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

  • Kei Morii1,

    Affiliated with

    • Tokushi Sato2,

      Affiliated with

      • Yoshihiro Sawano3Email author and

        Affiliated with

        • Hidemitsu Wadade4

          Affiliated with

          Boundary Value Problems20102010:584521

          DOI: 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584521/MediaObjects/13661_2009_Article_940_IEq1_HTML.gif fails for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F584521/MediaObjects/13661_2009_Article_940_IEq2_HTML.gif . 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.

          Publisher note

          To access the full article, please see PDF.

          Authors’ Affiliations

          (1)
          Heian Jogakuin St. Agnes' School
          (2)
          Mathematical Institute, Tohoku University
          (3)
          Department of Mathematics, Kyoto University
          (4)
          Department of Mathematics, Osaka City University

          Copyright

          © Kei Morii et al. 2010

          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.