Spectral projection operators of the Sub-Laplacian and Laguerre calculus on non-degenerate nilpotent Lie groups of step two

In this paper, we introduce the spectral projection operators ${\mathbb{P}}_m$ on non-degenerate nilpotent Lie groups $\mathcal{N}$ of step two, associated to the joint spectrum of sub-Laplacian and derivatives in step two. We construct their kernels $P_m(\mathbf{y},\mathbf{t})$ by using Laguerre calculus and find a simple integral representation formula for $\mathbf{y}\mathrm{\ne }0$. Then we show the kernels are Lipschitzian homogeneous functions on $\mathcal{N}\setminus \{\mathbf{0}\}$ by analytic continuation. Moreover, they are shown to be Calderón-Zygmund kernels, so that the spectral projection operator ${\mathbb{P}}_m$ can be extended to a bounded operator from $L^p(\mathcal{N})$ to itself. We also prove a convergence theorem of the Abel sum ${\lim }_{R\to 1^{-}}{\sum }_{m=0}^{\mathrm{\infty }}{R}^m{\mathbb{P}}_m\phi =\phi$ by estimating the $L^p(\mathcal{N})$-norms of ${\mathbb{P}}_m$. Furthermore, ${\mathbb{P}}_m$ are mutually orthogonal projection operators and ${\sum }_{m=0}^{\mathrm{\infty }}{\mathbb{P}}_m\phi =\phi$ for $\phi \in L^2(\mathcal{N})$.