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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: AN EXISTENCE THEOREM FOR AN IMPLICIT INTEGRAL EQUATION WITH DISCONTINUOUS RIGHT-HAND SIDE | EXTENSIONS OF HARDY INEQUALITY JUNYONG ZHANG Received 2 May 2006 Revised 2 August 2006 Accepted 13 August 2006 We study extended Hardy inequalities using Littlewood-Paley theory and nonlinear estimates method in Besov spaces. Our results improve and extend the well-known results of Cazenave 2003 . Copyright 2006 Junyong Zhang. 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 A remarkable result of Hardy-type inequality comes from the following proposition the proof ofwhich is given by Cazenave 2 . Proposition 1.1. Let 1 p X. If q n is such that 0 q p then I u p q e L1 Rn for every u e w1 p Rn . Furthermore Í u J dx yiiuiiL-qiivuiL 1.1 JRn I q n - qZ 11 Lp 11 Lp v 7 for every u e w 1 p Rn . It is easy to see that the proposition fails when s 1 where s q p. In this paper we are trying to find out what happens if s 1. We show that it does not only become true but obtains better estimates. The described result is stated and proved in Section 3. The method invoked is different from that by Cazenave in 2 it relies on some Littlewood-Paley theory and Besov spaces theory that are cited in Section 2. 2. Preliminaries In this section we introduce some equivalent definitions and norms for Besov space needed in this paper. The reader is referred to the well-known books of Runst and Sickel 5 Triebel 6 and Miao 4 for details. Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006 Article ID 69379 Pages 1-5 DOI 10.1155 JIA 2006 69379 2 Extensions of Hardy inequality We first introduce the following equivalent norms for the homogeneous Besov spaces B . p m ll h p m I m su dau m b SLlp yơ u p t c 1 m 2.1 where AyU TyU u da dĩ1 dĩ dann Tyu - u y di i 1 2 . n. dxi 2.2 a a1 a2 . an and s s Ơ with 0 Ơ 1 namely Ơ s s where s denotes the largest integer not larger than s. In the case
