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


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.


Probability Measure Sobolev Space Harmonic Oscillator Spectral Function Besov Space 
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  1. 1.
    Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Cyril, R., Scheffer, G.: Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses, 10. Société Mathématique de France, Paris (2000)Google Scholar
  2. 2.
    Burq, N., Lebeau, G.: Injections de Sobolev probabilistes et applications. Ann. Sci. Éc. Norm. Supér. (2014) (to appear)Google Scholar
  3. 3.
    Burq N., Tzvetkov N.: Random data Cauchy theory for supercritical wave equations I: local existence theory. Invent. Math. 173(3), 449–475 (2008)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Burq N., Tzvetkov N.: Random data Cauchy theory for supercritical wave equations II: A global existence result. Invent. Math. 173(3), 477–496 (2008)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Burq, N., Tzvetkov, N.: Probabilistic well-posedness for the cubic wave equation (2011) Preprint: arXiv:1103.2222 (to appear in JEMS)Google Scholar
  6. 6.
    Feng, R., Zelditch, S.: Median and mean of the Supremum of L 2 normalized random holomorphic fields Preprint: arXiv:1303.4096v1 [math.PR] (2013)Google Scholar
  7. 7.
    Hörmander L.: The spectral function of an elliptic operator. Acta Math. 121, 193–218 (1968)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Ivrii, V.: Microlocal Analysis and Precise Spectral Asymptotics. Springer Monographs in Mathematics (1998)Google Scholar
  9. 9.
    Kakutani S.: On equivalence of infinite product measures. Ann. Math. 49(1), 214–224 (1948)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Karadzhov G.E.: Riesz summability of multiple Hermite series in L p spaces. Math. Z. 219, 107–118 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Koch H., Tataru D.: L p eigenfunction bounds for the Hermite operator. Duke Math. J. 128(2), 369–392 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Koch H., Tataru D., Zworski M.: Semiclassical L p estimates. Ann. Henri Poincaré 8, 885–916 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Ledoux, M.: The concentration of measure phenomenon. Mathematical Surveys and Monographs, vol. 89, AMS (2001)Google Scholar
  14. 14.
    Poiret, A.: Solutions globales pour des équations de Schrödinger sur-critiques en toutes dimensions (2012) Preprint: arXiv:1207.3519Google Scholar
  15. 15.
    Poiret, A., Robert, D., Thomann, L.: Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator. (2013) Preprint arXiv:1309.0795Google Scholar
  16. 16.
    Robert D.: Autour de l’approximation semi-classique. Birkhäuser, Boston (1987)zbMATHGoogle Scholar
  17. 17.
    Robert, D., Thomann, L.: Random weighted Sobolev inequalities and application to quantum ergodicity. Preprint. hal-00919443 (2013)Google Scholar
  18. 18.
    Shiffman B., Zelditch S.: Random polynomials of high degree and Lévy concentration of measure. Asian J. Math. 7(4), 627–646 (2003)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Shiryaev A.N.: Probability. Graduate Texts in Mathematics, 95. Springer, New York (1996)Google Scholar
  20. 20.
    Talagrand M.: Concentration of measure and isoperimetric inequalties in product spaces. Publications Mathématiques de l’I.H.E.S. 81, 73–205 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
  22. 22.
    Thangavelu S.: Lectures on Hermite and Laguerre expansions. Mathematical Notes, 42. Princeton University Press, Princeton, NJ (1993)Google Scholar
  23. 23.
    Yajima, K., Zhang, G.: Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity. J. Differ. Equ. 1, 81–110 (2004)Google Scholar

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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