Spaces A,(X) and interpolation

we give necessary and sufficient conditions for pairs (X, Y) to be weak inter-mediate for o, i.e., each operator which is of weak types (Xi, YJ, i = 1, 2, also maps X boundedly to Y. Spaces A,(X) are introduced and are shown to have many of the properties that characterize Lorentz L*‘r spaces. Necessary and sufficient conditions in terms of a simple function F(s, t) are given in order that (d,(X), A,(Y)) be weak intermediate for o. Other properties of the function F(s, t) yield sufficient conditions and necessary conditions for interpolation theorems