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
1 O O
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
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