HARDY TYPE INEQUALITIES 23

For later reference we note the following consequences of (2.27):

( 2 . 2 9 ) | ? | S q ( l + | P ' | , | C ' | S q

2

( l

+

| p " | ) .

We next introduce a notion that allows us to estimate the term cSyv

in a natural way

Definition : Let V be the 1-semilocal closure of V . A norm | | . | | on P

is called admissible if , for some finite constant C and all u € V and

all n £ C (I) , one has :

(2-30) | | nu||

a

s CCII u||2 + | | nPu||+ | | (|n|2 + |V|)u||) .

Clearly we can ( and we shall ) assume C chosen so that | | u|| is

less than C|| u||

?

for all u € V .We shall be interested in taking | | . ||

as large as possible . It is clear that || . | | „1 is an admissible norm .

Hence we may assume , without loss of generality , that | | . | | ^ | | . j|wl

H

(because one may replace || . | | by | | . | | + | | . | | 1 ) .

Our estimates are valid without any direct assumption on the negative

part of Q . If , however , Q is only weakly unbounded below , there is a

standard choice of an admissible norm . This is the content of the next

lemma .

Lemma 2.8 : Assume that Q is the restriction to V of a 1-ultralocal opera-

tor , denoted also by Q , defined en V with values in H and satisfying

2

^

Q ^ -K(1+P ) as form on V for some K ^ 0 . Assume also that M is bounded

on I . Then

2 1/2

(2.3D ||u||

±

= [(1+K)|| u|| Hl + (u,L0u)3

is an admissible norm on V .

Proof : The last term on the r.h.s. of (2.31) is defined as