Pointwise convergence of the heat and subordinates of the heat semigroups associated with the Laplace operator on homogeneous trees and two weighted $L^p$ maximal inequalities

In this paper we consider the heat semigroup $\{W_t{\}}_{t>0}$ defined by the combinatorial Laplacian and two subordinated families of $\{W_t{\}}_{t>0}$ on homogeneous trees $X$. We characterize the weights $u$ on $X$ for which the pointwise convergence to initial data of the above families holds for every $f\in L^p(X,\mu ,u)$ with $1\le p<\mathrm{\infty }$, where $\mu$ represents the counting measure in $X$ . We prove that this convergence property in $X$ is equivalent to the fact that the maximal operator on $t\in (0,R)$, for some $R>0$, defined by the semigroup is bounded from $L^p(X,\mu ,u)$ into $L^p(X,\mu ,v)$ for some weight $v$ on $X$.