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Failure of the condition N below $$W^{1,n}$$. (English) Zbl 1027.26015
The Lusin condition N says that given a continuous mapping $$f$$ from a bounded subset $$\Omega$$ of $$\mathbb{R}^n$$ into $$\mathbb{R}^n$$, $$n\geq 2$$, then $$f$$ maps every subset of $$\Omega$$ of zero measure to a set of zero measure. The well-known result by M. Marcus and V. Mizel [Bull. Am. Math. Soc. 79, 790-795 (1973; Zbl 0275.49041)] states that $$f\in W^{1,1}(\Omega,\mathbb{R}^n)$$ satisfies the condition N under the assumption $$|Df|\in L^p(\Omega)$$ for some $$p>n$$. The condition N may fail if this assumption holds only for $$p\leq n$$ and then there is a question about the size of an exceptional set, i.e., the set outside of which the condition holds. J. Malý and O. Martio [J. Reine Angew. Math. 458, 19-36 (1995; Zbl 0812.30007)] proved that for $$f\in W^{1,n}(\Omega,\mathbb{R}^n)$$ exceptional sets have the Hausdorff dimension zero.
The author deals with the case below $$W^{1,n}(\Omega,\mathbb{R}^n)$$. He proves that for $$Q_0=\left[0,1\right]^n$$ and $$f\in W^{1,1}(Q_0,\mathbb{R}^n)$$ such that $$\underset{0<\varepsilon\leq n-1} \sup \varepsilon\int_{Q_0}|Df(x)|^{n-\varepsilon}dx<\infty$$ one cannot, in general, find an exceptional set of Hausdorff dimension smaller than $$n$$.

MSC:
 26B35 Special properties of functions of several variables, Hölder conditions, etc. 74B20 Nonlinear elasticity 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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