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

Abstract

Let $P(\Delta )$ be a polynomial of the Laplace operator $\Delta ={\sum }_{j=1}^n\frac{{\partial }^2}{\partial x_j^2}$ 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., $L^2({\mathbb{R}}^n,e^{-|x{|}^2})$.

arXiv admin note: text overlap with arXiv:1909.12477