# Rough traces of $BV$ functions in metric measure spaces

## TL;DR

This paper develops a Maz'ya-type approach to define and analyze rough traces of BV functions in metric measure spaces, establishing an integration by parts formula and comparing with classical trace notions.

## Contribution

It introduces a new framework for rough traces of BV functions in metric spaces and compares it with existing Lebesgue-point based methods.

## Key findings

- Established an integration by parts formula involving rough traces.
- Identified conditions for equivalence between rough trace and classical trace notions.
- Extended BV trace theory to doubling metric measure spaces supporting Poincaré inequalities.

## Abstract

Following a Maz'ya-type approach, we adapt the theory of rough traces of functions of bounded variation ($BV$) in the context of doubling metric measure spaces supporting a Poincar\'e inequality. This eventually allows for an integration by parts formula involving the rough trace of such a function. We then compare our analysis with the discussion done in a recent work by P. Lahti and N. Shanmugalingam, where traces of $BV$ functions are studied by means of the more classical Lebesgue-point characterization, and we determine the conditions under which the two notions coincide.