The spaces $\mathrm{H}^n(\mathfrak{osp}(1|2),M)$ for some weight modules $M$

We entirely compute the cohomology for a natural and large class of $\mathfrak{osp}(1|2)$ modules $M$. We study the restriction to the $\mathfrak{sl}(2)$ cohomology of $M$ and apply our results to the module $M={\mathfrak D}_{\lambda,\mu}$ of differential operators on the super circle, acting on densities.