A Caratheodory theorem for the bidisk via Hilbert space methods

TL;DR

This paper extends classical boundary behavior results of bounded analytic functions from the disk to the bidisk using Hilbert space techniques, establishing new boundary derivative properties and their representations.

Contribution

It introduces a Carathéodory-type theorem for the bidisk, relating boundary derivatives and angular gradients of bounded analytic functions via Hilbert space methods.

Findings

Boundary angular gradients are continuous nontangential limits.

Directional derivatives exist under finite liminf conditions.

Directional derivatives can be expressed through an associated Pick class function.

Abstract

If $\ph$ is an analytic function bounded by 1 on the bidisk $\D^2$ and $\tau\in\tb$ is a point at which $\ph$ has an angular gradient $\nabla\ph(\tau)$ then $\nabla\ph(\la) \to \nabla\ph(\tau)$ as $\la\to\tau$ nontangentially in $\D^2$. This is an analog for the bidisk of a classical theorem of Carath\'eodory for the disk. For $\ph$ as above, if $\tau\in\tb$ is such that the $\liminf$ of $(1-|\ph(\la)|)/(1-\|\la\|)$ as $\la\to\tau$ is finite then the directional derivative $D_{-\de}\ph(\tau)$ exists for all appropriate directions $\de\in\C^2$. Moreover, one can associate with $\ph$ and $\tau$ an analytic function $h$ in the Pick class such that the value of the directional derivative can be expressed in terms of $h$.