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

Abstract

Using Hörmander $L^2$ 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.