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

In this paper sufficient conditions for weighted weak and strong type (p, p) estimates with $$p>1$$ the centered maximal function on infinite rooted k-ary tree are provided. Here line initated by authors Safe in Ombrosi et al. (Int Math Res Not IMRN 4:2736–2762, 2021) providing a further extension of use techniques due to Naor Tao (J Funct Anal 259(3):731–779, 2010) is continued. The fact that class weights from wider $$L^p$$ than one obtained [16] established as well. Some results highlighting pathological nature theory setting settled. It shown $$A_p$$ condition no precise setting, since there exist such boundedness holds but not satisfied. also Sawyer testing either hold equivalent setting. will be well weight can extended two