28 AMREIN BOUTET DE MONVEL-BERTHIER GEORGESCU
(2.41) 2Im(Pv,tMp"v) = (v,-(tMptf)fv) = (v,[-tMcp"! -(M+tMf )p"]v)£
(v, [-tMcpIM-(l+2v) |p"| ]v) (v,[-tM(p , M -v(l+2v) |tcp"|]v) ,
(2.42) ^ V
= I
^ r ^ f ^tcp ,
( 2 . 4 3 ) | t c p " |
2
2 q
2
( l + c p '
2
)
Finally , let us show how to estimate second order derivatives
Lemma 2.12 : Let q 2 1 and assume that S and R are small at infinity in
the sense of Proposition 2.10 . Then there is a constant c such that for
all real t p C (I) satisfying
tp' 0 , cp'(t) ^ qp(t) and |cp"(t)| S q(l+tp'(t)) for t a+1 ,
for each v (0,1] and all v e P(r(v)) :
(2.44) | | tSv|| s vc||^ H
2
+ vc|| Pv||+ vc||(l+cp«)v|| .
Let p be another number in [ 1,°°) . Then there are numbers c and r , depen-
2
ding only on q and p , such that for all ( p as above , for all n G C (I)
with n 0 , |nf(t)| + |n"(t)| ^ pn(t) for t a+1 , and for all v P(r):
( 2 . 4 5 ) || nv| |
2
s c || nL(p)v|| + c | | (i+p')nPv|| + c | | ( l + c p ' )
2
n v | | .
Proof : (i) Let v (0,1] and v P(r(v)) . Then , by (2.33) , (2.30)
and (2.34) :
| | tS^H S v|| v||1 = v||(i+p.)-Z-f||1 S
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