Birth of a Theorem: A Mathematical Adventure (23 page)

Let
with
0
<
λ
*
<
λ
₀
.
Let
be a continuous function of t
≥
0,
valued in
such that

(7.22)

where c
₀
≥
0
, m
>
1
,
and
K
₀
(
t,
τ)
, K
₁
(
t,
τ)
are nonnegative kernels. Let
Then

(i) Assume
γ
>
1
and K
₁
=
cK
^(
α
),
γ
for some c
>
0,
where
K
^(
α
),
γ
is defined by

and
appears in Proposition 7.1. Then there are positive constants C and
χ
,
depending only on
γ,
λ
*,
λ
_0
,
κ
_,
c
_0
,
C
_W
,
m, uniform as
γ
→
1
, such that if

(7.23)

and

(7.24)

then for any