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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 inequalities for the Fourier transform

Author: Benjamin Muckenhoupt
Journal: Trans. Amer. Math. Soc. 276 (1983), 729-742
MSC: Primary 42A38; Secondary 26D15, 42B10, 44A15
MathSciNet review: 688974
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Abstract: Given $ p$ and $ q$ satisfying $ 1 < p \leqslant q < \infty $, sufficient conditions on nonnegative pairs of functions $ U,V$ are given to imply

$\displaystyle {\left[ {\int_{{R^n}}^{} {\vert\hat f(x){\vert^q}U(x)\,dx}} \righ... ...qslant c{\left[ {\int_{{R^n}}^{} {\vert f(x){\vert^p}V(x)\,dx}} \right]^{1/p}},$

where $ \hat f$ denotes the Fourier transform of $ f$, and $ c$ is independent of $ f$. For the case $ q = p^{\prime}$ the sufficient condition is that for all positive $ r$,

$\displaystyle \left[ {\int_{U(x) > Br} {U(x)\;dx}} \right]\left[ {\int_{V(x) < {r^{p - 1}}} {V{{(x)}^{- 1/(p - 1)}}\;dx}} \right] \leqslant A,$

where $ A$ and $ B$ are positive and independent of $ r$. For $ q \ne p^{\prime}$ the condition is more complicated but also is invariant under rearrangements of $ U$ and $ V$. In both cases the sufficient condition is shown to be necessary if the norm inequality holds for all rearrangements of $ U$ and $ V$. Examples are given to show that the sufficient condition is not necessary for a pair $ U,V$ if the norm inequality is assumed only for that pair.

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PII: S 0002-9947(1983)0688974-X
Article copyright: © Copyright 1983 American Mathematical Society

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