This paper is concerned with certain estimates on the asymptotic beha-
viour of functions u defined on an interval (a,°°) with values in a Hilbert
space H. More precisely, if L is a second order ordinary differential ope-
rator the coefficients of which are operators acting in H, we wish to obtain
inequalities allowing one to get information about the behaviour of a func-
tion u in a neighbourhood of infinity from the asymptotic behaviour of the
function Lu. For reasons that will be explained below, these inequalities
will be called Hardy type inequalities.
The principal application we have in mind is the study of the asympto-
tic behaviour of eigenfunctions of second order elliptic partial differen-
tial operators, i.e. of operators A of the form
(1.1) (Au)Cx) = [ - IJ f^a.^x)!- - iXj=1 b.U)^
By distinguishing the radial variable from the angular ones, the operator
A defined by (1.1) falls naturally into the more general class of ordinary
second order differential operators with operator-valued coefficients. In
our opinion the use of this latter framework leads to a simplification and
to better transparency of the estimates ( compare for example with  or
), allows a unified treatment of various cases of interest and gives
more possibilities of applications even to partial differential operators.
A. Description of some Hardy type inequalities. The property that we
try to establish is the following: if, in a certain weak sense, a function
u does not grow at infinity, then one can infer its asymptotic
Received by the Editors December 15, 1986. Research partially supported by
C.N.R.S. and the Swiss National Science Foundation .