Openness and Discreteness of Mappings of Finite Distortion on Carnot Groups

Abstract

We prove that a mapping of finite distortion \( f:\Omega\to 𝔾 \) in a domain \( \Omega \) of an \( H \)-type Carnot group \( 𝔾 \) is continuous, open, and discrete provided that the distortion function \( K(x) \) of \( f \) belongs to \( L_{p,\operatorname{loc}}(\Omega) \) for some \( p>\nu-1 \). In fact, the proof is suitable for each Carnot group provided it has a \( \nu \)-harmonic function of the form \( \log\rho \), where the homogeneous norm \( \rho \) is \( C^{2} \)-smooth.