Abstract and Applied Analysis
Volume 2012 (2012), Article ID 780132, 18 pages
http://dx.doi.org/10.1155/2012/780132
Research Article

A Weighted Variant of Riemann-Liouville Fractional Integrals on ℝ 𝑛

Zun Wei Fu,1 Shan Zhen Lu,2 and Wen Yuan2

1Department of Mathematics, Linyi University, Shandong, Linyi 276005, China
2School of Mathematical Sciences, Beijing Normal University and Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Received 14 June 2012; Accepted 21 August 2012

Academic Editor: Bashir Ahmad

Copyright © 2012 Zun Wei Fu 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

We introduce certain type of weighted variant of Riemann-Liouville fractional integral on ℝ 𝑛 and obtain its sharp bounds on the central Morrey and 𝜆 -central BMO spaces. Moreover, we establish a sufficient and necessary condition of the weight functions so that commutators of weighted Hardy operators (with symbols in 𝜆 -central BMO space) are bounded on the central Morrey spaces. These results are further used to prove sharp estimates of some inequalities due to Weyl and Cesàro.