Geometric properties of infinite graphs and the Hardy-Littlewood maximal operator

TL;DR

This paper investigates geometric properties of infinite graphs and their influence on the weak-type boundedness of the Hardy-Littlewood maximal operator, providing examples to illustrate key differences among these properties.

Contribution

It establishes connections between various geometric properties of infinite graphs and the boundedness of the maximal operator, with new examples illustrating these relationships.

Findings

Connections between doubling condition and maximal operator boundedness

Examples of infinite graphs illustrating property differences

Analysis of bounded degree and equidistant comparison property

Abstract

We study different geometric properties on infinite graphs, related to the weak-type boundedness of the Hardy-Littlewood maximal averaging operator. In particular, we analyze the connections between the doubling condition, having finite dilation and overlapping indices, uniformly bounded degree, the equidistant comparison property and the weak-type boundedness of the centered Hardy-Littlewood maximal operator. Several non-trivial examples of infinite graphs are given to illustrate the differences among these properties.