$L^2$ estimate for polynomials of the Laplace operator with Gaussian measure

Abstract

Let $P(\Delta)$ be a polynomial of the Laplace operator $\Delta=\sum_{j=1}^n\frac{\partial^2}{\partial x^2_j}$ on $\mathbb{R}^n$. We prove the existence of weak solutions of the equation $P(\Delta)u=f$ and the existence of a bounded right inverse of the differential operator $P(\Delta)$ in the weighted Hilbert space with Gaussian measure, i.e., $L^2(\mathbb{R}^n,e^{-|x|^2})$.