×

Weight characterizations for the discrete Hardy inequality with kernel. (English) Zbl 1090.26018

Summary: A discrete Hardy-type inequality \[ \left(\sum _{n=1}^{\infty}\left(\sum_{k=1}^n d_{n,k}a_{k}\right)q u_{n}\right)^{1/q}\leq C\left(\sum _{n=1}^{\infty}a_n^pv_n\right)^{1/p} \] is considered for a positive \` kernel\' \(d={d_{n,k}}\), \(n,k\in \mathbb Z_{+}\), and \(p\leq q\). For kernels of product type some scales of weight characterizations of the inequality are proved with the corresponding estimates of the best constant \(C\). 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 \(\{a_n\}_{n=1}^{\infty}\) are replaced by the nonincreasing sequences \(\{a_n^\ast\}_{n=1}^{\infty}\) are proved and discussed in the light of some other recent results of this type.

MSC:

26D15 Inequalities for sums, series and integrals
39A12 Discrete version of topics in analysis
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Andersen KF, Heinig HP: Weighted norm inequalities for certain integral operators.SIAM Journal on Mathematical Analysis 1983,14(4):834-844. 10.1137/0514064 · Zbl 0527.26010 · doi:10.1137/0514064
[2] Bennett G: Some elementary inequalities.The Quarterly Journal of Mathematics. Oxford. Second Series 1987,38(152):401-425. · Zbl 0649.26013 · doi:10.1093/qmath/38.4.401
[3] Bennett G: Some elementary inequalities. III.The Quarterly Journal of Mathematics. Oxford. Second Series 1991,42(166):149-174. · Zbl 0751.26007 · doi:10.1093/qmath/42.1.149
[4] Gol’dman, ML, Hardy type inequalities on the cone of quasi-monotone functions (1998), Khabarovsk
[5] 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. · Zbl 1066.47504
[6] Kufner A, Persson L-E: Weighted Inequalities of Hardy Type. World Scientific, New Jersey; 2003:xviii+357. · Zbl 1065.26018 · doi:10.1142/5129
[7] 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 print
[8] 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. · Zbl 1129.26016
[9] Opic B, Kufner A: Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series. Volume 219. Longman Scientific & Technical, Harlow; 1990:xii+333. · Zbl 0698.26007
[10] 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/S1025583402000371 · Zbl 1024.26008 · doi:10.1155/S1025583402000371
[11] 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 · Zbl 1093.26023
[12] 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. · Zbl 0649.26013
[13] 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 61 · Zbl 1064.26023
[14] 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 pages · Zbl 1064.26023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.