New Thoughts on Besov Spaces by Jaak Duke University. Peetre

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We want however to say a little bit more about how big the number d must be . First we state the following lemma that will do us great service in what follows too . Lemma l . Let f E: s• with supp f K ( r ) = { I � I � r } Then holds n (­pl • (12) (13) holds Assume that supp f (13 I R (r ) r < I � I <2 r} • Then ) If r > l we can as well substitute J for I . Remark . I f f E: S . to say that supp f is compact is by the Paley-Wiener theorem the same as to say that f is an entire function of exponential type .

S q= O f i s a polynomial ) . The same phenomena Bp we encoun te red alre ady in Chap . 2 in connection with the e xample with Lip s . Al so I s f cannot be de fined for al l f E S . Indeed we would l ike to have I s f ([;) =/ E; / s 1 ([;) as in the But the fact that [; = 0 i s a s ingularity i f s 0 case of Js • < i s an obstacle . The remedy for all thi s is to do the calcul us modulo polynomial s , of degree < d , where d is a suitab le n umber . Let us 52 give a complete analysis of the situation .

F s = ( 1 -8 ) s + 8 s o l (0 < 8 < 1) I t i s also possib le t o show that Bpsq i f s (0 < 8 < 1) . 30 The proo fs wil l be found in Chap . 3 . Th is example i s e ssentially a simpler special 0 case of the pre ce ding one . Let c be the space o f continuous Example 4. bounded functions in I I al l and let C lR = ( --oo , oo ) , with the norm sup I a ( x ) I co 1 be the space o f function s whose first derivative 0 exists and belongs to c , with I I al l c sup I a ' (x) I 1 ( He re we are cheating a l ittle bit , since this become s a true norm only after identi fication of functions which diffe r by a constant.