Pointwise convergence of the heat and subordinates of the heat semigroups associated with the Laplace operator on homogeneous trees and two weighted <i>L<SUP>p</SUP></i> maximal inequalities

Abstract

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 is an element of L-p(X, mu, u) with 1 <= p < infinity, 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 is an element of (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.