Mathematical Notes

, Volume 98, Issue 3–4, pp 550–560 | Cite as

Embedding of Sobolev space in the case of the limit exponent



We establish the embeddings of the Sobolev space W p s and the space B pq s (in the case of the limit exponent) in the spaces of locally summable functions of zero smoothness. This refines the embeddings of the Sobolev space in the Lorentz space and in the Lorentz–Zygmund space. The relationship between the Lorentz spaces and the corresponding spaces of functions of zero smoothness is established. Similar embeddings of the spaces of potentials are determined.


Sobolev space Wps the space Bpqs locally summable function of zero smoothness Lorentz space Lorentz–Zygmund space space of potentials 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. John and L. Nirenberg, “On functions of bounded mean oscillation,” Comm. Pure Appl. Math. 14, 415–426 (1961).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    A. Gogatishvili, P. Koskela, and Y. Zhou, “Characterization of Besov and Triebel–Lizorkin spaces on metric measure spaces,” Forum Math. 25 (4), 787–819 (2013).MathSciNetMATHGoogle Scholar
  3. 3.
    O. V. Besov, “To the Sobolev embedding theorem for the limiting exponent,” in Trudy Mat. Inst. Steklov, Vol. 284: Function Spaces and Related Questions of Analysis (MAIK, Moscow, 2014), pp. 89–104 [Proc. Steklov Inst. Math. 284, 81–96 (2014)].Google Scholar
  4. 4.
    O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Embedding Theorems (Nauka, Moscow, 1996) [in Russian].Google Scholar
  5. 5.
    V. I. Kolyada, “On relations between moduli of continuity in different metrics,” in Trudy Mat. Inst. Steklov, Vol. 181: Studies in the Theory of Differentiable Functions of Several Variables and Its Applications. Pt. 12 (Nauka, Moscow, 1988), pp. 117–136 [Proc. Steklov Inst. Math. 181, 127-148 (1989)].Google Scholar
  6. 6.
    R. O’Neil, “Convolution operators in L(p, q) spaces,” Duke Math. J. 30, 129–142 (1963).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    J. Peetre, “Espaces d’interpolation et théorème de Soboleff,” Ann. Inst. Fourier (Grenoble) 16 (1), 279–317 (1966).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    V. I. Yudovich, “Some estimates connected with integral operators and with solutions of elliptic equations,” Dokl. Akad. Nauk SSSR 138 (4), 805–808 (1961) [SovietMath. Dokl. 2 (4), 746–749 (1961)].MathSciNetGoogle Scholar
  9. 9.
    K. Hansson, “Imbedding theorems of Sobolev type in potential theory,” Math. Scand. 45 (1), 77–102 (1979).MathSciNetMATHGoogle Scholar
  10. 10.
    A. Cianchi, “Optimal Orlicz–Sobolev embeddings,” Rev. Mat. Iberoamericana 20 (2), 427–474 (2004).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    H. Triebel, Structure of Functions, in Monogr. Math. (Birhäuser Verlag, Basel, 2001), Vol. 97.CrossRefGoogle Scholar
  12. 12.
    V. I. Kolyada, “Estimates of rearrangements and imbedding theorems,” Mat. Sb. 136 (1), 3–23 (1988) [Math. USSR-Sb. 64 (1), 1–21 (1989)].Google Scholar
  13. 13.
    B. Muckenhoupt, “Hardy’s inequalities with weights,” Studia Math. 44, 31–38 (1972).MathSciNetMATHGoogle Scholar
  14. 14.
    A. Kufner, L. Maligranda, and L.-E. Persson, Hardy Inequality. About Its History and Some Related Results (Vydavatelský Servis, Pilsen, 2007).MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

Personalised recommendations