Strong and weak associativity and reflexivity of certain function classes

The paper provides an overview of recent results on the problem of character- ization of associated and doubly associated spaces to functional classes including both ideal and non-ideal structures. The latter include two-weight Sobolev spaces of the first order on the positive semi-axis [1]. It is shown that, unlike the con- cept of duality, associativity can be “strong” and “weak”. At the same time, the twice-associated spaces are divided into three more types. In this context, it is established that the Sobolev space of functions with a compact support has weakly associated reflexivity, and a strongly associated to a weakly associated space consists only of zero [2]. Similar properties are possessed by weight classes of Cesaro and Copson type, for which the problem has been fully studied and their connection with Sobolev spaces with power weights has been established [3]. As an application, the problem of the boubdedness of the Hilbert transformation from the Sobolev weight space to the Lebesgue weight space is considered [4].

References

1. 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                
2. V. D. Stepanov, E. P. Ushakova, “Strong and weak associativity of weighted Sobolev spaces of the first order”, Uspekhi Mat. Nauk, 78:1(469) (2023), 167–204  
3. V.D. Stepanov, “On Cesàro and Copson type function spaces. Reflexivity”, J. Math. Anal. Appl., 507:1 (2022), 125764, 18 pp.  
4. V.D. Stepanov, “On the boundedness of the Hilbert transform from weighted Sobolev space to weighted Lebesgue space”, J. Fourier Anal. Appl., 28 (2022), 46, 17 pp.