On the Fourier transform of the symmetric decreasing rearrangements

Jaming, Philippe

Annales de l'Institut Fourier, Tome 61 (2011) no. 1, pp. 53-77.

Le but de cet article est d’approfondir des travaux de Montgomery sur les séries de Fourier et de Donoho et Stark en traitement du signal sur la transformée de Fourier de la réarrangée d’une fonction. Plus précisément, nous montrons que le comportement $L^2$ sur un petit ensemble de la transformée de Fourier d’une fonction est contrôlé par le comportement $L^2$ de la transformée de Fourier de sa réarrangée symétrique. Dans le cas $L^1$ un résultat similaire est démontré pour les fonctions à support de mesure finie.

Par ailleurs, nous donnons une démonstration simple et une extension d’un résultat de Lieb sur la régularité d’une réarrangée. Finalement, nous donnons une application directe aux solutions de l’équation de Shrödinger.

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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