On the Harnack distance

TL;DR

This paper explores two new estimates for the Harnack distance in bounded Euclidean domains, utilizing geometric entropy of arcwise connectedness and a novel separatedness concept, advancing understanding of domain geometry.

Contribution

It introduces a new separatedness concept for subsets in domains and applies entropy of arcwise connectedness to estimate the Harnack distance.

Findings

Two types of estimates for Harnack distance are developed.

The separatedness concept can be more subtle than existing notions.

Applications to zero sets of holomorphic functions are discussed.

Abstract

In this note, we consider two types of estimates for the Harnack distance in bounded domains of a finite-dimensional Euclidean space. The first type is based on the geometric concept of the entropy of arcwise connectedness. We used this concept earlier to study zero sets of holomorphic functions in weighted classes of holomorphic functions. The second type is based on the concept of separatedness for subsets in domain, which is introduced in this note. The latter concept, generally speaking, can be more subtle.