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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Radial functions and invariant convolution operators

Author: Christopher Meaney
Journal: Trans. Amer. Math. Soc. 286 (1984), 665-674
MSC: Primary 43A22; Secondary 42B15
MathSciNet review: 760979
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Abstract: For $ 1 < p < 2$ and $ n > 1$, let $ {A_p}({{\mathbf{R}}^n})$ denote the Figà-Talamanca-Herz algebra, consisting of functions of the form $ ( \ast)$

$\displaystyle \sum\limits_{k = 0}^\infty {{f_k} \ast {g_k}} $

with $ \sum\nolimits_k {\vert\vert{f_k}\vert{\vert _p}\cdot\vert\vert{g_k}\vert{\vert _{p\prime}} < \infty } $. We show that if $ 2n/(n + 1) < p < 2$, then the subalgebra of radial functions in $ {A_p}({{\mathbf{R}}^n})$ is strictly larger than the subspace of functions with expansions $ ( \ast )$ subject to the additional condition that $ {f_k}$ and $ {g_k}$ are radial for all $ k$. This is a partial answer to a question of Eymard and is a consequence of results of Herz and Fefferman. We arrive at the statement above after examining a more abstract situation. Namely, we fix $ G \in [FIA]_{B}^{ - }$ and consider $ ^B{A_p}(G)$ the subalgebra of $ B$-invariant elements of $ {A_p}(G)$. In particular, we show that the dual of $ ^B{A_p}(G)$ is equal to the space of bounded, right-translation invariant operators on $ {L^{p}}(G)$ which commute with the action of $ B$.

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Keywords: Radial function, Figà-Talamanca-Herz algebra, $ [FIA]_{B}^{ - }$, $ B$-characters, convolution operator, compact semisimple Lie group, central function, multiplier, Fourier transform
Article copyright: © Copyright 1984 American Mathematical Society