Weighted Paley–Wiener theorem on the Hilbert transform

Elijah Liflyand; Sergey Tikhonov

doi:10.1016/j.crma.2010.10.028

Mathematical Analysis

Weighted Paley–Wiener theorem on the Hilbert transform
[Version avec poids du théorème de Paley–Wiener sur la transformée de Hilbert]

Présenté par : Jean-Pierre Kahane

Elijah Liflyand ¹ ; Sergey Tikhonov ²

¹ Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
² ICREA and Centre de Recerca Matemàtica (CRM), 08193 Bellaterra, Barcelona, Spain

Comptes Rendus. Mathématique, Volume 348 (2010) no. 23-24, pp. 1253-1258.

Résumé
Abstract

Nous prouvons des analogues avec poids du théorème de Paley–Wiener, à savoir l'intégrabilité de la transformée de Hilbert d'une fonction intégrable impaire décroissante sur $R^{+}$ . Nos résultats étendent au cas $p = 1$ ceux de Hardy–Littlewood et de Flett concernant l'intégrabilité avec poids de la transformée de Hilbert d'une fonction paire ou impaire sous la même condition de décroissance sur $R^{+}$ ou sous la condition moins restrictive de « monotonie généralisée ».

We prove weighted analogues of the Paley–Wiener theorem on integrability of the Hilbert transform of an integrable odd function which is monotone on $R_{+}$ . This extends Hardy–Littlewood's and Flett's results to the case $p = 1$ under the assumption of (general) monotonicity for an even/odd function.

Reçu le : 2010-09-17
Accepté le : 2010-10-18
Publié le : 2010-12-01

DOI : 10.1016/j.crma.2010.10.028

Affiliations des auteurs :

Elijah Liflyand ¹ ; Sergey Tikhonov ²

¹ Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
² ICREA and Centre de Recerca Matemàtica (CRM), 08193 Bellaterra, Barcelona, Spain

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     title = {Weighted {Paley{\textendash}Wiener} theorem on the {Hilbert} transform},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1253--1258},
     publisher = {Elsevier},
     volume = {348},
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     year = {2010},
     doi = {10.1016/j.crma.2010.10.028},
     language = {en},
}

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Elijah Liflyand; Sergey Tikhonov. Weighted Paley–Wiener theorem on the Hilbert transform. Comptes Rendus. Mathématique, Volume 348 (2010) no. 23-24, pp. 1253-1258. doi : 10.1016/j.crma.2010.10.028. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.10.028/

Bibliographie
Cité par

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[3] J. Garcia-Cuerva; J.L. Rubio de Francia Weighted Norm Inequalities and Related Topics, North-Holland, 1985

[4] G.H. Hardy; J.E. Littlewood Some more theorems concerning Fourier series and Fourier power series, Duke Math. J., Volume 2 (1936), pp. 354-382

[5] R. Hunt; B. Muckenhoupt; R. Wheeden Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc., Volume 176 (1973), pp. 227-251

[6] H. Kober A note on Hilbert's operator, Bull. Amer. Math. Soc., Volume 48 (1942) no. 1, pp. 421-426

[7] E. Liflyand; S. Tikhonov The Fourier transforms of general monotone functions, Analysis and Mathematical Physics, Trends in Mathematics, Birkhäuser, 2009, pp. 373-391

[8] E. Liflyand, S. Tikhonov, A concept of general monotonicity and applications, Math. Nachrichten., in press.

[9] R.E.A.C. Paley; N. Wiener Notes on the theory and application of Fourier transform, Note II, Trans. Amer. Math. Soc., Volume 35 (1933), pp. 354-355

[10] S. Tikhonov Trigonometric series with general monotone coefficients, J. Math. Anal. Appl., Volume 326 (2007), pp. 721-735

[11] A. Zygmund Some points in the theory of trigonometric and power series, Trans. Amer. Math. Soc., Volume 36 (1934), pp. 586-617

Cité par Sources :

^☆ The research was partially supported by the MTM 2008-05561-C02-02, 2009 SGR 1303, RFFI 09-01-00175, NSH-3252.2010.1, and ESF Network Programme HCAA.

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