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 Weighted Norm Inequalities for Solutions to the Nonhomogeneous A-Harmonic Equation | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 851236 15 pages doi 2009 851236 Research Article Weighted Norm Inequalities for Solutions to the Nonhomogeneous A-Harmonic Equation Haiyu Wen Department of Mathematics Harbin Institute of Technology Harbin 150001 China Correspondence should be addressed to Haiyu Wen wenhy@ Received 10 March 2009 Accepted 18 May 2009 Recommended by Shusen Ding We first prove the local and global two-weight norm inequalities for solutions to the nonhomoge-neous A-harmonic equation A x g du h d v for differential forms. Then we obtain some weighed Lipschitz norm and BMO norm inequalities for differential forms satisfying the different nonhomogeneous A-harmonic equations. Copyright 2009 Haiyu Wen. 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 In the recent years the A-harmonic equations for differential forms have been widely investigated see 1 and many interesting and important results have been found such as some weighted integral inequalities for solutions to the A-harmonic equations see 2-7 . Those results are important for studying the theory of differential forms and both qualitative and quantitative properties of the solutions to the different versions of A-harmonic equation. In the different versions of A-harmonic equation the nonhomogeneous A-harmonic equation A x g du h d v has received increasing attentions in 8 Ding has presented some estimates to such equation. In this paper we extend some estimates that Ding has presented in 8 into the two-weight case. Our results are more general so they can be used broadly. It is well-known that the Lipschitz norm supQCQ Q -1- k n u - UQII1 Q where the supremum is over all local cubes Q as k 0 is the BMO norm supQCQ Q -1 u - uq 1 Q so the natural .