Birth of a Theorem: A Mathematical Adventure (25 page)

Step 2:
L
²
bound.
This is the step where the smallness assumption (7.23) will be the most important. For all
we define

(7.30)

(7.31)

(7.32)

and we extend all these functions by 0 for negative values of
t.
Taking the Fourier transform in the time variable yields
since condition
(L)
implies
we deduce
i.e.,

(7.33)

Plugging (7.33) into (7.32), we deduce

(7.34)

Then

(7.35)

(Note: We bounded
by
which seems very crude; but the decay of
as a function of
k
will save us.) Next, we note that