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.

Abstract (VO)
Résumé

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.

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.

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/

Bibliographie
Cité par

[1] J.-Ph. Anker Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J., Volume 65 (1992) no. 2, pp. 257-297

[2] J.-Ph. Anker; L. Ji Comportement exact du noyau de la chaleur et de la fonction de Green sur les espaces symétriques non compacts, C. R. Acad. Sci. Paris, Sér. I, Volume 362 (1998), pp. 153-156

[3] S.C. Bagchi; S.K. Ray Uncertainty principles like Hardy's theorem on some Lie groups, J. Austral. Math. Soc. Ser. A, Volume 65 (1999), pp. 289-302

[4] S. Ben Farah, K. Mokni, K. Trimèche, An L^p–L^q version of Hardy's theorem for spherical Fourier transform on semi-simple Lie groups, Preprint, 2002

[5] M.G. Cowling; J.F. Price Generalizations of Heisenberg's inequality, Lecture Notes in Math., 992, Springer, Berlin, 1983, pp. 443-449

[6] M.G. Cowling; A. Sitaram; M. Sundari Hardy's uncertainty principle on semi-simple groups, Pacific J. Math., Volume 192 (2000) no. 2, pp. 293-296

[7] M. Eguchi; S. Koizumi; K. Kumahara An L^p version of the Hardy theorem for motion groups, J. Austral. Math. Soc. Ser. A, Volume 68 (2000), pp. 55-67

[8] L. Gallardo; K. Trimèche Un analogue d'un théorème de Hardy pour la transformation de Dunkl, C. R. Acad. Sci. Paris, Sér. I, Volume 334 (2002), pp. 849-854

[9] G.H. Hardy A theorem concerning Fourier transforms, J. London Math. Soc., Volume 8 (1933), pp. 227-231

[10] V. Havin; B. Jöricke The Uncertainty Principle in Harmonic Analysis, A Series of Modern Surveys in Math., 28, Springer-Verlag, Berlin, 1994

[11] S. Helgason Groups and Geometric Analysis, Integral Geometry, Invariant Differential Operators and Spherical Functions, Academic Press, New York, 1984

[12] E.K. Narayanan; S.K. Ray L^p version of Hardy's theorem on semi-simple Lie groups, Proc. Amer. Math. Soc., Volume 130 (2002) no. 6, pp. 1859-1866

[13] J. Sengupta An analogue of Hardy's theorem for semi-simple Lie groups, Proc. Amer. Math. Soc., Volume 128 (2000) no. 8, pp. 2493-2499

[14] A. Sitaram; M. Sundari An analogue of Hardy's theorem for very rapidly decreasing functions on semi-simple Lie groups, Pacific J. Math., Volume 177 (1997), pp. 187-200

[15] M. Sundari Hardy's theorem for the n-dimensional Euclidean motion group, Proc. Amer. Math. Soc.126 (1998), pp. 1199-1204

Hatem Mejjaoli; Khalifa Trimèche Qualitative uncertainty principles for the generalized Fourier transform associated to a Dunkl type operator on the real line, Analysis and Mathematical Physics, Volume 6 (2016) no. 2, pp. 141-162 | DOI:10.1007/s13324-015-0111-7 | Zbl:1352.43006
Latifa Bou Attour; Khalifa Trimèche Uncertainty principle and (Lp,Lq)sufficient pairs on Chébli–Trimèche hypergroups, Integral Transforms and Special Functions, Volume 16 (2005) no. 8, p. 625 | DOI:10.1080/10652460500110404

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