The Choquet and Kellogg properties for the fine topology when $p=1$ in metric spaces

TL;DR

This paper establishes key properties of the fine topology, including the Kellogg, quasi-Lindel"of, and Choquet properties, specifically for the case p=1 in complete metric spaces with doubling measures and Poincaré inequalities.

Contribution

It extends the theory of fine topology properties to the p=1 case in metric spaces with specific geometric and measure-theoretic conditions.

Findings

Proves the fine Kellogg property for p=1.

Establishes the quasi-Lindel"of principle for the fine topology.

Demonstrates the Choquet property for the fine topology.

Abstract

In the setting of a complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we prove the fine Kellogg property, the quasi-Lindel\"of principle, and the Choquet property for the fine topology in the case $p=1$.