The paper is a survey of the modern results devoted to the theory of elliptic operators and elliptic boundary–value problems on Hilbert scales that consist of Hormander function spaces. Theorems on the Fredholm property and local regularity of the solutions are established. Applications to a convergence of spectral expansions in eigenfunctions of elliptic operators are given.

[1] Mihaylets V.A., Murach A.A. Hermander spaces, interpolation and elliptic problems // Proceedings of the Institute of Mathematics. of the National Academy of Sciences of Ukraine. Vol. 84. - Kyiv, 2010.

[2] Mikhaylets V.A., Murach A.A. Refined scales of spaces and elliptic boundary value problems. I // Ukr. mat. zhurn. 2006. -58, № 2. - P. 217-235.

[3] Mikhaylets, V.A.; Murach, A.A. Refined scales of spaces and elliptic boundary value problems. - № 3. - P. 352-370.

[4] Mikhaylets, V.A.; Murach, A.A. Regular elliptic boundary value problem for a homogeneous equation in a two-sided refined space scale  // Ibidem. - № 11. - P. 1536-1555.

[5] Mikhaylets, V.A.; Murach, A.A. Elliptic operator with homogeneous regular boundary conditions in the bilateral refined scale of spaces // Ukr. mat. visnik. - 2006. -3, № 4. - P. 547-580.

[6] Mikhaylets, V.A.; Murach, A.A. Refined scales of spaces and elliptic boundary value problems. III // Ukr. mat. zhurn. - 2007. -59, № 5. - P. 679-701.

[7] Murach, A. A. Elliptic pseudodifferential operators in the refined scale of spaces on a closed manifold // Ibidem. -№ 6. - P. 798-814.

[8] Mikhailets V. A., Murach A. A. Interpolation with a function parameter and refined scale of spaces // Methods Funct. Anal. Topology. – 2008. – 14 , № 1. – P. 81–100.

[9] Murach A. A. Douglis–Nirenberg elliptic systems in the refined scale of spaces on a closed manifold // lbid. – 2008. – 14 , № 2. – P. 142–158.

[10] Mikhaylets, V.A.; Murach, A.A. Elliptic boundary value problem in the bilateral refined scale of spaces  // Ukr. mat. zhurn. 2008. -60, № 4. - P. 497-520.

[11] Mikhailets V. A., Murach A. A. Elliptic systems of pseudodifferential equations in a refined scale on a closed manifold // Bull. Pol. Acad. Sci. Math. – 2008. – 56 , № 3–4. – P. 213–224.

[12] Mikhailets V.A., Murach A.A. On elliptic operators on a closed manifold // Doklady. NAS of Ukraine. - 2009. - № 3. - P. 13-19.

[13] Hermander L. Linear differential operators with partial derivatives. - Moscow: Mir, 1965. (Translated view. Berlin, Springer-Verlag,1963.)

[14] Seneta E. Properly changing functions. - Moscow: Nauka, 1985.

[15] Volevich L.R., Paneyakh B.P. Some spaces of generalised functions and embedding theorems // Uspekhi mat. nauk.- 1965.-20, № 1. - P. 3-74.

[16] Hermander L. Analysis of linear differential operators with partial derivatives: In 4 vol. 2. - Moscow: Mir, 1986.

[17] Lizorkin, P.I. Spaces of generalised smoothness // H. Triebel. Theory of Functional Spaces. - Moscow: Mir, 1986. - P. 381-415.

[18] Triebel H. The structure of functions. – Basel: Birkhäser, 2001.

[19] Jacob N. Pseudodifferential operators and Markov processes (in 3 volumes). – London: Imperial College Press, 2001, 2002, 2005.

[20] Farkas W., Leopold H.-G. Characterisations of function spaces of generalized smoothness // Ann. Mat. Pura Appl. – 2006. – 185 , № 1. – P. 1–62.

[21] Haroske D.D., Moura S.D. Continuity envelopes of spaces of generalised smoothness, entropy and approximation numbers // J. Approximation Theory. – 2004. – 128 . – P. 151–174.

[22] Paneah B. The oblique derivative problem. The Poincaré problem. – Berlin: Wiley–VCH, 2000.

[23] Bingham N.H., Goldie C.M., Teugels J.L. Regular variation. – Cambridge: Cambridge Univ. Press, 1989.

[24] Lyons J.-L., Madgenes E. Inhomogeneous boundary value problems and their applications. - Moscow: Mir, 1971.

[25] Berg J., Löfström J. Interpolation spaces. Introduction. - Moscow: Mir, 1980.

[26] Hermander L. Analysis of linear differential operators with partial derivatives: In 4 vol. 3. - Moscow: Mir, 1987.

[27] Berezanskiy Y.M., Krein S.G., Roitberg Y.A. Theorem on homeomorphisms and local increase of smoothness up to the boundary of solutions of elliptic equations // Dokl. of the USSR Academy of Sciences. -1963. -148, № 4. - P. 745-748.

[28] Roitberg Ya. A. Elliptic boundary value problems in the spaces of distributions. – Dordrecht: Kluwer Acad. Publishers, 1996.

[29] Roitberg Ya. A. Elliptic problems with inhomogeneous boundary conditions and local increase of smoothness up to the boundary of generalised solutions // Dokl. of the USSR Academy of Sciences. - 1964. -157, № 4. - P. 798-801.

[30] Murach A. A. Extension of some Lions–Magenes theorems // Methods Funct. Anal. Topology. – 2009. – 15 , № 2. – P. 152–167.