Cohomology of $\mathfrak{a}\mathfrak{f}\mathfrak{f}(1)$ and $\mathfrak{a}\mathfrak{f}\mathfrak{f}(1\mathrm{\mid }1)$ acting on the space of $n$-ary differential operators on the superspace ${\mathbb{R}}^{1\mathrm{\mid }1}$

We consider the $\mu$-densities spaces ${\mathcal{F}}_{\mu }$ with $\mu \in \mathbb{R}$, we compute the space ${\mathrm{H}}_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^1(\mathfrak{a}\mathfrak{f}\mathfrak{f}(1),{\mathrm{D}}_{\lambda ,\mu })$ where $\lambda =({\lambda }_1,\dots ,{\lambda }_n)\in {\mathbb{R}}^n$ and ${\mathrm{D}}_{\lambda ,\mu }$ is the space of $n$-ary differential operators from ${\mathcal{F}}_{{\lambda }_1}\otimes \cdots \otimes {\mathcal{F}}_{{\lambda }_n}$ to ${\mathcal{F}}_{\mu }$. We also compute the super analog space ${\mathrm{H}}_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^1(\mathfrak{a}\mathfrak{f}\mathfrak{f}(1\mathrm{\mid }1),{\mathfrak{D}}_{\lambda ,\mu })$.