Approximate differentiability of mappings of Carnot-Carathéodory spaces
In 1951, Whitney obtained a result that can be considered as the final representation of a initial theorem first proved by Stepanoff (1923, 1925). The theorem of Whitney states that if \(P\subset\mathbb R^n\) is a measurable and bounded set, and \(f:P\to\mathbb R^m\) is a measurable function, then the following conditions are equivalent: (1) the mapping \(f\) is approximately differentiable almost everywhere in \(P\); (2) the mapping \(f\) has approximate derivatives with respect to each variable almost everywhere in \(P\); (3) there is a countable family of disjoint sets \(Q_1,Q_2,\dots\) such that \(|P\backslash\cup_iQ_i|=0\) and every restriction \(f|_{Q_i}\) is a Lipschitz mapping; (4) for every \(\varepsilon>0\), there are a closed set \(Q\subset P\) such that \(|P\backslash Q|<\varepsilon\), and a \(C^1\)-smooth mapping \(g:P\to\mathbb R^m\) such that \(g=f\) in \(Q\).NEWLINENEWLINEIn the paper under review, the authors obtain a partial generalization of the above theorem for mappings of Carnot-Carathéodory spaces. The paper is organized as follows. In the second section, the authors give the basic notions and structures concerning Carnot-Carathéodory spaces. In Subsections 2.2 and 2.4 they specify a metric and coordinate system in the Carnot-Carathéodory spaces. In Subsection 2.5 they build a special coordinate system of the second kind based on the compositions of the integral lines of the horizontal vector fields. As a consequence of this result, they obtain a Chow-Rashevsky theorem for \(C^1\)-smooth vector fields. They formulate also a local approximation theorem for Carnot-Carathéodory metric.NEWLINENEWLINEThe fourth section is devoted to the proof of the theorem on approximate differentiability; the authors state the main result and show trivial implications. In the last section, they present an area formula for approximately differentiable mappings.NEWLINENEWLINENEWLINELet us end this review with the statement of the main result of the paper: Let \(\mathcal M, \widetilde{\mathcal M}\) be Carnot-Carathéodory spaces, \(E\subset\mathcal M\) be a measurable subset of \(\mathcal M\) and \(f:E\to\widetilde{\mathcal M}\) be a measurable mapping. The following conditions are equivalent: (1) the mapping \(f\) is approximately differentiable almost everywhere in \(E\); (2) the mapping \(f\) has approximate derivatives along the basic horizontal vector fields almost everywhere in \(E\); (3) there is a sequence of disjoint sets \(Q_1,Q_2,\dots\) such that \(|E\backslash\cup_iQ_i|=0\) and every restriction \(f|_{Q_i}\) is a Lipschitz mapping.
scientific article; zbMATH DE number 6222642