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Sufficiency conditions for $ L\sp p$-multipliers with power weights


Authors: Benjamin Muckenhoupt, Richard L. Wheeden and Wo-Sang Young
Journal: Trans. Amer. Math. Soc. 300 (1987), 433-461
MSC: Primary 42A45; Secondary 42B15
DOI: https://doi.org/10.1090/S0002-9947-1987-0876461-9
MathSciNet review: 876461
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Abstract: Weighted norm inequalities in $ {R^1}$ are proved for multiplier operators with the multiplier function of Hörmander type. The operators are initially defined on the space $ {\mathcal{S}_{0,0}}$ of Schwartz functions whose Fourier transforms have compact support not including 0. This restriction on the domain of definition makes it possible to use weight functions of the form $ {\left\vert x \right\vert^\alpha }$ for $ \alpha $ larger than usually considered. For these weight functions, if $ (\alpha + 1)/p$ is not an integer, a strict inequality on $ \alpha $ is shown to be sufficient for a norm inequality to hold. A sequel to this paper shows that the weak version of this inequality is necessary.


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DOI: https://doi.org/10.1090/S0002-9947-1987-0876461-9
Article copyright: © Copyright 1987 American Mathematical Society

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