- UPPER BOUND FOR THE LOCAL ENTROPY f B v dμ, 203
Now provided A is chosen so that A 2: 1 + lO(n - l)Q (t -f), then (22.74)
implies that (22.73) holds on [t' - O", t] for some O" > 0. Since t -t ~ ~Q-^1 ,
by choosing A such that(22. 75) A 2: 1+5(n - l)a,
we conclude that (22.74) holds on the whole interval [i, t].
In particular, taking t' = i in (22.74), we obtain
d 9 (i) (x, xo) ~ dg(f) (x, xo) + lO(n - l)Q~ (t -f)
~ (2A + l)c:o + lO(n - l)A~·
Thus (22.70) will follow from showing that
(22. 76) (2A + l)c:o + lO(n - l)A~ + ~ ~ b - 200nVt.
Now since b = lOc:oA, A 2: 67n, and a ~ 1, we have
(22.77) b 2: (2A + 536n) c:o 2: (2A + 1 + lO(n - l)A + 1 + 200n) c:o.
This implies (22.76) since ~ ~ c: and Vt~ c:. The proof of (22.63) is
complete.
3.4. Proof of part (2) of Lemma 22.13.
We now prove (22.64).
STEP 1. A sufficient inequality for fI and J to satisfy (22.64).
By (22.69), we have/31 (1- l0~ 2 ) ~ JM(-v)hdμg(O)
(22.78) =-JM(t(R+2!:ij-JVJJ
2
)+!-n)fidμ 9 (o)·On the other hand,(22.79)
JM (2!:if - JV fJ2
) fI dμ 9 (o)=JM (2!:i (J + logh) - JV (J + logh) J^2 ) fI dμ 9 (o)
=JM (2/:if + 2/:i: - 2JVloghJ^2 ) fI dμ 9 (o)
- JM (\vf\
2
+ 2\V], Vlogh) + JVloghJ2
) fI dμg(O)·