Optimal heat kernel bounds and asymptotics on Damek–Ricci spaces

We give optimal bounds for the radial, space and time derivatives of arbitrary order of the heat kernel of the Laplace–Beltrami operator on Damek–Ricci spaces. In the case of symmetric spaces of rank one, these complete and actually improve conjectured estimates by Anker and Ji. We also provide asymptotics at infinity of all the radial and time derivates of the kernel. Along the way, we provide sharp bounds for all the derivatives of the Riemannian distance and obtain analogous bounds for those of the heat kernel of the distinguished Laplacian.