Strong and Weak Associativity of Weighted Sobolev Spaces of the First Order

A brief overview of the recent results on the problem of characterization of associative and double associative spaces of function classes, including both ideal and non-ideal structures, is presented. The latter include two-weighted Sobolev spaces of the first order on the positive semi- axis. It is shown that, in contrast to the notion of duality, associativity can be ‘strong’ or ‘weak’. In addition, double associative spaces are further divided into three types. In this context it is established that a weighted Sobolev space of functions with compact support possesses weak associative reflexivity, while the strong associative space of a weak associative space is trivial. Weighted classes of Cesàro and Copson type have similar properties; for these classes the problem us fully investigated, and their connections with Sobolev spaces with power weights are established. As an application, the problem of boundedness of the Hilbert transform from a weighted Sobolev space to a weighted Lebesgue space is considered. Bibliography: 49 titles.