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Beurling's Theorem and Characterization of Heat Kernel for Riemannian Symmetric Spaces of Noncompact Type

Published online by Cambridge University Press:  20 November 2018

Rudra P. Sarkar
Affiliation:
Stat-Math Unit, Indian Statistical Institute, 203 B. T. Road, Calcutta 700108, India e-mail: rudra@isical.ac.in
Jyoti Sengupta
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Rd., Mumbai 400005, India e-mail: sengupta@math.tifr.res.in
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Abstract

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We prove Beurling's theorem for rank 1 Riemannian symmetric spaces and relate its consequences with the characterization of the heat kernel of the symmetric space.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2007

References

[1] Anker, J.-P., Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces. Duke Math. J. 65(1992), no. 2, 257297.Google Scholar
[2] Anker, J.-P., A basic inequality for scattering theory on Riemannian symmetric spaces of the noncompact type. Amer. J. Math. 113(1991), no. 3, 391398.Google Scholar
[3] Bonami, A., Demange, B., and Jaming, P., Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms. Rev. Mat. Iberoamericana 19(2003), no. 1, 2355.Google Scholar
[4] Cowling, M., Sitaram, A., and Sundari, M., Hardy's uncertainty principle on semisimple groups. Pacific J. Math. 192(2000), no. 2, 293296.Google Scholar
[5] Demange, B., Principes d’incertitude associés à des formes quadratiques non dǵénérés. Thèse, Université d’Orléans, 2004.Google Scholar
[6] Ebata, M., Eguchi, M., Koizumi, S., and Kumahara, K., A generalization of the Hardy theorem to semisimple Lie groups. Proc. Japan Acad. Ser. A Math. Sci. 75(1999), no. 7, 113114.Google Scholar
[7] Folland, G. B. and Sitaram, A., The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3(1997), no. 3, 207238.Google Scholar
[8] Gangolli, R. and Varadarajan, V. S., Harmonic Analysis of Spherical Functions on Real Reductive Groups. Ergebnisse derMathematik und ihrer Grenzgebiete 101, Springer-Verlag, Berlin, 1988.Google Scholar
[9] Havin, V. and Jöricke, B., The Uncertainty Principle in Harmonic Analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete 28, Springer-Verlag, Berlin, 1994.Google Scholar
[10] Helgason, S., A duality for symmetric spaces with applications to group representations. Advances in Math. 5(1970), 1154.Google Scholar
[11] Helgason, S., Eigenspaces of the Laplacian; integral representations and irreducibility. J. Functional Analysis 17(1974), 328353.Google Scholar
[12] Helgason, S., Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions. Pure and Applied Mathematics 113, Academic Press, Orlando, FL, 1984.Google Scholar
[13] Helgason, S., Geometric Analysis on Symmetric Spaces. Mathematical Surveys and Monographs 39, American Mathematical Society, Providence, RI, 1994.Google Scholar
[14] Helgason, S., The Abel, Fourier and Radon transforms on symmetric spaces. Indag. Math. 16(2005), 531551.Google Scholar
[15] Helgason, S., Rawat, R., Sengupta, J., and Sitaram, A., Some remarks on the Fourier transform on a symmetric space. Technical Report January 1998, Indian Statistical Institute, Bangalore.Google Scholar
[16] Hörmander, L., A uniqueness theorem of Beurling for Fourier transform pairs. Ark. Mat. 29(1991), no. 2, 237240.Google Scholar
[17] Kostant, B., On the existence and irreducibility of certain series of representations. Bull. Amer. Math. Soc. 75(1969), 627642.Google Scholar
[18] Mohanty, P., Ray, S. K., Sarkar, R. P., and Sitaram, A., The Helgason-Fourier transform for symmetric spaces. II. J. Lie Theory 14(2004), no. 1, 227242.Google Scholar
[19] Narayanan, E. K. and Ray, S. K., The heat kernel and Hardy's theorem on symmetric spaces of noncompact type. Proc. Indian Acad. Sci. Math. Sci. 112(2002), no. 2, 321330.Google Scholar
[20] Narayanan, E. K. and Ray, S. K., Lp version of Hardy's theorem on semisimple Lie groups. Proc. Amer. Math. Soc. 130(2002), no. 6, 18591866 Google Scholar
[21] Ray, S. K. and Sarkar, R. P., Cowling-Price theorem and characterization of heat kernel on symmetric spaces. Proc. Indian Acad. Sci. Math. Sci. 114(2004), no. 2, 159180.Google Scholar
[22] Sengupta, J., The uncertainty principle on Riemannian symmetric spaces of the noncompact type. Proc. Amer. Math. Soc. 130 (2002), no. v, 10091017 (electronic).Google Scholar
[23] Sitaram, A. and Sundari, M., An analogue of Hardy's theorem for very rapidly decreasing functions on semi-simple Lie groups. Pacific J. Math. 177(1997), no. 1, 187200.Google Scholar
[24] Stein, E. M., Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Annals of Mathematics Studies 63, Princeton University Press, Princeton, 1970.Google Scholar
[25] Thangavelu, S., Hardy's theorem for the Helgason Fourier transform on noncompact rank one symmetric spaces. Colloq. Math. 94(2002), no. 2, 263280.Google Scholar