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
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