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Restrictions of $ L\sp{p}$ transforms


Author: Louis Pigno
Journal: Proc. Amer. Math. Soc. 29 (1971), 511-515
MSC: Primary 42.40; Secondary 46.00
DOI: https://doi.org/10.1090/S0002-9939-1971-0279531-5
Erratum: Proc. Amer. Math. Soc. 48 (1975), 515.
MathSciNet review: 0279531
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Abstract: Let G be a locally compact abelian group with dual $ \Gamma $, E a subset of $ \Gamma $, and $ \phi $ a complex-valued function defined on $ \Gamma $. Assume $ \phi $ has $ \sigma $-compact support. In this paper we prove that $ \phi $ is a multiplier of type $ ({L^1},{L^{{p_1}}} \cap {L^{{p_2}}},E)\;(1 \leqq {p_1} \leqq 2,1 < {p_2} \leqq \infty )$ if and only if $ \phi = \hat f$ a.e. on E for some $ f \in {L^{{p_1}}}(G) \cap {L^{{p_2}}}(G)$. We give applications of this result to the problems of restrictions, uniqueness, inversion and characterization of $ {L^p}$ transforms.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0279531-5
Keywords: Multiplier, quotient space, regular Toeplitz summation matrix
Article copyright: © Copyright 1971 American Mathematical Society

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