On prime end distortion estimates of mappings with the Poletsky condition in domains with the Poincar´e inequality
Abstract
This article is devoted to the study of mappings with bounded and
finite distortion defined in some domain of the Euclidean space. We
consider mappings that satisfy some upper estimates for the
distortion of the modulus of families of paths, where the order of
the modulus equals to $p,$ $n-1<p\leqslant n.$ The main problem
studied in the manuscript is the investigation of the boundary
behavior of such mappings, more precisely, the distortion of the
distance under mappings near boundary points. The publication is
primarily devoted to definition domains with ``bad boundaries'', in
which the mappings not even have a continuous extension to the
boundary in the Euclidean sense. However, we introduce the concept
of a quasiconformal regular domain in which the specified continuous
extension is valid and the corresponding distance distortion
estimates are satisfied; however, both must be understood in the
sense of the so-called prime ends. More precisely, such estimates
hold in the case when the mapping acts from a quasiconformal regular
domain to an Ahlfors regular domain with the Poincar\'e inequality.
The consideration of domains that are Ahlfors regular and satisfy
the Poincar\'e inequality is due to the fact that, lower estimates for
the modulus of families of paths through the diameter of the
corresponding sets hold in these domains. (There are the so-called
Loewner-type estimates). We consider homeomorphisms and mappings
with branching separately. The main analytical condition under which
the results of the paper were obtained is the finiteness of the
integral averages of some majorant involved in the defining modulus
inequality under infinitesimal balls. This condition includes the
situation of quasiconformal and quasiregular mappings, because for
them the specified majorant is itself bounded in a definition
domain. Also, the results of the article are valid for more general
classes for which Poletsky-type upper moduli inequalities are
satisfied, for example, for mappings with finite length distortion.
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