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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 itsright inverse witch is also a bounded operator.

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