Uncertainty principle and Lp–Lq-sufficient pairs on noncompact real symmetric spaces

Slaim Ben Farah; Kamel Mokni

doi:10.1016/S1631-073X(03)00220-6

Mathematical Analysis

Uncertainty principle and L^p–L^q-sufficient pairs on noncompact real symmetric spaces
[Principe d'incertitude et paires L^p–L^q-suffisantes sur les espaces symétriques réels non-compacts]

Présenté par : Jean-Pierre Kahane

Slaim Ben Farah ¹ ; Kamel Mokni ¹

¹ Faculté des sciences de Monastir, département de mathématiques, 5019 Monastir, Tunisia

Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 889-892.

Résumé
Abstract

On considère un groupe de Lie semi-simple réel G de centre fini et K un sous-groupe compact maximal de G. Soit $G = K \exp (\bar{𝔞_{+}}) K$ une décomposition de Cartan de G. Pour x∈G, on note ∥x∥ la norme de la composante de x dans $𝔞_{+}$ . Soient $a > 0, b > 0$ et 1⩽p,q⩽∞. Dans cette Note on donne une condition nécessaire et suffisante sur $a, b$ telle que pour toute fonction f mesurable et K-bi-invariante sur G, si e^a∥x∥²f∈L^p(G) et $e^{b ∥ λ ∥^{2}} ℱ (f) \in L^{q} (𝔞_{+}^{*})$ alors f=0 presque partout.

We consider a real semi-simple Lie group G with finite center and a maximal compact sub-group K of G. Let $G = K \exp (\bar{𝔞_{+}}) K$ be a Cartan decomposition of G. For x∈G denote ∥x∥ the norm of the $𝔞_{+}$ -component of x in the Cartan decomposition of G. Let $a > 0, b > 0$ and 1⩽p,q⩽∞. In this Note we give necessary and sufficient conditions on $a, b$ such that for all K-bi-invariant measurable function f on G, if e^a∥x∥²f∈L^p(G) and $e^{b ∥ λ ∥^{2}} ℱ (f) \in L^{q} (𝔞_{+}^{*})$ then f=0 almost everywhere.

Reçu le : 2003-02-25
Accepté le : 2003-04-22
Publié le : 2003-05-31

DOI : 10.1016/S1631-073X(03)00220-6

Affiliations des auteurs :

Slaim Ben Farah ¹ ; Kamel Mokni ¹

¹ Faculté des sciences de Monastir, département de mathématiques, 5019 Monastir, Tunisia

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     author = {Slaim Ben Farah and Kamel Mokni},
     title = {Uncertainty principle and {\protect\emph{L}\protect\textsuperscript{\protect\emph{p}}{\textendash}\protect\emph{L}\protect\textsuperscript{\protect\emph{q}}-sufficient} pairs on noncompact real symmetric spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {889--892},
     publisher = {Elsevier},
     volume = {336},
     number = {11},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00220-6},
     language = {en},
}

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Slaim Ben Farah; Kamel Mokni. Uncertainty principle and L^p–L^q-sufficient pairs on noncompact real symmetric spaces. Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 889-892. doi : 10.1016/S1631-073X(03)00220-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00220-6/

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