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 ´ ¨ Sharp Constants of Brezis-Gallouet-Wainger Type Inequalities with a Double Logarithmic Term on Bounded Domains in Besov and Triebel-Lizorkin Spaces | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 584521 38 pages doi 2010 584521 Research Article Sharp Constants of Brezis-Gallouet-Wainger Type Inequalities with a Double Logarithmic Term on Bounded Domains in Besov and Triebel-Lizorkin Spaces Kei Morii 1 Tokushi Sato 2 Yoshihiro Sawano 3 and Hidemitsu Wadade4 1 Heian Jogakuin St. Agnes School 172-2 Gochomecho Kamigyo-ku Kyoto 602-8013 Japan 2 Mathematical Institute Tohoku University Sendai 980-8578 Japan 3 Department of Mathematics Kyoto University Kyoto 606-8502 Japan 4 Department of Mathematics Osaka City University 3-3-138 Sugimoto Sumiyoshi-ku Osaka 558-8585 Japan Correspondence should be addressed to Yoshihiro Sawano yoshihiro-sawano@ Received 4 October 2009 Revised 15 September 2010 Accepted 12 October 2010 Academic Editor Veli B. Shakhmurov Copyright 2010 Kei Morii 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. The present paper concerns the Sobolev embedding in the endpoint case. It is known that the embedding W 1 n Rn LTO Rn fails for n 2. Brezis-Gallouet-Wainger and some other authors quantified why this embedding fails by means of the Holder-Zygmund norm. In the present paper we will give a complete quantification of their results and clarify the sharp constants for the coefficients of the logarithmic terms in Besov and Triebel-Lizorkin spaces. 1. Introduction and Known Results We establish sharp Brézis-Gallouêt-Wainger type inequalities in Besov and Triebel-Lizorkin spaces as well as fractional Sobolev spaces on a bounded domain Q c Rn. Throughout the present paper we place ourselves in the setting of Rn with n 2. We treat only real-valued functions. First we recall the Sobolev embedding theorem in the critical case. For 1 q TO it is well known that the embedding Wn q q Rn Lr Rn .
