Fefferman-Stein inequalities for the Hardy-Littlewood maximal function on the infinite rooted $k$-ary tree

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 $$w\left(\{x\in T:Mf(x)>\lambda \}\right)\le c_s\frac{1}{\lambda }{\int }_T\mathrm{\mid }f(x)\mathrm{\mid }M(w^s)(x{)}^{\frac{1}{s}}dxs>1$$ 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.