An extension and trace theorem for functions of H-bounded variation in Carnot groups of step 2

TL;DR

This paper extends functions of bounded H-variation in Carnot groups of step 2 to larger domains and establishes a trace theorem for such functions on H-admissible domains, broadening the understanding of BV functions in these geometric settings.

Contribution

It introduces an extension theorem for BV functions in Carnot groups of step 2 and proves a trace theorem for H-admissible domains, including non-characteristic and symmetric domains.

Findings

Extension of BV functions from H-admissible domains to the entire group.

Identification of classes of H-admissible domains, including non-characteristic and symmetric ones.

Counterexample of a domain that is not H-admissible despite being $C^{1,eta}$.

Abstract

This paper provides an extension for a function $u \in BV_H(\Omega)$ to a function $u_0 \in BV_H(G)$ when $\Omega$ is ``H-admissible,'' and G is a step 2 Carnot group. It is shown that H-admissible domains include non-characteristic domains and domains in groups of Heisenberg type which have a partial symmetry about characteristic points. An example is given of a domain that is $C^{1,\alpha}$, $\alpha <1$, that is not H-admissible. Further, when $\Omega$ is H-admissible a trace theorem is proved for $u \in BV_H(\Omega)$.