(T ,°°), T ^ -°°,and c is independent of T and v. This method for proving
absence of eigenvalues of A in certain intervals [A , A ] seems more power-
ful than other methods, since it allows in particular more singular coef-
ficients for the operator A.
The present paper is concerned solely with proving Hardy type inequali-
ties and with the specification of certain classes of ordinary and par-
tial differential operators for which these inequalities can be used.
Applications to more general partial differential operators and the deri-
vation of Carleman type inequalities will be the topic of subsequent com-
munications. We stress that Hardy type inequalities apply to a general
class of functions u, in particular that u need not be an eigenfunction
of the differential operator under consideration; in many cases, u is not
2 . .
even required to belong to L at infinity, although some local regularity
conditions will have to be satisfied. On the other hand, we are interes-
ted not only in allowing general behaviour at infinity of the coefficients
of the operator (1.1), but also ( and especially ) in allowing them to
be locally singular. Our final result will be valid for a., locally Lips-
chitz and for b. and c measurable functions such that the associated mul-
1 2
tiplication operators are well defined on H. . „(ft) and H, (Q) respective-
loc loc
ly with values in H, (ft).
C. Organization of the paper. In Section 2 we present the abstract
framework in which we shall work and prove some preliminary inequalities.
The introduction of a slightly unusual formalism (compared to other work on
differential operators with operator-valued coefficients) is justified by
the fact that it will permit us to obtain estimates for partial differential
operators of the form (1.1) with locally singular coefficients of the type
described above. Of special importance is the identity (2.14) which we call
the fundamental identity, since it is the basis of all our Hardy type ine-
qualities. Most of the preliminary inequalities derived in this section
are consequences of this identity under some special assumptions on the
coefficients of the differential operator L, and all of these inequali-
ties are " local " in the sense that they are only shown to hold for
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