Restriction Theorems On M\'etiver Groups Associated to Joint Functional Calculus

In this article, we get the spectral solution ${\mathcal{P}}_{\mu }^m$ of operators $m(\mathcal{L},-{\mathrm{\Delta }}_{\mathfrak{z}})$, the joint functional calculus of the sub-Laplacian and Laplacian on the centre of M'etivier group. Then, we give some group-analogues of the Thomas-Stein-type restriction theorem, asserting the mix-norm boundness of the restriction operators ${\mathcal{P}}_{\mu }^m$ for two classes of functions $m=(a^{\alpha }+b^{\beta }{)}^{\gamma }$ and $m=(1+a^{\alpha }+b^{\beta }{)}^{\gamma }$ with $\alpha, \beta>0, \gamma\neq0$.