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

Abstract:

In 1980, Roe proved that if a doubly-infinite sequence $\{f_k\}$ of functions on $\mathbb {R}$ satisfies $f_{k+1}=(df_{k}/dx)$ and $|f_{k}(x)|\leq M$ for all $k=0,\pm 1,\pm 2,\cdots$ and $x\in \mathbb {R}$, then $f_0(x)=a\sin (x+\varphi )$, where $a$ and $\varphi$ are real constants. This result was extended to $\mathbb {R}^n$ by Strichartz in 1993, where $d/dx$ was substituted by the Laplacian on $\mathbb {R}^n$. While it is plausible that this theorem extends to 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 indeed be extended to any Riemannian symmetric space of noncompact type. We observe that this failure is rooted in the $p$-dependence 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, and 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 can be used to characterize the Poisson transforms of $L^p$ functions on the boundary, which somewhat resembles the original theorem of Roe on $\mathbb {R}$.