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Transactions of the American Mathematical Society

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

 
 

 

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


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 $|x{|^{kp-1}}$, 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 $|x{|^{kp-1}}$. A necessary and sufficient condition is derived for a sequence $m(j)$ to be a multiplier on ${L^2}$ of the circle with weight $|x{|^{2k - 1}}$.


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Article copyright: © Copyright 1983 American Mathematical Society