Geodesic
Characterization of the function spaces associated with weighted Sobolev spaces of the first order on the real line
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 74 (2019) no. 6, pp. 1075-1115

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A brief survey of results on the characterization of the spaces associated with given classes of function spaces is presented. It is shown that the situation differs in general for ideal and non-ideal spaces. In the second case the notion of associated space splits into two. In the main body of the text a complete description is given of the function spaces associated with weighted Sobolev spaces of the first order on the real line. Bibliography: 54 titles.
Keywords: function spaces, associated spaces, duality principle, weighted Sobolev spaces, Oinarov–Otelbaev construction, Hardy–Steklov integral operators. 
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     author = {D. V. Prokhorov and V. D. Stepanov and E. P. Ushakova},
     title = {Characterization of the function spaces associated with weighted {Sobolev} spaces of the first order on the real line},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {1075--1115},
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D. V. Prokhorov; V. D. Stepanov; E. P. Ushakova. Characterization of the function spaces associated with weighted Sobolev spaces of the first order on the real line. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 74 (2019) no. 6, pp. 1075-1115. http://geodesic.mathdoc.fr/item/RM_2019_74_6_a2/