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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.
