Homeomorphisms of the Heisenberg group preserving BMO

TL;DR

This paper offers a novel geometric proof linking quasiconformal mappings to BMO function preservation, extending the classical Euclidean result to Heisenberg and Carnot groups.

Contribution

It introduces a new geometric proof of Reimann's theorem, applicable to Heisenberg and stratified nilpotent Carnot groups, broadening the theorem's scope.

Findings

New geometric proof of Reimann's theorem

Extension of the theorem to Heisenberg groups

Applicability to stratified nilpotent Carnot groups

Abstract

We provide a new geometric proof of Reimann's theorem characterizing quasiconformal mappings as the ones preserving functions of bounded mean oscillation. While our proof is new already in the Euclidean spaces, it is applicable in Heisenberg groups as well as in more general stratified nilpotent Carnot groups.