8 BAERNSTEIN AND SAWYER
Then f satisfies (1*1) and belongs to K for every
ae [1,»] and qp . Also f £ L but f £ LP . Hence f g HP .
Here is another way in which Theorem la is sharp. Let e (k) , kl ,
be a given non-increasing sequence with lim e (k) = 0 . Then there is
an f £ L satisfying (1.1), f = 0 in |x| 1 , with
SJ" |f|)P2kn(1-p) «(k). ,
1
\
but f ^ H . To obtain such an f , select a subsequence Sc Z fc
which T, (k) o o . The desired function is given by
k£S
f(x) = 2"nk/p fQ (x2'k) , x e \ , k e S ,
= 0 , otherwise .
There exists also a compactly supported example of this type. Define
f(x) = 2~nk/p fQ (x2"k) , x e \ , k e - S ,
= 0
9
otherwise.
Then
'z IS |f|)P2nk(1-p)e(-k)».
Take cpG Cn with compact support and J*c p = 1 ? and define cp*f via
(1.9) . From (1.11) it follows that lim (cp*f) (x) = f(x) , x ^ 0 .
t- 0
Since f £ LP , it follows that sup J cp*f| £ LP , and hence f £ HP .
t0
In §5 we construct examples f, which show that when N = n(— - 1)
1 1 p
K, P is not contained in H _ , no matter how small q is . To
state a replacement for Theorem la we introduce subspaces
Y(p) z Kn(P l) ? . Assume that N = n(- - 1) .
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