24 GEORGIOS K. ALEXOPOULOS
COROLLARY
3.1.2. For all a 1 there is c a, S 0 and m G N such that
for all r 1 and all u 0 satisfying
(--+L]u = Q in (0,(l +
a2)r2)
x £
cr
(e)
we /im;e
(3.1.2) inf{u;(l + a-
2
r
2
, l +
a2r2)
x Bar(e)} 6u(l,e) r"
m
.
Moreover, if for some 1 R r,
inf{u;(0,#
2
) xBR(e)} 1
then
(R\
~rn
(3.1.3) inf {u; (1 +
a~2r2,1
+
a2r2)
x £
a r
(e)} 5 ( - ] .
3.2. Proof of the first growth lemma. The following lemma is an imme-
diate consequence of the propositions (1.6.2) and (1.6.3).
LEMMA
3.2.1. For all a 0 there is S 0 and £ G (0,1) such that
(3.2.1) [ pt(x,y)dy5
J A
for all r 1, (t,x) G
(a~2r2,a2r2)
x Bar(e) and A C Br(e) satisfying
\A\Z\Br(e)\.
Let us denote by z(t) the diffusion process generated by the sub-Laplacian L.
Note that the transition function P(t,x,A) = P[z(£ -f s) G -A|z(s) = x] of z(£)
satisfies
P{t,x,A) = / pt(x,y)dy.
J
A
Let us also denote by Px, x G G the probabilities attached to the diffusion z(t) and
satisfying
P*[z(0) =x] = l and P ^ t ) GA]= f Pt(x,y)dy.
J A
For every ball Br(x), we consider the stopping time
r* = inf{t : *(t) i Br(x)}.
LEMMA
3.2.2. For all e 0 t/iere is a constant c c(e) 0 such that
(3.2.2) Px[Tcrr2]e, r 1.
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