4 AMREIN BOUTET DE MONVEL-BERTHIER GEORGESCU
for suitable weight functions a
s
o. : (a,b) 1, and such inequalities
have been used in the study of weighted Sobolev spaces (see [23] for a
general result and further references). For example, if c p : (a,b) M
is of class C , tpf(t) 0 for all t and p(t) -°°as t -* a, then one
has for all absolutely continuous functions u : (a,b) -* $ such that
lim inf |u(t) | = 0 as t - b:
(1.7) I I ^,)1/2ePu||L2((ajb).dt) S | |
(»')-1/2e«(»-X)u\\L2a&h);&ty
In an important publication [33 Agmon generalized the inequality
(1.4) corresponding to the "definite case" in the spirit of (1.6) (see
Theorem 1.5 of [3J)» Using this general inequality, Agmon was able to
obtain very precise information about the non-isotropic exponential decay
of the eigenfunctions associated to the eigenvalues below the bottom of
the essential spectrum of operators of the form -£.T _.3/3x.a.,(x)3/3x, +V
in L (]R ).A striking feature of the generalized Hardy type inequalities
corresponding to (1.4) is that in the term on the right-hand side the sim-
ple operator A has been replaced by the full differential operator the
spectrum of which one tries to study ( for instance by A - V(x) or by an
operator A of the form (1.1)). The reason for this is as follows: if the
weight that replaces pT grows for example exponentially, then the boot-
strap procedure used by Agmon in the proof of Theorem 3-3 in [2] will not
work any more, unless one imposes very stringent conditions on the per-
turbation V.
The main purpose of our present paper is to generalize in a similar
spirit the inequality (1.3) corresponding to the "indefinite case". Since
we are interested only in the behaviour at infinity of the functions, we
consider operators in an exterior domain, i.e. in an open neighbourhood
fi of infinity. As in AgmonTs work [315 the full differential operator
will appear in our inequalities. When applied to partial differential
operators A of the form (1.1), our general results will typically lead
to inequalities of the following type:
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