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By H. Bass

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The Bloch space functions in D such that sup (1-|z\) If' (z)l < z<*D ‘ The smallest such C $S is the class of all analytic C < Oo . is the Bloch norm Definition 2. The analytic function class BMOA in D, if aup T ^ T T T T T |B(z0 ,r)| z0 ^ D and belongs to the ("area" l) \ |f(z) )B (Zo>r) where supremum is taken over all disks with f || f l| dxdy < c < B(zOJr) = r < 1-lzQl. The smallest such z: C \z-z0\^ r ^ is the BMOA norm llf Hbmoa* We denote also the norm in the space Now we can formulate our main result.

Let f be analytic in the functional born by this function and D and let <%> be H p e U } ) * . Since A^ c_ L^ we have by the Hahn-Banach theorem that there exists a function from L~(D) with \\ ^ = so that for all g(z) b(z) b g e A^ (V-\z\2)* dxdy. 2 ) we get g(w)( 1 -\w|2)ei ( | ^ — dxdy) dudv and this means that <(z) = ^ Q^(z,w)b(w) (1 - \w\2)^ dudv. (b)— >(c). 3)« It suffices to prove that dQ^Czjw) (1 — |z \) Id-iwi2/' dudv < Co dz 'D We have iL Q (z,w) = (1 -zw)7T5 dz and hence ( w = p e1 ^ ) D — Q^(z,w)| (1 -|wl^)^ dudv < dz I 1 ,% ( 1 - p2 ) pdpdv^ < C 0 ■'-Tv.

Disk 0 1, where wfc = (1-|zk l) s o o we get that also D by the mapping w = 2z-1. The is analytic and bounded in |w\<1 , 2 zk- 1 . Then since z ^ s lie in an angle (l-z,l = O o . Now note that k 2 ,ii-wk \ = a = c~. But as it is easy to see the sequence angle and we get that Remembering now that lytic function h(w) -jwj^ also lies in an = O© • ^w^ is the zero set of a bounded ana­ we immediately get that h(w)l=0 . Theorem is proved. 2). It presents a necessary condition on A^ zero sets. We will see later in Chapter 4 that it is also suffici­ ent and hence will solve the problem of characterization of A^ zero sets completely.

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