Multidimensional Paley–Zygmund theorems and sharp $L^p$ estimates for some elliptic operators

Imekraz, Rafik

Annales de l'Institut Fourier, Tome 69 (2019) no. 6, pp. 2723-2809.

Le but de cet article est double. Premièrement, nous étudions des conditions suffisantes de convergence pour des séries aléatoires de fonctions propres dans $L^{\infty }$. Les fonctions propres sont considérées par rapport à un opérateur elliptique de référence tel que l’opérateur de Laplace–Beltrami ou un opérateur de Schrödinger avec un potentiel confinant de l’espace euclidien. Cela constitue une généralisation d’un vieux résultat de Paley et Zygmund. Dans un deuxième temps, nous obtenons quelques estimées $L^p$ optimales de fonctions propres incluant une généralisation de l’inégalité de Bernstein. Nous montrons que ces deux thèmes sont intimement liés.

The goal of the paper is twofold. Firstly we study sufficient conditions of convergence for random series of eigenfunctions in $L^{\infty }$. The eigenfunctions are considered with respect to a reference elliptic operator like the Laplace–Beltrami operator or a Schrödinger operator with a growing potential on the Euclidean space. That is a generalization of an old result due to Paley and Zygmund. Secondly, we obtain a few optimal $L^p$ bounds of eigenfunctions including a generalization of the Bernstein inequality. We show that the previous two themes are intimately linked.

Classification : 60G50, 15B52, 46B09
Keywords: Paley–Zygmund theorems, elliptic operators, wave equation, Sobolev embeddings
Mot clés : Théorèmes de Paley–Zygmund, opérateurs elliptiques, équations des ondes, injections de Sobolev

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