HARDY TYPE INEQUALITIES
25
Proof : We set n = \ and u = tyv in (2.30) . By using (2.29) and
l^'| |q$ |q (see (2.27)), we obtain (2.32).
We are now in a position to complete the estimate (2.16) . For this
it is necessary to put some conditions on S and R at infinity . As before,
we set i p = (1+tcp')"1/2 and $ = (l+t!pf)"1/2 .
Proposition 2.10 : Assume that S and R are "small at infinity" in the
following sense : there is an admissible norm | | .|| on D , an increasing
1 \,
function p : I -* [1,°°) and a decomposition of S on V into S = S1 + S
p
,
with S. : V - H 1-ultralocal , such that for each v £ (0,1] , for all
u £ fl(r(v)) and all v e P(r(v)):
(2.33) || t S
l U
| | S v | | u | |
1
(2.34) || ptS
2
v|| 2 v|| v||
2
(2.35) || t(SP+R)v|| v|| v||
2
Let q , q 1 be two real numbers and a £ JR . Then there are constants
v £ (0,1) and c £ (I,00) such that for all real functions t p , t p having
the properties
(a) c p £ C2(I) , cp » 0 , p'(t) qp(t) for t a+1 ,
(b) c p £ C3((a+1,«)), 0 ^ cpf qtp! and | tcp" | +1tcp"f | q(l+cp'),
for all v £ (0,v ) and all v £ P(r(v)) , the following is true :
^||(tip4)L(cp)v||2 k (v,[(2-a)P2+Qa]v)+2Im(Pv,tMr4)"v)+
(2.36) + (Pv,[4(l-vc)t(p» - vc(l+cp t 2 $ 2 )]Pv) +
+
(v,[a(pt2+2tcplip"-vc(l+cpf2+|tcp"|2+cp»i|$2+cp"25/2)]v)
.
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