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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 John-Nirenberg Type Inequalities for the Morrey-Campanato Spaces | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 239414 5 pages doi 10.1155 2008 239414 Research Article John-Nirenberg Type Inequalities for the Morrey-Campanato Spaces Wenming Li College of Mathematics and Information Science Hebei Normal University Shijiazhuang 050016 Hebei China Correspondence should be addressed to Wenming Li lwmingg@sina.com Received 17 April 2007 Accepted 3 December 2007 Recommended by Y. Giga We give John-Nirenberg type inequalities for the Morrey-Campanato spaces on R . Copyright 2008 Wenming Li. 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. Given a function f G L11oc R and a cube Q on R let fo denote the average of f on Q fo 1 Q JQ f x dx. We say that f has bounded mean oscillation if there is a constant C such that for any cube Q 1 f x - fo dx C. Q Q 1 The space of functions with this property is denoted by BMO. For f G BMO define the norm on BMO by 1 llf IIbmo sup Q I f x - fQ dx-Q 2 John and Nirenberg 1 obtained the following well-known John-Nirenberg inequality for bMo. Theorem 1. Let f G BMO and f IIbmo 0- Then there exist positive constants C1 and C2 depending only on the dimension such that for all cube Q and any Ằ 0 x G Q If x -Q I IIBMO Q . 3 2 Journal of Inequalities and Applications Suppose f is a locally integrable function on R Q is a cube and s is a nonnegative integer let PQf x be the unique polynomial of degree at most s such that f x - PQf x xadx 0 4 Q for all 0 a s. Moreover for any x G Q . A c . . pQ f x f x dx 5 Q Q where the constant A is independent of f and Q. Clearly A 1. For p 0 s 0 1 q OT we will say that a locally integrable function f x belongs to the Morrey-Campanato spaces L p q s if llf WLp qs sup Q -p ini i lf x - PQ f x qdxj OT 6 Q Q Q where Q is a cube. Then if f - g is a polynomial of degree at .
