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Hardy Inequalities on the Real Line

Published online by Cambridge University Press:  20 November 2018

Mohammad Sababheh
Mohammad Sababheh*
Affiliation:
Princess Sumaya University For Technology, Amman 11941-Jordane-mail: sababheh@psut.edu.josababheh@yahoo.com
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Abstract

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We prove that some inequalities, which are considered to be generalizations of Hardy's inequality on the circle, can be modified and proved to be true for functions integrable on the real line. In fact we would like to show that some constructions that were used to prove the Littlewood conjecture can be used similarly to produce real Hardy-type inequalities. This discussion will lead to many questions concerning the relationship between Hardy-type inequalities on the circle and those on the real line.

Keywords

42A0542A99Hardy's inequalityinequalities including the Fourier transform and Hardy spaces
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 1 , 01 March 2011 , pp. 159 - 171
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Fournier, J. J. F., Some remarks on the recent proofs of the Littlewood conjecture. In: Second Edmonton conference on approximation theory (Edmonton, Alta., 1982), CMS Conf. Proc., 3, American Mathematical Society, Providence, RI, 1983, pp. 157–170.Google Scholar
[2] Hardy, G. H. and Littlewood, J. E., A new proof of a theorem on rearrangements. J. London Math. Soc. 23(1948), 163–168. doi:10.1112/jlms/s1-23.3.163Google Scholar
[3] Klemes, I., A note on Hardy's inequality. Canad. Math. Bull. 36(1993), no. 4, 442–448.Google Scholar
[4] Konjagin, S. V., On the Littlewood problem. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 45(1981), no. 2, 243–265, 463.Google Scholar
[5] McGehee, O. C., Pigno, L., and Smith, B., Hardy's inequality and the L 1 norm of exponential sums. Ann. of Math. 113(1981), no. 3, 613–618. doi:10.2307/2007000Google Scholar
[6] Sababheh, M., Constructions of bounded functions related to two-sided Hardy inequalities. Ph. D. thesis, McGill University, 2006.Google Scholar
[7] Sababheh, M., Hardy-type inequalities on the real line. J. Inequal. Pure Appl. Math. 9(2008), no. 3, Article 72, 6 pp.Google Scholar
[8] Smith, B., Two trigonometric designs: one-sided Riesz products and Littlewood products. In: General Inequalities, 3, Internat. Schriftenreihe Numer. Math., 64, Birkhäuser, Basel, 1983, pp. 141–148.Google Scholar