Heat Diffusion on Homogeneous Trees

Let X be a homogeneous tree. We study the heat diffusion process associated with the nearest neighbour isotropic Markov operator on X. In particular it is shown that the heat maximal operator is weak type (1, 1) and strong type (p, p), for every 1 < p < ∞. We estimate the asymptotic behaviour of the heat maximal function. Moreover, we introduce a family of H^p spaces on X. It is proved that H^p=l^p(X) for 1 < p < ∞ and is conjectured that H^p for p less than 1, is trivial.