Cauchy integral equalities and applications

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 $\pi (0) = 0$.