$L^p$-boundedness of Riesz transforms on solvable extensions of Carnot groups

Let $G=N⋊\mathbb{R}$, where $N$ is a Carnot group and $\mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous left-invariant sub-Laplacians on $N$ and $\mathbb{R}$ can be lifted to $G$, and their sum is a left-invariant sub-Laplacian $\mathrm{\Delta }$ on $G$. We prove that the first-order Riesz transforms $X{\mathrm{\Delta }}^{-1\mathrm{/}2}$ are bounded on $L^p(G)$ for all $p\in (1,\mathrm{\infty })$, where $X$ is any horizontal left-invariant vector field on $G$. This extends a previous result by Vallarino and the first-named author, who obtained the bound for $p\in (1,2]$. The proof makes use of an operator-valued spectral multiplier theorem, recently proved by the authors, and hinges on estimates for products of modified Bessel functions and their derivatives.