SUB-LAPLACIANS WIT H DRIF T 27
We set
A1 = A n (3fc, 1 + r2 - 3fe) x Br(e) and A2 = A \ Ax.
Let a G (0,1) to be determined later.
Case I : \A±\ a\A\.
Then \A2\ (1 - a)\A\ and hence
\A\
1
I^bl
| ( l , r
2
) x £
r
( e ) | - 1 - a |(1,1 +
r2)
x Br(e)\
1 |(0, 3fc] x ffr(e)l + l[l + r 2 - 3 / c , l + r2) x Br(e)\
- r ^ |(l,l + r 2 ) x £
r
( e ) |
6k 1
1 a
r2
'
If A ^ 0 then there is (£,x) G (l, l + r 2 ) x 5
r
( e ) such that u(t, x) 1 and so (3.3.2)
follows from (3.1.2).
Case II : \A±\ a\A\.
By the local Harnack inequality (2.2.1), there is c r$ and 8\ 0 such that
(3.4.1) * (o(r0i 2k, x), {(fc, x)}, (0, 2A: + ^r 2 ) x £
cro
(x) J 5X.
Let
Afc - {(t,x) : (£-/c,x) G i i } .
Then
4* C (4fc, 1 + r2 - 2fc) x Sr(e) and \Ak\ = |^il a\A\.
We set
ASl = U
( M ) e A f c
Q(r
0
,£,x)n(l,r
2
) x £
r
(e).
Then of course Ak C A^ and by (3.4.1)
(3.4.2) ^ ( 4 , A ( 0 , l + r
2
) x 4 ( e ) ) $i.
Let us fix r\ 1.
Then, by lemma 3.4.1, there is £2,£ G (0,1) and c r\ such that
(3.4.3) *
{(rj-2s\ rfs2)
x ^
s
( e ) , V, (0, r?V) X Bca(e)) 52
for all 5 1 and every measurable subset V C (0, s2) x J5s(x), satisfying
| F | £ | ( 0 , s
2
) x £
s
( e ) | .
We consider the set of balls
Q= {Q(s,t,x) C (l,r 2 ) x Br(e) : s l,s + d(e,x) r and I Q n ^ J £|Q|}
We have put the technical condition s + d(e, x) r in order not to make use of any
further properties of the control distance d(.,.).
We set
W = u
Q G Q
Q .
Then of course Ak C W.
Previous Page Next Page