On convergence and compactness of space homeomorphisms

TL;DR

This paper proves theorems on the convergence and compactness of space homeomorphisms, especially ring $Q$-homeomorphisms, with implications for Sobolev mappings.

Contribution

It establishes new convergence and compactness results for classes of ring $Q$-homeomorphisms, including conditions for compactness based on mean oscillation.

Findings

Family of ring $Q$-homeomorphisms fixing two points is compact if $Q$ has finite mean oscillation.

Theorems on convergence of general space homeomorphisms are proved.

Results have potential applications to Sobolev's mappings.

Abstract

Various theorems on convergence of general space homeomorphisms are proved and, on this basis, theorems on convergence and compactness for classes of the so-called ring $Q$--homeomorphisms are obtained. In particular, it was established by us that a family of all ring $Q$--homeomorphisms $f$ in ${\Bbb R}^n$ fixing two points is compact provided that the function $Q$ is of finite mean oscillation. These results will have wide applications to Sobolev's mappings.