Random weighted Sobolev inequalities on $\mathbb{R}^d$ and application to Hermite functions

TL;DR

This paper extends randomization techniques to establish optimal weighted Sobolev inequalities on ^d with the harmonic oscillator, leading to bases of Hermite functions with improved decay properties in higher dimensions.

Contribution

It introduces a new probabilistic framework for Sobolev inequalities on ^d, utilizing spectral function estimates for the harmonic oscillator, and constructs Hermite bases with enhanced decay.

Findings

Established optimal weighted Sobolev estimates on ^d

Constructed measures satisfying concentration of measure property

Found Hermite bases with good decay in L^(^d) for d.

Abstract

We extend a randomisation 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 satisfy 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 $d\geq 2$.