a system $\Sigma$ of such surfaces, the image of the restriction of $f$ to “almost every” surface is again a surface. Moreover, on these surfaces, $f$ takes ${H^{n - 1}}$ (Hausdorff $(n - 1)$-dimensional) null sets to ${H^{n - 1}}$ null sets. “Almost every” surface is given a precise meaning via the concept of the module of a system of measures, a generalization of the concept of extremal length.


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Keywords: Quasiconformal mappings, set of finite perimeter, <!– MATH ${\operatorname {ACL} _n}$ –> <IMG WIDTH="58" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${\operatorname {ACL} _n}$"> homeomorphism, module of system of measures
Article copyright: © Copyright 1973 American Mathematical Society