HARDY TYPE INEQUALITIES 29
* vc( II T ^ r l l
2
+ || (!•*• )P
I
|^
r
||
+
|| (l*p
+
4 ^ 1 )v|| ) S
£ vc(n
w
H
2
+
n
p v
ii
+
" 1 ^ 1
+ (1+q)|1 ( l +
*
, ) v
i i
}
and
|| tS
2
v| | = || (l+vp')tS
2
(l+cp')"" 1 v|| £
S (l
+
q ) || ptS2(l+cp' )~ 1 v|| * v d + q H I ^ r H g
These two inequalities prove (2.44) .
(ii) To prove (2.45), we use Lemma 2.6. Under the present assumptions
on the function r\9 we may deduce from (2.22) that
(2.46) | | nv||2 c | | nL(p)v||+ cp||(i+p')nSv|| +
+ c ( i + p ) || (i+cp»)nPv|| + c ( i + p ) 11 [ ( i + t p ' ) 2 + |cp"| ] n v | | .
On the other hand , if we replace in (2.44) the function v by
(l+cpf)nv , we get that
|| (i+p»)nSv|| ^ || ts(i+p»)nv|| ^ vc|| nvj|
2
+
+ vc|| (i+tp» )nPv-i[(i+cpf )n 3' v || + vc|| (l+tp1 )2nv||
^ vc|| nv||
2
+ vc|| (i+cp')nPv|| + vc|| (|cp" | n + (
i+cpf)
|
nf
| )v|| +
+ vc || (i+p» )2nv|| .
(2.45) is now obtained by inserting this last inequality into (2.46) ,
by using the hypotheses made on cp" and on r\ , and by taking v small
enough .
Previous Page Next Page