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