Open Access

Weight characterizations for the discrete Hardy inequality with kernel

  • Christopher A. Okpoti1Email author,
  • Lars-Erik Persson1 and
  • Anna Wedestig1
Journal of Inequalities and Applications20062006:18030

DOI: 10.1155/JIA/2006/18030

Received: 16 August 2005

Accepted: 17 August 2005

Published: 27 April 2006

Abstract

A discrete Hardy-type inequality https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F18030/MediaObjects/13660_2005_Article_1573_IEq1_HTML.gif is considered for a positive "kernel" https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F18030/MediaObjects/13660_2005_Article_1573_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F18030/MediaObjects/13660_2005_Article_1573_IEq3_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F18030/MediaObjects/13660_2005_Article_1573_IEq4_HTML.gif . For kernels of product type some scales of weight characterizations of the inequality are proved with the corresponding estimates of the best constant https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F18030/MediaObjects/13660_2005_Article_1573_IEq5_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F18030/MediaObjects/13660_2005_Article_1573_IEq6_HTML.gif are replaced by the nonincreasing sequences https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F18030/MediaObjects/13660_2005_Article_1573_IEq7_HTML.gif are proved and discussed in the light of some other recent results of this type.

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Authors’ Affiliations

(1)
Department of Mathematics, Luleå University of Technology

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Copyright

© Okpoti et al. 2006

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.