Hardy--Littlewood Maximal Operator and Two-Layer Muckenhoupt Weights on Infinite Rooted $k$-Ary Trees

Let $k\geq 2$ be an integer, $T$ an infinite rooted $k$-ary tree, and $M$ the Hardy--Littlewood maximal operator on $T$. For any $p\in(0,\infty)$, we characterize the weight $w$ such that $M$ is bounded on $L^p(w)$. To this end, we introduce a two-layer Muckenhoupt weight class $\mathscr A_p$ and pr