SUB-LAPLACIANS WITH DRIFT
33
Then, for all B C A,
| U J

B 0 ( / ) | K | U /
6
S J | .
PROOF.
The proof is taken from [KS] and is given for reasons of completeness.
The set Li/egi" is open and hence it can be represented as the union Di^gl =
UnIn of certain nonintersecting intervals In. So we have
| UIeB 9{I)\ | Un UlCIn9(I)\ I
Un
9(In)\ Yl
\9(Jn)\
n
^ « | /
n
| = « ^ | U/GS7|
n n
which proves the lemma.
Now (3.6.1) follows from the above lemma, by taking
a2rj2
a2r)2 - 1
K
=
jY
I
,
B =
{R x
{x}
H
Q2
:
Q2
G
Q2}
and by setting
0((*l,*2) ) = ( * 2 - « ( * 2 - t l ) , * 2 ) .
3.7. The Oscillation of the heat functions. Proposition 1.6.5 is a conse-
quence of the following:
PROPOSITION
3.7.1. Let b,(3 and c be as in lemma 3.3.1. Then there is a G
(0,1) such that
(3.7.1) Osc U , (/3r2, (0 + b2)r2) x Bbr{e)\ a Osc U , (0, (0 + 62)r2) x £
cr
(e) J
/or all r 1 and all functions u satisfying
(^+L\U = 0
in (0,((3 +
b2)r2)xBcr(e).
PROOF.
Let us fix a function u satisfying
(^+L\U
= 0
i
n
(0,(/3 + & 2 )r 2 )x£
c r
(e).
and let
mi = inf {u; (0, (0 +
b2)r2)
x £
c r
(e)} , Mx = sup {u; (0, (0 +
62)r2)
x J5cr(e)}
ra2 = inf {u;
(/?r2,
(0 +
62)r2)
x £
6r
(e)} ,
M2 = sup {u;
(/?r2,
(/? +
62)r2)
x Bbr(e)} .
Let also
4 = \(t,x) G (l,r 2 ) x £
r
( e ) :w(t,x) - ( r a i + M i )
Case I : |A| ||(1,1 +
r2)
x J5r(e)|.
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