HARDY TYPE INEQUALITIES 19

Remark : If c p is of class C we can simplify further , because :

(2.15) 2Im(Pv,tMcp"v) = (v,-(tMplf)fv) .

We shall now derive some simple inequalities which will be useful

later on . We say that a function f : I -*- B is increasing if f * £ 0 , and

2

we shall systematically use the following notation : if c p £ C (I) is real

-1/2 1

and increasing , we setty(t)= (l+tcpf(t)) ; we then have i p € C (I)

and 0 i p S 1 . The inequalities will involve a parameter q , which is

always supposed to be greater than 1 . The constants c, v , r that appear

in these inequalities will depend on c p only through q , if the contrary

is not explicitly stated . To be more clear, we shall say , for example ,

"with c depending only on q" ( but of course c will depend also on L , a,

etc.). The constants , even if denoted by the same letters , are different

in different places . The essential idea of our proof is already contained

in the following easy consequence of the fundamental identity (2.14) :

2 ^

Lemma 2.5 : Let c p € C (I) be real and increasing . Let c p : (a+1,00) • B

be of class C , increasing and such that cpf qcpf on (a+1,00), for some

q 1 . Set $ = (l+tcp»)"1/2 and \ = cp'+i^"1 . Then , for all v € (0,1],

all a € B and all v € P(r(v)) , one has :

i | | tipL(cp)v|| 2 (v,[(2-a)P 2 +Q

a

]v ) + (Pv,[4(l-2vq)tcp»-v(8+|a|)q]Pv) +

(2.16) + (v,[acp'

2

+2tcp

?

cp"-vq(2+|a| )

2

cp»

2

-v|tcp"|

2

-6v

3

(l+a

2

)q]v) +

+ 2Im(Pv,tMp!fv)-i|| t(SP+R+icS)^v|| 2 .

Proof : (i) We estimate several terms in (2.14) as follows , using (2.1)

with v G (0,1] :

(2.17) 2Im(P1_av,tL(cp)v) E 2Im(i|;"1P1_av,ti|;L(cp)v)