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In this paper, we introduce a coherent subsheaf of Saito's $S$-sheaf, whichis a combination of the $S$-sheaf and the multiplier ideal sheaf. We constructits $L^2$-Dolbeault resolution, which generalizes MacPherson's conjecture onthe $L^2$ resolution of the Grauert-Riemenschneider sheaf. We also provevarious vanishing theorems for the $S$-sheaf (Saito's vanishing theorem,Kawamata-Viehweg vanishing theorem and some new ones like Nadel vanishingtheorem) transcendentally. Finally, we discuss some applications of our resultson the relative version of Fujita's conjecture (e.g. Kawamata's conjecture).

$L^2$-Dolbeault resolution of the lowest Hodge piece of a Hodge module