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) . •