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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Sharp Adams-type inequalities in $ \mathbb{R}^{n}$


Authors: Bernhard Ruf and Federica Sani
Journal: Trans. Amer. Math. Soc. 365 (2013), 645-670
MSC (2010): Primary 46E35; Secondary 26D15
Published electronically: July 25, 2012
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Abstract: Adams' inequality for bounded domains $ \Omega \subset \mathbb{R}^4$ states that the supremum of $ \int _{\Omega } e^{32 \pi ^2 u^2} \, dx$ over all functions $ u \in W_0^{2, \, 2}(\Omega )$ with $ \Vert \Delta u\Vert _2 \leq 1$ is bounded by a constant depending on $ \Omega $ only. This bound becomes infinite for unbounded domains and in particular for $ \mathbb{R}^4$.

We prove that if $ \Vert\Delta u\Vert _2$ is replaced by a suitable norm, namely $ \Vert u \Vert:=\Vert- \Delta u + u\Vert _2$, then the supremum of $ \int _{\Omega } (e^{32 \pi ^2 u^2} -1) \, dx$ over all functions $ u \in W_0^{2, \, 2}(\Omega )$ with $ \Vert u\Vert \leq 1$ is bounded by a constant independent of the domain $ \Omega $.

Furthermore, we generalize this result to any $ W_0^{m, \, \frac n m}(\Omega )$ with $ \Omega \subseteq \mathbb{R}^{n}$ and $ m$ an even integer less than $ n$.


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Additional Information

Bernhard Ruf
Affiliation: Dipartimento di Matematica, Università degli Studi di Milano, 20133 Milan, Italy
Email: bernhard.ruf@unimi.it

Federica Sani
Affiliation: Dipartimento di Matematica, Università degli Studi di Milano, 20133 Milan, Italy
Email: federica.sani@unimi.it

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05561-9
PII: S 0002-9947(2012)05561-9
Keywords: Limiting Sobolev embeddings, Trudinger-Moser inequalities, inequality of D. R. Adams, best constants
Received by editor(s): November 4, 2010
Received by editor(s) in revised form: February 2, 2011
Published electronically: July 25, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.