Meromorphic functions and differences of subharmonic functions in integrals and the difference characteristic of Nevanlinna. III. Estimates of integrals over fractal sets in terms of the Hausdorff measure and content

TL;DR

This paper extends previous work on estimates of the radial maximum growth characteristic of δ-subharmonic functions by relating integral bounds to Hausdorff measures and contents of fractal sets, including the d-dimensional case.

Contribution

It provides new estimates of integral quantities of subharmonic functions over fractal sets using Hausdorff measures, generalizing earlier results to include the d-dimensional case.

Findings

Estimates of integral of the positive part of the maximum growth characteristic in terms of Hausdorff measure and content.

Explicit bounds for the case of d-dimensional Hausdorff measure.

Application to sets where the integration function is constant on components of the complement.

Abstract

Let $U\not\equiv \pm\infty$ be a $\delta$-subharmonic function on a closed disc of radius $R$ centered at zero. In the previous two parts of our paper, we obtained general and explicit estimates of the integral of the positive part of the radial maximum growth characteristic ${\mathsf M}_U(t):= \sup\bigl\{U(z)\bigm| |z|=r\bigr\}$ over the increasing integration function $m$ on the segment $[0, r]\subset [0,R)$ through the difference characteristic of Nevanlinna and the quantities associated with the integration function $m$. The third part of our paper contains estimates of these quantities in terms of the Hausdorff $h$-measure and $h$-content of compact subset $S\subset [0, r]$ such that the integration function $m$ is constant on each open component of the connectivity of the complement $[0, r]\setminus S$. The case of the d-dimensional Hausdorff measure is highlighted separately.