26
AMREIN' BOUTET DE MONVEL-BERTHIER GEORGESCU
Proo f : ( i ) Let v G ( 0 , 1 ) and v G P ( r ( v ) ) . N o t i c e t h a t , by ( 2 . 2 9 ) ,
| c | ^ q(l+cp
f
) £ q ( l + q ) p . By u s i n g t h i s i n e q u a l i t y , ( 2 . 3 3 ) , (2.3
1
*) and
( 2 . 3 2 ) , one o b t a i n s t h a t
CS^v|| || t c S ? v | | || t S ^ v H + q ( l + q ) | | p t S ^ v j
( 2 . 3 7 ) S v| | C$v||
1
+ v q ( l + q ) | | h\\
2
*
vc || $ v | |
2
+ vc || (l+tp ! )if/Pv|| + vc || (l+(p T ^+|cp" | )5v|
This implies together with Lemma 2.7 that there are constants v 0 and
c such that for all v (0,v ) and all v G P(r(v)) :
( 2 . 3 8 ) || $ v | | c ( | | ?L(cp)v|| + || (1+cp- )$Pv| | + || ( l + c p '
2
+
| c p " | ) ^ v | | )
( i i ) We now u s e ( 2 . 3 5 ) , ( 2 . 3 7 ) and ( 2 . 3 8 ) t o e s t i m a t e t h e l a s t
ter m i n ( 2 . 1 6 ) , w i t h v G ( 0 , v ) , a s f o l l o w s :
i | | t(SP
+
R
+
i^S)^v|| 2 | | | t ( S P + R ) $ v | | 2 + | | | t^v|| 2
2
£ 2v| | 3?v||
2
+6vc
2
( || $ v | |
2
+ | | (l+p')fav|| +|| (l+cp'
2
+|(p" | )3»v||
2
) ^
2 2
£ v c ( | | 3;L(p)v|| 2 + || (l+ip')3Pv|| 2 + I! (l+cp» 2 +|cp"|)^v| | 2 ) .
Upon inserting this last inequality into (2.16) , one arrives at (2.36) .
The inequality (2.36) will be used in several contexts . It clearly
"\
shows the role of the auxiliary function i p : it serves to moderate the
h
growth of tp' which cannot be dominated by the other terms , unless
(2-a)P2
+ Q gives a strong positive contribution . If cp1 is bounded , we
can take i p = 1 ,i.e. c p = 0 , and the arguments are slightly simpler :
for example , there is no need for the norm | | . | | , one can take p = 1
and S = S . The term 2Im(Pv,tMcpT,v) can be estimated in two ways , depen-
3 2
ding on whether ( p belongs to C or only to C . The second case is con-
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