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

TL;DR

This paper uses H"ormander's $L^2$-method to analyze a class of differential operators in weighted Hilbert spaces, establishing the existence of bounded right inverses and exploring specific cases with applications to complex analysis.

Contribution

It provides a general framework for the invertibility of a family of complex differential operators in weighted spaces, extending previous results and analyzing special cases.

Findings

Existence of bounded right inverse for the operator

Analysis of cases with specific parameter constraints

Extension of $L^2$-methods to higher-order operators

Abstract

By H\"ormander'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 $(\alpha, \beta, \gamma) \neq(0,0,0)$ 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 $\vert \beta \vert \geq 1$. (2) Case where $\beta= \gamma=0$: The operator $\alpha \partial^{k} \bar{\partial}^{k} + c$ with $\vert \alpha \vert \geq 1$.