## 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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