By K. Barbey, H. König

ISBN-10: 3540082522

ISBN-13: 9783540082521

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St(u,v). 1+tct(u,v) 3) L i k e w i s e we o b t a i n from I) ct(u,v)+St(w,v) If(u) t for f6A with llfll~1 and If(w) l~t, 1+Ct(U,v)St(w,v) c t (u,v) +c t (w,v) ct(u,v)+St(w,v) or St(w,u) ct(w,u) ~< I +C t (U,V) C t (W,V) l+ct(u,v)St(w,v) and hence ct(w,u) ~ c t ( u , v ) + c t ( w , v ) . 5. 3. fll~1. I+Ct(U,V) If(v) I Fix now k 6 A with K: = Re k > O. For each s>O then e - S K 6 A and hence l(1-e-SK(u)) s For s%O it follows > 1-Ct(U'V) l+ct(u,v)e-SK(v) that I I e -sK(v) ~( - ). 3. Let f6A w i t h sume that f(v) = l+f If(u) l~t, and as- I-If I 2 k := 6 A with K := Re k > 0 .

2. 3 for t=O to y i e l d IIf~<1 and c o m b i n e the e l e m e n t a r y ll the a s s e r t i o n . a-b! 2 It f o l l o w s V a,b6~ with lal~1 . that 4 If(u)-f(v) I 4+ If(u)-f(v)l 2 < So(U'V) and h e n c e the a s s e r t i o n . a := f(v) and d e f i n e even for all f6A w i t h To p r o v e ~ let f6A w i t h tlflI 21b 1 = so t h a t £ A satisfies h(u) lh(u)-h(v) I ~ G(u,v). 8 R E T U R N c C(DUS) f~flS. i) So(U,Z) easiest then the functions functions F := Re and for z6S.

G6ReC(X) :g~f}cReC(X) with the p r i m i t i v e on a real v e c t o r while is the s e c o n d m e m b e r is QED. a useful modification lemma w h i c h w i l l be a p p l i e d p r o o f of the a b s t r a c t F . a n d M . R i e s z theorem in the next section. 5 LEMMA: a sequence Assume that O ~ f n 6 U S C ( X ) of f u n c t i o n s Vn6A with with IVnI~1 %(fn)~O. T h e n there and % ( l l - V n l ) ~ O exists such that IIVnfnlI~o. Proof: Choose u n6A with numbers Re u n => f and tn>O w i t h ~(Un) of $.

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Abstract Analytic Function Theory And Hardy Algebras by K. Barbey, H. König

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