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Abstract and Applied Analysis
Volume 2012, Article ID 929381, 14 pages
Research Article

Necessary and Sufficient Conditions for Boundedness of Commutators of the General Fractional Integral Operators on Weighted Morrey Spaces

1School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China
2Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 20 March 2012; Revised 5 July 2012; Accepted 20 July 2012

Academic Editor: Giovanni Galdi

Copyright © 2012 Zengyan Si and Fayou Zhao. 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.


We prove that 𝑏 is in Lip𝛽(πœ”) if and only if the commutator [𝑏,πΏβˆ’π›Ό/2] of the multiplication operator by 𝑏 and the general fractional integral operator πΏβˆ’π›Ό/2 is bounded from the weighted Morrey space 𝐿𝑝,π‘˜(πœ”) to πΏπ‘ž,π‘˜π‘ž/𝑝(πœ”1βˆ’(1βˆ’π›Ό/𝑛)π‘ž,πœ”), where 0<𝛽<1, 0<𝛼+𝛽<𝑛,1<𝑝<𝑛/(𝛼+𝛽), 1/π‘ž=1/π‘βˆ’(𝛼+𝛽)/𝑛, 0β‰€π‘˜<𝑝/π‘ž, πœ”π‘ž/π‘βˆˆπ΄1, and π‘Ÿπœ”>(1βˆ’π‘˜)/(𝑝/(π‘žβˆ’π‘˜)), and here π‘Ÿπœ” denotes the critical index of πœ” for the reverse HΓΆlder condition.