Trasferring $L^p$ eigenfunction bounds from $S^{2n+1}$ to $h^n$

TL;DR

This paper transfers $L^p$ eigenfunction bounds from the complex sphere to the Heisenberg group using Lie group contraction, deriving new estimates and a restriction theorem for the sub-Laplacian on $h^n$.

Contribution

It introduces a method to transfer spectral estimates from spheres to the Heisenberg group via Lie group contraction, providing new bounds and a restriction theorem.

Findings

Derived $L^p-L^2$ estimates for $h^n$ from sphere bounds.

Established a discrete restriction theorem for the sub-Laplacian on $h^n$.

Connected spectral bounds on spheres with those on the Heisenberg group.

Abstract

By using the notion of contraction of Lie groups, we transfer $L^p-L^2$ estimates for joint spectral projectors from the unit complex sphere $\sfera$ in ${{\mathbb{C}}}^{n+1}$ to the reduced Heisenberg group $h^{n}$. In particular, we deduce some estimates recently obtained by H. Koch and F. Ricci on $h^n$. As a consequence, we prove, in the spirit of Sogge's work, a discrete restriction theorem for the sub-Laplacian $L$ on $h^n$.