Widths of Sobolev weight classes on a domain with cusp
Sbornik. Mathematics, Tome 206 (2015) no. 10, pp. 1375-1409 Cet article a éte moissonné depuis la source Math-Net.Ru

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Order estimates for the Kolmogorov, Gelfand and linear widths of the unit ball in weighted Sobolev space on a domain with cusp in a weighted Lebesgue space are obtained. Certain limit conditions on the parameters are considered for which the estimates for widths may differ from those in the case of weight one and a domain with Lipschitz boundary. Bibliography: 52 titles.
Keywords: width, domain with cusp, weighted Sobolev space.
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A. A. Vasil'eva. Widths of Sobolev weight classes on a domain with cusp. Sbornik. Mathematics, Tome 206 (2015) no. 10, pp. 1375-1409. http://geodesic.mathdoc.fr/item/SM_2015_206_10_a1/

[1] O. V. Besov, “Integral representations of functions, and imbedding theorems for a region with flexible horn condition”, Proc. Steklov Inst. Math., 170 (1987), 11–31 | MR | Zbl

[2] D. A. Labutin, “Integral representations of functions and embeddings of Sobolev spaces on cuspidal domains”, Math. Notes, 61:2 (1997), 164–179 | DOI | DOI | MR | Zbl

[3] D. A. Labutin, “Embedding of Sobolev spaces on Hölder domains”, Proc. Steklov Inst. Math., 227 (1999), 163–172 | MR | Zbl

[4] P. Hajłasz, P. Koskela, “Isoperimetric inequalities and imbedding theorems in irregular domains”, J. London Math. Soc. (2), 58:2 (1998), 425–450 | DOI | MR | Zbl

[5] T. Kilpeläinen, J. Malý, “Sobolev inequalities on sets with irregular boundaries”, Z. Anal. Anwendungen, 19:2 (2000), 369–380 | DOI | MR | Zbl

[6] V. G. Maz'ya, S. V. Poborchi, “Imbedding theorems for Sobolev spaces on domains with peak and on Hölder domains”, St. Petersburg Math. J., 18:4 (2007), 583–605 | DOI | MR | Zbl

[7] O. V. Besov, “Kolmogorov widths of Sobolev classes on an irregular domain”, Proc. Steklov Inst. Math., 280 (2013), 34–45 | DOI | MR | Zbl

[8] H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978, 528 pp. | MR | MR | Zbl | Zbl

[9] A. Kufner, Weighted Sobolev spaces, Teubner-Texte Math., 31, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980, 151 pp. | MR | Zbl

[10] D. E. Edmunds, H. Triebel, Function spaces, entropy numbers, differential operators, Cambridge Tracts in Math., 120, Cambridge Univ. Press, Cambridge, 1996, xii+252 pp. | DOI | MR | Zbl

[11] H. Triebel, Theory of function spaces, v. III, Monogr. Math., 100, Birkhäuser Verlag, Basel, 2006, xii+426 pp. | MR | Zbl

[12] D. E. Edmunds, W. D. Evans, Hardy operators, function spaces and embeddings, Springer Monogr. Math., Springer-Verlag, Berlin, 2004, xii+326 pp. | DOI | MR | Zbl

[13] L. D. Kudryavtsev, S. M. Nikol'skij, “Spaces of differentiable functions of several variables and imbedding theorems”, Analysis III. Spaces of differentiable functions, Encyclopaedia Math. Sci., 26, 1991, 1–140 | DOI | MR | MR | Zbl | Zbl

[14] V. G. Maz'ja, “Classes of domains and imbedding theorems for function spaces”, Soviet Math. Dokl., 1 (1960), 882–885 | MR | Zbl

[15] Yu. G. Reshetnyak, “Integral representations of differentiable functions in domains with nonsmooth boundary”, Siberian Math. J., 21:6 (1981), 833–839 | DOI | MR | Zbl

[16] Yu. G. Reshetnyak, “Zamechanie ob integralnykh predstavleniyakh differentsiruemykh funktsii mnogikh peremennykh”, Sib. matem. zhurnal, 25:5 (1984), 198–200 | MR | Zbl

[17] B. Bojarski, “Remarks on Sobolev imbedding inequalities”, Complex analysis (Joensuu, 1987), Lecture Notes in Math., 1351, Springer, Berlin, 1988, 52–68 | DOI | MR | Zbl

[18] O. V. Besov, “Imbedding of Sobolev spaces on domains with the splitting flexible cone condition”, Proc. Steklov Inst. Math., 173 (1987), 13–30 | MR | Zbl

[19] W. D. Evans, D. J. Harris, “Fractals, trees and the Neumann Laplacian”, Math. Ann., 296:3 (1993), 493–527 | DOI | MR | Zbl

[20] O. V. Besov, “On the compactness of embeddings of weighted Sobolev spaces on a domain with irregular boundary”, Proc. Steklov Inst. Math., 232 (2001), 66–87 | MR | Zbl

[21] O. V. Besov, “Sobolev's embedding theorem for a domain with irregular boundary”, Sb. Math., 192:3 (2001), 323–346 | DOI | DOI | MR | Zbl

[22] O. V. Besov, “On the compactness of embeddings of weighted Sobolev spaces on a domain with an irregular boundary”, Dokl. Math., 63:1 (2001), 95–100 | MR | Zbl

[23] O. V. Besov, “Integral estimates for differentiable functions on irregular domains”, Sb. Math., 201:12 (2010), 1777–1790 | DOI | DOI | MR | Zbl

[24] O. V. Besov, “Sobolev's embedding theorem for anisotropically irregular domains”, Eurasian Math. J., 2:1 (2011), 32–51 | MR | Zbl

[25] L. Caso, R. D'Ambrosio, “Weighted spaces and weighted norm inequalities on irregular domains”, J. Approx. Theory, 167 (2013), 42–58 | DOI | MR | Zbl

[26] A. Pietsch, “$s$-numbers of operators in Banach spaces”, Studia Math., 51 (1974), 201–223 | MR | Zbl

[27] S. Heinrich, “On the relation between linear $n$-widths and approximation numbers”, J. Approx. Theory, 58:3 (1989), 315–333 | DOI | MR | Zbl

[28] D. E. Edmunds, J. Lang, “Gelfand numbers and widths”, J. Approx. Theory, 166 (2013), 78–84 | DOI | MR | Zbl

[29] V. M. Tikhomirov, Nekotorye voprosy teorii priblizhenii, Izd-vo MGU, M., 1976, 304 pp. | MR

[30] V. M. Tikhomirov, “Teoriya priblizhenii”, Analiz – 2, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 14, VINITI, M., 1987, 103–260 | MR | Zbl

[31] A. Pinkus, $n$-widths in approximation theory, Ergeb. Math. Grenzgeb. (3), 7, Springer-Verlag, Berlin, 1985, x+291 pp. | DOI | MR | Zbl

[32] V. M. Tikhomirov, “Diameters of sets in function spaces and the theory of best approximations”, Russian Math. Surveys, 15:3 (1960), 75–111 | DOI | MR | Zbl

[33] R. S. Ismagilov, “Diameters of sets in normed linear spaces and the approximation of functions by trigonometric polynomials”, Russian Math. Surveys, 29:3 (1974), 169–186 | DOI | MR | Zbl

[34] B. S. Kashin, “Diameters of some finite-dimensional sets and classes of smooth functions”, Math. USSR-Izv., 11:2 (1977), 317–333 | DOI | MR | Zbl

[35] E. D. Gluskin, “Norms of random matrices and widths of finite-dimensional sets”, Math. USSR-Sb., 48:1 (1984), 173–182 | DOI | MR | Zbl

[36] R. A. DeVore, R. C. Sharpley, S. D. Riemenschneider, “$n$-widths for $C^\alpha_p$ spaces”, Anniversary volume on approximation theory and functional analysis (Oberwolfach, 1983), Internat. Schriftenreihe Numer. Math., 65, Birkhäuser, Basel, 1984, 213–222 | MR | Zbl

[37] M. Sh. Birman, M. Z. Solomyak, “Piecewise-polynomial approximations of functions of the classes $W_p^\alpha$”, Math. USSR-Sb., 2:3 (1967), 295–317 | DOI | MR | Zbl

[38] A. El Kolli, “$n$-ième épaisseur dans les espaces de Sobolev”, J. Approximation Theory, 10:3 (1974), 268–294 | DOI | MR | Zbl

[39] I. V. Bojkov, “Approximation of some classes of functions by local splines”, Comput. Math. Math. Phys., 38:1 (1998), 21–29 | MR | Zbl

[40] I. V. Boykov, Optimal approximation and Kolmogorov widths estimates for certain singular classes related to equations of mathematical physics, 2013, arXiv: 1303.0416v1

[41] H. Triebel, “Entropy and approximation numbers of limiting embeddings; an approach via Hardy inequalities and quadratic forms”, J. Approx. Theory, 164:1 (2012), 31–46 | DOI | MR | Zbl

[42] A. A. Vasil'eva, “Widths of weighted Sobolev classes on a John domain: strong singularity at a point”, Rev. Mat. Complut., 27:1 (2014), 167–212 | DOI | MR | Zbl

[43] M. S. Aitenova, L. K. Kusainova, “Ob asimptotike raspredeleniya approksimativnykh chisel vlozhenii vesovykh klassov Soboleva. I”, Matem. zhurn., 2:1(3) (2002), 3–9 (electronic) | MR | Zbl

[44] M. S. Aitenova, L. K. Kusainova, “Ob asimptotike raspredeleniya approksimativnykh chisel vlozhenii vesovykh klassov Soboleva. II”, Matem. zhurn., 2:2(4) (2002), 7–14 (electronic) | MR | Zbl

[45] P. I. Lizorkin, M. O. Otelbaev, “Estimates of approximate numbers of the imbedding operators for spaces of Sobolev type with weights”, Proc. Steklov Inst. Math., 170 (1987), 245–266 | MR | Zbl

[46] M. Otelbaev, “Estimates of the Kolmogorov diameters for a class of Besov spaces”, Soviet Math. Dokl., 18:4 (1978), 1159–1163 | MR | Zbl

[47] O. V. Besov, V. P. Il'in, S. M. Nikol'skii, Integral representations of functions and imbedding theorems, v. I, II, Scripta Series in Mathematics, V. H. Winston Sons, Washington, D.C.; Halsted Press [John Wiley Sons], New York–Toronto, Ont.–London, 1978, 1979, viii+345 pp., viii+311 pp. | MR | MR | MR | Zbl | Zbl

[48] V. D. Stepanov, “Two-weighted estimates of Riemann–Liouville integrals”, Math. USSR-Izv., 36:3 (1991), 669–681 | DOI | MR | Zbl

[49] S. Sobolev, “On a theorem of functional analysis”, Amer. Math. Soc. Transl. (2), 34, 1963, 39–68 | Zbl

[50] S. L. Sobolev, Applications of functional analysis in mathematical physics, Transl. Math. Monogr., 7, Amer. Math. Soc., Providence, R.I., 1963, vii+239 pp. | MR | MR | Zbl

[51] A. A. Vasil'eva, “Kolmogorov and linear widths of the weighted Besov classes with singularity at the origin”, J. Approx. Theory, 167 (2013), 1–41 | DOI | MR | Zbl

[52] A. A. Vasil'eva, “Widths of weighted Sobolev classes on a John domain”, Proc. Steklov Inst. Math., 280 (2013), 91–119 | DOI | MR | Zbl