Weight characterizations for the discrete Hardy inequality with kernel
© Okpoti et al. 2006
Received: 16 August 2005
Accepted: 17 August 2005
Published: 27 April 2006
A discrete Hardy-type inequality is considered for a positive "kernel" , , and . For kernels of product type some scales of weight characterizations of the inequality are proved with the corresponding estimates of the best constant . A sufficient condition for the inequality to hold in the general case is proved and this condition is necessary in special cases. Moreover, some corresponding results for the case when are replaced by the nonincreasing sequences are proved and discussed in the light of some other recent results of this type.
- Andersen KF, Heinig HP: Weighted norm inequalities for certain integral operators. SIAM Journal on Mathematical Analysis 1983,14(4):834–844. 10.1137/0514064MATHMathSciNetView ArticleGoogle Scholar
- Bennett G: Some elementary inequalities. The Quarterly Journal of Mathematics. Oxford. Second Series 1987,38(152):401–425.MATHMathSciNetView ArticleGoogle Scholar
- Bennett G: Some elementary inequalities. III. The Quarterly Journal of Mathematics. Oxford. Second Series 1991,42(166):149–174.MATHMathSciNetView ArticleGoogle Scholar
- Gol'dman ML: Hardy type inequalities on the cone of quasi-monotone functions. In Research report 98/31. Russian Academy of Sciences Far-Eastern Branch Computer Center, Khabarovsk; 1998. 70 pages 70 pagesGoogle Scholar
- Gol'dman ML: Estimates for the norms of integral and discrete operators of Hardy type on cones of quasimonotone functions. Doklady Akademii Nauk 2001,377(6):733–738.MathSciNetMATHGoogle Scholar
- Kufner A, Persson L-E: Weighted Inequalities of Hardy Type. World Scientific, New Jersey; 2003:xviii+357.MATHView ArticleGoogle Scholar
- Okpoti CA: Weight characterization of discrete Hardy and Carleman type inequalities, Licentiate thesis. Department of Mathematics, Luleå University of Technology, Luleå; 2005. in print in printGoogle Scholar
- Okpoti CA, Persson L-E, Wedestig A: Scales of weight characterizations for the discrete Hardy and Carleman inequalities. In Proceedings of Function Spaces, Differential Operators and Nonlinear Analysis (FSDONA '04), 2004, Milovy. Academy of Sciences of the Czech Republic; 236–258.
- Opic B, Kufner A: Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series. Volume 219. Longman Scientific & Technical, Harlow; 1990:xii+333.MATHGoogle Scholar
- Persson L-E, Stepanov VD: Weighted integral inequalities with the geometric mean operator. Journal of Inequalities and Applications 2002,7(5):727–746. an abbreviated version can also be found in Russian Academy of Sciences. Doklady. Mathematics 63 (2001), 201–202 an abbreviated version can also be found in Russian Academy of Sciences. Doklady. Mathematics 63 (2001), 201–202 10.1155/S1025583402000371MATHMathSciNetGoogle Scholar
- Persson L-E, Stepanov VD, Ushakova EP: Equivalence of Hardy-type inequalities with general measures on the cones of non-negative respective non-increasing functions. to appear in Proceedings of the American Mathematical Society to appear in Proceedings of the American Mathematical Society
- Sinnamon G: Hardy's inequality and monotonicity. In Proceedings of Function Spaces, Differential Operators and Nonlinear Analysis (FSDONA '04), 2004, Milovy. Academy of Sciences of the Czech Republic; 292–310.
- Wedestig A: Some new Hardy type inequalities and their limiting inequalities. JIPAM. Journal of Inequalities in Pure and Applied Mathematics 2003,4(3):15. article 61 article 61MathSciNetMATHGoogle Scholar
- Wedestig A: Weighted inequalities of Hardy-type and their limiting inequalities, M.S. thesis. Department of Mathematics, Luleå University of Technology, Luleå; 2003. 106 pages 106 pagesMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.