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Stability in the \(C\)-norm and \(W^1_{\infty}\)-norm of classes of Lipschitz functions of one variable. (Russian, English) Zbl 1014.26009

Sib. Mat. Zh. 43, No. 5, 1026-1045 (2002); translation in Sib. Math. J. 43, No. 5, 827-842 (2002).
In the framework of Kopylov’s \(\omega\)-stability concept, see [A. P. Kopylov, Sib. Math. J. 25, 274-283 (1984; Zbl 0546.30019)], the author studies some stable classes of Lipschitz functions of one real variable and gives an exhaustive classification for \(\omega\)-stable classes of Lipschitz mappings of intervals of the real axis \(\mathbb R\) with values in \(\mathbb R^m\), \(m \geq 1\). It turns out that each \(\omega\)-stable class is generated by some compact set \(G \subset\mathbb R^m\) and partial preorder \(\pi\) on \(G\) by the following rule: the class consists of all Lipschitz mappings \(g\:\Delta\subset\mathbb R\to\mathbb R^m\) such that \(g'(x) \in G\) a.e. and the derivative \(g'\) increases with respect to \(\pi\) (Theorem 1).
Using this complete description for the \(\omega\)-stable classes of mappings of intervals in \(\mathbb R\) to \(\mathbb R^m\), the author proves that, for all such classes, stability estimates hold in the \(C\)-norm as well as in the \(W_{\infty}^1\)-norm (Theorem 2).

MSC:

26A16 Lipschitz (Hölder) classes
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26A21 Classification of real functions; Baire classification of sets and functions
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems

Citations:

Zbl 0546.30019
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