In this paper weighted endpoint estimates for the Hardy-Littlewood maximal function on {the infinite rooted} $k$-ary tree are provided. Motivated by Naor and Tao the following Fefferman-Stein estimate \[ w\left(\left\{ x\in T\,:\,Mf(x)>\lambda\right\} \right)\leq c_{s}\frac{1}{\lambda}\int_{T}|f(x)|M(w^{s})(x)^{\frac{1}{s}}dx\qquad s>1 \] is settled and moreover it {is shown it} is sharp, in the sense that it does not hold in general if $s=1$. Some examples of non trivial weights such that the weighted weak type $(1,1)$ estimate holds are provided. A {strong} Fefferman-Stein type estimate and as a consequence some vector valued extensions are obtained. In the Appendix a weighted counterpart of the abstract {theorem} of Soria and Tradacete on infinite trees is established.

中文翻译：

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无限根 k 叉树上 Hardy-Littlewood 极大函数的 Fefferman-Stein 不等式

在本文中，提供了 {the infinite rooted} $k$-ary 树上 Hardy-Littlewood 极大函数的加权端点估计。受 Naor 和 Tao 的启发，以下 Fefferman-Stein 估计 \[ w\left(\left\{ x\in T\,:\,Mf(x)>\lambda\right\} \right)\leq c_{s} \frac{1}{\lambda}\int_{T}|f(x)|M(w^{s})(x)^{\frac{1}{s}}dx\qquad s>1 \]是固定的，而且它{显示它}是尖锐的，因为如果 $s=1$，它一般不成立。提供了一些非平凡权重的例子，使得加权弱类型 $(1,1)$ 估计成立。{strong} Fefferman-Stein 类型估计，因此获得了一些向量值扩展。在附录中，建立了 Soria 和 Tradacete 在无限树上的抽象{定理}的加权对应物。