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

.