2

AMREIN BOUTET DE MONVEL-BERTHIER GEORGESCU

behaviour from that of Lu . The example that we consider typical for this

kind of property is the classical inequality of Hardy ([151,Theorem 330)

which , with the notation D = -id/dt , states that

(1.2)

||tTul|

9

S

(x+1/2)"1

J)

tT+1(D-X)u||

?

IT(IR+;dt) L (]R+;dt)

for all x -1/2 , all X € B and all absolutely continuous u : 3R -* • $

+

the derivative of which is square-integrable on each interval (0,b),

b °°3 and which satisfy lim inf |u(t)| = 0 as t -» • ». Remark that the

general solution of the equation (D-X)v = f has the form v(t) = v (t)+

a.exp(iXt) , where a is an arbitrary complex number , and the purpose

of the condition lim inf |u(t)| = 0 as t - » in (1.2)is to select the

unique solution that tends to zero at infinity . For comparison with

Theorem 1.1 below , it is also of interest to observe that this condi-

2-1/4 2

tion is implied by the stronger one that (1+t ) * u € L (B ;dt).

The importance of the above type of inequalities in the study of

eigenvalues and eigenvectors of partial differential operators in L

(]Rn)

was first emphasized by Agmon [2] . The natural framework for general

inequalities of the type (1.2)is described in Appendix B of his paper [2].

As an application of these ideas to second order partial differential

operators we point out the following theorem , in which

Hs(3Rn)

denotes

2 1/2

the usual Sobolev space of order s, p = p(x) = (1+x ) and

A = ln . d2/dx2

Theorem 1.1 : (a) For each T -1/2 and each e 0 there is a constant

c = c(x,e) such that for all X G

[e.e'1]

and all u with

p"1/2u

€

L2(]Rn);

T l/

(1.3) II P T u||

H

2

( I R

n

)

* o M

pT+

+ 1(A+X)u||

L

2

( ] R

n

(b) For each T £ ]R and each e 0 ther e i s a constan t c = C ( T , E ) such

tha t fo r a l l X € ( p with dist(X,3R

+

) ^ e and a l l u € Sf (3Rn) :

( 1 ' 4 ) II P T u||

H

2

(

^ n

}

c || pT(A

+

X)u||

L

2

( B

n

}

.