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The Helgason Fourier transform for semisimple Lie groups I: The Case of SL2(ℝ)

Published online by Cambridge University Press:  17 April 2009

Rudra P. Sarkar  and
Alladi Sitaram
Rudra P. Sarkar
Affiliation:
Stat-Math Unit, Indian Statistical Institute, 203 B. T. Road, Calcutta 700108, India, e-mail: [email protected]
Alladi Sitaram
Affiliation:
Stat-Math Unit, Indian Statistical Institute, 8th Mile, Mysore Rd, Bangalore 560059, India, e-mail: [email protected]
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We consider a Helgason-type Fourier transform on SL2(ℝ) and prove various results on L1-harmonic analysis on the full group analogous to those on symmetric spaces.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 73 , Issue 3 , June 2006 , pp. 413 - 432
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Andersen, N.B., ‘Paley-Wiener Theorems for hyperbolic spaces’, J. Funct. Anal. 179 (2001), 66–119.Google Scholar
[2]Barker, W.H., ‘L P harmonic analysis on SL(2, R)’, Mem. Amer. Math. Soc. 393 76 (1988).Google Scholar
[3]Benedicks, M., ‘On Fourier transforms of functions supported on sets of finite Lebesgue measure’, J. Math. Anal. Appl. 106 (1985), 180–183.CrossRefGoogle Scholar
[4]Benyamini, Y. and Weit, Y., ‘Harmonic analysis of spherical functions on SU(1, 1)’, Ann. Inst. Fourier (Grenoble) 42 (1992), 671–694.Google Scholar
[5]Camporesi, R., ‘The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces’, Pacific J. Math. 179 (1997), 263–300.Google Scholar
[6]Cowling, M., Price, J.F. and Sitaram, A., ‘A qualitative uncertainty principle for semisimple Lie groups’, J. Austral. Math. Soc. Ser. A 45 (1988), 127–132.Google Scholar
[7]Cowling, M., Sitaram, A. and Sundari, M., ‘Hardy's uncertainty principle on semisimple groups’, Pacific J. Math. 192 (2000), 293–296.Google Scholar
[8]Echterhoff, S., Kaniuth, E., Kumar, A., ‘A qualitative uncertainty principle for certain locally compact groups’, Forum Math. 3 (1991), 355–369.CrossRefGoogle Scholar
[9]Ehrenpreis, L. and Mautner, F.I., ‘Some properties of the Fourier transform on semi-simple Lie groups. I’, Ann. of Math. 61 (1955), 406–439.CrossRefGoogle Scholar
[10]Ehrenpreis, L. and Mautner, F.I., ‘Some properties of the Fourier-transform on semisimple Lie Groups. III’, Trans. Amer. Math. Soc. 90 (1959), 431–484.Google Scholar
[11]Gangolli, R. and Varadarajan, V.S., Harmonic analysis of spherical functions on real reductive groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 101 (Springer-Verlag, Berlin, 1988).CrossRefGoogle Scholar
[12]Helgason, S., Non-Euclidean analysis,(Proc. J. Bolyai Memorial Conference,Budapest,July 2002).Google Scholar
[13]Helgason, S., ‘The Abel, Fourier and Radon transforms on symmetric spaces’, Indag. Math. (to appear).Google Scholar
[14]Helgason, S., and Johnson, K., ‘The bounded spherical functions on symmetric spaces’, Advances in Math. 3 (1969), 586–593.CrossRefGoogle Scholar
[15]Helgason, S., Rawat, R., Sengupta, J. and Sitaram, A., ‘Some remarks on the Fourier transform on a symmetric space’, (Indian Statistical Institute, Bangalore Centre, Technical Report, 1998).Google Scholar
[16]Lang, S., SL2(ℝ) (Addison-Wesley Publishing Co., Massachusetts, London, Amsterdam, 1975).Google Scholar
[17]Mohanty, P., Ray, S.K., Sarkar, R.P. and Sitaram, A., ‘The Helgason Fourier transform for Symmetric spaces II’, J. Lie Theory 14 (2004), 227–242.Google Scholar
[18]Narasimhan, R., Analysis on real and complex manifolds, North-Holland Mathematical Library 35 (North-Holland, Amsterdam, 1985).Google Scholar
[19]Price, J.F. and Sitaram, A., ‘Functions and their Fourier transforms with supports of finite measure for certain locally compact groups’, J. Funct. Anal. 79 (1988), 166–182.CrossRefGoogle Scholar
[20]Rossmann, W., ‘Analysis on real hyperbolic spaces’, J. Funct. Anal. 30 (1978), 448–477.Google Scholar
[21]Sarkar, R.P., ‘Wiener Tauberian Theorems for SL2(R)’, Pacific J. Math. 177 (1997), 291–304.CrossRefGoogle Scholar
[22]Sarkar, R.P., ‘Wiener Tauberian Theorem for rank one symmetric spaces’, Pacific J. Math. 186 (1998), 349–358.CrossRefGoogle Scholar
[23]Sarkar, R.P., ‘A complete analogue of Hardy's Theorem on SL 2(ℝ) and characterization of the heat kernel’, Proc. Indian Acad. Sci. Math. Sci. 112 (2002), 579–593.CrossRefGoogle Scholar
[24]Sarkar, R.P. and Sitaram, A., ‘The Helgason Fourier transform for symmetric spaces’, (A Tribute to C.S. Seshadri (Chennai, 2002)), Trends in Mathematics 24 (Birkhauser Verlag, Basel), pp. 467–473.Google Scholar
[25]Sitaram, A., ‘On an analogue of the Wiener Tauberian Theorem for symmetric spaces of the noncompact type’, Pacific J. Math. 133 (1988), 197–208.Google Scholar
[26]Stanton, R.J. and Tomas, P.A., ‘Pointwise inversion of the spherical transform on L p(G/K), 1 ≤ p < 2’, Proc. Amer. Math. Soc. 73 (1979), 398–404.Google Scholar
[27]Strichartz, R.S., ‘Harmonic analysis on hyperboloids’, J. Funct. Anal. 12 (1973), 341–383.CrossRefGoogle Scholar