Boundary Value Problems
Volume 2010 (2010), Article ID 584521, 38 pages
Research Article

Sharp Constants of Brézis-Gallouët-Wainger Type Inequalities with a Double Logarithmic Term on Bounded Domains in Besov and Triebel-Lizorkin Spaces

1Heian Jogakuin St. Agnes' School, 172-2, Gochomecho, Kamigyo-ku, Kyoto 602-8013, Japan
2Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
3Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
4Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

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 ( 𝑛 ) 𝐿 ( 𝑛 ) fails for 𝑛 2 . Brézis-Gallouët-Wainger and some other authors quantified why this embedding fails by means of the Hölder-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.