Uncertainty principle and L^p–L^q-sufficient pairs on noncompact real symmetric spaces

Ben Farah, Slaim; Mokni, Kamel

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

On considère un groupe de Lie semi-simple réel G de centre fini et K un sous-groupe compact maximal de G. Soit $\mathrm{G}=\mathrm{K}\exp \left(\overline{{𝔞}_{+}}\right)\mathrm{K}$ une décomposition de Cartan de G. Pour x∈G, on note ∥x∥ la norme de la composante de x dans ${𝔞}_{+}$. Soient $\mathrm{a}>0,\mathrm{b}>0$ et 1⩽p,q⩽∞. Dans cette Note on donne une condition nécessaire et suffisante sur $\mathrm{a},b$ telle que pour toute fonction f mesurable et K-bi-invariante sur G, si e^a∥x∥2f∈L^p(G) et ${\mathrm{e}}^{\mathrm{b}\parallel \lambda {\parallel }^2}ℱ\left(\mathrm{f}\right)\in {\mathrm{L}}^q\left({𝔞}_{+}^{*}\right)$ 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 $\mathrm{G}=\mathrm{K}\exp \left(\overline{{𝔞}_{+}}\right)\mathrm{K}$ 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 $\mathrm{a}>0,\mathrm{b}>0$ and 1⩽p,q⩽∞. In this Note we give necessary and sufficient conditions on $\mathrm{a},b$ such that for all K-bi-invariant measurable function f on G, if e^a∥x∥2f∈L^p(G) and ${\mathrm{e}}^{\mathrm{b}\parallel \lambda {\parallel }^2}ℱ\left(\mathrm{f}\right)\in {\mathrm{L}}^q\left({𝔞}_{+}^{*}\right)$ then f=0 almost everywhere.

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

[2] Anker, J.-Ph.; Ji, L. 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] Bagchi, S.C.; Ray, S.K. 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] Cowling, M.G.; Price, J.F. Generalizations of Heisenberg's inequality, Lecture Notes in Math., 992, Springer, Berlin, 1983, pp. 443-449

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

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

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

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

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

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

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