A new Brézis‐Gallouët‐Wainger inequality from the viewpoint of the real interpolation functors. Issue 2 (25th August 2013)

• Record Type: Journal Article
• Authors: Sawano, Yoshihiro
• Is Part Of: Mathematische Nachrichten. Volume 287:Issue 2/3(2014)
• Page Start: 352
• Page End: 358
• Publication Date: 2013-08-25
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201100324
• Languages: English
• ISSNs: 0025-584X

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>The Brézis‐Gallouët‐Wainger inequality describes a subtle embedding property into <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4sq49b7z" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:dummy:mana201100324:equation:mana201100324-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mi>∞</mml:mi></mml:msup></mml:math></alternatives>. The relation between the Brézis‐Gallouët‐Wainger inequality and the real interpolation functor together with the sharpness of the results is discussed in the present paper. As our first main results shows, it turns out that there are two intermediate terms between <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4sq49bbm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:dummy:mana201100324:equation:mana201100324-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mi>∞</mml:mi></mml:msup></mml:math></alternatives> and the logarithmic boundedness, which is supposed to be the right‐hand side of the Brézis‐Gallouët‐Wainger inequality. As the second result, the first result is extended to inequalities which reflect the meaning of the second index of Besov spaces and the interpolation theorem.</p> </abstract>