Generalized study of the operator $\hat{I}\pm {\partial }^k{\bar{\partial }}^k+{\hat{I}}^2{\bar{\partial }}^k+{\hat{I}}^3{\partial }^k+c$ in weighted Hilbert space $L^2(\mathbb{C},{\mathrm{e}}^{-|z{|}^2})$

Abstract

By Hörmander's $L^2$-method, we study the operator $\hat{I}\pm {\partial }^k{\bar{\partial }}^k+{\hat{I}}^2{\bar{\partial }}^k+{\hat{I}}^3{\partial }^k+c$ for any order $k$ with $\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3\in \mathbb{R}$ such that $(\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3)\ne (0,0,0)$ in the weighted Hilbert space $L^2(\mathbb{C},{\mathrm{e}}^{-|z{|}^2})$. We prove the existence of its right inverse which is also a bounded operator. Subsequently we will study two cases that arise from this operator, namely: (1) Case where $\hat{I}\pm ={\hat{I}}^3=0$: The operator ${\hat{I}}^2{\bar{\partial }}^k+c$ with $|{\hat{I}}^2|\ge 1$. (2) Case where ${\hat{I}}^2={\hat{I}}^3=0$: The operator $\hat{I}\pm {\partial }^k{\bar{\partial }}^k+c$ with $|\hat{I}\pm |\ge 1$.