Analysis & PDE

  • Anal. PDE
  • Volume 11, Number 2 (2018), 263-350.

Concentration et randomisation universelle de sous-espaces propres

Rafik Imekraz

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Abstract

Nous étudions des conditions nécessaires et suffisantes de convergence pour des séries aléatoires de fonctions propres dans Lp, avec p fini. De façon précise, nous montrons des résultats optimaux pour les harmoniques sphériques sur Sd et l’oscillateur harmonique sur d (cela améliore des résultats de Ayache–Tzvetkov, Grivaux et Imekraz–Robert–Thomann). Dans le cas multidimensionnel, nous utiliserons des séries aléatoires faisant intervenir des matrices aléatoires. Cela nous permettra de donner un éclairage sur la construction d’une famille de mesures construites par Burq–Lebeau sur l’espace de Hilbert d’une variété riemannienne compacte. En fait, nous montrons que c’est précisément parce que Lp est de cotype fini que cette construction est possible (il s’agit d’une version multidimensionnelle du théorème de Maurey–Pisier).

We study necessary and sufficient conditions of convergence for random series of eigenfunctions in Lp, for finite p. More precisely, we get optimal results for the spherical harmonics on Sd and for the harmonic oscillator on d (this improves results by Ayache–Tzvetkov, Grivaux and Imekraz–Robert–Thomann). In the multidimensional framework, our random series involve random matrices. This illuminates a construction, made by Burq–Lebeau, of a family of specific measures on the Hilbert space of a Riemannian boundaryless compact manifold. Actually, we show that the latter construction is possible because Lp has finite cotype (this is nothing but a multidimensional version of the Maurey–Pisier theorem).

Article information

Source
Anal. PDE, Volume 11, Number 2 (2018), 263-350.

Dates
Received: 29 March 2016
Revised: 6 March 2017
Accepted: 17 June 2017
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513774507

Digital Object Identifier
doi:10.2140/apde.2018.11.263

Mathematical Reviews number (MathSciNet)
MR3724490

Zentralblatt MATH identifier
1378.15021

Subjects
Primary: 15B52: Random matrices 46B09: Probabilistic methods in Banach space theory [See also 60Bxx] 60G50: Sums of independent random variables; random walks

Keywords
random matrix eigenfunctions Sobolev embedding

Citation

Imekraz, Rafik. Concentration et randomisation universelle de sous-espaces propres. Anal. PDE 11 (2018), no. 2, 263--350. doi:10.2140/apde.2018.11.263. https://projecteuclid.org/euclid.apde/1513774507


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