Random-Weighted Sobolev Inequalities on $${\mathbb{R}^d }$$ R d and Application to Hermite Functions

Abstract

We extend a randomization method, introduced by Shiffman–Zelditch and developed by Burq–Lebeau on compact manifolds for the Laplace operator, to the case of \({\mathbb{R}^d}\) with the harmonic oscillator. We construct measures, thanks to probability laws which satisfies the concentration of measure property, on the support of which we prove optimal-weighted Sobolev estimates on \({\mathbb{R}^d}\). This construction relies on accurate estimates on the spectral function in a non-compact configuration space. As an application, we show that there exists a basis of Hermite functions with good decay properties in \({L^{\infty}(\mathbb{R}^{d})}\), when d ≥ 2.