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Open Access Research Article

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

Kei Morii1, Tokushi Sato2, Yoshihiro Sawano3* and Hidemitsu Wadade4

Author Affiliations

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

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Boundary Value Problems 2010, 2010:584521  doi:10.1155/2010/584521

Published: 18 October 2010

Abstract

The present paper concerns the Sobolev embedding in the endpoint case. It is known that the embedding fails for . 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.