The Gauss-Green theorem in stratified groups

TL;DR

This paper develops a new theoretical framework for divergence-measure fields in stratified Lie groups, enabling Gauss-Green formulas for low-regularity vector fields on finite perimeter sets.

Contribution

It introduces divergence-measure fields in stratified groups, generalizing BV fields and establishing foundational Gauss-Green theorems in this noncommutative setting.

Findings

Established properties of divergence-measure fields in stratified groups

Proved Gauss-Green formulas for low-regularity vector fields

Extended classical divergence theory to noncommutative geometric context

Abstract

We lay the foundations for a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Such vector fields form a new family of function spaces, which generalize in a sense the $BV$ fields. They provide the most general setting to establish Gauss-Green formulas for vector fields of low regularity on sets of finite perimeter. We show several properties of divergence-measure fields in stratified groups, ultimately achieving the related Gauss-Green theorem.