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 behaviour 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 $\{\rho _{\varepsilon} \}$, $\varepsilon \in (0,\,1)$, we devise sufficient and necessary conditions on $\{\rho _{\varepsilon} \}$ for the associated nonlocal functionals to converge as $\varepsilon \to 0$ to a variant of the Dirichlet integral. Finally, some comparison between our result and the existing literature is provided.