HARDY TYPE INEQUALITIES 5

(1.8) ||(l+«p»)m+1-se%||H s

( n )

S K||t(l+(p')m(l+tcp')"1/2e(P(A-A)u||

L

2

( Q )

.

Here c p : (0,°°) - B is a (smooth) weight function which typically tends to

+°° at infinity, and in (1.8) c p and cpf mean multiplication by cp(|x|) and

cpT ( | x | ) respectively (cp1 is the derivative of cp); X is a positive constant,

m a non-negative integer, s € [0,2], K a constant depending on X, m and

cp, and the inequality will hold for all u G H

(JRn)

that satisfy u(x) = 0

for |x| r, where r is a finite constant depending on X, m and c p and

such that ft contains the set { x £ 1 ||x| ^ r}. Precise conditions on

the coefficients of A and on the weight functions c p implying the validity

of (1.8) will be specified in the last part of this introduction. Here

we shall add some general comments on this type of inequality.

Hardy type inequalities are particularly useful for obtaining upper

bounds ( in the sense of L ) for eigenfunctions of differential operators.

For example, in the context of the inequality (1.8), assume that u is an

eigenfunction of A associated to the eigenvalue X, i.e. (A-X)u = 0 . Fix

an admissible weight function c p and an integer m, which determines the

constants K and r. Then choose a function r\ € C

(]Rn)

such that n(x) = 0

if |x| r and n(x) = 1 if |x| r+1. By replacing u by nu in (1.8), one

obtains in this way that (1+cp1 )uelpu G tfS for each admissible cp, each

u ^ 0 and each s G [0,2] .

A further interesting application is the proof of the non-existence

of non-trivial eigenfunctions of A for X in certain intervals [X , X.],

by applying the well known Carleman method. For this one has to know, in

addition to the upper bounds just mentioned, also an inequality of Carle-

man type, i.e. an inequality of the form

||eT(pv|l Hs S c ||8eT,p(A-A)v||L2 ,

2

where 6 means multiplication by a suitable smooth function 9(x), v G H (ft)

has compact support in ft, x is a parameter varying over an interval