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

Let ∆ = ∇∗∇ be the distinguished Laplacian on a Damek–Ricci space. We prove the Lp-boundedness of the vector of first-order Riesz transforms ∇∆−1/2 in the full range p ∈ (1,∞). The most demanding part of the proof is establishing the boundedness for p ∈ (2,∞); 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.