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 ${\{W_t\}}_{t>0}$ 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 $1\le p<\infty$, 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 $X$.