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Hardy Inequalities on the Real Line

  • Mohammad Sababheh (a1)

Abstract

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.

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References

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[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. 157170.
[2] Hardy, G. H. and Littlewood, J. E., A new proof of a theorem on rearrangements. J. London Math. Soc. 23(1948), 163168. doi:10.1112/jlms/s1-23.3.163
[3] Klemes, I., A note on Hardy's inequality. Canad. Math. Bull. 36(1993), no. 4, 442448.
[4] Konjagin, S. V., On the Littlewood problem. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 45(1981), no. 2, 243265, 463.
[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, 613618. doi:10.2307/2007000
[6] Sababheh, M., Constructions of bounded functions related to two-sided Hardy inequalities. Ph. D. thesis, McGill University, 2006.
[7] Sababheh, M., Hardy-type inequalities on the real line. J. Inequal. Pure Appl. Math. 9(2008), no. 3, Article 72, 6 pp.
[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. 141148.
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Hardy Inequalities on the Real Line

  • Mohammad Sababheh (a1)

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