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Transactions of the American Mathematical Society

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

 
 

 

$ L\sp{p}$ multipliers with weight $ x\sp{kp-1}$


Authors: Benjamin Muckenhoupt and Wo Sang Young
Journal: Trans. Amer. Math. Soc. 275 (1983), 623-639
MSC: Primary 42A45
DOI: https://doi.org/10.1090/S0002-9947-1983-0682722-5
MathSciNet review: 682722
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Abstract: Let $ k$ be a positive integer and $ 1 < p < \infty $. It is shown that if $ T$ is a multiplier operator on $ {L^p}$ of the line with weight $ \vert x{\vert^{kp-1}}$, then $ Tf$ equals a constant times $ f$ almost everywhere. This does not extend to the periodic case since $ m(j) = 1/j, j \ne 0$, is a multiplier sequence for $ {L^p}$ of the circle with weight $ \vert x{\vert^{kp-1}}$. A necessary and sufficient condition is derived for a sequence $ m(j)$ to be a multiplier on $ {L^2}$ of the circle with weight $ \vert x{\vert^{2k - 1}}$.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0682722-5
Article copyright: © Copyright 1983 American Mathematical Society

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