tailieunhanh - báo cáo hóa học:" Research Article Notes on Interpolation Inequalities Jiu-Gang Dong1 and Ti-Jun Xiao2"

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 Notes on Interpolation Inequalities Jiu-Gang Dong1 and Ti-Jun Xiao2 | Hindawi Publishing Corporation Advances in Difference Equations Volume 2011 Article ID 913403 6 pages doi 2011 913403 Research Article Notes on Interpolation Inequalities Jiu-Gang Dong1 and Ti-Jun Xiao2 1 Department of Mathematics University of Science and Technology of China Hefei 230026 China 2 Shanghai Key Laboratory for Contemporary Applied Mathematics School of Mathematical Sciences Fudan University Shanghai 200433 China Correspondence should be addressed to Ti-Jun Xiao xiaotj@ Received 3 October 2010 Accepted 16 November 2010 Academic Editor Toka Diagana Copyright 2011 . Dong and . Xiao. 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. An easy proof of the John-Nirenberg inequality is provided by merely using the Calderon-Zygmund decomposition. Moreover an interpolation inequality is presented with the help of the John-Nirenberg inequality. 1. Introduction It is well known that various interpolation inequalities play an important role in the study of operational equations partial differential equations and variation problems see . 1-6 . So it is an issue worthy of deep investigation. Let Qo be either Rn or a fixed cube in Rn. For f e Ll1oc Q0 write f IIbmo sup QcQ0 If 1Q1J Qlf - fQ lte where the supremum is taken over all cubes Q c Q0 and fQ 1 Q Jq fdx. Recall that BMO Q0 is the set consisting of all locally integrable functions on Q0 such that f BMO TO which is a Banach space endowed with the norm II BMO. It is clear that any bounded function on Q0 is in BMO Q0 but the converse is not true. On the other hand the BMO space is regarded as a natural substitute for LTO in many studies. One of the important features of the space is the John-Nirenberg inequality. There are several versions of its proof see for example 2 7-9 . Stimulated by these works we give in this .