Theorem 5.16.26 (Ohsawa–Takegoshi)151D57BCA31B

Proof. This is local on $Y$. In coordinates in which $Y$ is a coordinate subspace, take a germ $f\in\mathcal I(\varphi|_Y)$. The weighted restriction theorem extends $f$ to a holomorphic germ $F$ in the ambient coordinates with $|F|^2\exp(-\varphi)$ integrable. Thus $F\in\mathcal I(\varphi)$ and its restriction is $f$. This is precisely the inclusion in the image ideal $\operatorname{Res}_Y\mathcal I(\varphi)$. The hypothesis that the restricted weight is not identically $-\infty$ ensures that its multiplier ideal has the intended nontrivial local definition; the earlier theorem also covers the zero-ideal case.