Boundary measures of holomorphic functions on the imaginary domain

TL;DR

The paper extends the Herglotz-Nevanlinna integral framework to boundary measures for holomorphic functions on the imaginary domain, linking boundary limits to representing measures via two local coordinates (on $\mathbb{R}$ and under inversion). It develops boundary distributions and inversion formulas that connect limits along circles and lines, and establishes Vladimirov-type estimates to ensure well-defined boundary behavior. It then analyzes how Möbius (fractional-linear) transformations affect representing measures, proving that $\varphi(A.z)$ preserves representability and transforms its measure by a explicit push-forward $\lambda^A$. Through a comprehensive set of examples, the work characterizes when boundary measures exist (e.g., for powers $z^p$ with $-1<\Re p<1$) and provides explicit boundary densities, illustrating the practical reach of the approach for generalized Pick/Herglotz representations.

Abstract

In connection with the Herglotz-Nevanlinna integral representation of so-called Pick functions, we introduce the notion of boundary measure of holomorphic functions on the imaginary domain and elucidate some of basic properties.

Figures (3)

• Figure 1: Arc to Line
• Figure 2: $p$ region of $z^p$
• Figure 3: $p$ region of $z^p(\log z)^{\pm 1}$

Theorems & Definitions (38)

• Example 1.1
• Theorem 2.1
• Lemma 4.1
• proof
• Corollary 4.2
• Lemma 4.3
• proof
• Lemma 4.4
• proof
• Lemma 4.5