SUB-LAPLACIANS WITH DRIFT
31
If
v(0)2
t - 1, then we set
(t -
v{a)2,
t) x Bv{a) (c(a)), for
v{a)2
t-l
(1,1 +
v(a)2)
x B
v ( a )
(c(a)), for
v(a)2
t - 1.
We observe that
d(c(a), y) d(c{a),x) -f d(x, ?/) d(c(cr), x) -f s
T - a + 5 = v(a)
and hence (r,y) G
£a
for all cr G [0,T].
Also by (3.5.3) and (3.5.4) there is a0 G [0, T] such that
£\Dao\ \DaonASl\ (l + e^D*0]
and the lemma follows.
If Q Q(s,t,x) = (t
\s2,t
-f
\s2)
x Bs(x), then we shall denote
OK
o^
Q* = Q*(s,t,x) = (t-
-s2,t
+ y
5
2
) x £
5 s
(x).
LEMMA 3.5.2. There is a finite sequence of balls
Q1, Q2, Q3,...,
Q
n
s^c/i £/m£
1.
Qz
G Q°, 1 i n,
2.
Ql
n (^ = 0, i ^ j , 1 i, j n and
3. I f C U ^ .
PROOF. Let
A0
= and set
s0 = = sup{radius(Q) '• Q e A0}
where radius (Q(s, t, x)) = 5.
We chose as Q1 any ball in satisfying
(3.5.5) n = radius^ 1 ) y .
Assume that we have chosen the balls
Q1, Q2,..., Qfc-
Then in order to chose
the ball
Qk+l
let us set
^ = { Q G Q ° : Q n Q
2
= 0, z = l,2,...,fc}.
If
^4fc
= 0 then we stop there. If not, then we set
Sk = sup{radius(Q) : Q G Ak}
and we chose as Qk+l any ball in *4fc satisfying
(3.5.6) r
f c + 1
=radius(Q / c + 1 ) y .
Note that since the chosen balls are disjoint and of radius 1, we will eventually
have
An
0 and this proccess will stop. We obtain in this way a finite sequence
Q\Q
2
,...,Q
n
of balls satisfying (1) and (2).
It remains to prove that (3) is also satisfied, i.e. that W C
U^=1Q2*.
For this,
it is enough to prove that if Q is any ball Q G then there is i$ G {1,2, ...,n}
such that
(3.5.7) QQQ***.
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