EMBEDDING AND MULTIPLIER THEOREMS FOR
HP(Rn)
Definition f £ Y(p) if
(1-12) ||f||Y(p) * Z (L |f|)P2kn(1-p)(1+|k|)»
k =
-a
K
and
(1.13)
J*f(x)xP
dx = 0 , for every |p| = N .
The integrals in (1.13) are absolutely convergent, by (1.8) . For
f £ Y(p) the embeddings (1.9) and (1.10) of f in S! agree.
THEOREM lb. Suppose that N = n( 1) and p 1 . Then Y(p) d
HP
_and
! \
P
*
c
»
f
"Y(P)
The theorem applies in particular when f £ Y(p) n L and f satisfies
(1.1) . It's easy to show that K,'qc Y(p) when Ctn( 1) . so this
1
P
theorem generalizes TW for 0pl , N = n( 1) . The examples f--
and f in §5 show that the weights 1+ |k| in (1,12) cannot be replaced by
essentially smaller ones.
Theorem lb becomes false when p = 1 . In place of Y(l) we have to
consider a smaller space Y
Definition f £ Y if f £ L' , Jf = 0 , and
a-
1
*)
iifii
* = ;i
f
(
x
i (
L
+
i o
§
+
1 &
1
1 +
g
+
ix| dx
Y
llf
I I 1
THEOREM l c . Y*
c
H
1
and ||f||
±
C ||f|| * .
H Y
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