On equivalence of the analytical and geometrical definitions of mappings with an $s$-averaged characteristic

In this paper we continue to develop the geometric method of studying properties of space mappings with an $s$-averaged characteristic. The method is based on the characteristic distortion law for modules of families of curves.
In recent decades, the theory of mappings with bounded distortion in the $n$-dimensional Euclidean space is one of the most meaningful and intensively developed branches of the function theory. These mappings were introduced and systematically studied in works by Yu. G. Reshetnyak, published since 1966. A part of Yu. G. Reshetnyak's results is contained in [1].
The most powerful tools used in the study of space mapping properties are methods that study invariance properties of conformal capacity or the module of families of curves. The equivalence of analytical and geometrical (expressed in terms of the conformal capacity of condensers) definitions of mappings with bounded distortion was introduced by O. Martio, S. Rickman and J. Väisälä [2]. In [3], the equivalence was proved for definitions of homeomorphic mappings with distortion bounded on the average. The equivalence for mappings nonhomeomorphic with distortion bounded on the average was considered in [4].
In the presented paper, we give a geometric definition of mappings with an $s$-averaged characteristic using the concept of a spherical $p$-module of a family of curves. This definition can also be interpreted as a generalization of the method of modules for mappings with an $s$-averaged characteristic. We also study these mappings properties and prove the equivalence of the geometric and analytic definitions using the distortion theorems.