Study of the operator $\partial^{k} \bar{\partial}^{k} + c$ in the weighted Hilbert space $L^2(\mathbb{C}, {\rm e}^{-\vert z \vert^2})$

TL;DR

This paper investigates the properties of a specific differential operator in a weighted Hilbert space, establishing the existence of a bounded right inverse using H"ormander's $L^2$-method.

Contribution

It demonstrates the existence of a bounded right inverse for the operator $ extstyle rac{ar{ ext{d}}^k}{ ext{d}z^k} + c$ in a weighted space, extending previous understanding.

Findings

Proves the existence of a bounded right inverse for the operator.

Utilizes H"ormander's $L^2$-method in the analysis.

Applies to operators of arbitrary order $k$.

Abstract

By the H\"ormander's $L^2$-method, we study the operator $\partial^k \bar{\partial}^{k} + c$ for any order $k$ in the weighted Hilbert space $L^2(\mathbb{C}, {\rm e}^{-\vert z \vert^2})$. We prove the existence of its right inverse witch is also a bounded operator.