Discrete convolutions of $$\mathrm {BV}$$ functions in quasiopen sets in metric 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 ($$\mathrm {BV}$$ function) can be approximated in the $$\mathrm {BV}$$ and $$L^{\infty }$$ norms by $$\mathrm {BV}$$ 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 Poincare inequality.