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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An optimal limiting $ 2D$ Sobolev inequality


Author: Andrei Biryuk
Journal: Proc. Amer. Math. Soc. 138 (2010), 1461-1470
MSC (2000): Primary 52A40, 46E35; Secondary 46E30, 26D10
Published electronically: November 13, 2009
MathSciNet review: 2578540
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Abstract: The main goal of this paper is to prove an optimal limiting Sobolev inequality in two dimensions for Hölder continuous functions. Additionally, from this inequality we derive the double logarithmic inequality

$\displaystyle \Vert u\Vert _{L^{\infty}} \leqslant \frac{\Vert\nabla u\Vert _{L... ...{{\left.\rm\dot C\right.^{\alpha}}}} {\Vert\nabla u\Vert _{L^{2}}} )} \Bigr)} $

for functions $ u\in W^{1,2}_0(B_1)$ on the unit disk $ B_1$ in $ \mathbb{R}^2$, $ \alpha\in(0,1].$


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Additional Information

Andrei Biryuk
Affiliation: Departamento de Matematica, Instituto Superior Técnico, Centro de Análise Mate- mática, Geometria e Sistemas Dinâmicos, Lisbon, Portugal

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10159-4
PII: S 0002-9939(09)10159-4
Keywords: Limiting Sobolev embedding theorems, double logarithmic inequality.
Received by editor(s): March 3, 2009
Received by editor(s) in revised form: August 10, 2009
Published electronically: November 13, 2009
Additional Notes: The author is supported in part by CAMGSD and FCT/POCTI-POCI/FEDER. Part of the research (Theorem 3) was done while the author was a member of McMaster University in 2004. The author was supported in part by a CRC postdoctoral fellowship of McMaster University.
Communicated by: Walter Craig
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.