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)
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