  We consider the fractional Laplacian operator $(-\Delta)^s$ (let $ s \in
(0,1) $) on Euclidean space and investigate the validity of the classical
integration-by-parts formula that connects the $ L^2(\mathbb{R}^d) $ scalar
product between a function and its fractional Laplacian to the nonlocal norm of
the fractional Sobolev space $ \dot{H}^s(\mathbb{R}^d) $. More precisely, we
focus on functions belonging to some weighted $ L^2 $ space whose fractional
Laplacian belongs to another weighted $ L^2 $ space: we prove and disprove the
validity of the integration-by-parts formula depending on the behaviour of the
weight $ \rho(x) $ at infinity. The latter is assumed to be like a power both
near the origin and at infinity (the two powers being possibly different). Our
results have direct consequences for the self-adjointness of the linear
operator formally given by $ \rho^{-1}(-\Delta)^s $. The generality of the
techniques developed allows us to deal with weighted $ L^p $ spaces as well.
