WEAK-TYPE INTERPOLATION AND ORBITS

Combining two theorems of Ya.A. Brudnyi-N.Ya. Krugliak and R. Sharpley , we get the result that in weak-type interpolation orbit and K-orbit spaces of a point a coincide as sets and have equivalent norms. But, f rom these theorems one only can deduce that the equivalence of the nor ms depends upon a, that is to say at least one of the constants of emb edding depends upon a. In our paper, we will show, introducing ''S-orb its'' as a tool, that both constants of embeddings, in fact, are indep endent of a and the spaces involved. This leads us to the fact that in weak-type interpolation, also the orbit, K-orbit and S-orbit spaces o f a space X coincide. Using a result which is analogous to a theorem o f N. Aronszajn-E. Gagliardo on strong-type interpolation, namely that a space X has the interpolation property if and only if X coincide wit h its orbit space, we obtain characterizations of weak-type interpolat ion spaces by means of the above orbit spaces.