Weighted $L^{p}$ estimates on the infinite rooted $k$-ary tree

In this paper, building upon ideas of Naor and Tao and continuing the study initiated in by the authors and Safe, sufficient conditions are provided for weighted weak type and strong type $(p,p)$ estimates with $p>1$ for the centered maximal function on the infinite rooted $k$-ary tree to hold. Consequently a wider class of weights for those strong and weak type $(p,p)$ estimates than the one obtained in by the authors and Safe in a previous work is provided. Examples showing that the Sawyer type testing condition and the $A_p$ condition do not seem precise in this context are supplied as well. We also prove that strong and weak type estimates are not equivalent, highlighting the pathological nature of the theory of weights in this setting. Two weight counterparts of our conditions will be obtained as well.