Weighted Sobolev spaces and embedding theorems
HTML articles powered by AMS MathViewer
- by V. Gol’dshtein and A. Ukhlov PDF
- Trans. Amer. Math. Soc. 361 (2009), 3829-3850 Request permission
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.References
- Alois Kufner, Weighted Sobolev spaces, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 31, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980. With German, French and Russian summaries. MR 664599
- Bengt Ove Turesson, Nonlinear potential theory and weighted Sobolev spaces, Lecture Notes in Mathematics, vol. 1736, Springer-Verlag, Berlin, 2000. MR 1774162, DOI 10.1007/BFb0103908
- Petr Gurka and Bohumír Opic, Continuous and compact imbeddings of weighted Sobolev spaces. I, Czechoslovak Math. J. 38(113) (1988), no. 4, 730–744. MR 962916
- Bohumír Opic and Petr Gurka, Continuous and compact imbeddings of weighted Sobolev spaces. II, Czechoslovak Math. J. 39(114) (1989), no. 1, 78–94 (English, with Russian and Czech summaries). MR 983485
- Petr Gurka and Bohumír Opic, Continuous and compact imbeddings of weighted Sobolev spaces. III, Czechoslovak Math. J. 41(116) (1991), no. 2, 317–341. MR 1105449
- Francesca Antoci, Some necessary and some sufficient conditions for the compactness of the embedding of weighted Sobolev spaces, Ricerche Mat. 52 (2003), no. 1, 55–71. MR 2091081
- O. V. Besov, The Sobolev embedding theorem for a domain with irregular boundary, Dokl. Akad. Nauk 373 (2000), no. 2, 151–154 (Russian). MR 1788313
- Guy David and Stephen Semmes, Strong $A_\infty$ weights, Sobolev inequalities and quasiconformal mappings, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 101–111. MR 1044784
- V. Gol′dshteĭn and L. Gurov, Applications of change of variables operators for exact embedding theorems, Integral Equations Operator Theory 19 (1994), no. 1, 1–24. MR 1271237, DOI 10.1007/BF01202289
- Vladimir Gol′dshtein and Alexander G. Ramm, Compactness of the embedding operators for rough domains, Math. Inequal. Appl. 4 (2001), no. 1, 127–141. MR 1809846, DOI 10.7153/mia-04-10
- S. K. Vodop′yanov and A. D. Ukhlov, Set functions and their applications in the theory of Lebesgue and Sobolev spaces. II [Translation of Mat. Tr. 7 (2004), no. 1, 13–49; MR2068275], Siberian Adv. Math. 15 (2005), no. 1, 91–125. MR 2141791
- Victor I. Burenkov, Sobolev spaces on domains, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 137, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998. MR 1622690, DOI 10.1007/978-3-663-11374-4
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Tero Kilpeläinen, Weighted Sobolev spaces and capacity, Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), no. 1, 95–113. MR 1246890
- Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207810
- Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
- V. M. Gol′dshteĭn and Yu. G. Reshetnyak, Quasiconformal mappings and Sobolev spaces, Mathematics and its Applications (Soviet Series), vol. 54, Kluwer Academic Publishers Group, Dordrecht, 1990. Translated and revised from the 1983 Russian original; Translated by O. Korneeva. MR 1136035, DOI 10.1007/978-94-009-1922-8
- S. K. Vodop′yanov and A. D. Ukhlov, Set functions and their applications in the theory of Lebesgue and Sobolev spaces. I [Translation of Mat. Tr. 6 (2003), no. 2, 14–65; MR2033646], Siberian Adv. Math. 14 (2004), no. 4, 78–125 (2005). MR 2125997
- Vodop’yanov S. K., Ukhlov A. Mappings associated with weighted Sobolev spaces, Contemporary Mathematics, 2007, vol. 424 (to appear)
- Riesz F. Sur les opérations fonctionelles linéaires, C. R. Acad. Sci. Paris, 1909, vol. 149, pp. 974–977.
- Wayne Smith and David A. Stegenga, Hölder domains and Poincaré domains, Trans. Amer. Math. Soc. 319 (1990), no. 1, 67–100. MR 978378, DOI 10.1090/S0002-9947-1990-0978378-8
Additional Information
- V. Gol’dshtein
- Affiliation: Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, 84105 Beer Sheva, Israel
- MR Author ID: 197069
- A. Ukhlov
- Affiliation: Department of Mathematics, Ben Gurien Unniversity of the Negev, P.O. Box 653, 84105 Beer Sheva, Israel
- MR Author ID: 336276
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 3829-3850
- MSC (2000): Primary 46E35, 30C65
- DOI: https://doi.org/10.1090/S0002-9947-09-04615-7
- MathSciNet review: 2491902