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Transactions of the American Mathematical Society

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



Fourier inequalities with nonradial weights

Author: C. Carton-Lebrun
Journal: Trans. Amer. Math. Soc. 333 (1992), 751-767
MSC: Primary 42B10; Secondary 26D10, 47G10
MathSciNet review: 1132433
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Abstract: Let $ \mathcal{F}\;f(\gamma ) = {\smallint _{{\mathbb{R}^n}}}({e^{ - 2i\pi \gamma \bullet x}} - 1)f(x)\,dx,n > 1$, and $ u$, $ v$ be nonnegative functions. Sufficient conditions are found in order that $ \left\Vert \mathcal{F}\;f\right\Vert _{q,u} \leq C\left\Vert f\right\Vert _{p,v}$ for all $ f \in L_v^p({\mathbb{R}^n})$. Pointwise and norm approximations of $ \mathcal{F}\;f$ are derived. Similar results are obtained when $ u$ is replaced by a measure weight. In the case $ v(x) = \vert x{\vert^{n(p - 1)}}$, a counterexample is given which shows that no Fourier inequality can hold for all $ f$ in $ L_{c,0}^\infty $. Spherical restriction theorems are established. Further conditions for the boundedness of $ \mathcal{F}$ are discussed.

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