Real Analysis Exchange

An equivalence theorem for integral conditions related to Hardy's inequality.

Amiram Gogatishvili, Alois Kufner, Lars-Erik Persson, and Anna Wedestig

Full-text: Open access


Let $1<p\leq q<\infty .$ Inspired by some recent results concerning Hardy type inequalities we state and prove directly the equivalence of four scales of integral conditions. By applying our result to the original Hardy type inequality situation we obtain a new proof of a number of characterizations of the Hardy inequality and obtain also some new weight characterizations. As another application we prove some new weight characterizations for embeddings between some Lorentz spaces.

Article information

Real Anal. Exchange, Volume 29, Number 2 (2003), 867-880.

First available in Project Euclid: 7 June 2006

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26D10: Inequalities involving derivatives and differential and integral operators 26D15: Inequalities for sums, series and integrals
Secondary: 47B07: Operators defined by compactness properties 47B38: Operators on function spaces (general)

Inequalities Hardy's inequality Weights Scales of weight characterizations Hardy operator Continuity equivalent integral conditions comparisons


Gogatishvili, Amiram; Kufner, Alois; Persson, Lars-Erik; Wedestig, Anna. An equivalence theorem for integral conditions related to Hardy's inequality. Real Anal. Exchange 29 (2003), no. 2, 867--880.

Export citation


  • G. Bennett, Some elementary inequalities, Quart. J. Math. Oxford Ser. (2), 38 (1987), No. 152, 401–425.
  • A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, World Scientific Publishing Co, Singapore, New Jersey, London, Hong Kong, 2003.
  • A. Kufner, L. E. Persson and A. Wedestig, A study of some constants characterizing the weighted Hardy inequality, Proceedings of the The Wadyslaw Orlicz Centenary Conference and Function Spaces VII, to appear.
  • B. Opic and P. Gurka, Weighted inequalities for geometric means, Proc. Amer. Math. Soc., 3 (1994), 771–779.
  • B. Opic and A. Kufner, Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series, Vol 211, Longman Scientific and Technical Harlow, 1990.
  • L. E. Persson and V. D. Stepanov, Weighted integral inequalities with the geometric mean operator, J. Inequal. Appl. 7 (2002), no. 5, 727–746 (An abbreviated version can also be found in Russian Akad. Sci. Dokl. Math. 63 (2001), 201–202).
  • E. T. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math., 96 (1990), 145–158.
  • V. D. Stepanov, The weighted Hardy's inequality for nonincreasing functions, Trans. Amer. Math. Soc., 338 (1993), no. 1, 173–186.