22 AMREIN BOUTET DE MONVEL-BERTHIER GEORGESCU
(SP+R)v|| £ |||t(SP+R)v|| S |||v||2
which clearly implies that
( 2 . 2 5 ) || v | |
2
2 | | (L(p)-ip'S)v|| +c| | (1+|P' | ) P v | | +c || (l+|p» I 2+|P" I M l
To arrive at (2.22) , it suffices to replace v by nv in (2.25) and to
apply (2.24) .
Lemma 2.7 : Assume that | | t(SP+R)v|| £ | | v||2 for v P(r(l)) , and let
q £ 1 . Then there is a constant c such that , if i p C (I) and
t p : (a+lj00) - * 3R is an increasing function of class C satisfying
| tip" | + |tip"'| q(l+p») , one has for all v E P(r(l)) :
(2.26) |ftvH? i S c(|| $L(p)v|| HI CS?v|| + | | (l+|tp'|)5;Pv|| +
(l+|cp» |2+ | co" | )$v|| ) ,
^ .
O/
^ ^
^-1
where i p =
(l+tcpT)-1/2
and c = cp'+^
Proof : We first observe that
$' _ l^j-t^pV $1 _ ir2+t$^»
+
l^'+tip"^
^ 1 + tcp' if / d 1+tcp1 l+ttp!
This implies that , on (a + l,°°) :
(2.27) | $ . | S § ^ , i f e ^ - r 1 ) ! S ^
+
| 3 l 2 SQ 2 ,
(2.28) |$"| = K ^ - 1 ) ^ + * ' 2 r 2 * l S 252*
The inequality (2.26) follows from (2.22) by taking n = i p and using (2.27)
and (2.28) .
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