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

Abstract

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 $\mathbf{y}\neq 0$. Then we show the kernels are Lipschitzian homogeneous functions on $\mathcal{N}\setminus \{\mathbf{0}\}$ by analytic continuation. Moreover, they are shown to be Calder\'{o}n-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 $\lim _{R \rightarrow 1^-} \sum_{m=0}^{\infty} R^{m}\mathbb{P}_{m}\phi=\phi$ by estimating the $L^p(\mathcal{N})$-norms of $\mathbb{P}_m$. Furthermore, $\mathbb{P}_m$ are mutually orthogonal projection operators and $\sum_{m=0}^{\infty} \mathbb{P}_{m}\phi=\phi$ for $\phi\in L^2(\mathcal{N})$.