Capacity estimates, Liouville's theorem, and singularity removal for mappings with bounded $(p,q)$-distortion

The mappings with bounded weighted $(p,q)$-distortion are natural generalizations of the class of mappings with bounded distortion which appears as a doubly indexed scale for $p=q=n$ in the absence of weight functions. In case $n-1<q\le p=n$, the mappings with bounded $(p,q)$-distortion were studied previously in a series of articles under the additional assumption that the mapping enjoys Luzin's $\mathscr N$-property. In this article we present the first facts of the theory of mappings with bounded $(p,q)$-distortion which are obtained without additional analytical assumptions. The core of the theory consists of the new analytical properties of pushforward functions; in particular, we prove that the gradient of the pushforward function vanishes almost everywhere on the image of the branch set. Some estimates are given on the capacity of the images of condensers under mappings with bounded $(p,q)$-distortion. We obtain Liouville-type theorems and the singularity removal theorems for the mappings of this class, and we apply these theorems to classifying manifolds.