Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article Aλ3 λ1 , λ2 , Ω -Weighted Inequalities with Lipschitz r and BMO Norms | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 589040 14 pages doi 2010 589040 Research Article Ar3 Ằ1 Ằ2 Q -Weighted Inequalities with Lipschitz and BMO Norms Yuxia Tong 1 Juan Li 2 and Jiantao Gu1 1 College of Science Hebei Polytechnic University Tangshan 063009 China 2 Department of Mathematics Ningbo University Ningbo 315211 China Correspondence should be addressed to Yuxia Tong tongyuxia@ Received 29 December 2009 Revised 25 March 2010 Accepted 31 March 2010 Academic Editor Shusen Ding Copyright 2010 Yuxia Tong 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. We first define a new kind of A 3 Ằ1 Ằ2 Q two-weight then obtain some two-weight integral inequalities with Lipschitz norm and BMO norm for Green s operator applied to differential forms. 1. Introduction Green s operator G is often applied to study the solutions of various differential equations and to define Poisson s equation for differential forms. Green s operator has been playing an important role in the study of PDEs. In many situations the process to study solutions of PDEs involves estimating the various norms of the operators. Hence we are motivated to establish some Lipschitz norm inequalities and BMO norm inequalities for Green s operator in this paper. In the meanwhile there have been generally studied about Ar Q -weighted 1 2 and A Q -weighted 3 4 different inequalities and their properties. Results for more applications of the weight are given in 5 6 . The purpose of this paper is to derive the new weighted inequalities with the Lipschitz norm and BMO norm for Green s operator applied to differential forms. We will introduce Ar3 A1 A2Z Q -weight which can be considered as a further extension of the A Q -weight. We keep using the traditional notation. Let Q be a .
