On the structure of Nevanlinna measures

TL;DR

This paper investigates the structural properties of Nevanlinna measures, providing characterizations, support descriptions, and measure estimates, with applications to the polydisc setting.

Contribution

It offers new characterizations of Nevanlinna measures via Fourier transforms and detailed analysis of their support and singular parts, including extremal measures.

Findings

Characterization of measures through Fourier transform.

Description of measures supported on hyperplanes.

Estimates for measures of expanding and shrinking cubes.

Abstract

In this paper, we study the structural properties of Nevanlinna measures, i.e. Borel measures that arise in the integral representation of Herglotz-Nevanlinna functions. In particular, we give a characterization of these measures in terms of their Fourier transform, characterize measures supported on hyperplanes including extremal measures, describe the structure of the singular part of the measures when some variable are set to a fixed value, and provide estimates for the measure of expanding and shrinking cubes. Corresponding results are stated also in the setting of the polydisc where applicable, and some of our proofs are actually perfomed via the polydisc.