Advertisement

Siberian Mathematical Journal

, Volume 58, Issue 3, pp 445–460 | Cite as

Integral representation and embedding theorems for n-dimensional multianisotropic spaces with one anisotropic vertex

  • G. A. Karapetyan
Article

Abstract

We prove embedding theorems for the multianisotropic Sobolev spaces generated by the completely regular Newton polyhedron. Under study is the case of the polyhedron with one anisotropic vertex. We obtain a special integral representation of functions in terms of the tuple of multi-indices of the Newton polyhedron.

Keywords

embedding theorems multianisotropic space completely regular polyhedron integral representation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Sobolev S. L., “On one theorem of functional analysis,” Mat. Sb., vol. 4, no. 3, 471–497 (1938).Google Scholar
  2. 2.
    Sobolev S. L., Some Applications of Functional Analysis in Mathematical Physics, Amer. Math. Soc., Providence (1991).zbMATHGoogle Scholar
  3. 3.
    Nikol’skiĭ S. M., “On a problem by S. L. Sobolev,” Sib. Mat. Zh., vol. 3, no. 6, 845–857 (1962).Google Scholar
  4. 4.
    Smith K. T., “Inequalities for formally positive integro-differential forms,” Bull. Amer. Math., vol. 67, 368–370 (1961).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Il’in V. P., “Integral representations of differentiable functions and their application to questions of continuation of functions of classes W p l(G),” Sib. Math. J., vol. 8, no. 3, 421–432 (1967).CrossRefGoogle Scholar
  6. 6.
    Besov O. V., “On coercivity in nonisotropic Sobolev spaces,” Math. USSR-Sb., vol. 2, no. 4, 521–534 (1967).CrossRefzbMATHGoogle Scholar
  7. 7.
    Reshetnyak Yu. G., “Some integral representations of differentiable functions,” Sib. Math. J., vol. 12, no. 2, 285–290 (1971).CrossRefzbMATHGoogle Scholar
  8. 8.
    Besov O. V., Il’in V. P., and Nikol’skiĭ S. M., Integral Representations of Functions and Embedding Theorems, John Wiley and Sons, New York etc. (1978).zbMATHGoogle Scholar
  9. 9.
    Karapetyan G. A., “The integral representation and embedding theorems for multianisotropic spaces in the plane with one anisotropic vertex,” Izv. NAN RA Mat., vol. 51, no. 6, 23–42 (2016).Google Scholar
  10. 10.
    Karapetyan G. A., “The integral representation and embedding theorems for multianisotropic spaces in the plane,” Izv. NAN RA Mat. (to be published).Google Scholar
  11. 11.
    Karapetyan G. A., “On stabilization at infinity to polynomial of solutions of a certain class of regular equations,” Proc. Steklov Inst. Math., vol. 187, 131–145 (1990).zbMATHGoogle Scholar
  12. 12.
    Uspenskiĭ S. V., “The representation of functions defined by a certain class of hypoelliptic operators,” Proc. Steklov Inst. Math., vol. 117, 343–352 (1972).MathSciNetGoogle Scholar
  13. 13.
    Hörmander L., “On the theory of general partial differential operators,” Acta. Math., vol. 94, 161–248 (1975).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Nikol’skiĭ S. M., “Stable boundary-value problems of a differentiable function of several variables,” Mat. Sb., vol. 61, no. 2, 224–252 (1963).MathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Russian-Armenian UniversityYerevanArmenia

Personalised recommendations