Pointwise convergence of the heat and subordinates of the heat semigroups associated with the Laplace operator on homogeneous trees and two weighted Lp maximal inequalities

In this paper, we consider the heat semigroup Formula: see text defined by the combinatorial Laplacian and two subordinated families of Formula: see text on homogeneous trees Formula: see text. We characterize the weights Formula: see text on Formula: see text for which the pointwise convergence to initial data of the above families holds for every Formula: see text with Formula: see text, where Formula: see text represents the counting measure in Formula: see text. We prove that this convergence property in Formula: see text is equivalent to the fact that the maximal operator on Formula: see text, for some Formula: see text, defined by the semigroup is bounded from Formula: see text into Formula: see text for some weight Formula: see text on Formula: see text.