Weighted Sobolev spaces and embedding theorems

• Published in: Transactions of the American Mathematical Society 2009-07, Vol.361 (7), p.3829-3850
• Main Authors: V. Gol'dshtein, A. Ukhlov
• Format: Article
• Language: English
• Subjects: Banach space; Embeddings; Euclidean space; Exact sciences and technology; Functional analysis; Global analysis, analysis on manifolds; Homeomorphism; Manifolds and cell complexes; Mathematical analysis; Mathematical functions; Mathematical inequalities; Mathematical theorems; Mathematics; Sciences and techniques of general use; Sobolev spaces; Sufficient conditions; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-09-04615-7

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.