Riesz transforms on solvable extensions of stratified groups

Let G be the semidirect product of N and A, where N is a stratified group and A = R acts on N via automorphic dilations. Homogeneous sub-Laplacians on N and A can be lifted to left-invariant operators on G and their sum is a sub-Laplacian Delta on G. Here we prove weak type (1, 1), L-p-boundedness for p is an element of (1, 2] and H-1 -> L-1 boundedness of the Riesz transforms Y Delta^{-1/2} and Y Delta^{-1} Z, where Y and Z are any horizontal left-invariant vector fields on G, as well as the corresponding dual boundedness results. At the crux of the argument are large-time bounds for spatial derivatives of the heat kernel, which are new when Delta is not elliptic.