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Lusin’s condition (N) and mappings of the class \(W^{1,n}\). (English) Zbl 0812.30007

It is shown that every continuous \(W^{1,n}\)-mapping \(f: G\to \mathbb{R}^ n\) satisfies Lusin’s condition (N) provided that \(f\) is either Hölder continuous or an open mapping. Moreover, every \(W^{1,n}\)-mapping satisfies (N) outside a set of Hausdorff dimension zero. This yields Øksendal’s theorem on the boundary behavior of continuous \(W^{1,n}\)- mappings. Also new results on the boundary behavior of Dirichlet finite and quasiconformal mappings in \(\mathbb{R}^ n\), \(n\geq 2\), are proved. A Peano type example of a continuous \(W^{1,n}\)-mapping not satisfying (N) is presented.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
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