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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