## Cauchy integral equalities and applications

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- by Boo Rim Choe PDF
- Trans. Amer. Math. Soc.
**315**(1989), 337-352 Request permission

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

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## Additional Information

- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**315**(1989), 337-352 - MSC: Primary 32A35
- DOI: https://doi.org/10.1090/S0002-9947-1989-0935531-9
- MathSciNet review: 935531