In a beautiful paper published in 2002, V. Maz'ya and I. Verbitsky have characterised distribution V such that the multiplication by V if continuous from the first homogeneous Sobolev space to its dual. With M. Lansade, we have study the same question and find that the same answer is valid on weighted Riemannian manifold satisfying to Poincaré inequalities and the doubling condition.
