Birth of a Theorem: A Mathematical Adventure (24 page)

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Authors: Cédric Villani

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BOOK: Birth of a Theorem: A Mathematical Adventure
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(7.25)

 

where

(7.26)

 

(ii) Assume
for some
, where
appears in Proposition 7.1; then there is a numeric constant
Γ
>
0
such that whenever

 

one has, with the same notation as in (i),

(7.27)

 

where

 

Proof of Theorem 7.7.
We only treat (i), since the reasoning for (ii) is rather similar; and we only establish the conclusion as an
a priori
estimate, skipping the continuity/approximation argument needed to turn it into a rigorous estimate. Then the proof is done in three steps.

 

Step 1:
Crude pointwise bounds.
From (7.22) we have

(7.28)

 

We note that for any
and
t

0,

 

where (here as below)
C
stands for a numeric constant which may change from line to line. Assuming

K
0
(
t,
τ
)
d
τ

1/2, we deduce from (7.28)

 

and by Gronwall’s lemma

(7.29)

 

where

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