A characterization of BV and Sobolev functions via nonlocal functionals in metric spaces

We study a characterization of BV and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincaré inequality. Compared with previous works, we consider more general functionals. We also give a counterexample in the case p=1 demonstrating that, unlike in Euclidean spaces, in metric measure spaces the limit of the nonlocal functionals is only comparable, not necessarily equal, to the variation measure ‖Df‖(Ω).