  We use methods from birational geometry to study the Hodge and weight
filtrations on the localization along a hypersurface. We focus on the lowest
piece of the Hodge filtration of the submodules arising from the weight
filtration. This leads to a sequence of ideal sheaves called weighted
multiplier ideals. The last ideal of this sequence is a multiplier ideal (and a
Hodge ideal), and we prove that the first is the adjoint ideal. We also study
the local and global properties of weighted multiplier ideals and their
applications to singularities of hypersurfaces of smooth varieties.
