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)

.