Journal of Inequalities and Applications
Volume 2007 (2007), Article ID 26765, 14 pages
doi:10.1155/2007/26765
Research Article

L2-Boundedness of Marcinkiewicz Integrals along Surfaces with Variable Kernels: Another Sufficient Condition

Qingying Xue1 and Kôzô Yabuta2

1School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
2School of Science and Technology, Kwansei Gakuin University, Gakuen 2-1, Sanda 669-1337, Japan

Received 18 December 2006; Accepted 23 April 2007

Academic Editor: Shusen Ding

Copyright © 2007 Qingying Xue and Kôzô Yabuta. 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 give the L2 estimates for the Marcinkiewicz integral with rough variable kernels associated to surfaces. More precisely, we give some other sufficient conditions which are different from the conditions known before to warrant that the L2-boundedness holds. As corollaries of this result, we show that similar properties still hold for parametric Littlewood-Paley area integral and parametric gλ∗ functions with rough variable kernels. Some of the results are extensions of some known results.