  Let $X$ be a complex space and $M$ a pure Hodge module with strict support
$X$. The purpose of this paper is to introduce a coherent subsheaf
$S(M,\varphi)$ of M. Saito's $S(M)$ which is a combination of $S(M)$ and the
multiplier ideal sheaf $\mathscr{I}(\varphi)$ while constructing a resolution
of $S(M,\varphi)$ by differential forms with certain $L^2$-boundary conditions.
This could be viewed as a wide generalization of MacPherson's conjecture on the
$L^2$-representation of the Grauert-Riemenschneider sheaf. As applications,
various vanishing theorems for $S(M)$ (Saito's vanishing, Kawamata-Viehweg
vanishing and some new ones like Nadel vanishing, partial vanishing) are proved
via standard differential geometrical arguments.
