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Weighted norm inequalities for the Fourier transform on certain totally disconnected groups


Author: T. S. Quek
Journal: Proc. Amer. Math. Soc. 101 (1987), 113-121
MSC: Primary 43A70
DOI: https://doi.org/10.1090/S0002-9939-1987-0897080-X
MathSciNet review: 897080
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Abstract: Let $ G$ be a locally compact totally disconnected Abelian group with dual group $ \Gamma $. Let $ U$ and $ V$ be nonnegative measurable functions on $ \Gamma $ and $ G$, respectively. In this paper we give, in terms of $ U$ and $ V$, a necessary condition and some sufficient conditions for the inequality $ \vert\vert\hat fU\vert{\vert _q} \leq C\vert\vert fV\vert{\vert _p}$ to hold for all $ f$ in $ {L_1}\left( G \right)$, where $ \hat f$ denotes the Fourier transform of $ f$ and $ 1 < p \leq q < \infty $. If $ U$ and $ V$ are both radial, we give a necessary and sufficient condition for the above norm inequality to hold for all $ f$ in $ {L_1}\left( G \right)$.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0897080-X
Article copyright: © Copyright 1987 American Mathematical Society

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