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 Hypersingular Marcinkiewicz Integrals along Surface with Variable Kernels on Sobolev Space and Hardy-Sobolev Space | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 479576 14 pages doi 2011 479576 Research Article Hypersingular Marcinkiewicz Integrals along Surface with Variable Kernels on Sobolev Space and Hardy-Sobolev Space Wei Ruiying and Li Yin School of Mathematics and Information Science Shaoguan University Shaoguan 512005 China Correspondence should be addressed to Wei Ruiying weiruiying521@ Received 30 June 2010 Revised 5 December 2010 Accepted 20 January 2011 Academic Editor Andrei Volodin Copyright 2011 W. Ruiying and L. Yin. 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. Let a 0 the authors introduce in this paper a class of the hypersingular Marcinkiewicz integrals along surface with variable kernels defined by yffa f x 1 2 í 1 n-1 fív-fàí H la 2idf f3 2a n vh Ol fc n 1 V ĩq IS n-11 1 0 l Otx yl yl f x - woy 1ay at t II where X Z fc L K X L p with q max 1 2 n - 1 n 2a . The authors prove that the operator pOa is bounded from Sobolev space La V to ĩp ĩ space for 1 p 2 and from Hardy-Sobolev space Hp K to ư ĩln space for n a p 1. As corollaries of the result they also prove the LOfR - L2 Rn boundedness of the Littlewood-Paley type operators pO a s and pOa X which relate to the Lusin area integral and the Littlewood-Paley gX function. 1. Introduction Let R n 2 be the n-dimensional Euclidean space and s -1 be the unit sphere in R equipped with the normalized Lebesgue measure dơ dơ . For x fc R 0 let x x x . Before stating our theorems we first introduce some definitions about the variable kernel Q x z . A function Q x z defined on R X R is said to be in L R X Lq S -1 q 1 if Q x z satisfies the following two conditions 1 Q Xz O z for any x z fc R and any X 0 2 II IIl R XLỈ S -1 supr 0 yđt ƠS -1 rz y z qaơ z 1 q . In 1955 Calderon and Zygmund
