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Transactions of the American Mathematical Society
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
MathSciNet review: 0282210
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1970-0282210-5
PII: S 0002-9947(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