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

Kei Morii; Tokushi Sato; Yoshihiro Sawano; Hidemitsu Wadade

doi:10.1155/2010/584521

Research Article
Open Access

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 Sawano³Email author and
Hidemitsu Wadade⁴

Boundary Value Problems20102010:584521

https://doi.org/10.1155/2010/584521

Received: 4 October 2009
Accepted: 12 October 2010
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.

Keywords

Differential Equation
Partial Differential Equation
Ordinary Differential Equation
Functional Equation
Bounded Domain

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Authors’ 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

Copyright

This article is published under license to BioMed Central Ltd. 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.