Analysis of $L^p$-type estimates of Poisson transform on Homogeneous Trees

TL;DR

This paper establishes restriction theorems for the Helgason-Fourier transform on homogeneous trees, providing new insights into eigenfunctions of the Laplacian related to Poisson transforms and boundary functions.

Contribution

It proves restriction theorems for the Helgason-Fourier transform on homogeneous trees and characterizes eigenfunctions as Poisson transforms of boundary functions.

Findings

Restriction theorem for Helgason-Fourier transform established

Characterization of eigenfunctions as Poisson transforms of boundary functions

Norm estimates of Poisson transform derived

Abstract

In this article we prove the restriction theorem for Helgason-Fourier transform on homogeneous tree. Our proof is based on the duality argument and the norm estimates of Poisson transform. We also characterize all eigenfunctions of the laplacian on homogeneous tree which are Poisson transform of $L^p$ functions defined on the boundary.