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 s 0 and s 1 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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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.)
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).
L. Hörmander, Linear Partial Differential Operators, Springer, Berlin (1963).
L. Hörmander, The Analysis of Linear Partial Differential Operators. II: Differential Operators with Constant Coefficients, Springer, Berlin (1983).
L. R. Volevich and B. P. Paneah, “Certain spaces of generalized functions and embedding theorems,” Rus. Math. Surv., 20, No. 1, 1–73 (1965).
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).
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).
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).
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).
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).
V. A. Mikhailets and A. A. Murach, “Elliptic problems and H¨ormander spaces,” Oper. Theory: Adv. Appl., 191, 447–470 (2009).
B. Paneah, The Oblique Derivative Problem. The Poincaré Problem, Wiley, Berlin (2000).
E. Seneta, Regularly Varying Functions, Springer, Berlin (1976).
N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge (1989).
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin (1976).
C. Foias¸ and J.-L. Lions, “Sur certains théorèmes d’interpolation,” Acta Sci. Math. (Szeged), 22, No. 3–4, 269–282 (1961).
J. Peetre, “On interpolation functions. II,” Acta Sci. Math. (Szeged), 29, No. 1–2, 91–92 (1968).
V. I. Ovchinnikov, “The methods of orbits in interpolation theory,” Math. Rep. Ser. 1, No. 2, 349–515 (1984).
M. S. Agranovich, “Elliptic operators on closed manifolds,” in: Encyclopedia of Mathematical Sciences, Partial Differential Equations, VI, 63, Springer, Berlin (1994), pp. 1–130.
H. Triebel, The Structure of Functions, Birkhäuser, Basel (2001).
N. Jacob, Pseudodifferential Operators and Markov Processes (in 3 volumes), Imperial College Press, London (2001, 2002, 2005).
F. Nicola and L. Rodino, Global Pseudodifferential Calculus on Euclidean Spaces, Birkhäuser, Basel (2010).
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).
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).
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).
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).
A. A. Murach, “On elliptic systems in Hörmander spaces,” Ukr. Math. J., 61, No 3, 467–477 (2009).
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).
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).
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).
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).
P. Mathé and U. Tautenhahn, “Interpolation in variable Hilbert scales with application to inverse problems,” Inverse Problems, 22, No. 6, 2271–2297 (2006).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 3, pp. 392–404, March, 2013.
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Mikhailets, V.A., Murach, A.A. Extended Sobolev Scale and Elliptic Operators. Ukr Math J 65, 435–447 (2013). https://doi.org/10.1007/s11253-013-0787-5
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DOI: https://doi.org/10.1007/s11253-013-0787-5