# Discrete convolutions of BV functions in quasiopen sets in metric spaces

## TL;DR

This paper develops a new approximation technique for BV functions in quasiopen sets within metric spaces, extending classical methods and providing new insights into the structure of BV functions.

## Contribution

It introduces a discrete convolution method in quasiopen sets and demonstrates approximation of BV functions with controlled jump sets, in a general metric space setting.

## Key findings

- BV functions can be approximated by BV functions with finite measure jump sets
- The technique applies in metric spaces with doubling measure and Poincaré inequality
- Results are new even in Euclidean spaces

## Abstract

We study fine potential theory and in particular partitions of unity in quasiopen sets in the case $p=1$. Using these, we develop an analog of the discrete convolution technique in quasiopen (instead of open) sets. We apply this technique to show that every function of bounded variation (BV function) can be approximated in the BV and $L^{\infty}$ norms by BV functions whose jump sets are of finite Hausdorff measure. Our results seem to be new even in Euclidean spaces but we work in a more general complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality.