Open Access

On Boundedness of Weighted Hardy Operator in and Regularity Condition

Journal of Inequalities and Applications20102010:837951

https://doi.org/10.1155/2010/837951

Received: 22 September 2010

Accepted: 26 November 2010

Published: 6 December 2010

Abstract

We give a new proof for power-type weighted Hardy inequality in the norms of generalized Lebesgue spaces . Assuming the logarithmic conditions of regularity in a neighborhood of zero and at infinity for the exponents , necessary and sufficient conditions are proved for the boundedness of the Hardy operator from into . Also a separate statement on the exactness of logarithmic conditions at zero and at infinity is given. This shows that logarithmic regularity conditions for the functions at the origin and infinity are essentially one.

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Authors’ Affiliations

(1)
Education Faculty, Dicle University
(2)
Institute of Mathematics and Mechanics of National Academy of Science

Copyright

© Aziz Harman and Farman Imran Mamedov. 2010

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.