Riesz transforms for the distinguished Laplacian on Damek-Ricci spaces and operator-valued multivariate spectral multipliers

Let $\Delta = \nabla^* \nabla$ be the distinguished Laplacian on a Damek-Ricci space. We prove the $L^{p}$-boundedness of the vector of first-order Riesz transforms $\nabla \Delta^{-1/2}$ in the full range $p\in(1,\infty)$. The most demanding part of the proof is establishing the boundedness for $p \in (2,\infty)$; this is obtained as a consequence of an operator-valued spectral multiplier theorem for the joint functional calculus of a commuting system of self-adjoint operators, which we prove here and may be of independent interest.