Weighted Inequalities for Fractional maximal functions on the infinite rooted $k$-ary tree

TL;DR

This paper introduces a fractional maximal function on infinite rooted k-ary trees, analyzing its weighted boundedness and providing examples of weights satisfying strong type estimates.

Contribution

It extends the theory of fractional maximal functions to the setting of infinite rooted k-ary trees and explores their weighted boundedness properties.

Findings

Identified conditions for weighted boundedness of the fractional maximal function.

Provided explicit examples of weights satisfying strong type (p, q) estimates.

Extended classical maximal function theory to a tree structure setting.

Abstract

In this article we introduce the fractional Hardy-Littlewood maximal function on the infinite rooted $k$-ary tree and study its weighted boundedness. We also provide examples of weights for which the fractional Hardy-Littlewood maximal function satisfies strong type $(p, q)$ estimates on the infinite rooted $k$-ary tree.