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

 

Compactness properties of Sobolev imbeddings for rearrangement invariant norms


Authors: Guillermo P. Curbera and Werner J. Ricker
Journal: Trans. Amer. Math. Soc. 359 (2007), 1471-1484
MSC (2000): Primary 46E35, 46E30; Secondary 47G10
Published electronically: October 17, 2006
MathSciNet review: 2272134
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Abstract: Compactness properties of Sobolev imbeddings are studied within the context of rearrangement invariant norms. Attention is focused on the extremal situation, namely, when the imbedding is considered as defined on its optimal Sobolev domain (with the range space fixed). The techniques are based on recent results which reduce the question of boundedness of the imbedding to boundedness of an associated kernel operator (of just one variable).


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Additional Information

Guillermo P. Curbera
Affiliation: Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160, Sevilla 41080, Spain
Email: curbera@us.es

Werner J. Ricker
Affiliation: Math.–Geogr. Fakultät, Katholische Universität Eichstätt–Ingolstadt, D–85072 Eichstätt, Germany
Email: werner.ricker@ku-eichstaett.de

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04203-6
PII: S 0002-9947(06)04203-6
Keywords: Sobolev imbeddings, compactness, rearrangement invariant spaces, optimal domains
Received by editor(s): January 3, 2005
Published electronically: October 17, 2006
Additional Notes: The authors gratefully acknowledge the support of the Katholische Universität Eichstätt–Ingolstadt (Germany) and D.G.I. #BFM2003–06335–C03–01 (Spain).
The results of this paper were presented at the 7th International Conference on Harmonic Analysis and Partial Differential Equations, held at El Escorial, Spain, in June 2004.
Article copyright: © Copyright 2006 American Mathematical Society