Cohomology of $\mathfrak {osp}(2|2)$ Acting on Spaces of Linear Differential Operators on the Superspace $\mathbb{R}^{1|2}$

We compute the first differential cohomology of the orthosymplectic Lie superalgebra $\mathfrak{osp}(2|2)$ with coefficients in the superspace of linear differential operators acting on the space of weighted densities on the $(1,2)$-dimensional real superspace. We also compute the same, but $\mathfrak{osp}(1|2)$-relative, cohomology. We explicitly give $1$-cocycles spanning these cohomologies. This work is the simplest generalization of a result from [1].