polation space with respect to
and L°° if and only if it has the following property
(of being a
If f X and some element g G
+ 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]).
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