Laplace-Beltrami算子几乎$L^p$ -本征函数的特征
Characterization of almost $L^p$-eigenfunctions of the Laplace-Beltrami operator
摘要：
在\cite{Roe}罗证明，如果一个双重无限序列$\{f_k\}$的职能$\R$满足$f_{k+1}=(df_{k}/dx)$和$|f_{k}(x)|\leq M$的所有$k=0,\pm 1,\pm 2,...$和$x\in \R$，然后$f_0(x)=a\sin(x+φ)$其中$a$和$φ$是真正的常数。这一结果被Elhartz \cite{Str}推广到$\R^n$，其中$d/dx$被$\R^n$上的拉普拉斯算子取代。虽然将这个定理推广到其他的黎曼流形或李群是合理的，但李哈茨证明了这个结果对海森堡群成立，但对双曲三维空间失败。这个否定结果可以推广到任何非紧型的黎曼对称空间。我们观察到，这种失败是植根于$p$ -依赖的$L^p$ -谱的Laplacian的双曲空间。考虑到这一点，我们将证明，对于所有秩为一的非紧型黎曼对称空间，或更一般的调和$NA$群，定理实际上是真的，当一致有界性被替换为一致的“几乎$L^p$有界性”。此外，我们将看到，对于对称空间，这个定理能够表征$L^p$函数在边界上的泊松变换，这与Roe关于$\R$的原始定理有些相似。
英文摘要：
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 $f_0(x)=a\sin(x+φ)$ where $a$ and $φ$ 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 $\R$.
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