EMBEDDING AND MULTIPLIER THEOREMS FOR HP(Rn)
so the distribution defined by f in the usual way coincides with the one
defined by (1.9) and (1.10)
The Taibleson-Weiss (p,q.,N,e) molecules are jus t the functions in
K y^ for some OC n( —) which satisf y (1.1) . By (1-6) , each of
q P q
these molecules belongs to a space K, for some a n ( 1) .
Our sharp version of the Taibleson-Weiss theorem requires statement in
three separate cases., according as Nn( 1) , N = n( 1) 1 , or
p = 1 . The theorems are proved in §4 .
THEOREM la. Suppose that Nn( 1) . Then K ^ P ~ ^ P c HP and
11*1 1 pSC||f|| ,
The theorem applies in particula r to functions in K P which
1
l
satisfy (1.1). By (1.4) we have K^P " 1 ^P
= D
K^q when an(- - 1),
so this theorem generalizes TW for 0pl , N ^ n(— - 1) . Note also
that by stating the theorem for K we have dispensed with the global
integrability hypothesis.
Theorem la is sharp in the sense that no larger K or K space is
embedded in H . Choose f L°° , f £ 0 , such that supp f c:An
and,, denoting surface area by da^
(1-11) J " f0(x)xp do(x) = 0
Ixl = r
for every |p| N and r £ (0,=) . Define f by
f(x) = (2"nk k_1)1/p fQ(x2"k), x \ , k0,
= 0 , x 1
Previous Page Next Page