Friday, July 19, 2013

1307.4976 (Aurélien Poiret et al.)

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

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 $d\geq 2$.
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