## 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

Zbl 0546.30019
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