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
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.
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.