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Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
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Some new scales of characterization of Hardy’s inequality; pp. 7–18

(Full article in PDF format) doi: 10.3176/proc.2010.1.03


Authors

Amiran Gogatishvili, Alois Kufner, Lars-Erik Persson

Abstract

Let 1 < p ≤ q < ∞. Inspired by some recent results concerning Hardy-type inequalities where the equivalence of four scales of integral conditions was proved, we use related ideas to find ten new equivalence scales of integral conditions. By applying this result to the original Hardy-type inequality, we obtain a new proof of a number of characterizations of the Hardy inequality and also some new weight characterizations.

Keywords

Hardy operator, Hardy’s inequality.

References

  1. Gogatishvili , A. , Kufner , A. , Persson , L. E. , and Wedestig , A. An equivalence theorem for some scales of integral conditions related to Hardy’s inequality with applications. Real Anal. Exchange , 2004 , 29 , 867–880.

  2. Gogatishvili , A. , Kufner , A. , and Persson , L. E. An Equivalence Theorem with Application to Hardy’s Inequality. Research Report 8 , Department of Mathematics , Luleå University of Technology , 2007.

  3. Gurka , P. Generalized Hardy’s inequality. Časopis Pĕst. Mat. , 1984 , 109 , 194–203.

  4. Kufner , A. and Persson , L. E. Weighted Inequalities of Hardy Type. World Scientific Publishing Co , Singapore , 2003.

5. Kufner , A. , Persson , L. E. , and Wedestig , A. A study of some constants characterizing the weighted Hardy inequality. Banach Center Publ. , Orlicz Centenary Volume , 2004 , 64 , 135–146.
doi:10.4064/bc64-0-11

  6. Kufner , A. , Maligranda , L. , and Persson , L. E. The Hardy Inequality. About Its History and Some Related Results. Vydavatelský servis Publishing House , Pilsen , 2007.

  7. Muckenhoupt , B. Hardy's inequality with weights. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity , I. Studia Math. , 1972 , 44 , 31–38.

  8. Opic , B. and Kufner , A. Hardy-Type Inequalities. Pitman Research Notes in Mathematics Series , Vol. 211 , Longman Scientific and Technical , Harlow , 1990.

  9. Persson , L.-E. and Stepanov , V. Weighted integral inequalities with the geometric mean operator. J. Inequal. Appl. , 2002 , 7 , 727–746.
doi:10.1155/S1025583402000371

10. Tomaselli , G. A class of inequalities. Boll. Un. Mat. Ital. , 1969 , 2 , 622–631.
 
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Current Issue: Vol. 68, Issue 4, 2019




Publishing schedule:
No. 1: 20 March
No. 2: 20 June
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