10
M. CWIKEL AND P. G. NILSSON
expression in the style of Holmstedt's well known formula [H] for (LP,L9), i.e.
K(t, x; X) - \\xxEt \\x0 + t\\x - xxEt \\x±
for some increasing family of measurable sets {Et}t0 depending on x. In fact Brudnyi
and Krugljak have also obtained a result similar to this one. This "Holmstedt-like"
formula, together with some additional remarks, also enables us to present a simplified
and strengthened version of a theorem of [Cw4] which shows that (relative) decompos-
ability is a sufficient condition for couples to be (relatively) CM.
For the second main result of this section we need to assume that the underlying
measure space is nonatomic. We show that under further suitable (mild) hypotheses,
each element x G S(X) can be expressed as a sum of two disjointly supported elements
each of whose if-functionals is equivalent to K(t, x; X).
Section 5 begins by establishing a "quantitative" and "^-localized" form of one of
the main results: If 0 G (0,1) and X and Y are suitable couples of lattices then [Xw]#
and [Yv]# are relative X-K spaces for some fixed constant A and all choices of weight
functions if and only if [X]# and \Y\Q are relatively decomposable. This result enables
us to prove quantitative versions of the other main theorems which in turn are to be
generalized to their final form in Section 6.
To indicate the flavour of Section 6 we must first recall a theorem of Aronszajn and
Gagliardo ([AG p. 73 Theorem 6.XI]) which states that if AQ and A\ are compatible
Banach spaces and the Banach space A is an interpolation space with respect to AQ
and Ai, i.e. if every operator T which is bounded on AQ and A\ is also bounded on A,
then there exists some positive constant c such that A is a c-interpolation space, i.e. all
such operators T satisfy
| | T | U ^ A
cmax{||Tm
0
_^
0
, | | T | |
A l
_
A l
} .
Most of the results of Section 6 are also "uniform boundedness principles" of this type
for couples of lattices (XQ, XI). For example we show that to each K space X can be
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