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

2024/04/04 by Kang, Qianqian, Chang, Der-Chen, Wang, Wei
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2404.03378

Abstract

In this paper, we introduce the spectral projection operators ℙ_m on non-degenerate nilpotent Lie groups N of step two, associated to the joint spectrum of sub-Laplacian and derivatives in step two. We construct their kernels P_m(y,t) by using Laguerre calculus and find a simple integral representation formula for y≠ 0. Then we show the kernels are Lipschitzian homogeneous functions on N∖ \0\ by analytic continuation. Moreover, they are shown to be Calderón-Zygmund kernels, so that the spectral projection operator ℙ_m can be extended to a bounded operator from L^p(N) to itself. We also prove a convergence theorem of the Abel sum lim _{R → 1^-} ∑_m=0^∞ R^mℙ_mϕ=ϕ by estimating the L^p(N)-norms of ℙ_m. Furthermore, ℙ_m are mutually orthogonal projection operators and ∑_m=0^∞ ℙ_mϕ=ϕ for ϕ∈ L²(N).

Related

• The Laguerre calculus on the nilpotent Lie groups of step two
• An Update On The L^p-L^q Norms of Spectral Multipliers on Unimodular Lie Groups
• Subelliptic pseudo-differential operators and Fourier integral operators\n on compact Lie groups
• Spectral multipliers for wave operators
• Contraction of the \mathfraksl₂ -triple associated to the (k, a) -generalized Fourier transform