HARDY TYPE INEQUALITIES

9

a-v(x) = 6., . The general case needs a special coordinate transformation

JK

JK

and will be presented in a forthcoming separate publication .

Our first version of the inequality (1.8) was a mixed Hardy-Carleman

type inequality valid for logarithmic weights only , i.e. p(t) = ylogt

with y G ]R (see [5] , Section 5 ) . The conditions that we had to impose

on c. and cp were of the same order of generality as here , c. being

treated by the bootstrap argument of Agmon. However, the fact that ( p was of

a very restricted form was inconvenient in applications: in order to

apply the Carleman method for proving absence of positive eigenvalues of

-A + c. + c0 , we had to require that c„ maps H. (Bn) into L, (Bn) , a

1 2 ' 1 loc loc '

condition that is much stronger than what one had expected (see also Re-

mark 4.3 ) •

The generalization of the above-mentioned Hardy type inequalities

with logarithmic weights to those given here was obtained by using the

idea , contained in the series of papers by Froese et al [10] - [12] , of

approximating a weight function by a sequence of more slowly growing

functions , together with an appropriate iteration procedure ( see the

discussion at the beginning of Section 3 and under point (y) of the proof

of Theorem 3.4 ) . Some of these results were announced in [4] , and a

preliminary version appeared as a preprint [51 .

As already pointed out , an important role is played in our estimates

by the fundamental identity (2.14) which is a generalization of an identity

obtained in the work of Eidus [8] . More or less similar identities have

been used in various research on absence of positive eigenvalues of Schro-

dinger operators ; we refer especially to the papers by Roze [28] , Mochi-

zuki [26] and the references given in [26] to other Japanese research .

The identities (2.2) of [11] and (3.1) of [10] are also of a similar

nature .

The derivation of our initial Hardy-Carleman type inequalities ( cf .

above) was based on an identity of the type (2.14) for functions v that