36
GEORGIOS K. ALEXOPOULOS
By (3.8.5) we have
Osc (u, Dc/cr(s0,2/0)) - Osc (u, D^Cfcr(s0, 2/0))
_ _ _/^m/-—km
~ a a S
and hence there is (ti,xi) G D^kr(so,yo) such that
^i^i)---ic
m
r
f e r o
-
a ad
If ^C f c + l r 1 then we stop there. If not, then we consider a ball
I£Cfc+ir(si,2/i) C D^kr(s0,y0)
such that
(ti,xi) G J0^Cfc+ir(3i,yi).
Again by (3.8.4)
inf{u;D
icfc+lr
(
Sl
,2/i)} lc™C ( f c + 1 ) m
and hence
Osc (u,£c i *+!,.(*!,!&)) - - --CmCkm- -CmC{k+l)m.
s a a 0 0
By (3.8.5)
Osc (w, DCfc+ir(si, yi)) - Osc (u, Z?^Cfc+r(5i, 2/1))
a 2 a 2 5 a 5
and hence there is (£2,^2) £
^cfc+1^(Sl'^1) s u c
^
^^a^
u(t2,x2) A _
A T ^ ' C " ^
-
IIcmr(fe+1)m.
If ^C
f e + 2 r
1 then we stop there. If not we continue and if v G N is such that
(3.8.9) ±C,k+vr 1 ^k+v+1r,
then we obtain, in this way, finite sequences of points
(£0,x0),(£i,a?!),...,(£,,,£,,)
a n d (5o,2/o),
(si,2/i),.--, (^-1,2/^-1)
such that
( ^ M ' ^ M )
G
^C
f c + / z
r(
5
M-l'2/M-l) '
^ i
C
f c + M + i
r
( 5 M ' ^ ) ^ A : f c + / L i r ( 5 M - i ^ - i )
and
(3.8.10) u(^,a;M) \ - -^?
m
T
f c m
- - ^ C T ^
a^ a^ 0 a^
x
o
*~ *• rim/-—(fc+2)ra _ _ _ _rvm/-—(fc+/i,— l)m
a^~2
5 a ( 5
for all fi = 1,2, ...,JA
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