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**A characteristic property of a class of space homeomorphisms.**
*(Russian)*
Zbl 0725.30010

Questions of analysis and approximation, Collect. Sci. Works, Kiev, 81-88 (1989).

[For the entire collection see Zbl 0685.00003.]

This is a short note on particular properties of some classes of homeomorphisms f: \(D\to D^*\) between bounded open sets in \({\mathbb{R}}^ n\). A pair (E,G) consisting of an open set G in \({\mathbb{R}}^ n\) and a subset \(E\subset G\) compact in \({\mathbb{R}}^ n\) is called a condensator. For any \(\alpha\) with \(1\leq \alpha \leq n\), such a condensator has a real- valued \(\alpha\)-capacity which is defined by an infimum of integrals over G, taken over a certain class of functions \(\phi: G\to [0,1]\). Theorem 1 of this paper says that a homeomorphism f: \(D\to D^*\) is almost everywhere differentiable in D if f satisfies certain inequalities with respect to the \(\alpha\)-capacities of condensators (E,G) in D, i.e., \(E\subset G\subseteq D\). The second part of the paper deals with characteristic values attached to homeomorphisms f: \(D\to D^*\), which are defined by particular integrals (over D) of functions constructed from (pointwise) generalized derivatives of f. Theorem 2 gives then a characterization of those homeomorphisms that have prescribed bounded characteristic integrals in the above sense. The equivalent conditions are expressed in terms of (then existing) quasi-additive set-valued functions on D, which satisfy boundedness conditions for condensators in D.

This is a short note on particular properties of some classes of homeomorphisms f: \(D\to D^*\) between bounded open sets in \({\mathbb{R}}^ n\). A pair (E,G) consisting of an open set G in \({\mathbb{R}}^ n\) and a subset \(E\subset G\) compact in \({\mathbb{R}}^ n\) is called a condensator. For any \(\alpha\) with \(1\leq \alpha \leq n\), such a condensator has a real- valued \(\alpha\)-capacity which is defined by an infimum of integrals over G, taken over a certain class of functions \(\phi: G\to [0,1]\). Theorem 1 of this paper says that a homeomorphism f: \(D\to D^*\) is almost everywhere differentiable in D if f satisfies certain inequalities with respect to the \(\alpha\)-capacities of condensators (E,G) in D, i.e., \(E\subset G\subseteq D\). The second part of the paper deals with characteristic values attached to homeomorphisms f: \(D\to D^*\), which are defined by particular integrals (over D) of functions constructed from (pointwise) generalized derivatives of f. Theorem 2 gives then a characterization of those homeomorphisms that have prescribed bounded characteristic integrals in the above sense. The equivalent conditions are expressed in terms of (then existing) quasi-additive set-valued functions on D, which satisfy boundedness conditions for condensators in D.

Reviewer: W.Kleinert (Berlin)

### MSC:

30C65 | Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations |

26D10 | Inequalities involving derivatives and differential and integral operators |

26E25 | Set-valued functions |

54C30 | Real-valued functions in general topology |