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

TL;DR

This paper extends $L^2$-estimates to the $d$-operator on super forms within the framework of $R$-K"ahler metrics, providing existence results analogous to classical complex analysis estimates.

Contribution

It introduces $R$-K"ahler metrics on $R^n$ and establishes $L^2$-estimates for the $d$-operator on super forms, generalizing H"ormander's estimates.

Findings

Established existence theorems for $deta=eta$ on super forms.

Derived $L^2$-estimates analogous to those in complex K"ahler geometry.

Extended classical analysis techniques to the setting of super forms.

Abstract

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.