Sharp conditions for the validity of the Bourgain-Brezis-Mironescu formula

Following the seminal paper by Bourgain, Brezis and Mironescu, we focus on the asymptotic behavior of some nonlocal functionals that, for each $u\in L^2(\mathbb{R}^N)$, are defined as the double integrals of weighted, squared difference quotients of $u$. Given a family of weights $\{\rho_\epsilon\}$, $\epsilon\in (0,1)$, we devise sufficient and necessary conditions on $\{\rho_\epsilon\}$ for the associated nonlocal functionals to converge as $\epsilon\to 0$ to a variant of the Dirichlet integral. Finally, some comparison between our result and the existing literature is provided.