A theorem of Roe and Strichartz for Riemannian symmetric spaces of noncompact type

TL;DR

This paper extends a classical theorem relating bounded sequences and Laplacians from Euclidean space to Riemannian symmetric spaces of noncompact type, showing the theorem holds under modified boundedness conditions.

Contribution

It generalizes Roe and Strichartz's theorem from Euclidean space to noncompact symmetric spaces with appropriate boundedness modifications.

Findings

The theorem fails on hyperbolic 3-space without modifications.

Modified boundedness conditions ensure the theorem's validity on all noncompact symmetric spaces.

The result clarifies the scope of the theorem in different geometric contexts.

Abstract

Generalizing a result of Roe \cite{Roe} Strichartz proved in \cite{Str} that if a doubly-infinite sequence $\{f_k\}$ of functions on $\R^n$ satisfies $f_{k+1}=\Delta f_k$ and $|f_{k}(x)|\leq M$ for all $k=0,\pm 1,\pm 2,...$ and $x\in \R^n$, then $\Delta f_0(x)= -f_0$. Strichartz also showed that the result fails for hyperbolic 3-space. This negative result can be indeed extended to any Riemannian symmetric space of noncompact type. Taking this into account we shall prove that for all Riemannian symmetric spaces of noncompact type the theorem actually holds true when uniform boundedness is modified suitably.