Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A compact embedding of a Sobolev space is equivalent to an embedding into a better space


Authors: Piotr Hajłasz and Zhuomin Liu
Journal: Proc. Amer. Math. Soc. 138 (2010), 3257-3266
MSC (2010): Primary 46E35; Secondary 46E30
DOI: https://doi.org/10.1090/S0002-9939-10-10390-6
Published electronically: April 15, 2010
MathSciNet review: 2653955
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 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)
  • 5. 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)
  • 9. RAO, M. M., REN, Z. D.: Theory of Orlicz spaces. Monographs and Textbooks in Pure and Applied Mathematics, 146. Marcel Dekker, Inc., New York, 1991. MR 1113700 (92e:46059)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46E35, 46E30

Retrieve articles in all journals with MSC (2010): 46E35, 46E30


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: https://doi.org/10.1090/S0002-9939-10-10390-6
Received by editor(s): March 17, 2009
Received by editor(s) in revised form: December 18, 2009
Published electronically: April 15, 2010
Additional Notes: The first author was supported by NSF grants DMS-0500966 and DMS-0900871.
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society