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ple operator A has been replaced by the full differential operator the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: