Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The norm of the $ L\sp{p}$-Fourier transform on unimodular groups


Author: Bernard Russo
Journal: Trans. Amer. Math. Soc. 192 (1974), 293-305
MSC: Primary 43A15
DOI: https://doi.org/10.1090/S0002-9947-1974-0435731-X
MathSciNet review: 0435731
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We discuss sharpness in the Hausdorff Young theorem for unimodular groups. First the functions on unimodular locally compact groups for which equality holds in the Hausdorff Young theorem are determined. Then it is shown that the Hausdorff Young theorem is not sharp on any unimodular group which contains the real line as a direct summand, or any unimodular group which contains an Abelian normal subgroup with compact quotient as a semidirect summand. A key tool in the proof of the latter statement is a Hausdorff Young theorem for integral operators, which is of independent interest. Whether the Hausdorff Young theorem is sharp on a particular connected unimodular group is an interesting open question which was previously considered in the literature only for groups which were compact or locally compact Abelian.


References [Enhancements On Off] (What's this?)

  • [1] A. Benedek and R. Panzone, The spaces $ {L^p}$ with mixed norm, Duke Math. J. 28 (1961), 301-324. MR 23 #A3451. MR 0126155 (23:A3451)
  • [2] J. Dixmier, Les algèbres d'opérateurs dans l'espace Hilbertien, Cahiers scientifiques, fasc. 25, Gauthier-Villars, Paris, 1957. MR 20 #1234. MR 0094722 (20:1234)
  • [3] P. Eymard, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181-236. MR 37 #4208. MR 0228628 (37:4208)
  • [4] S. Grosser and M. Moskowitz, On central topological groups, Trans. Amer. Math. Soc. 127 (1967), 317-340. MR 35 #292. MR 0209394 (35:292)
  • [5] R. Godement, Sur la transformation de Fourier dans les groupes discrets, C. R. Acad. Sci. Paris 228 (1949), 627-628. MR 10, 429. MR 0028323 (10:429e)
  • [6] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups analysis on locally compact Abelian groups, Die Grundlehren der math. Wissenschaften, Band 152, Springer-Verlag, Berlin and New York, 1970. MR 41 #7378. MR 0262773 (41:7378)
  • [7] E. Hewitt and K. Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, 2nd printing corrected, Springer-Verlag, Berlin and New York, 1969. MR 43 #428. MR 0274666 (43:428)
  • [8] I. I. Hirschman, A maximal problem in harmonic analysis. II, Pacific J. Math. 9 (1959), 525-540. MR 24 #A1572. MR 0131724 (24:A1572)
  • [9] R. A. Kunze, $ {L^p}$-Fourier transforms on locally compact unimodular groups, Trans. Amer. Math. Soc. 89 (1958), 519-540. MR 20 #6668. MR 0100235 (20:6668)
  • [10] R. A. Kunze and E. M. Stein, Uniformly bounded representations and harmonic analysis of the $ 2 \times 2$ real unimodular group, Amer. J. Math. 82 (1960), 1-62. MR 29 # 1287. MR 0163988 (29:1287)
  • [11] R. L. Lipsman, Harmonic analysis on $ SL(n,C)$, J. Functional Analysis 3 (1969), 126-155. MR 38 #5997. MR 0237716 (38:5997)
  • [12] I. E. Segal, An extension of Plancherel's formula to separable unimodular groups, Ann. of Math. (2) 52 (1950), 272-292. MR 12, 157. MR 0036765 (12:157f)
  • [13] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Series, no. 30, Princeton Univ. Press, Princeton, N.J., 1970. MR 44 #7280. MR 0290095 (44:7280)
  • [14] W. F. Stinespring, Integration theorems for gages and duality for unimodular groups, Trans. Amer. Math. Soc. 90 (1959), 15-56. MR 21 #1547. MR 0102761 (21:1547)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 43A15

Retrieve articles in all journals with MSC: 43A15


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0435731-X
Keywords: Unimodular group, dual gage space, convolution, Hausdorff Young theorem, $ {L^p}$-maximal function, subcharacter, $ {L^p}$-Fourier transform, regular representation, direct integral, semidirect product, integral operator
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society