On the Fourier transform of the symmetric decreasing rearrangements
Annales de l'Institut Fourier, Volume 61 (2011) no. 1, p. 53-77
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.
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.
DOI : https://doi.org/10.5802/aif.2597
Classification:  42A38,  42B10,  42C20,  33C10
Keywords: Fourier transform, rearrangement inequalities, Bessel functions
@article{AIF_2011__61_1_53_0,
     author = {Jaming, Philippe},
     title = {On the Fourier transform of the symmetric decreasing rearrangements},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {1},
     year = {2011},
     pages = {53-77},
     doi = {10.5802/aif.2597},
     zbl = {1242.42005},
     mrnumber = {2828126},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2011__61_1_53_0}
}
Jaming, Philippe. On the Fourier transform of the symmetric decreasing rearrangements. Annales de l'Institut Fourier, Volume 61 (2011) no. 1, pp. 53-77. doi : 10.5802/aif.2597. https://aif.centre-mersenne.org/item/AIF_2011__61_1_53_0/

[1] Askey, R.; Steinig, J. Some positive trigonometric sums, Trans. Amer. Math. Soc., Tome 187 (1974), pp. 295-307 | Article | MR 338481 | Zbl 0244.42002

[2] Benedetto, John J.; Heinig, Hans P. Weighted Fourier inequalities: new proofs and generalizations, J. Fourier Anal. Appl., Tome 9 (2003) no. 1, pp. 1-37 | Article | MR 1953070 | Zbl 1034.42010

[3] Burchard, A. Steiner symmetrization is continuous in W 1,p , Geom. Funct. Anal., Tome 7 (1997) no. 5, pp. 823-860 | Article | MR 1475547 | Zbl 0912.46034

[4] Cianchi, Andrea Second-order derivatives and rearrangements, Duke Math. J., Tome 105 (2000) no. 3, pp. 355-385 | Article | MR 1801766 | Zbl 1017.46023

[5] Cooke, R. G. Gibbs’ phenomenon in Fourier-Bessel series and integrals, Proc. London Math. Soc., Tome 27 (1927), pp. 171-192 | Article

[6] Cooke, R. G. A monotonic property of Bessel functions, J. London Math. Soc., Tome 12 (1937), pp. 180-185 | Article

[7] Donoho, David L.; Stark, Philip B. Rearrangements and smoothing, Tech. Rep. (1988) (Dept. of Statist., Univ. of California, Berkeley)

[8] Donoho, David L.; Stark, Philip B. A note on rearrangements, spectral concentration, and the zero-order prolate spheroidal wavefunction, IEEE Trans. Inform. Theory, Tome 39 (1993) no. 1, pp. 257-260 | Article | MR 1211505 | Zbl 0767.33018

[9] Feller, William An introduction to probability theory and its applications. Vol. II., John Wiley & Sons Inc., New York, Second edition (1971) | MR 270403 | Zbl 0138.10207

[10] Gasper, George Positive integrals of Bessel functions, SIAM J. Math. Anal., Tome 6 (1975) no. 5, pp. 868-881 | Article | MR 390318 | Zbl 0313.33013

[11] Grafakos, Loukas Classical and modern Fourier analysis, Pearson Education, Inc., Upper Saddle River, NJ (2004) | MR 2449250 | Zbl 1148.42001

[12] Havin, Victor; Jöricke, Burglind The uncertainty principle in harmonic analysis, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 28 (1994) | MR 1303780 | Zbl 0827.42001

[13] Jaming, Philippe Nazarov’s uncertainty principles in higher dimension, J. Approx. Theory, Tome 149 (2007) no. 1, pp. 30-41 | Article | MR 2371612 | Zbl pre05237721

[14] Jodeit, M.; Tochinsky, A. Inequalities for the Fourier Transform, Studia Math., Tome 37 (1971), pp. 245-276 | MR 300073 | Zbl 0224.46037

[15] Jurkat, W. B.; Sampson, G. On maximal rearrangement inequalities for the Fourier transform, Trans. Amer. Math. Soc., Tome 282 (1984) no. 2, pp. 625-643 | Article | MR 732111 | Zbl 0537.42029

[16] Jurkat, W. B.; Sampson, G. On rearrangement and weighted inequalities for the Fourier transform, Indiana Univ. Math. J., Tome 33 (1984), pp. 257-270 | Article | MR 733899 | Zbl 0536.42013

[17] Kawohl, Bernhard Rearrangements and convexity of level sets in PDE, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1150 (1985) | MR 810619 | Zbl 0593.35002

[18] Lieb, Elliott H. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), Tome 118 (1983) no. 2, pp. 349-374 | Article | MR 717827 | Zbl 0527.42011

[19] Lieb, Elliott H.; Loss, Michael Analysis, American Mathematical Society, Providence, RI, Graduate Studies in Mathematics, Tome 14 (1997) | MR 1415616

[20] Makai, E. On a monotonic property of certain Sturm-Liouville functions, Acta Math. Acad. Sci. Hungar., Tome 3 (1952), pp. 165-172 | Article | MR 54103 | Zbl 0048.32302

[21] Misiewicz, Jolanta K.; Richards, Donald St. P. Positivity of integrals of Bessel functions, SIAM J. Math. Anal., Tome 25 (1994) no. 2, pp. 596-601 | Article | MR 1266579 | Zbl 0799.33003

[22] Montgomery, Hugh L. A note on rearrangements of Fourier coefficients, Ann. Inst. Fourier (Grenoble), Tome 26 (1976) no. 2, pp. v, 29-34 | Article | Numdam | MR 407517 | Zbl 0318.42009

[23] Nazarov, F. L. Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type, Algebra i Analiz, Tome 5 (1993) no. 4, pp. 3-66 | MR 1246419 | Zbl 0801.42001

[24] Pisier, Gilles The volume of convex bodies and Banach space geometry, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 94 (1989) | Article | MR 1036275 | Zbl 0698.46008

[25] Steinig, John On a monotonicity property of Bessel functions, Math. Z., Tome 122 (1971) no. 4, pp. 363-365 | Article | MR 447654 | Zbl 0208.32802

[26] Tao, Terence Nonlinear dispersive equations, Published for the Conference Board of the Mathematical Sciences, Washington, DC, CBMS Regional Conference Series in Mathematics, Tome 106 (2006) (Local and global analysis) | MR 2233925 | Zbl 1106.35001

[27] Vretblad, Anders Fourier analysis and its applications, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 223 (2003) | MR 1992764 | Zbl 1032.42001

[28] Watson, G. N. A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, Cambridge Mathematical Library (1995) (Reprint of the second (1944) edition) | MR 1349110 | Zbl 0849.33001

[29] Widder, David Vernon The Laplace Transform, Princeton University Press, Princeton, N. J., Princeton Mathematical Series, v. 6 (1941) | MR 5923 | Zbl 0063.08245

[30] Zygmund, Antoni Trigonometrical series, Dover Publications, New York (1955) | MR 72976 | Zbl 0065.05604