Journal of Inequalities and Applications 
Volume 2006 (2006), Article ID 91541, 17 pages
doi:10.1155/JIA/2006/91541

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

Abstract

We establish a weighted Lp boundedness of a parametric Marcinkiewicz integral operator ℳΩ,hp if Ω is allowed to be in the block space Bq(0,-1/2)(Sn−1) 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 ℳΩ,h,λ∗,p and ℳΩ,h,sp related to the Littlewood-Paley gλ∗-function and the area integral S, respectively. It is known that the condition Ω∈Bq(0,−1/2)(Sn−1) is optimal for the L2 boundedness of ℳΩ,11.