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 + |
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