Vanishing Theorem for Hodge Ideals on smooth hypersurfaces

We use a Koszul-type resolution to prove a weak version of Bott’s vanishing theorem for smooth hypersurfaces in ${\mathbb{P}}^n$ and use this result to prove a vanishing theorem for Hodge ideals associated with an effective Cartier divisor on a hypersurface. This extends an earlier result of Mustata and Popa.