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) .