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.