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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Cauchy integral equalities and applications

Author: Boo Rim Choe
Journal: Trans. Amer. Math. Soc. 315 (1989), 337-352
MSC: Primary 32A35
MathSciNet review: 935531
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Abstract: We study bounded holomorphic functions $ \pi $ on the unit ball $ {B_n}$ of $ {\mathbb{C}^n}$ satisfying the following so-called Cauchy integral equalities:

\begin{displaymath}\begin{array}{*{20}{c}} {C[{\pi ^{m + 1}}\bar \pi ] = {\gamma _m}{\pi ^m}} & {(m = 0,1,2, \ldots)} \\ \end{array} \end{displaymath}

for some sequence $ {\gamma _m}$ depending on $ \pi $. Among the applications are the Ahern-Rudin problem concerning the composition property of holomorphic functions on $ {B_n}$, a projection theorem about the orthogonal projection of $ {H^2}({B_n})$ onto the closed subspace generated by holomorphic polynomials in $ \pi $, and some new information about the inner functions. In particular, it is shown that if we interpret $ {\text{BMOA}}({B_n})$ as the dual of $ {H^1}({B_n})$, then the map $ g \to g \circ \pi $ is a linear isometry of $ {\text{BMOA}}({B_1})$ into $ {\text{BMOA}}({B_n})$ for every inner function $ \pi $ on $ {B_n}$ such that $ \pi (0) = 0$.

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Keywords: Cauchy Integral Equalities, the Ahern-Rudin problem, projection
Article copyright: © Copyright 1989 American Mathematical Society