INTERPOLATION OF WEIGHTED BANACH LATTICES

7

Related results are given in [Ov3] for relative interpolation spaces with respect to two

couples of LP spaces which are not relative CM couples. Cf. also [Dm3].

The other path is to seek more general procedures for determining whether or

not other couples are CM or relatively CM. Some steps in this direction have been

taken for example in [Cw4], [CNl] and [N2] with crucial aid from the important work

of Brudnyi and Krugljak [BK1, BK2]. Kalton [Ka2] has made very significant progress

towards finding criteria to determine in general which couples of rearrangement in-

variant spaces are CM couples. In all known examples of CM couples, some or all

of the spaces involved have a lattice structure or something resembling it. This sug-

gests the natural, but apparently still extremely difficult problem of finding necessary

and sufficient conditions characterizing (relative) CM couples of Banach lattices. Our

main result solves a variant of this problem. We characterize those couples of weighted

Banach lattices which are (relatively) CM for all choices of weight functions.

Perhaps the best and easiest way for the reader to proceed from here with getting

a feel for what we shall do in this paper, is by looking at a much simpler "miniature

model" of the big machine which we are going to set up. This can be found in the

eleven pages of our earlier paper [CNl], where we establish a special case of the main

result to be presented here.

The Banach lattices which we consider are "concrete" Banach lattices of measur-

able functions. They are arbitrary except for the requirements that they should have

the (weak) Fatou property, and be saturated and cr-order continuous. In fact not all

of these requirements are needed for all results. Furthermore, we suspect that the

requirement of cr-order continuity can be removed in quite a number of cases. We

hope to deal with this in a future paper. The requirement of saturation is simply a

matter of convenience, since a rather straightforward theorem enables our results to

be interpreted in the context of couples which are not saturated. As for the Fatou

property, even if it is not imposed it seems to be rather unavoidable, except in some

trivial cases. For example, if for some couple of Banach lattices XQ and X\ and for all