1. EMBEDDING THEOREMS
I f f £
HP
the n | f ( £ ) | C | ^ |
n (
p "
1 )
[ TW, p . 105], s o we expec t t h a t
f w i l l s a t i s f y th e v a n i s h i n g moment c o n d i t i o n
(1.1) Jf(x)x P dx = 0 , 0 |P | N ,
whenever the integrals make sense. Recall that N isthe greatest integer
less than orequal to n(— - 1) and integrals without limits are over all
of R . Suppose that f does satisfy the necessary cancellation (1.1) .
What kind ofsize condition on f will guarantee that f £ H ?
One such condition has been found byTaibleson and Weiss [ TW, Theorem
2.9] , who followed uponearlier work byCoifman and Weiss [CW] . Suppose
that 0plqoo and that ql when p = 1 . Taibleson and Weiss
define a (P^q^N, e) molecule centered atthe origin tobea function
f £ L (R) which satisfies (1-1) and also
aq
JlfCx)!4 | x P dx=o ,
where OC and e are relatedby
1 K A
a = n( ) - n(- - 1) +
c
p q p
and i t is assumed that
e
n( 1) . Thus OC n( ) . Theorem ( 2 . 9 )
P P q
of TWasserts that (p^q^N, e) molecules belong to H .
We are going toprove a stronger embedding theorem which canbe
regarded asa critical endpointcase of the Taibleson-Weiss theorem. To
formulate this theorem, and some others inthis paper, wemust introduce
4
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