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.