# On the Fourier transform of the symmetric decreasing rearrangements

Abstract : Inspired by work of Montgomery on Fourier series and Donoho-Strak in signal processing, we investigate two families of rearrangement inequalities for the Fourier transform. More precisely, we show that the $L^2$ behavior of a Fourier transform of a function over a small set is controlled by the $L^2$ behavior of the Fourier transform of its symmetric decreasing rearrangement. In the $L^1$ case, the same is true if we further assume that the function has a support of finite measure. As a byproduct, we also give a simple proof and an extension of a result of Lieb about the smoothness of a rearrangement. Finally, a straightforward application to solutions of the free Shrödinger equation is given.
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https://hal.archives-ouvertes.fr/hal-00316922
Contributor : Philippe Jaming <>
Submitted on : Wednesday, September 3, 2008 - 12:34:20 PM
Last modification on : Wednesday, April 1, 2020 - 12:20:06 AM
Document(s) archivé(s) le : Thursday, June 3, 2010 - 6:29:46 PM

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### Identifiers

• HAL Id : hal-00316922, version 1
• ARXIV : 0809.0604

### Citation

Philippe Jaming. On the Fourier transform of the symmetric decreasing rearrangements. Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2011, 61, pp.53-77. ⟨hal-00316922⟩

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