SUB-LAPLACIANS WIT H DRIF T 29

So there is a ball

Q2 = (t + s2, t + a2rj2s2) x £

s

(x) G Q

such that

2 2 2 ^ 2

a r\ s wyr .

Now, by corollary 3.1.2, for all a\ 0, there is c ai, £4 0 and m G N such that

for all R crrys,

# f (* +

R2,

t + (1 +

a?)i?2)

x BaiR{x),

g2,(t,

* + (1 +

a?)#2)

x 5CjR(x)

(3.4.9) ^(T)"

^"" (£)

7

,m/2

The lemma follows from (3.4.9) above, by taking a\ large enough and by replacing

R by an appropriate multiple of r.

Case lib :

\W2

\ (1,1 +

r2)

x J5r(e)| u\A\.

Let us first observe that since W C VF1 we have

(3.4.10) |W| \Wl\.

The following lemma is the analog of the lemma 2.3 of [ KS p. 158].

LEMMA 3.4.2.

(3.4.11)

\wx\-^—AW2\2.2^

azr]z

— 1

The proof of the above lemma will be given later.

Combining (3.4.11) with (3.4.10) and (3.4.7) we have

\w2\ a2r]l

n

1\w\

^ ^ ( i + ^iHi|

r2^2 _

X

a2r\2

We set

Then we have

a2rj2

(l + | ) a | A | .

A0 = W2 n (1,1 + r2) xBr(e).

\A0\=\W2f) (1,1+r2)xBr(e)\

= \W2\-\W2\(l,l + r2)xBr{e)\

\W2\ - w\A\ (1 + j)a\A\ - u\A\

[(l + | ) a -

W

] | ^ |.

It follows that if we chose a G (0,1) so that

(1 + | ) « 1 + |