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-