HARDY TYPE INEQUALITIES

3

Part (a) follows from Theorem 3.2 in [2],and part (b) is easily

proved by making a Fourier transformation . We observe that the two cases

covered by Theorem 1.1 are quite different . We say that (a) is the

"indefinite case", because for strictly positive X the operator A + A

does not have a definite sign in any neighbourhood of infinity , while

(b) will be called the "definite case" , since A+A A 0 in all ]Rn

for strictly negative X . It is easy to show (see the proof of Theorem

3.1 in [2]) that (a) (or (b)) implies discreteness of the positive ( or

negative) point spectrum of the operator - A + V(x) in L (!I R ) if V is

a real-valued function such that |x|V (or V respectively) tends to zero

at infinity in some weak sense . Also (a) and (b) imply that the eigen-

functions corresponding to non-zero eigenvalues decay at infinity more

rapidly than any inverse power of |x| .

In the above context it is interesting to note that the situation

is completely different for X - 0 : if a,b and T are fixed real numbers,

then it is impossible to have an inequality of the form

(1.5) I I

PTu||

S O(T)(||

pT+aAu||

+

||pbu||

)

for all u for which the right-hand side is finite and for all x £ x

This may be explained by the fact that there are functions that are har-

monic in the domain {x € B n | |x| 1} and that decay at infinity more

rapidly than a given inverse power of |x| . For example , take u € C (]R )

such that u(x) = |x|2~n for |x| * 1 ; then u G L2(]Rn) if n 5 and

Au(x) = 0 if |x| 1 , but | | pTu| | 2(]Rn, = ° ° for each x ^ (n-4)/2

However the inequality (1.5) holds for each u such that pTu € L (IRn) if

a £ 2 ; we refer to [25] for a detailed study of the case X - 0 .

B. Generalizations of the above Hardy type inequalities . Hardy's

original inequality (1.2) has been generalized by various authors to the

form

(1.6) | | aou||L 2

( ( b);dt)

S const.||a1(D-X)u||L 2

( ( b) dt)