A note on rearrangements of Fourier coefficients
Annales de l'Institut Fourier, Volume 26 (1976) no. 2, p. 29-34

Let $f\left(x\right)\sim \Sigma {a}_{n}{e}^{2\pi inx},f*\left(x\right)\sim {\sum }_{n=0}^{\infty }a{*}_{n}\phantom{\rule{0.166667em}{0ex}}\mathrm{cos}\phantom{\rule{0.166667em}{0ex}}2\pi nx$, where the $a{*}_{n}$ are the numbers $|{a}_{n}|$ rearranged so that ${a}_{n}^{*}↘0$. Then for any convex increasing $\psi$, $\parallel \psi \left(|f{|}^{2}{\parallel }_{1}\le \parallel \psi \left(20|f*{|}^{2}{\parallel }_{1}$. The special case $\psi \left(t\right)={t}^{q/2}$, $q\ge 2$, gives $\parallel f{\parallel }_{q}\le 5\parallel f*{\parallel }_{q}$ an equivalent of Littlewood.

Soit $f\left(x\right)\sim \Sigma {a}_{n}{e}^{2\pi inx},f*\left(x\right)\sim {\sum }_{n=0}^{\infty }a{*}_{n}\phantom{\rule{0.166667em}{0ex}}\mathrm{cos}\phantom{\rule{0.166667em}{0ex}}2\pi nx$, où la suite $a{*}_{n}$ est le réarrangement décroissant de la suite $|{a}_{n}|$. Pour toute fonction $\psi$ positive, convexe et croissante, on a $\parallel \psi \left(|f{|}^{2}{\parallel }_{1}\le \parallel \psi \left(20|f*{|}^{2}{\parallel }_{1}$. Dans le cas particulier $\psi \left(t\right)={t}^{q/2}$, $q\ge 2$, on obtient l’inégalité de Littlewood $\parallel f{\parallel }_{q}\le 5\parallel f*{\parallel }_{q}$.

@article{AIF_1976__26_2_29_0,
author = {Montgomery, Hugh L.},
title = {A note on rearrangements of Fourier coefficients},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Durand},
volume = {26},
number = {2},
year = {1976},
pages = {29-34},
doi = {10.5802/aif.612},
zbl = {0318.42009},
mrnumber = {53 \#11292},
language = {en},
url = {http://www.numdam.org/item/AIF_1976__26_2_29_0}
}

Montgomery, Hugh L. A note on rearrangements of Fourier coefficients. Annales de l'Institut Fourier, Volume 26 (1976) no. 2, pp. 29-34. doi : 10.5802/aif.612. http://www.numdam.org/item/AIF_1976__26_2_29_0/

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