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
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