HARDY TYPE INEQUALITIES 21
a(p,2+2t(pV-vtpt2( |a|+4q+a2q)-v|ttp"|2-6v3(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||\pv||p , 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|| ^ | | v||2 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|+|^"n|-t-Icp*nt | )v|| .
Proof : From (2.11) and (2.9)we get
(2.23) L =
L((p)-icpTS-2iM(p!P-(Mtp!
)
!+MtpT2-3P-R
and
(2.24) (L((p)-icp
,
s)n =
nL(cp)-i((pfn+nT )S-2infMP-(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 ) ) :
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