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Lipschitz and bi-Lipschitz functions. (English) Zbl 0782.26007
Let \(f\) be a Lipschitz mapping of a unit cube \(Q_ 0\subset\mathbb{R}^ n\) into \(\mathbb{R}^ m\). The author proves that for each \(\delta>0\) there exist \(M\in\mathbb{R}\) and sets \(K_ 1,\dots,K_ M\subset Q_ 0\) such that the Hausdorff content of \(f\Bigl(Q_ 0\backslash \bigcup^ M_{j=1} K_ j\Bigr)\) is less than \(\delta\) and \(| f(x)- f(y)|<(\delta/2)| x-y|\), \(x,y\in K_ j\), \(j=1,\dots,M\). The result strengthens a theorem of G. David [same journal 4, No. 1, 73-114 (1988; Zbl 0696.42011)].

MSC:
26B35 Special properties of functions of several variables, Hölder conditions, etc.
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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