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MATHEMATICS
On the module of continuity of mappings with an $s$-averaged characteristic
A. N. Malyutina,
U. K. Asanbekov Tomsk State University, Tomsk, Russian Federation
Abstract:
We continue studying analytical properties of non-homeomorphic mappings with an
$s$-averaged characteristic. O. Martio proposed the theory of
$\mathcal{Q}$-homeomorphisms (2001). The concept of
$\mathcal{Q}$-homeomorphisms was extended to maps with branching (2004). In this paper, we study analytical properties of non-homeomorphic mappings with an
$s$-averaged characteristic and consider the question of continuity of mappings with an
$s$-averaged characteristic. By the well-known Sobolev theorem, a function of class
$W^1_{s,loc}(R^n)$ for is equivalent to a continuous function. This property does not hold when
$s<n$. The authors presented such example for mappings with an
$s$-averaged characteristic in 2016.
In this paper, we generalize the result obtained earlier to a more general class of mappings with an
$s$-averaged characteristic. Relevant examples are built. The purpose of this paper is to indicate the necessary conditions under which mappings from classes and subclasses of mappings with an
$s$-averaged characteristic
$1<s<n$ will be continuous. Here,
$n$ is the dimension of the space, and
$s$ is the averaging parameter. We proved a theorem in which we obtain necessary conditions for the continuity of such mappings that are with the abovementioned
$s$. Earlier, such a result was obtained for functions of the class
$W^1_{s,loc}(R^n)$. The theorem is an analogue of the Mori lemma.
Keywords:
spatial mappings with an $s$-averaged characteristic, modulus of continuity, mapping class.
UDC:
517.54
MSC: 26B30,
26B35 Received: 17.03.2019
DOI:
10.17223/19988621/59/2
Fulltext:
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