Symmetric differentials on weighted hypersurfaces

We study symmetric differentials on weighted hypersurfaces. Let $k$ be an algebraically closed field and let $$ X_d\subset \mathbb{P}_k(a_0,\ldots,a_N),\qquad N\ge3, $$ be a weighted hypersurface, with $a_{\min}:=\min_i a_i$. In characteristic zero we prove twisted vanishing in the range $t<m a_{\mi