On the Fourier transform of the symmetric decreasing rearrangements

Philippe Jaming (1, 2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 IMB - Institut de Mathématiques de Bordeaux

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.

Mots clés: Fourier transform; rearrangement inequalities; Bessel functions

Domaines: Analyse classique [math.CA]

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