EMBEDDING AND MULTIPLIER THEOREMS FOR HP(Rn)
certain function spaces K and K which, as far as we know, were first
considered by Herz [H]. Suppose that
laoo , 0aa , 0bc» .
Definition (a) K ' consists of a l l functions f ^ L, (R \ J0()
for which the norm or quasi-norm
d-2) P!!.a!b=i Z (J " IflV/^^!1^
is finite.
n
, a,b a *a,b
(b) K = L H K , with norm or quasi-norm
a a
a a
Recall that A, = \2 |x| 2 j . The usual modifications are made
when a = Q O or b = » .
The K spaces appear in [H] , where they are denoted K , . F l e t t [F]
ab
gave a characterization of the Herz spaces which is easily seen to be
equivalent to (1.2). They have been previously applied in H theory by
Johnson [JO 2].
Elementary considerations show that the following inclusion relations
are valid.
(1.4) (3a = Ka'bcKP'C ,
a a
(1.5) b c = » Ka'bcKa'C ,
a a '
(1.6) a a = K^b c KY'b , where
y
= a- n(— - —)
1
l
a2 al al a2
Relations (1.5) and (1.6) are valid for the K spaces, but (1.4) is
not.
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