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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 Some Estimates of Integrals with a Composition Operator | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 928150 10 pages doi 10.1155 2010 928150 Research Article Some Estimates of Integrals with a Composition Operator Bing Liu Department of Mathematical Science Saginaw Valley State University University Center MI 48710 USA Correspondence should be addressed to Bing Liu sgbing987@hotmail.com Received 27 December 2009 Revised 11 March 2010 Accepted 16 March 2010 Academic Editor Yuming Xing Copyright 2010 Bing Liu. 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. We give some estimates of integrals with a composition operator namely composition of homotopy differential and Green s operators T d G with the Lipschitz and BMO norms. We also have estimates of those integrals with a singular factor. 1. Introduction The purpose of this paper is to establish the Poincaré-type inequalities for the composition of the homotopy operator T differential operator d and Green s operator G under Lipschitz and BMO norms. One of the reasons that we consider this composition operator is due to the Hodge theorem. It is well known that Hodge decomposition theorem plays important role in studying harmonic analysis and differential forms see 1-3 . It gives a relationship of the three key operators in harmonic analysis namely Green s operator G the Laplacian operator A and the harmonic projection operator H. This relationship offers us a tool to apply the composition of the three operators under the consideration to certain harmonic forms and to obtain some estimates for certain integrals which are useful in studying the properties of the solutions of PDEs. We also consider the integrals of this composition operator with a singular factor because of their broad applications in solving differential and integral equations see 4 . We first give
