4 B. JAWERTH AND M. MILMAN

f

t rt

(1.1) (Tf)*(s) ds f*(s) ds, t 0.

J

0 0

By integrating both sides and using that / / • ds— = / log(r/s) • ds, we see that

(1.1) implies

rt rt

(1.2) log(t/s)(Tf)*(s) ds log(t/s)f*(s) ds, t 0.

Jo Jo

We claim that (1.2) also implies (1.1) (at least for f e L n L00) so-that they are

(essentially) equivalent. To see this we consider

lim [ log(t/s)(Tf)*(s)ds / [ log(t/s)f*(s)ds.

By assumption this limit, if exists, must be less than or equal to 1. Both the

denominator and numerator tend to oo as t - oo (since they are integrals with

respect to ds/s of increasing functions) and, hence, by I'Hopitals rule (sic), we have

that the limit equals

lim f W ( s ) d s / [V(s)ds = ||Tf||Li/||f||Li.

t-»ao J o J 0

This shows that T is bounded on L with norm 1. Similarly we find that T is

bounded on L® with norm 1. As we pointed out before, this is equivalent to (1.1).

The inequality (1.1) thus says that T is bounded on L and on L® while (1.2)

implies that T is bounded on LfogL and on L®; the fact that these are equivalent

means that we may cancel logarithms. If we want to find an abstract analog of this,

or if we just have a different number of logarithms on the left—hand side and

right—hand side, we need to find a substitute for differentiation and I'Hopitals rule.

The substitute we use are the KJ—inequalities mentioned above. These are also

studied in this section. In § 4 we indicate how to generalize the results of previous

sections to quasi—Banach spaces and point out some connections with so called

semi—modulars. In §5 we consider applications and examples involving extrapolation

theory, including those outlined above. The strong maximal theorem of Jessen,

Marcinkiewicz and Zygmund [32] states that the strong maximal operator Ms on

IRn