On a Right Inverse of a Polynomial of the Laplace in the Weighted Hilbert Space $L^2 (\mathbb{R}^n ,e^{−|x|^2} )$

Abstract

Let $P(∆)$ be a polynomial of the Laplace operator $$∆ = \sum\limits^n_{j=1}\frac{∂^2}{∂x^2_j} \ \  on  \ \  \mathbb{R}^n.$$ We prove the existence of a bounded right inverse of the differential operator $P(∆)$ in the weighted Hilbert space with the Gaussian measure, i.e., $L^2(\mathbb{R}^n ,e^{−|x|^2}).$