Dependence on Parameters in Interpolation Methods Associated to Polygons

Abstract

A theorem due to S. Janson, P. Nilsson, J. Peetre and M. Zafran [Proc. Lond. Math. Soc., III. Ser. 48, 283-299 (1984; Zbl 0532.46046)] states that for a Banach couple A such that _(A) is not closed in _(A) the real interpolation spaces A_,q and A_,p coincide if and only if _ = _ and p = q. Here the analogous problem for N-tuples of Banach spaces is investigated. It is assumed that the N-tuple A satisfies a certain condition "(A) which ensures that the J- and the K-methods with respect to a polygon _ coincide. Also, it is assumed that _(A) is not closed in _(A).
The authors prove the following results:
(1) If P,Q 2 int _ and AP,p = AQ,q, then p = q.
(2) If P, Q,R 2 int _ and AP,q = AQ,q = AR,q, then P, Q and R are affinely dependent.