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

OMBROSI, SHELDY; RIVERA-RÍOS, ISRAEL P; SAFE, MARTÍN D

INTERNATIONAL MATHEMATICS RESEARCH NOTICES

OXFORD UNIV PRESS

Año: 2021 vol. 2021 p. 2736 - 2762

1073-7928

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\[w\left(\left\{ x\in T\,:\,Mf(x)>\lambda\right\} \right)\leq c_{s}\frac{1}{\lambda}\int_{T}|f(x)|M(w^{s})(x)^{\frac{1}{s}}dx\qquad s>1\]is settled, and moreover, it is shown 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.