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

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.