On the Gehring type condition and properties of mappings

The goal of this work is to obtain an analytical description of mappings satisfying some capacity inequality (so called $G_p$-condition): we study mappings for which the $G_p$-condition holds for a cubical ring. In other words, we replace rings with concentric spheres in the $G_p$-condition by rings with concentric cubes. We obtain new analytic properties of homeomophisms in $\mathbb R^n$ meeting Gehring type capacity inequality. In this paper the capacity inequality means that the capacity of the image of a cubical ring is controlled by the capacity of the given ring. From the analytic properties we conclude some geometric properties of mappings under consideration. The method is new and is based on an equivalent analytical description of such mappings previously established by the author. Our arguments are based on assertions and methods discovered in author's recent papers [1] and [2] (see also some references inside). Then we obtain geometric properties of these mappings.