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 Norm Comparison Inequalities for the Composite Operator | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 212915 13 pages doi 2009 212915 Research Article Norm Comparison Inequalities for the Composite Operator Yuming Xing1 and Shusen Ding2 1 Department of Mathematics Harbin Institute of Technology Harbin 150001 China 2 Department of Mathematics Seattle University Seattle WA 98122 USA Correspondence should be addressed to Yuming Xing xyuming@ Received 2 August 2008 Accepted 15 January 2009 Recommended by Andras Ronto We establish norm comparison inequalities with the Lipschitz norm and the BMO norm for the composition of the homotopy operator and the projection operator applied to differential forms satisfying the A-harmonic equation. Based on these results we obtain the two-weight estimates for Lipschitz and BMO norms of the composite operator in terms of the Ls-norm. Copyright 2009 Y. Xing and S. Ding. 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 purpose of this paper is to establish the Lipschitz norm and BMO norm inequalities for the composition of the homotopy operator T and the projection operator H applied to differential forms in Rn n 2. The harmonic projection operator H one of the key operators in the harmonic analysis plays an important role in the Hodge decomposition theory of differential forms. In the meanwhile the homotopy operator T is also widely used in the decomposition and the Lp-theory of differential forms. In many situations we need to estimate the various norms of the operators and their compositions. We always assume that M is a bounded convex domain and B is a ball in Rn n 2 throughout this paper. Let ơB be the ball with the same center as B and with diam ơB ơ diam B Ơ 0. We do not distinguish the balls from cubes in this paper. For any subset E c Rn
