Lipschitz and bi-Lipschitz functions. (English) Zbl 0782.26007

Jones, Peter W.

Rev. Mat. Iberoam. 4, No. 1, 115-121 (1988).

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

Keywords:

Hausdorff measure; singular integrals; BMO; harmonic functions; Sobolev space; bi-Lipschitz functions; Lipschitz mapping; Hausdorff content