Averaged mixed Julia-Fatou type theory with applications to spectral foliation

TL;DR

This paper introduces an averaged Julia-Fatou type theory to analyze boundary regularity of complex functions, with applications to spectral theory, extending classical boundary behavior theorems.

Contribution

It develops a new averaged approach to Julia-Fatou theorems using spectral foliation and Nevanlinna measures, bridging complex analysis and spectral theory.

Findings

Established boundary regularity criteria via averaged Julia-Fatou quotients.

Connected boundary behavior with spectral properties through Nevanlinna measures.

Extended classical theorems to a broader class of functions with spectral applications.

Abstract

Classically, theorems of Fatou and Julia describe the boundary regularity of functions in one complex variable. The former says that a complex analytic function on the disk has non-tangential boundary values almost everywhere, and the latter describes when a function takes an extreme value at a boundary point and is differentiable there non-tangentially. We describe a class of intermediate theorems in terms of averaged Julia-Fatou quotients. Boundary regularity is related to integrability of certain quantities against a special measure, the so-called Nevanlinna measure. Applications are given to spectral theory.