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A compact embedding of a Sobolev space is equivalent to an embedding into a better space
Author(s):
Piotr
Hajłasz;
Zhuomin
Liu
Journal:
Proc. Amer. Math. Soc.
138
(2010),
3257-3266.
MSC (2010):
Primary 46E35;
Secondary 46E30
Posted:
April 15, 2010
MathSciNet review:
2653955
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Abstract:
We prove that the compact embedding of the Orlicz-Sobolev space is equivalent to the existence of a bounded embedding into a higher Orlicz space.
References:
-
- 1.
- ADAMS, R. A., FOURNIER, J. J. F.: Sobolev spaces. Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. MR 2424078 (2009e:46025)
- 2.
- CIANCHI, A.: A sharp embedding theorem for Orlicz-Sobolev spaces. Indiana Univ. Math. J. 45 (1996), 39-65. MR 1406683 (97h:46044)
- 3.
- CONWAY, J. B.: A course in functional analysis. Second edition. Graduate Texts in Mathematics, 96. Springer-Verlag, New York, 1990. MR 1070713 (91e:46001)
- 4.
- DELLACHERIE, C., MEYER, P.-A.: Probabilities and potential. North-Holland Mathematics Studies, 29. North-Holland Publishing Co., Amsterdam-New York, 1978. MR 521810 (80b:60004)
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- HAJłASZ, P., KOSKELA, P.: Isoperimetric inequalities and imbedding theorems in irregular domains. J. London Math. Soc. 58 (1998), 425-450. MR 1668136 (99m:46079)
- 6.
- KAłAMAJSKA, A.: On compactness of embedding for Sobolev spaces defined on metric spaces. Ann. Acad. Sci. Fenn. Math. 24 (1999), 123-132. MR 1677969 (2000b:46061)
- 7.
- KERMAN, R., PICK, L.: Compactness of Sobolev imbeddings involving rearrangement-invariant norms. Studia Math. 186 (2008), 127-160. MR 2407972 (2009g:46061)
- 8.
- NECAS, J.: Les méthodes directes en théorie des équations elliptiques (French). Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967. MR 0227584 (37:3168)
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Additional Information:
Piotr
Hajłasz
Affiliation:
Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
Email:
hajlasz@pitt.edu
Zhuomin
Liu
Affiliation:
Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
Email:
liuzhuomin@hotmail.com
DOI:
10.1090/S0002-9939-10-10390-6
PII:
S 0002-9939(10)10390-6
Received by editor(s):
March 17, 2009
Received by editor(s) in revised form:
December 18, 2009
Posted:
April 15, 2010
Additional Notes:
The first author was supported by NSF grants DMS-0500966 and DMS-0900871.
Communicated by:
Nigel J. Kalton
Copyright of article:
Copyright
2010,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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