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Some imbedding theorems for the function spaces − fr,ρ,θ λ,ϕ,bs (G)

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One considers imbedding type theorems for the spaces − fr,ρ,θ λ,ϕ,bs (G) of real functions, defined on a domain G of then -dimensional Euclidean space En. As opposed to the known spaces of this type, the power function ta, characterizing the degree of the smoothness of the functions, is replaced here by a function ϕ(t), arbitrary in a certain sense.

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Literature cited

  1. 1.

    O. V. Besov, V. P. Il'in, and S. M. Nikol'skii, Integral Representations of Functions and Imbedding Theorems, Vols. I and II, Wiley, New York (1978–1979).

  2. 2.

    V. P. Il'in, “The function spaces − fr,ρ,G λ,a,bs (G),” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,30, 51–75 (1972).

  3. 3.

    S. M. Nikol'skii, “Inequalities for entire functions of finite degree and their applications to the theory of differentiable functions of several variables,” Tr. Mat. Inst. Akad. Nauk SSSR,38, 244–278 (1951).

  4. 4.

    O. V. Besov, “The investigation of a family of function spaces in connection with embedding and extension theorems,” Tr. Mat. Inst. Akad. Nauk SSSR,60, 42–81 (1961).

  5. 5.

    P. L. Ul'yanov, “On the imbedding of certain classes of functions,” Mat. Zametki,1, No. 4, 405–414 (1967).

  6. 6.

    P. L. Ul'yanov, “The imbedding of certain classes H fp ω of functions,” Izv. Akad. Nauk SSSR,32, 649–686 (1968).

  7. 7.

    V. A. Andrienko, “Necessary conditions for the imbedding of the function classes H fp ω ,” Mat. Sb.,78, No. 2, 280–300 (1969).

  8. 8.

    N. Temirgaliev, “The connection of imbedding theorems with the uniform convergence of multiple Fourier series,” Mat. Zametki,12, No. 2, 139–148 (1972).

  9. 9.

    S. Campanato, “Proprieta di una famiglia di spazi funzionali,” Ann. Scuola Norm. Sup. Pisa,18, No. 1, 137–160 (1964).

  10. 10.

    F. John and L. Nirenberg, “On functions of bounded mean oscillation,” Commun. Pure Appl. Math.,14, 415–426 (1961).

  11. 11.

    S. J. Berman, Characterizations of bounded mean oscillation,” Proc. Am. Math. Soc.,51, No. 1, 117–122 (1975).

  12. 12.

    S. Spanne, “Some function spaces defined using the mean oscillation over cubes,” Ann. Scuola Norm. Sup. Pisa,19, No. 4, 593–608 (1965).

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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 70, pp. 49–75, 1977.

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Il'in, V.P. Some imbedding theorems for the function spaces − fr,ρ,θ λ,ϕ,bs (G). J Math Sci 23, 1909–1929 (1983). https://doi.org/10.1007/BF01093274

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  • Euclidean Space
  • Function Space
  • Power Function
  • Real Function
  • Type Theorem