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

Abstract

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