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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
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, $ {\operatorname{ACL} _n}$ homeomorphism, module of system of measures
Article copyright: © Copyright 1973 American Mathematical Society