- UPPER BOUND FOR THE LOCAL ENTROPY fsvdμ 205
Using the definition (22.44) of hand the inequality (<// (s))^2 :=:; 10¢ (s),
we compute- f IVhl^2 - f (¢' (s (x, o)))^2 1
t JM -h-H dμg(O) = t JM ¢ (s (x, 0)) b2H dμg(O)
(22.84)lOc-^2 1
<--=--- b2 10A2·It remains to estimate the integral JM -h log h H dμ 9 (o). We have by
definition
h log h = 0 when h = 0 or h = 1and we have -hlogh::; e-^1 ::; 1 for all h 2:: 0. Recall that if h (x, ·) E (0, 1),
then by (22.49),Thus- r h log h H dμg(O) ::; r H dμg(O)
JM J B 9 (o) (xo,20e:oA)-B 9 (o) (xo,lOe:oA)
(22.85) ::; 1 - { H dμ 9 (o).
J B 9 (o) (xo,IOe:oA)
The result now follows from the claim that(22.86) { H dμ 9 (o) 2:: { h (x, 0) H dμ 9 (o) (x) 2:: 1-
9
A 2 ,
JBg(o)(xo,lOe:oA) JM 10where h : M x [O, c-^2 ] ---+ [O, 1] is defined, similarly to (22.44), as
-( ) _ (dg(t) (x, xo) + 200nVt)
h x, t - ¢ - '
bwhere b ~ 5c-oA. Note that supp h( ·, 0) c B 9 (o) (xo, lOc-oA). Indeed, (22.82)
follows from combining (22.83), (22.84), (22.85), and (22.86).
STEP 3. Completion of the proof of part (2) of Lemma 22.13.
Now it remains to prove (22.86). Assuming that A 2:: 67n, by (22.47) we
have- 10-
Dh (x, t) ::; b 2 h(x, t)
on M x [o, c-^2 ]. We compute
! JM H h dμ 9 (t) = JM ( H Oh - h D* H) dμg(t)
= JM HDh dμg(t)
10 r -
::; b 2 JMH hdμg(t)'