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, from these theorems one only can deduce that the equivalence of the norms depends upon a, that is to say at least one of the constants of embedding depends upon a. In our paper, we will show, introducing “S-orbits” as a tool, that both constants of embeddings, in fact, are independent 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 of a space X coincide. Using a result which is analogous to a theorem of N. Aronszajn-E. Gagliardo on strong-type interpolation, namely that a space X has the interpolation property if and only if X coincide with its orbit space, we obtain characterizations of weak-type interpolation spaces by means of the above orbit spaces.