tailieunhanh - Báo cáo hóa học: "Research Article Boundedness on Hardy-Sobolev Spaces for Hypersingular Marcinkiewicz Integrals with Variable Kernels"

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 Boundedness on Hardy-Sobolev Spaces for Hypersingular Marcinkiewicz Integrals with Variable Kernels | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 835938 17 pages doi 2008 835938 Research Article Boundedness on Hardy-Sobolev Spaces for Hypersingular Marcinkiewicz Integrals with Variable Kernels Xiangxing Tao 1 Xiao Yu 1 2 and Songyan Zhang1 1 Department of Mathematics Faculty of Science Ningbo University Ningbo Zhejiang Province 315211 China 2 Department of Mathematics Faculty of Science Zhejiang University Hangzhou Zhejiang Province 315211 China Correspondence should be addressed to Songyan Zhang zhangsongyan@ Received 3 August 2008 Accepted 20 October 2008 Recommended by Nikolaos Papageorgiou The existence and boundedness on Sobolev spaces and Hardy-Sobolev spaces for the hypersingular Marcinkiewicz integrals with variable kernels are derived. Copyright 2008 Xiangxing Tao et al. 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 The function Q x z defined on R X R is said to belong to L R X Lq S -1 if it satisfies the following two conditions 1 Q x Az Q x z for any x z e R and any A. 0 2 IIQIIl R xL S -1 supr 0 ydt JS -1 Q rz V A q ơ z 1 q o . Let a 0 and q max 1 2 - 1 2a and let Q e L R X Lq S -1 satisfy the following cancellation property Q x z Ym z dơ z 0 S -1 for all spherical harmonic polynomials Ym z with degree a and for any x e R . We consider the hypersingular Marcinkiewicz integral Q af x defined by 2 Journal of Inequalities and Applications TO dữÁf x 0 J x-y t ữ x x - ý x - y n 1 f ỳ dy 7 dt V 2 ị3 2a 1-2 for f e S Rn the Schwartz class. In general the way in which the integrals were given has to be interpreted. Let H h t h H J h t 2 dt t 1 2 to and let hf t x t-1-aJ t Q x x-y x - y n-1 f y dy. We have to show hf - x e H for every x e Rn and f e S and so that pQ a f x hf - x H can be regarded as a .