Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 837951, 14 pages
doi:10.1155/2010/837951
Research Article

On Boundedness of Weighted Hardy Operator in 𝐿 𝑝 ( ⋅ ) and Regularity Condition

Aziz Harman1 and Farman Imran Mamedov1,2

1Education Faculty, Dicle University, 21280 Diyarbakir, Turkey
2Institute of Mathematics and Mechanics of National Academy of Science, Azerbaijan

Received 22 September 2010; Accepted 26 November 2010

Academic Editor: P. J. Y. Wong

Copyright © 2010 Aziz Harman and Farman Imran Mamedov. 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

We give a new proof for power-type weighted Hardy inequality in the norms of generalized Lebesgue spaces 𝐿 𝑝 ( ⋅ ) ( ℝ 𝑛 ) . Assuming the logarithmic conditions of regularity in a neighborhood of zero and at infinity for the exponents 𝑝 ( 𝑥 ) ≤ 𝑞 ( 𝑥 ) , 𝛽 ( 𝑥 ) , necessary and sufficient conditions are proved for the boundedness of the Hardy operator ∫ 𝐻 𝑓 ( 𝑥 ) = | 𝑦 | ≤ | 𝑥 | 𝑓 ( 𝑦 ) 𝑑 𝑦 from 𝐿 𝑝 ( ⋅ ) | 𝑥 | 𝛽 ( ⋅ ) ( ℝ 𝑛 ) into 𝐿 𝑞 ( ⋅ ) | 𝑥 | ′ 𝛽 ( ⋅ ) − 𝑛 / 𝑝 ( ⋅ ) − 𝑛 / 𝑞 ( ⋅ ) ( ℝ 𝑁 ) . 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 𝛽 , 𝑝 at the origin and infinity are essentially one.