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

TL;DR

This paper establishes the existence of weak solutions and bounded right inverses for polynomial Laplace operators within Gaussian-weighted Hilbert spaces, advancing the understanding of differential operators under Gaussian measures.

Contribution

It proves the existence of solutions and right inverses for polynomial Laplace operators in Gaussian-weighted spaces, a novel result in this context.

Findings

Existence of weak solutions for $P( riangle)u=f$

Bounded right inverse of $P( riangle)$ in Gaussian $L^2$ space

Extension of classical PDE results to Gaussian-weighted Hilbert spaces

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., $L^2(\mathbb{R}^n,e^{-|x|^2})$.