HARDY TYPE INEQUALITIES 21
a(p,2+2t(pVvtpt2( a+4q+a2q)vttp"26v3(l+a2)q
. •
We next wish to dominate the last term in(2.16) bythe other ones .
Of course , the hypotheses weput until now are not sufficient for this.
We have a natural norm on V namely   .   « , and we could assume that
the last term of (2.16) is dominated byv\pvp , But then we haveto
know how to estimate this new term by the positive terms in(2.16) . This
is the purpose ofthe following two lemmas .
Lemma 2.6 : Assume that   t(SP+R)v ^   v2 for v e P(r(l)) . Then the
re is a constant c , depending only on M , such that for all v G P(r(l))
and all cp,n e
C2(I)
:
(2.22)   nv 
2
^ c  
n
L(cp)v + c  
((p»n+nT)Sv
+ c(n + h
f

+ lPfn)Pv
+
+ c   C n + nT  + n"i+l^T2n+^"ntIcp*nt  )v .
Proof : From (2.11) and (2.9)we get
(2.23) L =
L((p)icpTS2iM(p!P(Mtp!
)
!+MtpT23PR
and
(2.24) (L((p)icp
,
s)n =
nL(cp)i((pfn+nT )S2infMP(Mn*)f+2M(pfnf
.
Using the first identity , w e obtain that
 v
2
I! v 
+
  Pv
+
l V
1 1
M l v  + Pv + (L(cp)icp'S)v +
+ 2  Mcp'Pv + (M'cp'+M(cp"cp'2))v + (SP+R)v .
Now, sinc e r ( l ) ^ a+1 ^ 2 , we have fo r v G P ( r ( l ) ) :