Ukrainian Mathematical Journal

, Volume 65, Issue 3, pp 435–447 | Cite as

Extended Sobolev Scale and Elliptic Operators

  • V. A. Mikhailets
  • A. A. Murach
Article

We obtain a constructive description of all Hilbert function spaces that are interpolation spaces with respect to a pair of Sobolev spaces \( \left[ {{H^{{\left( {{s_0}} \right)}}}\left( {{{\mathbb{R}}^n}} \right),{H^{{\left( {{s_1}} \right)}}}\left( {{{\mathbb{R}}^n}} \right)} \right] \) of some integer orders s0 and s1 and form an extended Sobolev scale. We propose equivalent definitions of these spaces with the use of uniformly elliptic pseudo-differential operators positive-definite in \( {L_2}\left( {{{\mathbb{R}}^n}} \right) \). Possible applications of the introduced scale of spaces are indicated.

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References

  1. 1.
    V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation, and Elliptic Problems [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2010). (Available at arXiv: 1106.3214.)Google Scholar
  2. 2.
    V. A. Mikhailets and A. A. Murach, “The refined Sobolev scale, interpolation, and elliptic problems,” Banach J. Math. Anal., 6, No. 2, 211–281 (2012).MathSciNetMATHGoogle Scholar
  3. 3.
    L. Hörmander, Linear Partial Differential Operators, Springer, Berlin (1963).CrossRefMATHGoogle Scholar
  4. 4.
    L. Hörmander, The Analysis of Linear Partial Differential Operators. II: Differential Operators with Constant Coefficients, Springer, Berlin (1983).CrossRefMATHGoogle Scholar
  5. 5.
    L. R. Volevich and B. P. Paneah, “Certain spaces of generalized functions and embedding theorems,” Rus. Math. Surv., 20, No. 1, 1–73 (1965).CrossRefGoogle Scholar
  6. 6.
    V. A. Mikhailets and A. A. Murach, “Elliptic operators in a refined scale of function spaces,” Ukr. Math. J., 57, No. 5, 817–825 (2005).MathSciNetCrossRefGoogle Scholar
  7. 7.
    V. A. Mikhailets and A. A. Murach, “Improved scale of spaces and elliptic boundary-value problems. II,” Ukr. Math. J., 58, No. 3, 398–417 (2006).MathSciNetCrossRefGoogle Scholar
  8. 8.
    V. A. Mikhailets and A. A. Murach, “A regular elliptic boundary-value problem for a homogeneous equation in a two-sided improved scale of spaces,” Ukr. Math. J., 58, No. 11, 1748–1767 (2006).MathSciNetCrossRefGoogle Scholar
  9. 9.
    V. A. Mikhailets and A. A. Murach, “Refined scale of spaces and elliptic boundary-value problems. III,” Ukr. Math. J., 59, No. 5, 744–765 (2007).MathSciNetCrossRefGoogle Scholar
  10. 10.
    V. A. Mikhailets and A. A. Murach, “An elliptic boundary-value problem in a two-sided refined scale of spaces,” Ukr. Math. J., 60, No. 4, 574–597 (2008).MathSciNetCrossRefGoogle Scholar
  11. 11.
    V. A. Mikhailets and A. A. Murach, “Elliptic problems and H¨ormander spaces,” Oper. Theory: Adv. Appl., 191, 447–470 (2009).MathSciNetGoogle Scholar
  12. 12.
    B. Paneah, The Oblique Derivative Problem. The Poincaré Problem, Wiley, Berlin (2000).MATHGoogle Scholar
  13. 13.
    E. Seneta, Regularly Varying Functions, Springer, Berlin (1976).CrossRefMATHGoogle Scholar
  14. 14.
    N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge (1989).MATHGoogle Scholar
  15. 15.
    J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin (1976).CrossRefMATHGoogle Scholar
  16. 16.
    C. Foias¸ and J.-L. Lions, “Sur certains théorèmes d’interpolation,” Acta Sci. Math. (Szeged), 22, No. 3–4, 269–282 (1961).MathSciNetMATHGoogle Scholar
  17. 17.
    J. Peetre, “On interpolation functions. II,” Acta Sci. Math. (Szeged), 29, No. 1–2, 91–92 (1968).MathSciNetMATHGoogle Scholar
  18. 18.
    V. I. Ovchinnikov, “The methods of orbits in interpolation theory,” Math. Rep. Ser. 1, No. 2, 349–515 (1984).Google Scholar
  19. 19.
    M. S. Agranovich, “Elliptic operators on closed manifolds,” in: Encyclopedia of Mathematical Sciences, Partial Differential Equations, VI, 63, Springer, Berlin (1994), pp. 1–130.CrossRefGoogle Scholar
  20. 20.
    H. Triebel, The Structure of Functions, Birkhäuser, Basel (2001).CrossRefMATHGoogle Scholar
  21. 21.
    N. Jacob, Pseudodifferential Operators and Markov Processes (in 3 volumes), Imperial College Press, London (2001, 2002, 2005).CrossRefGoogle Scholar
  22. 22.
    F. Nicola and L. Rodino, Global Pseudodifferential Calculus on Euclidean Spaces, Birkhäuser, Basel (2010).CrossRefGoogle Scholar
  23. 23.
    A. A. Murach, “Elliptic pseudodifferential operators in a refined scale of spaces on a closed manifold,” Ukr. Math. J., 59, No. 6, 874–893 (2007).MathSciNetCrossRefGoogle Scholar
  24. 24.
    V. A. Mikhailets and A. A. Murach, “Interpolation with a function parameter and refined scale of spaces,” Meth. Funct. Anal. Topol., 14, No. 1, 81–100 (2008).MathSciNetMATHGoogle Scholar
  25. 25.
    A. A. Murach, “Douglis–Nirenberg elliptic systems in the refined scale of spaces on a closed manifold,” Meth. Funct. Anal. Topol., 14, No. 2, 142–158 (2008).MathSciNetMATHGoogle Scholar
  26. 26.
    V. A. Mikhailets and A. A. Murach, “Elliptic systems of pseudodifferential equations in the refined scale on a closed manifold,” Bull. Pol. Acad. Sci. Math., 56, No. 3–4, 213–224 (2008).MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    A. A. Murach, “On elliptic systems in Hörmander spaces,” Ukr. Math. J., 61, No 3, 467–477 (2009).MathSciNetCrossRefGoogle Scholar
  28. 28.
    T. N. Zinchenko and A. A. Murach, “Douglis–Nirenberg elliptic systems in Hörmander spaces,” 64, No. 11, 1477–1491 (2012); English translation: Ukr. Math. J., 64, No. 11, 1672–1687 (2012).MathSciNetCrossRefGoogle Scholar
  29. 29.
    V. A. Mikhailets and A. A. Murach, “On the unconditional almost-everywhere convergence of general orthonormal series,” Ukr. Math. J., 63, No. 10, 1543–1550 (2012).CrossRefMATHGoogle Scholar
  30. 30.
    V. A. Mikhailets and A. A. Murach, “General forms of the Menshov–Rademacher, Orlicz, and Tandori theorems on orthogonal series,” Meth. Funct. Anal. Topol., 17, No. 4, 330–340 (2011).MathSciNetMATHGoogle Scholar
  31. 31.
    M. Hegland, “Error bounds for spectral enhancement which are based on variable Hilbert scale inequalities,” J. Integral Equat. Appl., 22, No. 2, 285–312 2010).MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    P. Mathé and U. Tautenhahn, “Interpolation in variable Hilbert scales with application to inverse problems,” Inverse Problems, 22, No. 6, 2271–2297 (2006).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • V. A. Mikhailets
    • 1
  • A. A. Murach
    • 1
  1. 1.Institute of Mathematics, Ukrainian National Academy of SciencesKievUkraine

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