32
GEORGIOS K. ALEXOPOULOS
Since
An
= 0 and Q G .4° = Q°, there is i0 G {1,2,..., n} such that Q G
^A*0-1
and Q ^
Aio.
Let 5 =radius(Q). Then, by (3.5.5) and (3.5.6) we have Q n
Qio
^ 0
and 5 2ri0 and hence Q G Qz°*. This proves (3.5.7) and the lemma follows.
PROO F OF LEMMA
3.4.1. Let us first observe that by (1.2) there is a constant
c 1 such that
(3-5.8) I '£,(«'*'*)' c
y
' c~ \Q(s,t,x)\ -
for all s 1.
We have
|W| _\Ak\ + \W\Ak\ _i+\W\A5l\
\Ak\ \Ak\ \Ak\
\W\AsA \W\AsA
(3.5.9)
\w\ - |uf=1g»|
i
+
^ : ^ ? ' i
+
| w
\ ^
Since
ci:r=iiQ*i cEr=1iQ'i
we have
(3.5.10) |Q* \ ^ | =|Q*| - |Q f)A5l\ |Q«| - (1 + e)£|Q«| = [1 - (1 + e)£] \Q%
Combining (3.5.9) and (3.5.10) we have that
\w\
... , E ^ J i - q + e^lQ*! _ ! , i - ( i + e)$
- " * " ^Y™ I/Oi _ l i "
i4k i -
cEr=iiQ4l
which proves the lemma.
3.6. Proof of lemma 3.4.2. We set
^ - ^ n R x {x} and W2 = W2 H i x {x}
for x G Br(e).
It is enough to prove that
(3-6.1) \wl\ -^f^lV
We shall need the following lemma from [KS]:
LEMMA 3.6.1 (cf. [KS lemma 2.2 on p. 157]). Let K 1, let
A = {(ti, t2) C R : -c o ti t2 oo}
and /e£ # a function g : A-+ A satisfying
1. \g(I)\ K\I\, I G A and
2. g(h)Qg{I2) , ifhQh-
Previous Page Next Page