24 AMREIN BOUTET DE MONVEL-BERTHIER GEORGESCU
(u,L u) = (Pu,MPu) + (u,Qu) .
In particular (U,LQU) ^ (u,Qu) . Hence | | . | | is a norm on V and
II -II ! * II -II
H 1

If u V and n C1(I)is real , then
(nu,LQnu) = (Pnu,MPriu) + (n2u,Qu) =
= (Pn2u5MPu) - i(n'nu,MPu) - i(Pnu,Mn!u) + (n2u,Qu) =
= (n2u,Lou) - 2iRe(nfu,nMPu) + | | M1/2nfu| | 2 .
By using (2.1), we obtain from this identity that
|(nu,Lonu)| | | n2u|| -I I LOU|| + | | M||L » .(||nPu||2+2| |
n
fu||2) .
Consequently
|| nu||
1
* /I+7|| nu||
H
i + | (nu,LQnu) | 1 / 2
/1+T ( || nu|| +|| n f u | | +|| nPu|| ) +
+ | | n 2 u | | 1 / 2 II
L
O
LI||
1 / 2 + U M l ^ - C H nPixll + / 2 | | n ' u | | ) . -
The interest of (2.30) resides in the following estimate :
Lemma 2.9 : If | | . | | is an admissible norm , then under the assumptions
and with the notation of Lemma 2.7 , one also has
(2.32) | | "&h\\
1
S CjCII *v||2+|| (l+|p-|)3fPv||+||(l+k'l^k"! )*v|| ).
( If C denotes the constant in (2.30) , then one may take for example
C1 = 6q2C) .
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