On the rearrangement-invariant hull of the Calderón space

The authors study the isotropic Calderón space Lambda(E,F) consisting of functions f belonging to a rearrangement-invariant space E for which the quantity e_t(f)_Ecoloneq inf{|f-q|_Ecolon ,qin M_{tsp{1/n},E}(Bbb Rsp n)} satisfies e_t(f)_Ein F (here F is a certain ideal space), where t>0 and M_{tsp{1/n},E}(Bbb Rsp n) is the space of entire functions of n complex variables of exponential type with power 1/n in every variable whose restriction to Bbb Rsp n belongs to E. An important example of the Calderón space is the generalized Besov space defined by the best approximations. The authors obtain sharp sufficient conditions for the embedding Lambda(E,F)subset X, where X is another rearrangement-invariant space, which are "almost" necessary. As a consequence, they construct the optimal rearrangement-invariant range space for this embedding. The results are applied to generalized Besov spaces. Proofs are not included.