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)
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