Such pairs of couples may be termed relative CM couples.
This series of results might have tempted one to conjecture that all (relative)
interpolation spaces with respect to all couples can be characterized in this same way.
However there are also many known examples of couples which are not CM including
some which occur "naturally" in various applications of interpolation theory. The
known examples of non CM couples include:
(i) (LP, W1*) where W1* is a Sobolev space and 2 ± p £ (1, oc) ([Cwl]).
(h) (^e^f^er
) ([Cw2]).
(hi) ( L i n l / ^ Z ^ + L00) ([Ovl]).
(iv) The couple (C([0, l]),Lip([0,1])) of continuous, respectively Lipshitz functions
on an interval ([By], [CM], see also [BStl], [BSt2] for further study of this and related
(v) Couples (Lip(u)o), Lip(ui)) of Lipschitz spaces with respect to suitable functions
UJQ and u\. [BStl]. In fact in [BStl] it is shown that such couples are CM if and
only if c^o, OJ\ satisfy certain simple conditions. Other more general couples are also
(vi) The couples
and also the couples (Lipa(Xo)iLipa(Xi)), where
p = 1 or oo and a £ (0,1), for a large class of Banach couples (XQ, X I ) , ([Ms3], [MX]).
For other related results about non (relative) CM couples we refer also to [Msl].
A discussion of CM couples can also be found in Sections 4.4 and 4.7.2 of [BK2].
It could be claimed that one of the ultimate goals of interpolation theory should
be to provide means for describing all interpolation spaces for all couples (or families)
of (Banach) spaces. But such a goal seems utterly Utopian, at least at this stage. If we
attempt, nevertheless, to advance towards it, then there are currently two natural paths
of investigation to be pursued. One of them is to seek a better understanding of the
structure of non—CM couples, and to find some alternative way of characterizing their
interpolation spaces. Here one probably has to first study particular examples, such
as those just listed. This has been done in [MgO] for the couple (L1 D L°°, L1 + L°°).
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