Approach regions for domains in $\CC^2$ of finite type

TL;DR

This paper demonstrates that certain approach regions are optimal for boundary behavior of bounded analytic functions in finite type domains in e2b2, showing no Fatou theorem holds for broader regions.

Contribution

It establishes the optimality of specific approach regions for boundary limits in finite type domains, extending Fatou theorem considerations.

Findings

Approach regions studied are optimal for boundary behavior.

No Fatou theorem exists for broader complex tangential approach regions.

Boundary behavior is constrained by the geometry of finite type domains.

Abstract

Recall the Fatou theorem for the unit disc in $\CC$. Consider a domain in $\CC^2$ of finite type. In this paper we will show that the approach regions studied by Nagel, Stein, Wainger and Neff are the best possible ones for the boundary behavior of bounded analytic functions, and there is no Fatou theorem for complex tangentially broader approach regions.