Fefferman–Stein Inequalities for the Hardy–Littlewood Maximal Function on the Infinite Rooted k-ary Tree

Abstract

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∈T:Mf(x)>λ})≤cs1λ∫T|f(x)|M(ws)(x)1sdxs>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.