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

 

On maximal rearrangement inequalities for the Fourier transform


Authors: W. B. Jurkat and G. Sampson
Journal: Trans. Amer. Math. Soc. 282 (1984), 625-643
MSC: Primary 42B10; Secondary 26D15
MathSciNet review: 732111
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Abstract: Suppose that $ w$ is a measurable function on $ {{\mathbf{R}}^n}$ and denote by $ W = {w^ \ast }$ the decreasing rearrangement of $ \left\vert w \right\vert$ (provided that it exists). We show that the $ n$-dimensional Fourier transform $ \hat f$ satisfies (1)

$\displaystyle {\left\Vert {w\hat f} \right\Vert _q} \leqslant {\left\Vert {W{{(... ...t)\int_0^{1/t} {{f^ \ast }} } \right\Vert\quad (C\ {\text{absolute constant}}),$

if $ 1 < q < \infty $ and $ {t^{2/q - 1}}W(t) \searrow $ for $ t > 0$. We also show that (2)

$\displaystyle {\left\Vert {w\hat f} \right\Vert _q} \geqslant {c_{n,q}}{\left\V... ...\vert x \right\vert} {f(y)} dy} \right\Vert _q}\quad (f\ {\text{nonnegative),}}$

if $ 1 < q < \infty $ and $ w$ is nonnegative and symmetrically decreasing. Inequality (2) implies that (1) is maximal in the sense that the left side reaches the right side if $ f$ is nonnegative and symmetrically decreasing. Hence, (1) implies all other possible estimates in terms of $ W$ and $ {f^ \ast }$. The cases $ q \ne 2$ of (1) can be derived from the case $ q = 2$ (and same $ f$) by a convexity principle which does not involve interpolation. The analogue of (1) for Fourier series is due to H. L. Montgomery if $ q \geqslant 2$ (then the extra condition on $ W$ is automatically satisfied).

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DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0732111-0
PII: S 0002-9947(1984)0732111-0
Article copyright: © Copyright 1984 American Mathematical Society