Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 91541, 17 pages

On weighted inequalities for parametric Marcinkiewicz integrals

H. M. Al-Qassem

Department of Mathematics, Yarmouk University, Irbid-Jordan, Jordan

Received 25 February 2005; Revised 30 May 2005; Accepted 3 July 2005

Copyright © 2006 H. M. Al-Qassem. 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 establish a weighted Lp boundedness of a parametric Marcinkiewicz integral operator Ω,hp if Ω is allowed to be in the block space Bq(0,-1/2)(Sn1) for some q>1 and h satisfies a mild integrability condition. We apply this conclusion to obtain the weighted Lp boundedness for a class of the parametric Marcinkiewicz integral operators Ω,h,λ,p and Ω,h,sp related to the Littlewood-Paley gλ-function and the area integral S, respectively. It is known that the condition ΩBq(0,1/2)(Sn1) is optimal for the L2 boundedness of Ω,11.