Riesz transforms on solvable extensions of stratified groups

Let G=N⋊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 Δ on G. Here we prove weak type (1,1), Lp-boundedness for p∈(1,2] and H1→L1 boundedness of the Riesz transforms YΔ−1/2 and YΔ−1Z, 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 Δ is not elliptic.