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Annales Henri Poincaré

, Volume 16, Issue 2, pp 651–689 | Cite as

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

  • Aurélien Poiret
  • Didier Robert
  • Laurent ThomannEmail author
Article

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.

Keywords

Probability Measure Sobolev Space Harmonic Oscillator Spectral Function Besov Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Aurélien Poiret
    • 1
  • Didier Robert
    • 2
  • Laurent Thomann
    • 2
    Email author
  1. 1.Laboratoire de Mathématiques, UMR 8628 du CNRSUniversité Paris SudOrsay CedexFrance
  2. 2.Laboratoire de Mathématiques J. Leray, UMR 6629 du CNRSUniversité de NantesNantes Cedex 03France

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