INTERPOLATION OF WEIGHTED BANACH LATTICES

5

polation space with respect to

L1

and L°° if and only if it has the following property

(of being a

UK

space"):

If f € X and some element g G

L1

+ L°° satisfies

K(t, g\ L\L°°) K(t, /; L\L°°) for all t 0,

then g € X.

Subsequent research showed that an analogous description of all interpolation

spaces can also be given for many other couples in terms of their respective K-

functionals. These include couples of (weighted) Lp spaces ([Se, SS, LS, Sp, Cwl, AC]),

of trace ideals ([Ga]), of Hilbert spaces ([Se]), of Lorentz Lpq spaces, Besov spaces or,

more generally, couples of real interpolation spaces of the form (X0Ojpo,X0ljPl) ([Cw3]

cf. also [DO]), couples of Hardy spaces [Jo], [XI] and mixed couples of Hardy and

Lebesgue spaces [Sr], [X2] and also various other couples containing Lorentz A(/)

spaces or Marcinkiewicz spaces ([Cw3,CN3]). All these results are enhanced by a theo-

rem of Brudnyi and Krugljak [BK1] according to which the norm of any given normed

K space is equivalent to a norm obtained simply by applying an appropriate lattice

norm to the K-functional.

We have found it appropriate to refer to couples of Banach spaces whose inter-

polation spaces can be characterized in this way as Calder on-Mity agin couples or CM

couples. Many authors refer to them using a number of alternative names, such as

Calderon couples or K-adequate or K-monotone or C-couples.

Several results of a similar nature have also been obtained for describing pairs of

relative interpolation spaces, which are relevant for operators mapping from one couple

of spaces to a different couple. For example there is a simple description in terms

of X-functionals for describing all relative interpolation spaces for operators mapping

an arbitrary compatible couple of Banach spaces into a couple of weighted L°° spaces

([Pel, CP]) or from a couple of weighted L1 spaces into an (almost) arbitrary Banach

couple (implicit in [BK1] via results of [CP] cf. also [Ov2 Section 5.2]) or between

suitable different couples of Lp spaces or Lorentz spaces ([Dml, Dm2, Cw3, Cw4]).