It was proven in BBD (see Corollary 5.3.2) that for an open immersion $j$ the functor $j_{!*}$ preserves weights of mixed sheaves. The proof relies on several previous results; it is especially complicated in the case when $j$ is not affine. Does an easier proof (or a plan of it:)) exist? I would like to have a proof that (mostly) relies on the properties of $j_{!*}$ (and on the 'formal' properties of weights).