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 =;(cpPNcp)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 = JfCcpP^) , cp € S