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-