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

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 $L^{\mathrm{\infty }}({\mathbb{R}}^d$)$,when$d\geq 2$.