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

 

Weighted Sobolev spaces and embedding theorems


Authors: V. Gol'dshtein and A. Ukhlov
Journal: Trans. Amer. Math. Soc. 361 (2009), 3829-3850
MSC (2000): Primary 46E35, 30C65
Published electronically: March 4, 2009
MathSciNet review: 2491902
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Abstract: In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt $ A_p$- condition. Sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities. The proposed method is based on the concept of `generalized' quasiconformal homeomorphisms (homeomorphisms with bounded mean distortion). The choice of the homeomorphism type depends on the choice of the corresponding weighted Sobolev space. Such classes of homeomorphisms induce bounded composition operators for weighted Sobolev spaces. With the help of these homeomorphism classes the embedding problem for non-smooth domains is reduced to the corresponding classical embedding problem for smooth domains. Examples of domains with anisotropic Hölder singularities demonstrate the sharpness of our machinery comparatively with known results.


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Additional Information

V. Gol'dshtein
Affiliation: Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, 84105 Beer Sheva, Israel

A. Ukhlov
Affiliation: Department of Mathematics, Ben Gurien Unniversity of the Negev, P.O. Box 653, 84105 Beer Sheva, Israel

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04615-7
PII: S 0002-9947(09)04615-7
Received by editor(s): April 23, 2007
Received by editor(s) in revised form: August 16, 2007
Published electronically: March 4, 2009
Additional Notes: The second author was partially supported by the Israel Ministry of Immigrant Absorption
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.