# Tame ${L}^{p}$-multipliers

Colloquium Mathematicae (1993)

- Volume: 64, Issue: 2, page 303-314
- ISSN: 0010-1354

## Access Full Article

top## Abstract

top## How to cite

topHare, Kathryn. "Tame $L^p$-multipliers." Colloquium Mathematicae 64.2 (1993): 303-314. <http://eudml.org/doc/210194>.

@article{Hare1993,

abstract = {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.},

author = {Hare, Kathryn},

journal = {Colloquium Mathematicae},

keywords = {multipliers; tame measures; Fourier transform; Rajchman sets; Hardy space; complex Lie group},

language = {eng},

number = {2},

pages = {303-314},

title = {Tame $L^p$-multipliers},

url = {http://eudml.org/doc/210194},

volume = {64},

year = {1993},

}

TY - JOUR

AU - Hare, Kathryn

TI - Tame $L^p$-multipliers

JO - Colloquium Mathematicae

PY - 1993

VL - 64

IS - 2

SP - 303

EP - 314

AB - 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.

LA - eng

KW - multipliers; tame measures; Fourier transform; Rajchman sets; Hardy space; complex Lie group

UR - http://eudml.org/doc/210194

ER -

## References

top- [1] G. Brown, Riesz products and generalized characters, Proc. London Math. Soc. 30 (1975), 209-238. Zbl0325.43003
- [2] P. J. Cohen, On a conjecture of Littlewood and idempotent measures, Amer. J. Math. 82 (1960), 191-212. Zbl0099.25504
- [3] J. Diestel and J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., 1977.
- [4] R. E. Edwards, Fourier Series, Vol. 2, Springer, New York 1982. Zbl0599.42001
- [5] C. Graham, K. Hare and D. Ritter, The size of ${L}^{p}$-improving measures, J. Funct. Anal. 84 (1989), 472-495. Zbl0678.43001
- [6] C. C. Graham and O. C. McGehee, Essays in Commutative Harmonic Analysis, Springer, New York 1979. Zbl0439.43001
- [7] A. Grothendieck, Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74 (1952), 168-186. Zbl0046.11702
- [8] K. Hare, A characterization of ${L}^{p}$-improving measures, Proc. Amer. Math. Soc. 102 (1988), 295-299. Zbl0664.43001
- [9] K. Hare, Arithmetic properties of thin sets, Pacific J. Math. 131 (1988), 143-155. Zbl0603.43003
- [10] K. Hare, Properties and examples of $({L}^{p},{L}^{q})$ multipliers, Indiana Univ. Math. J. 38 (1989), 211-227. Zbl0655.43003
- [11] K. Hare, Union results for thin sets, Glasgow Math. J. 32 (1990), 241-254. Zbl0714.43010
- [12] K. Hare, The size of $({L}^{2},{L}^{p})$ multipliers, Colloq. Math. 63 (1992), 249-262. Zbl0795.43005
- [13] B. Host et F. Parreau, Ensembles de Rajchman et ensembles de continuité, C. R. Acad. Sci. Paris 288 (1979), 899-902. Zbl0422.43009
- [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. Zbl0466.43005
- [15] I. Klemes, Idempotent multipliers of ${H}^{1}\left(T\right)$, Canad. J. Math. 39 (1987), 1223-1234.
- [16] R. Larson, An Introduction to the Theory of Multipliers, Grundlehren Math. Wiss. 175, Springer, New York 1971.
- [17] J. F. Méla, Mesures ε-idempotentes de norme bornée, Studia Math. 72 (1982), 131-149. Zbl0503.43004
- [18] D. Oberlin, A convolution property of the Cantor-Lebesgue measure, Colloq. Math. 47 (1982), 113-117. Zbl0501.42007
- [19] A. Rajchman, Une classe de séries trigonométriques qui convergent presque partout vers zéro, Math. Ann. 101 (1929), 686-700. Zbl55.0162.04
- [20] L. T. Ramsey and B. B. Wells, Jr., Fourier-Stieltjes transforms of strongly continuous measures, Michigan Math. J. 24 (1977), 13-19. Zbl0346.43004
- [21] C. Rickart, The General Theory of Banach Algebras, Van Nostrand, Princeton 1960.
- [22] D. Ritter, Most Riesz product measures are ${L}^{p}$-improving, Proc. Amer. Math. Soc. 97 (1986), 291-295. Zbl0593.43002

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.