Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 18030, 14 pages
doi:10.1155/JIA/2006/18030
Weight characterizations for the discrete Hardy inequality with kernel
Department of Mathematics, Luleå University of Technology, Luleå 971 87, Sweden
Received 16 August 2005; Accepted 17 August 2005
Copyright © 2006 Christopher A. Okpoti et al. 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.
Abstract
A discrete Hardy-type inequality (∑n=1∞(∑k=1ndn,kak)qun)1/q≤C(∑n=1∞anpvn)1/p is considered for a positive “kernel” d={dn,k}, n,k∈ℤ+, and p≤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 {an}n=1∞ are replaced by the nonincreasing sequences {an*}n=1∞ are proved and discussed in the light of some other recent results of this type.