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Hausdorff–Young Inequalities for Group Extensions

Published online by Cambridge University Press:  20 November 2018

Hartmut Führ
Hartmut Führ*
Affiliation:
Institute of Biomathematics and Biometry, GSF Research Center for Environment and Health, D-85764 Neuherberg, Germany e-mail: fuehr@gsf.de
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Abstract

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This paper studies Hausdorff–Young inequalities for certain group extensions, by use of Mackey's theory. We consider the case in which the dual action of the quotient group is free almost everywhere. This result applies in particular to yield a Hausdorff–Young inequality for nonunimodular groups.

Keywords

43A3043A15.
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 4 , 01 December 2006 , pp. 549 - 559
Copyright
Copyright © Canadian Mathematical Society 2006

References

[1] Babenko, K. I., An inequality in the theory of Fourier integrals. Izv. Akad. Nauk. SSSR Ser. Mat. 25(1961), 531–542.Google Scholar
[2] Baklouti, A., Smaoui, K. and Ludwig, J., Estimate of the Lp-Fourier transform norm on nilpotent Lie groups. J. Funct. Anal. 199(2003), no. 2, 508–520.Google Scholar
[3] Beckner, W., Inequalities in Fourier analysis. Ann. of Math. 102(1975), no. 1, 159–182.Google Scholar
[4] Corwin, L. and Greenleaf, F. P., Representations of Nilpotent Lie Groups and Their Applications. Part I. Basic Theory and Examples. Cambridge Studies in Advanced Mathematics 18, Cambridge University Press, Cambridge, 1990.Google Scholar
[5] Dixmier, J., C*-Algebras. North Holland, Amsterdam, 1977.Google Scholar
[6] Duflo, M. and Moore, C. C., On the regular representation of a nonunimodular locally compact group, J. Funct. Anal. 21(1976), no. 2, 209–243.Google Scholar
[7] Eymard, P. and Terp, M., La transformation de Fourier et son inverse sur le groupe des ax+b d’un corps local. In; Analyse harmonique sur les groupes de Lie, Lecture Notes in Mathematics 739, Springer, 1979, pp. 207–248.Google Scholar
[8] Folland, G. B., A Course in abstract harmonic analysis. CRC Press, Boca Raton, FL, 1995.Google Scholar
[9] Fournier, J. J. F. and Russo, B., Abstract interpolation and operator-valued kernels. J. London Math. Soc. 16(1977), no. 2, 283–289.Google Scholar
[10] Inoue, J., L p-Fourier transforms on nilpotent Lie groups and solvable Lie groups acting on Siegel domains. Pacific J. Math. 155(1992), 295–318.Google Scholar
[11] Khalil, I., Sur l’analyse harmonique du groupe affine de la droite. Studia Math. 51(1974), 139–167.Google Scholar
[12] Kleppner, A. and Lipsman, R. L., The Plancherel formula for group extensions, I and II, Ann. Sci. Ecole Norm. Sup. 5(1972), 459–516; ibid. 6(1973), 103–132.Google Scholar
[13] Kunze, R. A., L p-Fourier transforms on locally compact unimodular groups. Trans. Amer. Math. Soc. 89(1958), 519–540.Google Scholar
[14] Kunze, R. A. and Stein, E. M., Uniformly bounded representations and harmonic analysis of the 2 × 2 real unimodular group. Amer. J. Math. 82(1960), 1–62.Google Scholar
[15] Mackey, G., Borel structure in groups and their duals. Trans. Amer. Math. Soc. 85(1957), 134–165.Google Scholar
[16] Russo, B., On the Hausdorff-Young theorem for integral operators. Pacific J. Math. 68(1977), no. 1, 241–253.Google Scholar
[17] Russo, B., The norm of the L p-Fourier transform on unimodular groups. Trans. Am. Math. Soc. 192(1974), 293–305.Google Scholar
[18] Russo, B., The norm of the L p p-Fourier transform on unimodular groups. II. Canad. J. Math. 28(1976), no. 6, 1121–1131.Google Scholar
[19] Tatsuuma, N., Plancherel formula for non-unimodular locally compact groups. J. Math. Kyoto Univ. 12(1972), 179–261.Google Scholar
[20] Terp, M., Lp Fourier transform on non-unimodular locally compact groups, Preprint.Google Scholar