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