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

Abstract

By Hörmander's $L^2$-method, we study the operator $\alpha {\partial }^k{\bar{\partial }}^k+\beta {\bar{\partial }}^k+\gamma {\partial }^k+c$ for any order $k$ with $\alpha ,\beta ,\gamma \in \mathbb{R}$ such that $(\alpha ,\beta ,\gamma )\ne (0,0,0)$ 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 $\alpha =\gamma =0$: The operator $\beta {\bar{\partial }}^k+c$ with $|\beta |\ge 1$. (2) Case where $\beta =\gamma =0$: The operator $\alpha {\partial }^k{\bar{\partial }}^k+c$ with $|\alpha |\ge 1$.