Let \({\cal L}\) be either the Laplace–Beltrami operator, its shift without spectral gap, or the distinguished Laplacian on a symmetric space of noncompact type \({\mathbb X}\) of arbitrary rank. We consider the heat equation, the fractional heat equation, and the Caffarelli–Silvestre extension problem associated with \({\cal L}\) , and in each of these cases we characterize the weights υ on \({\mathbb X}\) for which the solution converges pointwise a.e. to the initial data when the latter is in Lp(υ), 1 ≤ p < ∞. As a tool, we also establish vector-valued weak type (1, 1) and Lp estimates (1 < p < ∞) for the local Hardy–Littlewood maximal function on \({\mathbb X}\) .
