EXTRAPOLATION THEORY 5

satisfies

(1.3) t|{x:Msf t}| J-LJJ-

(l+log+-MP-)n—1dx.

An easy way to prove this is to majorize Ms by the iterate of n one—dimensional

Hardy-Littlewood maximal operators. (Let us parenthetically observe that (1.3)

combined with Lemma 4.4 below provides an easy approach to recent results for M

(cf. [3]).) The Hardy—Littlewood maximal operator is a typical example of a

weak—type (1,1) operator, bounded on

L00.

The first result in § 6 implies that an

operator satisfying an inequabty like (1.3) must in fact always be the composition of

n operators of weak—type (1,1) and bounded on L°°, at least "locally" (see §6 for

precise statements). In this section we also study certain monotonicity results for

the £ — inequalities. The motivation for this comes from the classical result of

Calderon to the effect that all interpolation spaces with respect to L and L^can be

obtained by using the K—functional and real interpolation. The main result in this

section states that if the operator norm (on the real interpolation spaces) blows up

sufficiently fast near the endpoints, then all extrapolation spaces can be obtained by

employing the £ — method. In §7 we return to the extremal interpolation methods

of Aronszajn— Gagliardo. Janson [27] has shown that the real as well as the

complex method can be described in terms of these. We show that the results from

§2 and §3 can be used to establish a general principle which produces precise

extrapolation results for all interpolation methods which can be described by the

Aronszajn—Gagliardo scheme.

Acknowledgements. We would like to take this opportunity to thank professors M.

Cwikel, E. M. Stein and G. Weiss for some helpful comments and for initiating our

collaboration.

Notations. A function f(t), t 0, is quasi-concave if f(t) is increasing and f(t)/t is

decreasing. If an operator T is bounded from A to B, we write T: A » B; if A