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}) ≤ cs 1 λ Z T |f(x)|M(w s )(x) 1 s dx 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 [38] is established.