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Báo cáo hóa học: " The integral estimate with Orlicz norm in L(x)averaging domain"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : The integral estimate with Orlicz norm in L (x)averaging domain | Wen Journal of Inequalities and Applications 2011 2011 90 http www.journalofinequalitiesandapplications.eom content 2011 1 90 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access The integral estimate with Orlicz norm in L x -averaging domain Haiyu Wen Correspondence wenhy@hit.edu.cn Department of Mathematics Harbin Institute of Technology Harbin 150001 PR China Springer Abstract In this article we obtain some local and global integral inequalities with Orlicz norm for the A-harmonic tensors in D x -averaging domain where x satisfies the ộp condition. These estimates indicate that many existing inequalities with Lp-norms are special cases of our results. Keywords A-harmonic equations differential forms Luxemburg norms homotopy operator 1 Introduction In recent years there are many remarkable results about the solutions of the nonho-mogeneous A-harmonic equation d A x dm B x dm have been made see 1-7 . For example in 2 the following Caccioppoli inequality has been established. II du p Q C Q -1 n II u p ơQ p 1 ơ 1. 1.1 In 3 we can find the general Poincaré inequality II u - UQ p Q C Q diam Q II du p ơQ p 0 Ơ 1. 1.2 In 4-6 many inequalities for the classical operators applied to the differential forms have been studied. These integral inequalities play a crucial role in studying PDE and the properties of the solutions of PDE. However most of these inequalities are developed with the Lp-norms. Meanwhile we know the Orlicz spaces is the important tool in studying PDE see 8 . So in this article the normalized Lp-norms are replaced by large norms in the scale of Orlicz spaces. We first introduce the ộp condition which is a particular class of the Young functions then using the result that the maximal operator Mộ is bounded on Lp R see 9 we establish some integral estimates with Orlicz norms. In the global case we also expand the local results to a relative large class of domains the L -averaging domain. Applying our results we can easily .