This article studies pseudo-differential operators on Grélaud’s group \({\cal G}_{w}\) , a nonunimodular exponentially solvable Lie group arising from Bianchi’s 3-dimensional Lie algebras classification. We study the group structure of Grélaud’s group, \({\cal G}_{w}\) , and then investigate harmonic analysis results, namely, Fourier inversion, Plancherel formula, Hausdorff–Young Theorem, etc., for the group \({\cal G}_{w}\) . The construction of pseudo-differential operators and their boundedness are discussed, and we obtain the necessary and sufficient conditions on the symbol σ such that the corresponding pseudo-differential operator Tσ on \({\cal G}_{w}\) is a Hilbert–Schmidt operator. Then, we characterize the trace class pseudo-differential operators on \({\cal G}_{w}\) and find their trace formula. Finally, we introduce the Weyl transform associated with the Wigner transform for Grélaud’s group \({\cal G}_{w}\) . Furthermore, we derive the boundedness or unboundedness of these transforms.
