A note on second order Riesz transforms in 3-dimensional Lie groups

We prove explicit $$L^p$$ bounds for second order Riesz transforms of the sub-Laplacian and Laplacian in Lie groups $${\mathbb {H}}$$ , $$\mathbb {SU}(2)$$ $$\widetilde{\mathbb {SL}}(2)$$ . Our proof makes use martingale transform techniques specific commutation properties between complex gradient those groups.