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

Let $k\ge 2$ be an integer, $T$ an infinite rooted $k$-ary tree, and $M$ the Hardy–Littlewood maximal operator on $T$. For any $p\in (0,\mathrm{\infty })$, 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 ${\mathcal{A}}_p$ and prove that, for any $p\in (\frac{1}{2},\mathrm{\infty })$, the boundedness of $M$ on $L^p(w)$, $w\in {\mathcal{A}}_p$, and the exponential decay boundedness of spherical averaging operators on $L^p(w)$ are mutually equivalent, and that, when $p\in (0,\frac{1}{2}]$, there exists no weight $w$ such that $M$ is bounded on $L^p(w)$. Moreover, for any $p\in (\frac{1}{2},\mathrm{\infty })$, we establish the quantitative estimate, with the optimal exponent $\frac{1}{p}$ of the weight constant, for the boundedness of $M$ on $L^p(w)$. For any $p\in (1,\mathrm{\infty })$, we also obtain two further equivalent characterizations of the boundedness of $M$ on $L^p(w)$, respectively, in terms of a global Sawyer-type testing condition and an estimate for the weighted product measure of distance incidence sets. As applications, for any $p\in (\frac{1}{2},\mathrm{\infty })$, under the assumption that $M$ is bounded on $L^p(w)$, we establish the boundedness of exponentially decaying kernel operators on $L^p(w)$ and weighted Fefferman–Stein vector-valued inequalities.