The Binary $\mathfrak{a}\mathfrak{f}\mathfrak{f}(n\mathrm{\mid }1)$-Invariant Differential Operators On Weighted Densities On The Superspace ${\mathbb{R}}^{1\mathrm{\mid }n}$ And $\mathfrak{a}\mathfrak{f}\mathfrak{f}(n\mathrm{\mid }1)$-Relative Cohomology

We consider the $\mathfrak{a}\mathfrak{f}\mathfrak{f}(n\mathrm{\mid }1)-$module structure on the spaces of differential bilinear operators acting on the superspaces of weighted densities. We classify $\mathfrak{a}\mathfrak{f}\mathfrak{f}(n\mathrm{\mid }1)-$invariant binary differential operators acting on the spaces of weighted densities. This result allows us to compute the first $\mathfrak{a}\mathfrak{f}\mathfrak{f}(n\mathrm{\mid }1)-$relative differential cohomology of $\mathcal{K}(n)$ with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities.