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) .