$L^2$-estimates for the $d$-operator acting on super forms

In the setting of super forms developed in a previous article by the author, we introduce the notion of $\mathbb{R}$-K\"ahler metrics on $\mathbb{R}^{n}$. We consider existence theorems and $L^{2}-$estimates for the equation $d\alpha=\beta$, where $\alpha$ and $\beta$ are super forms, in the spirit of H\"ormander's $L^{2}-$estimates for the $\bar{\partial}-$equation on a complex K\"ahler manifold.