SUB-LAPLACIANS WIT H DRIF T
and that by (3.8.2)
(3.8.4) inf{u; D£r{t, x)}
\e-m
o
for every ball Der(t,x) C (l,r
2
) x Br(e).
Furthermore, by (3.7.2), there is C 1 and a £ (0,1) such that
(3.8.5) Osc (u,Ds(t,x)) aOsc(u,DCs{t,x))
for all DCs(t,x) C
(0,r2)
x Br(e).
Let us fix ( G (a, 1) satisfying
(3.8.6) C m a C m + 1 .
Let also fix k G N such that
(3.8.7) C * ^ \
We set
\ _ Z^logC/logC i _ ^ m / - f c m _ _ _ l _ _
We shall prove that
(3.8.8) supL;(^r 2 ,r 2 ) x £
r / 2
( e ) j A.
Case I : ±(kr 1.
Then r
CC~fc
and hence, by (3.8.3),
sup{u;(l,r 2 ) x Br(e)} ^CmC~krn
which proves (3.8.8).
Case II :
±(kr
1.
Let us assume the contrary, i.e. that
sup | u; (-r 2 , r2) x Br/2(e) | A.
We shall reach a contradiction.
Let (t0,xo) £ (|r
2
,r
2
) x Br/2(e) such that
u(t0,x0) A.
We consider a ball Di^
r
(s
0
,yo ) £ (f^ 2 ^ 2 ) x Br/2{e) such that
(t0,x0) G ££Cfcr(so,2/o)-
Then by (3.8.4)
inf{u;^
c f c r
(
5 o
,yo)
-5CmCk7n
and hence
Osc (u, DMkr(s0, yo)) A - ic
m
C"
f e m
.
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