HARDY TYPE INEQUALITIES 17

More explicitly , for any v G V :

(2.8) (v,Q v) E (l-a)(v,Qv) + 2Re(v',tQv) .

2

Let us note the following commutation relation

5

valid for n E C (I)

on the domain Pn :

loc

(2.9) Lr i = nL - 2±nfMP - inTS - (Mn')1 .

2

To each real function c p G C (I) we associate the following operators

from £ into tf, :

loc loc

(2.10) L (cp) E e^L e_cp = LQ + 2icp'MP + (Mcp')T - M^P'2 ,

( 2 . 1 1 ) L(tp) E e^Le" ^ = L +

2itp!MP

+ icp'S +

(IVftp1

)

f

- M p

f 2

.

2

Lemma 2.3 : If h , h £ C (I) are real and v G V9 then :

2Im((h1P+iho)v,Lo((p)v) = (PvjM(h^-2ho)-MTh^b^cp'M]Pv)+

(2.12) +(v,[-(h^+2ho)Q-h1Q']v)+2Im(Pv:)h:L(MTcp'+Mcp,?)v)+

+(v,[M(h'+2h +h.f-)cp'2+h1MT(p'2+2b'Mcp'+(h,M)f ]v) .

1 O

iQo

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Proof: By using (2.6), this identity can be obtained by a simple inte-

gration by parts . We shall , however , give the details . Of course we

can assume Q = 0 , and then we have :

2Im((h1P+iho)v,PMPv) = 2Im(Pv,h PMPv)-2Re(v,h PMPv) =

= -i(Pv,(h1PM-MPh1)Pv) - (v,(hoPMP+PMPh )v)=

= -i(Pv,[-ih M'+iMh']Pv) - (v,[2Ph MP+in'MP-iPMh1]v) =

- L

1 O O O