20 AMREIN BOUTET DE MONVEL-BERTHIER GEORGESCU
(2.18) 2Im(Pv,tM'ipfv) - v||Pv||2 - i||tM'ip'vH 2 ,
(2.19) 2Im(P1^av,t(SP+R+itpTS)v) = 2Im(5/"1P1^av,t(SP+R+i^S)^v)
- vq || ^ P ^ v H
2
- i | | t(SP+R + i c S ) $ v | |
2
,
\,-2 -2
b e c a u s e ip ^ qip . O n t h e o t h e r hand , by ( 2 . 2 ) :
2
( 2 . 2 0 ) v| | i p " 1 P
1 - a
v | | 2 ( P v
3
2 v ( l
+
t c p M P v ) 4 - ( v
3
v ( ^ " a ) ( ^ + ^ - ) v ) .
Inserting (2 .17)-(2.20 ) into (2.14) and taking into account that
q k 1 , we obtain for all v e P :
i | | tipL(p)v||
2
( v
5
[ ( 2 - a ) P
2
+ Q
a
l v ) +
( 2 . 2 1 ) + ( P v , [ 4 ( M - v q ) t c p ' + ( 2 - a ) ( M - l ) - t M
,
- 5 v q ] P v ) +
+ ( v , [ M ( a ( p '
2
+ 2 t c p V ) + t M ' ( l - ) p
f 2
- v ( l - a )
2
q ( - ^ + ^ - ) ] v ) +
v
t
d z
+ 2Im(Pv,tMp"v) - 1 1 t(SP+R+icS)I|/v||2 .
(ii) Let us denote by J and J the square bracket in the second
and third term on the r.h.s. of (2.21) respectively . By using the assump-
tions made on the function r(v) , we get for v€ P(r(v)) :
* 4(l-vq-|l-M|g(H))tcp^-|2-a|!M-l|B(H)-|tM'|B(H)-5vq 2
4(l-vq-v)t(pt-|2-a|v-v-5vq Z 4 (l-2vq)tcp* -8vq-| a | v,
J
Q
acp'
2
+ 2t(p
,
cp"-v|a|cp
| 2
-v2cp
T
| tcp" |-2vtp
T 2
- v ( l - a )
2
q ( 3 t ~
2
+ ^ p '
2
)
acp' 2 + 2tcp , cp t, -vcp' 2 ( | a | + 3 + | ( l - a ) 2 q ) - v | t c p " | 2 - v ( l - a ) 2 - 3 q v 2
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