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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Weighted norm estimates for the Fourier transform with a pair of weights


Authors: Jan-Olov Strömberg and Richard L. Wheeden
Journal: Trans. Amer. Math. Soc. 318 (1990), 355-372
MSC: Primary 42B10; Secondary 42B30
MathSciNet review: 1002924
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Abstract: We prove weighted norm inequalities of the form

$\displaystyle {\left\Vert {\hat f} \right\Vert _{L_u^q}} \leq C{\left\Vert f \right\Vert _{H_\upsilon ^p}},\quad 0 < p \leq q < \infty ,$

for the Fourier transform on $ {{\mathbf{R}}^n}$. For some weight functions $ \upsilon $, the Hardy space $ H_\upsilon ^p$ on the right can be replaced by $ L_\upsilon ^p$. The proof depends on making an atomic decomposition of $ f$ and using cancellation properties of the atoms.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1990-1002924-1
PII: S 0002-9947(1990)1002924-1
Article copyright: © Copyright 1990 American Mathematical Society