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Sufficiency conditions for $ L\sp p$ multipliers with general weights


Authors: Benjamin Muckenhoupt, Richard L. Wheeden and Wo-Sang Young
Journal: Trans. Amer. Math. Soc. 300 (1987), 463-502
MSC: Primary 42A45; Secondary 42B15
DOI: https://doi.org/10.1090/S0002-9947-1987-0876462-0
MathSciNet review: 876462
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Abstract: Weighted norm inequalities in $ {R^1}$ are proved for multiplier operators with the multiplier function satisfying Hörmander type conditions. 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 a larger class of weight functions than usually considered; weight functions used here are of the form $ {\left\vert {g(x)} \right\vert^p}V(x)$ where $ g(x)$) is a polynomial of arbitrarily high degree and $ V(x)$ is in $ {A_p}$. For weight functions in $ {A_p}$, the results hold for all Schwartz functions. The periodic case is also considered.


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DOI: https://doi.org/10.1090/S0002-9947-1987-0876462-0
Article copyright: © Copyright 1987 American Mathematical Society

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