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