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

Aurélien Poiret, Didier Robert, Laurent Thomann
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 $d\geq 2$.
View original: http://arxiv.org/abs/1307.4976