A right inverse of Cauchy-Riemann operator $\bar{\partial}^k+a$ in weighted Hilbert space $L^2(\mathbb{C},e^{-|z|^2})$

TL;DR

This paper constructs a bounded right inverse for a differential operator involving the Cauchy-Riemann operator of any order in a weighted Hilbert space, extending complex analysis techniques.

Contribution

It introduces a method to find a bounded right inverse for the operator ar^k + a in weighted Hilbert spaces, generalizing previous results.

Findings

Existence of a bounded right inverse for ar^k + a

Extension of Hf6rmander L^2 method to higher-order operators

Application to weighted Hilbert space L^2({C}, e^{-|z|^2})

Abstract

Using H\"{o}rmander $L^2$ method for Cauchy-Riemann equations from complex analysis, we study a simple differential operator $\bar{\partial}^k+a$ of any order (densely defined and closed) in weighted Hilbert space $L^2(\mathbb{C},e^{-|z|^2})$ and prove the existence of a right inverse that is bounded.