Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Some Orlicz norms inequalities for the composite operator T ∘ d ∘ H | Dai et al. Journal of Inequalities and Applications 2011 2011 105 http content 2011 1 105 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Some Orlicz norms inequalities for the composite operator T d H Zhimin Dai Yong Wang and Gejun Bao Correspondence zmdai@yahoo. cn Department of mathematics Harbin Institute of Technology Harbin 150001 China Springer Abstract In this article we first establish the local inequality for the composite operator T d H with Orlicz norms. Then we extend the local result to the global case in the L u -averaging domains. Keywords composite operator Orlicz norms L domains 1 Introduction Recently as generalizations of the functions differential forms have been widely used in many fields such as potential theory partial differential equations quasiconformal mappings and nonlinear analysis see 1-4 . With the development of the theory of quasiconformal mappings and other relevant theories a series of results about the solutions to different versions of the A-harmonic equation have been found see 5-9 . Especially the research on the inequalities of the various operators and their compositions applied to the solutions to different sorts of the A-harmonic equation has made great progress 5 . The inequalities equipped with the Lp-norm for differential forms have been very well studied. However the inequalities with Orlicz norms have not been fully developed 9 10 . Also both Lp-norms and Orlicz norms of differential forms depend on the type of the integral domains. Since Staples introduced the Ls-averaging domains in 1989 several kinds of domains have been developed successively including Ls p -averaging domains see 11-13 . In 2004 Ding 14 put forward the concept of the L p -averaging domains which is considered as an extension of the other domains involved above and specified later. The homotopy operator T the exterior derivative operator d and the .