tailieunhanh - Báo cáo hóa học: " Research Article L2 -Boundedness of Marcinkiewicz Integrals along Surfaces with Variable Kernels: Another Sufficient Condition"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article L2 -Boundedness of Marcinkiewicz Integrals along Surfaces with Variable Kernels: Another Sufficient Condition | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 26765 14 pages doi 2007 26765 Research Article L2-Boundedness of Marcinkiewicz Integrals along Surfaces with Variable Kernels Another Sufficient Condition Qingying Xue and Kozo Yabuta Received 18 December 2006 Accepted 23 April 2007 Recommended by Shusen Ding 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 gỵ functions with rough variable kernels. Some of the results are extensions of some known results. Copyright 2007 Q. Xue and K. 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. 1. Introduction In order to study the elliptic partial differential equations of order two with variable coefficients Calderón and Zygmund 3 defined and studied the L2-boundedness of singular integral T with variable kernels. In 1980 Aguilera and Harboure 4 studied the problem of pointwise convergence of singular integral and the L2-bounds of Hardy-Littlewood maximal function with variable kernels. In 2002 Tang and Yang 1 gave the L2 boundedness of the singular integrals with rough variable kernels associated to surfaces of the form x 0 1 y y where y y yl for any y e Rn 0 n 2 . That is they considered the variable Calderon-Zygmund singular integral operator To defined by To f x . k x y f x - o yl y dy Rn and its truncated maximal operator To defined by T0 f x sup I k x y f x-o yl y dy 0 ly I 2 Journal of Inequalities and Applications for f e CM Rn where k x y n x y y n Rn