Pointwise convergence to initial data for some evolution equations on symmetric spaces

Let $\mathcal{L}$ 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 $\mathcal{L}$, and in each of these cases we characterize the weights $v$ on $\mathbb{X}$ for which the solution converges pointwise a.e. to the initial data when the latter is in $L^p(v)$, $1\le p<\mathrm{\infty }$. As a tool, we also establish vector-valued weak type $(1,1)$ and $L^p$ estimates ($1<p<\mathrm{\infty }$) for the local Hardy–Littlewood maximal function on $\mathbb{X}$.