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)

 
 

 

Some strict inclusions between spaces of $ L\sp{p}$-multipliers


Author: J. F. Price
Journal: Trans. Amer. Math. Soc. 152 (1970), 321-330
MSC: Primary 46.35; Secondary 42.00
DOI: https://doi.org/10.1090/S0002-9947-1970-0282210-5
MathSciNet review: 0282210
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that the Hausdorff topological group $ G$ is either compact or locally compact abelian and that $ {C_c}$ denotes the set of continuous complex-valued functions on $ G$ with compact supports. Let $ L_p^q$ denote the restrictions to $ {C_c}$ of the continuous linear operators from $ {L^p}(G)$ into $ {L^q}(G)$ which commute with all the right translation operators.

When $ 1 \leqq p < q \leqq 2$ or $ 2 \leqq q < p \leqq \infty $ it is known that

$\displaystyle (1)\quad L_p^p \subset L_q^q.$

The main result of this paper is that the inclusion in (1) is strict unless $ G$ is finite. In fact it will be shown, using a partly constructive proof, that when $ G$ is infinite

$\displaystyle \bigcup\limits_{1 \leqq q < p} {L_q^q \subsetneqq } L_p^p \subsetneqq \bigcap\limits_{p < q \leqq 2} {L_q^q} $

for $ 1 < p < 2$, with the first inclusion remaining strict when $ p = 2$ and the second inclusion remaining strict when $ p = 1$. (Similar results also hold for $ 2 \leqq p \leqq \infty $.)

When $ G$ is compact, simple relations will also be developed between idempotent operators in $ L_p^q$ and lacunary subsets of the dual of $ G$ which will enable us to find necessary conditions so that inclusion (1) is strict even if, for example, $ L_p^p$ and $ L_q^q$ are replaced by the sets of idempotent operators in $ L_p^p$ and $ L_q^q$ respectively.


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

  • [1] B. Brainerd and R. E. Edwards, Linear operators which commute with translations. I: Representation theorems, J. Austral. Math. Soc. 6 (1966), 289-327. MR 34 #6542. MR 0206725 (34:6542)
  • [2] R. E. Edwards, Functional analysis: Theory and applications, Holt, Rinehart and Winston, New York, 1965. MR 36 #4308. MR 0221256 (36:4308)
  • [3] -, Changing signs of Fourier coefficients, Pacific J. Math. 15 (1965), 463-475. MR 34 #564. MR 0200676 (34:564)
  • [4] -, Fourier series: A modern introduction. Vol. II, Holt, Rinehart and Winston, New York, 1967. MR 36 #5588. MR 0222538 (36:5588)
  • [5] 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)
  • [6] A. Figà-Talamanca, Translation invariant operators in $ {L^p}$, Duke Math. J. 32 (1965), 495-502. MR 31 #6095. MR 0181869 (31:6095)
  • [7] A. Figà-Talamanca and G. I. Gaudry, Density and representation theorems for multipliers of type $ (p,q)$, J. Austral. Math. Soc. 7 (1967), 1-6. MR 35 #666. MR 0209770 (35:666)
  • [8] -, Multipliers and sets of uniqueness of $ {L^p}$, Michigan Math. J. (to appear).
  • [9] A. Figà-Talamanca and D. Rider, A theorem of Littlewood and lacunary series for compact groups, Pacific J. Math. 16 (1966), 505-514. MR 34 #6444. MR 0206626 (34:6444)
  • [10] 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, 1970. MR 0262773 (41:7378)
  • [11] W. Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203-227. MR 22 #6972. MR 0116177 (22:6972)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46.35, 42.00

Retrieve articles in all journals with MSC: 46.35, 42.00


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0282210-5
Keywords: Lebesgue function spaces, locally compact abelian groups, compact groups, translations, multiplier operators, idempotent multiplier operators
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society