On the second cohomology of the affine superalgebra

We compute the second cohomology of the affine Lie superalgebra $\mathfrak{aff}(1|1)$ on the $(1,1)$-dimensional real superspace with coefficients in the superspace $\mathfrak{D}_{\lambda,\nu;\mu}$ of bilinear differential operators acting on weighted densities and with coefficients in the superspace $\mathfrak{D}_{\lambda_1,\lambda_2,\lambda_3;\mu}$ of trilinear differential operators acting on weighted densities. We explicitly give $2$-cocycles spanning these cohomology.