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 [23], the following Fefferman-Stein estimate w({x is an element of T : Mf(x) > lambda}) <= c(s) 1/lambda integral(T)vertical bar f(x)vertical bar M(W-s)(X)(1/s)dx s > 1 is settled, and moreover, it is shown that it is sharp, in the sense that it does not hold in general if s = 1. Some examples of nontrivial 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 [38] on infinite trees is established.