Abstract:
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.
The work of the first author was supported by the Russian Science Foundation under grant no. 14-11-00443 and carried out at the Steklov Mathematical Institute of the Russian Academy of Sciences. The work of the second and third authors (Sections 3-5) was supported by the Russian Science Foundation under grant no. 19-11-00087 and also carried out at the Steklov Institute.
Citation:
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”, Russian Math. Surveys, 74:6 (2019), 1075–1115
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\paper Characterization of the function spaces associated with weighted Sobolev spaces of the first order on the real line
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\pages 1075--1115
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Linking options:
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https://doi.org/10.1070/RM9873
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This publication is cited in the following 9 articles:
D. V. Prokhorov, “On Interpolation of Cesàro Function Spaces with Polynomial Weight”, Sib Math J, 66:2 (2025), 315
D. V. Prokhorov, “On the associated spaces of the weighted altered Cesàro space”, Eurasian Math. J., 15:1 (2024), 55–64
D. V. Prokhorov, “O prostranstvakh, assotsiirovannykh k prostranstvu Khardi”, Dalnevost. matem. zhurn., 24:1 (2024), 96–106
V. D. Stepanov, E. P. Ushakova, “Strong and weak associativity of weighted Sobolev spaces of the first order”, Russian Math. Surveys, 78:1 (2023), 165–202
V. D. Stepanov, “On Cesaro and Copson type function spaces. Reflexivity”, J. Math. Anal. Appl., 507:1 (2022), 125764
D. V. Prokhorov, “On the dual spaces for weighted altered Cesàro and Copson spaces”, Journal of Mathematical Analysis and Applications, 514:2 (2022), 126325
D. V. Prokhorov, “On the associated spaces for the altered Cesàro space”, Anal. Math., 48:4 (2022), 1169–1183
V. D. Stepanov, “On the boundedness of the Hilbert transform from weighted Sobolev space to weighted Lebesgue space”, J. Fourier Anal. Appl., 28:3 (2022), 46
D. V. Prokhorov, “On a Class of Functionals on a Weighted First-Order Sobolev Space on the Real Line”, Proc. Steklov Inst. Math., 312 (2021), 226–240
Citation in format AMSBIB
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