@MISC{Polydiscs71aremark, author = {On Polydiscs and Morisuke Hasumi}, title = {A Remark on Extreme Points in If}, year = {1971} }

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Abstract

The purpose of this short note is to give a simple remark on a recent paper by Yabuta [4]. Let U be the open unit disc and T the unit circumference which is the topological boundary of U. A function f in the Hardy class H1(Un) on the unit polydisc Un of dimension n is called extremal if f_??_0 and f/•af•ais an extreme point of the unit ball of H1(Un). For the case n=1, a well-known result of deLeeuw and Rudin [1] states that f is extremal if and only if it is an outer function in the sense of Beurling. For the case n•†2, Rudin [3] has defined outer functions as a direct generalization of Beurling's. However, it is not true that every extremal function in H1(Un)(n•†2) is an outer function in Rudin's sense. This was shown by Yabuta [4]. He observed that the function z1+z2 gives such an example. Here we give a proof to this fact. We should mention that we have a proof which is different from Yabuta's proof as well as the present one. 1. Let (X, u) be a probability measure space and let E be a subspace of L1(u). The following proposition was used by deLeeuw and Rudin [1] for the usual Hardy class H1(U) and by Gamelin [2] for the Hardy class H1 connected with a certain uniform algebra. PROPOSITION 1. Let f be nonzero element of E. Then f is not extremal in E if and only if there exists a bounded real-valued measurable function g on X such that fg e E, fg _??_0 and •çs_??_g du=0, where S(f) denotes the support of the function f (determined up to a set of measure zero).