Local behavior of homeomorphisms with a finite mean over spheres

Abstract

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.