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 Mat. Sb. (N.S.), 1980, Volume 112(154), Number 1(5), Pages 56–85 (Mi msb2712)

Imbedding theorems and compactness for spaces of Sobolev type with weights. II

P. I. Lizorkin, M. Otelbaev

Abstract: In this article theorems are established on imbedding and compactness for spaces of functions which are $p$th power summable with weight $\nu$ over the region $\Omega\subset\mathbf R^n$ and whose $m$th derivatives are $p$-summable with weight $\mu$ over $\Omega$. Moreover, necessary and sufficient conditions for the boundedness and compactness of the imbedding operator are obtained in terms of properties of the weight functions. The case of functions vanishing on the boundary is also considered. This article represents a continuation of previous research of the authors.
Bibliography: 2 titles.

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English version:
Mathematics of the USSR-Sbornik, 1981, 40:1, 51–77

Bibliographic databases:

UDC: 517.518.23
MSC: 46E35

Citation: P. I. Lizorkin, M. Otelbaev, “Imbedding theorems and compactness for spaces of Sobolev type with weights. II”, Mat. Sb. (N.S.), 112(154):1(5) (1980), 56–85; Math. USSR-Sb., 40:1 (1981), 51–77

Citation in format AMSBIB
\Bibitem{LizOte80} \by P.~I.~Lizorkin, M.~Otelbaev \paper Imbedding theorems and compactness for spaces of Sobolev type with weights.~II \jour Mat. Sb. (N.S.) \yr 1980 \vol 112(154) \issue 1(5) \pages 56--85 \mathnet{http://mi.mathnet.ru/msb2712} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=575932} \zmath{https://zbmath.org/?q=an:0465.46031|0447.46027} \transl \jour Math. USSR-Sb. \yr 1981 \vol 40 \issue 1 \pages 51--77 \crossref{https://doi.org/10.1070/SM1981v040n01ABEH001635} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1981MM63900003} 

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This publication is cited in the following articles:
1. Kusainova L., Mynbaev K., “The Embedding and Compactness Theorems for Sobolev Anisotropic Weight Spaces”, 263, no. 5, 1982, 1050–1053
2. Pietsch A., “Eigenvalues of Integral-Operators .2.”, Math. Ann., 262:3 (1983), 343–376
3. Feichtinger H., “Compactness in Translation Invariant Banach-Spaces of Distributions and Compact Multipliers”, J. Math. Anal. Appl., 102:2 (1984), 289–327
4. Opic B., Kufner A., “Remark on Compactness of Imbeddings in Weighted Spaces”, Math. Nachr., 133 (1987), 63–70
5. Gurka P., Opic B., “Continuous and Compact Imbeddings of Weighted Sobolev Spaces .1.”, Czech. Math. J., 38:4 (1988), 730–744
6. Desiatskova N., “Theorems of Embedding and Diameters of Some Weight Classes of Smooth Functions”, 302, no. 6, 1988, 1296–1300
7. Bulabaev A., Shuster L., “On the Theory of Stourm-Liouville Difference-Equations”, 309, no. 3, 1989, 521–524
8. Opic B., Rakosnik J., “Estimates for Mixed Derivatives of Functions From Anisotropic Sobolev-Slobodeckij Spaces with Weights”, Q. J. Math., 42:167 (1991), 347–363
9. Brown R., Opic B., “Embeddings of Weighted Sobolev Spaces Into Spaces of Continuous-Functions”, Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci., 439:1906 (1992), 279–296
10. L. K. Kusainova, “Embedding the weighted Sobolev space $W^l_p(\Omega;v)$ in the space $L_p(\Omega;\omega)$”, Sb. Math., 191:2 (2000), 275–290
11. S. G. Pyatkov, “Interpolation of Weighted Sobolev Spaces”, Siberian Adv. Math., 10:3 (2000), 83–132
12. Kordan Ospanov, “Coercive estimates for a degenerate elliptic system of equations with spectral applications”, Applied Mathematics Letters, 2011
13. Vasil'eva A.A., “Kolmogorov Widths of Weighted Sobolev Classes on a Domain for a Special Class of Weights. II”, Russian Journal of Mathematical Physics, 18:4 (2011), 465–504
14. Vasil'eva A.A., “Kolmogorov widths of weighted Sobolev classes on a domain for a special class of weights”, Russian Journal of Mathematical Physics, 18:3 (2011), 353–385
15. A. A. Vasil'eva, “Widths of weighted Sobolev classes on a John domain”, Proc. Steklov Inst. Math., 280 (2013), 91–119
16. Vasil'eva A.A., “Embedding Theorem for Weighted Sobolev Classes on a John Domain with Weights That Are Functions of the Distance to Some H-Set”, Russ. J. Math. Phys., 20:3 (2013), 360–373
17. Vasil'eva A.A., “Widths of Weighted Sobolev Classes on a John Domain: Strong Singularity at a Point”, Rev. Mat. Complut., 27:1 (2014), 167–212
18. Vasil'eva A.A., “Embedding Theorem for Weighted Sobolev Classes with Weights That Are Functions of the Distance to Some H-Set”, Russ. J. Math. Phys., 21:1 (2014), 112–122
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