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 EquationPartial Differential EquationOrdinary Differential EquationFunctional EquationBounded Domain

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Authors’ Affiliations

(1)

Heian Jogakuin St. Agnes' School, Kyoto, Japan

(2)

Mathematical Institute, Tohoku University, Sendai, Japan

(3)

Department of Mathematics, Kyoto University, Kyoto, Japan

(4)

Department of Mathematics, Osaka City University, Osaka, 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.