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

Abstract

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].