We prove weighted analogues of the Paley–Wiener theorem on integrability of the Hilbert transform of an integrable odd function which is monotone on . This extends Hardy–Littlewood's and Flett's results to the case under the assumption of (general) monotonicity for an even/odd function.
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 . Nos résultats étendent au cas 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 ou sous la condition moins restrictive de « monotonie généralisée ».
Accepted:
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@article{CRMATH_2010__348_23-24_1253_0,
author = {Liflyand, Elijah and Tikhonov, Sergey},
title = {Weighted {Paley{\textendash}Wiener} theorem on the {Hilbert} transform},
journal = {Comptes Rendus. Math\'ematique},
pages = {1253--1258},
publisher = {Elsevier},
volume = {348},
number = {23-24},
year = {2010},
doi = {10.1016/j.crma.2010.10.028},
language = {en},
url = {https://www.numdam.org/articles/10.1016/j.crma.2010.10.028/}
}
TY - JOUR AU - Liflyand, Elijah AU - Tikhonov, Sergey TI - Weighted Paley–Wiener theorem on the Hilbert transform JO - Comptes Rendus. Mathématique PY - 2010 SP - 1253 EP - 1258 VL - 348 IS - 23-24 PB - Elsevier UR - https://www.numdam.org/articles/10.1016/j.crma.2010.10.028/ DO - 10.1016/j.crma.2010.10.028 LA - en ID - CRMATH_2010__348_23-24_1253_0 ER -
%0 Journal Article %A Liflyand, Elijah %A Tikhonov, Sergey %T Weighted Paley–Wiener theorem on the Hilbert transform %J Comptes Rendus. Mathématique %D 2010 %P 1253-1258 %V 348 %N 23-24 %I Elsevier %U https://www.numdam.org/articles/10.1016/j.crma.2010.10.028/ %R 10.1016/j.crma.2010.10.028 %G en %F CRMATH_2010__348_23-24_1253_0
Liflyand, Elijah; Tikhonov, Sergey. 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://www.numdam.org/articles/10.1016/j.crma.2010.10.028/
[1] Weighted norm inequalities for Hilbert transforms and conjugate functions of even and odd functions, Proc. Amer. Math. Soc., Volume 56 (1976) no. 1, pp. 99-107
[2] Some theorems on odd and even functions, Proc. London Math. Soc. (3), Volume 8 (1958), pp. 135-148
[3] Weighted Norm Inequalities and Related Topics, North-Holland, 1985
[4] Some more theorems concerning Fourier series and Fourier power series, Duke Math. J., Volume 2 (1936), pp. 354-382
[5] Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc., Volume 176 (1973), pp. 227-251
[6] A note on Hilbert's operator, Bull. Amer. Math. Soc., Volume 48 (1942) no. 1, pp. 421-426
[7] 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] Notes on the theory and application of Fourier transform, Note II, Trans. Amer. Math. Soc., Volume 35 (1933), pp. 354-355
[10] Trigonometric series with general monotone coefficients, J. Math. Anal. Appl., Volume 326 (2007), pp. 721-735
[11] Some points in the theory of trigonometric and power series, Trans. Amer. Math. Soc., Volume 36 (1934), pp. 586-617
Cited by 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.
