It is well known that the modulus method is one of the
most powerful tools for studying mappings. Distortion estimates of
the modulus of paths families are established in many known classes,
in particular, the modulus does not change under conformal mappings,
is finitely distorted under qu\-a\-si\-con\-for\-mal mappings, at
the same time, its behavior under mappings with finite distortion
depends on the dilatation coefficient. One common case is the study
of mappings for which this coefficient is integrable in the domain.
In the context of our research, this case has been studied in detail
in our previous publications and its consideration has mostly been
completed. In particular, we obtained results on the local,
boundary, and global behavior of homeomorphisms, the inverse of
which satisfy the weight Poletsky inequality, provided that the
corresponding majorant is integrable. In contrast, the focus in this
paper is on mappings for which a similar inequality may contain non
integrable weights. Study of the situation of non integrable
majorants, in turn, is associated with the specific behavior of the
weight modulus of the annulus, which is achieved on a certain
function and up to constant is equal to $(n-1)$-degree of the Lehto
integral. To the same extent, these results are also related to
finding the extremal in the weight modulus of the ring. The basic
theorem contains the result about equicontinuity of homeomorphisms
with the inverse Poletsky inequality, when the corresponding weight
has finite integrals on some set of spheres, and the set of
corresponding radii of these spheres must have a positive Lebesgue
measure. According to Fubini's theorem, the mentioned result
summarizes the corresponding statement for any integrable majorants
and is fundamental in the sense that it is easy to give examples of
non integrable functions with finite integrals by spheres. In
addition, since conformal and quasiconformal mappings satisfy the
Poletsky inequality with a constant majorant in the forward and
inverse directions, the basic theorem may be considered as a
generalization of previously known statements in these classes. Note
that the main result does not contain any geometric constraints on
the definition and image domains of the mappings, in particular, the
definition domain is assumed to be arbitrary, and the image domain
is supposed to be only a bounded domain in Euclidean $n$-dimensional
space. The proof of the main theorem is given by the contradiction,
namely, we assume that the statement about equicontinuity of the
corresponding family of mappings is incorrect, and we obtain a
contradiction to this assumption due to upper and lower estimates of
the modulus of families of paths.
3x + 1 Problem: Iterative Operations Neither Cause Loops nor Diverge to Infinity
