Tame $L^p$-multipliers

Hare, Kathryn

Colloquium Mathematicae, Tome 66 (1993), p. 303-314 / Harvested from The Polish Digital Mathematics Library

Résumé

We call an $L^{p}$-multiplier m tame if for each complex homomorphism χ acting on the space of $L^{p}$ multipliers there is some $γ_{0} ∈ Γ$ and |a| ≤ 1 such that $χ(γm) = am(γ_{0}γ)$ for all γ ∈ Γ. Examples of tame multipliers include tame measures and one-sided Riesz products. Tame multipliers show an interesting similarity to measures. Indeed we show that the only tame idempotent multipliers are measures. We obtain quantitative estimates on the size of $L^{p}$-improving tame multipliers which are similar to those obtained for measures, but are false for non-tame multipliers. One-sided Riesz products are seen to play a similar role in the study of tame multipliers as Riesz products do in the study of measures.

Détail

Publié le : 1993-01-01
EUDML-ID : urn:eudml:doc:210194

Bibliographie

[000] [1] G. Brown, Riesz products and generalized characters, Proc. London Math. Soc. 30 (1975), 209-238. | Zbl 0325.43003

[001] [2] P. J. Cohen, On a conjecture of Littlewood and idempotent measures, Amer. J. Math. 82 (1960), 191-212. | Zbl 0099.25504

[002] [3] J. Diestel and J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., 1977.

[003] [4] R. E. Edwards, Fourier Series, Vol. 2, Springer, New York 1982. | Zbl 0599.42001

[004] [5] C. Graham, K. Hare and D. Ritter, The size of $L^p$-improving measures, J. Funct. Anal. 84 (1989), 472-495. | Zbl 0678.43001

[005] [6] C. C. Graham and O. C. McGehee, Essays in Commutative Harmonic Analysis, Springer, New York 1979. | Zbl 0439.43001

[006] [7] A. Grothendieck, Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74 (1952), 168-186. | Zbl 0046.11702

[007] [8] K. Hare, A characterization of $L^p$-improving measures, Proc. Amer. Math. Soc. 102 (1988), 295-299. | Zbl 0664.43001

[008] [9] K. Hare, Arithmetic properties of thin sets, Pacific J. Math. 131 (1988), 143-155. | Zbl 0603.43003

[009] [10] K. Hare, Properties and examples of $(L^p, L^q)$ multipliers, Indiana Univ. Math. J. 38 (1989), 211-227. | Zbl 0655.43003

[010] [11] K. Hare, Union results for thin sets, Glasgow Math. J. 32 (1990), 241-254. | Zbl 0714.43010

[011] [12] K. Hare, The size of $(L^2, L^p)$ multipliers, Colloq. Math. 63 (1992), 249-262. | Zbl 0795.43005

[012] [13] B. Host et F. Parreau, Ensembles de Rajchman et ensembles de continuité, C. R. Acad. Sci. Paris 288 (1979), 899-902. | Zbl 0422.43009

[013] [14] B. Host et F. Parreau, Sur les mesures dont la transformée de Fourier-Stieltjes ne tend pas vers 0 à l'infini, Colloq. Math. 41 (1979), 285-289. | Zbl 0466.43005

[014] [15] I. Klemes, Idempotent multipliers of $H^1(T)$, Canad. J. Math. 39 (1987), 1223-1234.

[015] [16] R. Larson, An Introduction to the Theory of Multipliers, Grundlehren Math. Wiss. 175, Springer, New York 1971.

[016] [17] J. F. Méla, Mesures ε-idempotentes de norme bornée, Studia Math. 72 (1982), 131-149. | Zbl 0503.43004

[017] [18] D. Oberlin, A convolution property of the Cantor-Lebesgue measure, Colloq. Math. 47 (1982), 113-117. | Zbl 0501.42007

[018] [19] A. Rajchman, Une classe de séries trigonométriques qui convergent presque partout vers zéro, Math. Ann. 101 (1929), 686-700. | Zbl 55.0162.04

[019] [20] L. T. Ramsey and B. B. Wells, Jr., Fourier-Stieltjes transforms of strongly continuous measures, Michigan Math. J. 24 (1977), 13-19. | Zbl 0346.43004

[020] [21] C. Rickart, The General Theory of Banach Algebras, Van Nostrand, Princeton 1960.

[021] [22] D. Ritter, Most Riesz product measures are $L^p$-improving, Proc. Amer. Math. Soc. 97 (1986), 291-295. | Zbl 0593.43002