BAERNSTEIN AND SAWYER
The main case of interest to us occurs when a = 1 , QC = n(— - 1) ,
b = p . Note t h a t
l*llp
i - z (J ,
f ) )
p
2
^ C I - P )
9
^M n
(-p - 1 ) , P k = -co Ak
from which, fo r P 1 , follow s
( i - 7 ) ||f|| G | | f | |
x
L kn(P
-
1 }
'
p
and
n(p " 1)
(i.8) jifooi
ixi
^ c Hf||
L
^n(p - l),p
When pl functions in K.. P "
P
need not be locally integrable
near the origin, and thus may not define distributions in S' in the usual
way. However, if for f £ K^P"
1);,P,
cp€ S , and N = [n(^ - 1)] we
define
(1.9) cp,f =;(cp-PNcp)f , when Nn(±- 1) ,
(1.10) cp,f = /(¥-PN. 1cp)f, when N = n(i - 1) 1
where P cp denotes the N'th Taylor polynomial of cp at x = 0 , then
(1.8) shows that f defines a continuous linear functional on S1 , and
J, _
hence for 0pl we may regard K P as being continuously
embedded in S* via the definitions (1.9) and (1.10) .
Note that if f £ K*? p ''Pc L and f satisfies the vanishing
moment condition (1.1) , then
Jfcp = JfCcp-P^) , cp S
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