Birth of a Theorem: A Mathematical Adventure (26 page)

so

Plugging this into (7.35) and using (7.22) again, we obtain

(7.36)

We separate this (by Minkowski’s inequality) into various contributions which we estimate separately. First, of course,

(7.37)

Next, for any
T
≥
1, by Step 1 and

(7.38)

Invoking Jensen and Fubini, we also have

(7.39)

(Basically we copied the proof of Young’s inequality.) Similarly,

(7.40)

The last term is also split, this time according to
τ
≤
T
or
τ
>
T
:

(7.41)

and

(7.42)

Gathering estimates (7.37) to (7.42), we deduce from (7.36)

(7.43)

where

Using Propositions 7.1 (case
γ
>
1) and 7.5, as well as assumptions (7.23) and (7.24), we see that
a
≤
1/2 for
χ
small enough and
T
satisfying (7.26). Then from (7.43) it follows that

Step 3:
Refined pointwise bounds.
Let us use (7.22) a third time, now for
t
≥
T
: