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
}
.
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