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Lp-improving measures on compact non-abelian groups

Part of: Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Kathryn E. Hare
Kathryn E. Hare
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
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Abstract

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A Borel measure μ on a compact group G is called Lp-improving if the operator Tμ: L2(G) → L2(G), defined by Tμ(f) = μ * f, maps into Lp(G) for some P > 2. We characterize Lp-improving measures on compact non-abelian groups by the eigenspaces of the operator Tμ if |Tμ|. This result is a generalization of our recent characterization of Lp-improving measures on compact abelian groups.

Two examples of Riesz product-like measures are constructed. In contrast with the abelian case one of these is not Lp-improving, while the other is a non-trivial example of an Lp improving measure.

MSC classification

Secondary: 43A05: Measures on groups and semigroups, etc. 43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 3 , June 1989 , pp. 402 - 414
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Bonami, A., ‘Étude des coefficients de Fourier des fonctions de Lp (G)’, Ann. Inst. Fourier (Grenoble) 20 (1970), 335–402.CrossRefGoogle Scholar
[2]Benke, G., ‘On the hypergroup structure of central Λ(p) sets’, Pacific J. Math. 50 (1974), 19–27.CrossRefGoogle Scholar
[3]Cartwright, D. and McMullen, J., ‘A structural criterion for the existence of infinite Sidon sets’, Pacific J. Math. 96 (1981), 301–317.CrossRefGoogle Scholar
[4]Figa-Talamanca, A. and Rider, D., ‘A theorem of Littlewood and lacunary series for compact groups’, Pacific J. Math. 16 (1966), 505–514.CrossRefGoogle Scholar
[5]Graham, D., Hare, K. and Ritter, D., ‘The size of Lp-improving measures’, J. Funct. Anal., to appear.Google Scholar
[6]Hare, K. E., ‘A characterization of Lp-improving measures’, Proc. Amer. Math. Soc. 102 (1988), 295–299.Google Scholar
[7]Hewitt, E. and Ross, K., Abstract harmonic analysis, Vol. II (Springer-Verlag, Berlin-Heidelberg-New York, 1979).CrossRefGoogle Scholar
[8]Lopez, J. and Ross, K., Sidon sets, (Lecture Notes in Pure and Applied Mathematics, 13, Marcel Dekker, New York, 1975).Google Scholar
[9]Oberlin, D., ‘A convolution property of the Cantor-Lebesgue measure’, Colloq. Math. 67 (1982), 113–117.CrossRefGoogle Scholar
[10]Parker, W., ‘Central Sidon and central Λ(p) sets’, J. Austral. Math. Soc. 14 (1972), 62–74.CrossRefGoogle Scholar
[11]Rider, D., ‘Central lacunary sets’, Monatsh. Math. 76 (1972), 328–338.CrossRefGoogle Scholar
[12]Ritter, D., ‘Most Riesz product measures are Lp-improving’, Proc. Amer. Math. Soc. 97 (1986), 291–295.Google Scholar
[13]Rudin, W., Functional analysis, (McGraw-Hill, New York, 1973).Google Scholar
[14]Rudin, W., ‘Trigonometric series with gaps’, J. Math. Mech. 9 (1960), 203–227.Google Scholar
[15]Stein, E. M., ‘Harmonic analysis on Rn’, Studies in harmonic analysis, pp. 97–135 (M.A.A. Studies Series 13, Ash, J. M. ed., 1976).Google Scholar
[16]Stein, E. M. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces (Princeton University Press, Princeton, N.J., 1971).Google Scholar