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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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