Characterization of almost $L^p$-eigenfunctions of the Laplace-Beltrami operator

TL;DR

This paper extends Roe's theorem on eigenfunctions to certain noncompact symmetric spaces, showing that under almost $L^p$ boundedness conditions, eigenfunctions are characterized as Poisson transforms of boundary functions.

Contribution

It proves that Roe's theorem holds for rank one symmetric spaces of noncompact type under almost $L^p$ boundedness, and characterizes eigenfunctions as Poisson transforms.

Findings

The theorem extends to harmonic $NA$ groups.

Eigenfunctions are characterized as Poisson transforms of boundary functions.

Failure of the original theorem on hyperbolic space is due to $p$-dependence of the spectrum.

Abstract

In \cite{Roe} Roe proved that if a doubly-infinite sequence $\{f_k\}$ of functions on $\R$ satisfies $f_{k+1}=(df_{k}/dx)$ and $|f_{k}(x)|\leq M$ for all $k=0,\pm 1,\pm 2,...$ and $x\in \R$, then $f_0(x)=a\sin(x+\varphi)$ where $a$ and $\varphi$ are real constants. This result was extended to $\R^n$ by Strichartz \cite{Str} where $d/dx$ is substituted by the Laplacian on $\R^n$. While it is plausible to extend this theorem for other Riemannian manifolds or Lie groups, Strichartz showed that the result holds true for Heisenberg groups, but fails for hyperbolic 3-space. This negative result can be indeed extended to any Riemannian symmetric space of noncompact type. We observe that this failure is rooted in the $p$-dependance of the $L^p$-spectrum of the Laplacian on the hyperbolic spaces. Taking this into account we shall prove that for all rank one Riemannian symmetric spaces of noncompact type, or more generally for the harmonic $NA$ groups, the theorem actually holds true when uniform boundedness is replaced by uniform "almost $L^p$ boundedness". In addition we shall see that for the symmetric spaces this theorem is capable of characterizing the Poisson transforms of $L^p$ functions on the boundary, which some what resembles the original theorem of Roe on $\R$.