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Transactions of the American Mathematical Society

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

 

 

Extension of Fourier $ L\sp{p}---L\sp{q}$ multipliers


Author: Michael G. Cowling
Journal: Trans. Amer. Math. Soc. 213 (1975), 1-33
MSC: Primary 43A22
DOI: https://doi.org/10.1090/S0002-9947-1975-0390652-7
MathSciNet review: 0390652
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Abstract: By $ M_p^q(\Gamma )$ we denote the space of Fourier $ {L^p} - {L^q}$ multipliers on the LCA group $ \Gamma $. K. de Leeuw [4] (for $ \Gamma = {R^a}$), N. Lohoué [16] and S. Saeki [19] have shown that if $ {\Gamma _0}$ is a closed subgroup of $ \Gamma $, and $ \phi $ is a continuous function in $ M_p^p(\Gamma )$, then the restriction $ {\phi _0}$ of $ \phi $ to $ {\Gamma _0}$ is in $ M_p^p({\Gamma _0})$, and $ {\left\Vert {{\phi _0}} \right\Vert _{M_p^p}} \leqslant {\left\Vert \phi \right\Vert _{M_p^p}}$. We answer here a natural question arising from this result: we show that every continuous function $ \psi $ in $ M_p^p(\Gamma )$ is the restriction to $ {\Gamma _0}$ of a continuous $ M_p^p(\Gamma )$ function whose norm is the same as that of $ \psi $. A Figà-Talamanca and G. I. Gaudry [8] proved this with the extra condition that $ {\Gamma _0}$ be discrete: our technique develops their ideas. An extension theorem for $ M_p^q({\Gamma _0})$ is obtained: this complements work of Gaudry [11] on restrictions of $ M_p^q(\Gamma )$-functions to $ {\Gamma _0}$.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0390652-7
Keywords: LCA groups, closed subgroups, convolution operator, Fourier transform, restrictions of multipliers to closed subgroups
Article copyright: © Copyright 1975 American Mathematical Society