A Theorem of Roe and Strichartz on homogeneous trees

Roe- Strichartz Theorem on Two Step Nilpotent Lie Groups

Strichartz characterized eigenfunctions of the Laplacian on Euclidean spaces by boundedness conditions which generalized a result of Roe for the one-dimensional case. He also proved an analogous statement for the sublaplacian on the Heisenberg groups. In this paper, we extend this result to connected, simply connected two step nilpotent Lie groups.

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

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 $f_{k+1}=Δf_k$ and $|f_{k}(x)|\leq M$ for all $k=0,\pm 1,\pm 2,...$ and $x\in \R^n$, then $Δf_0(x)= -f_0$. 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 suitabl…

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

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 ex…

齐次树上乘子的Strichartz定理

A theorem of Strichartz for multipliers on homogeneous trees

本文将欧氏空间上的Strichartz定理推广至齐次树，用缓增分布和乘子算子替代有界函数与Laplacian，并探讨了热算子、Schrödinger算子及球面平均等具体乘子算子情形。

具有 Sobolev 正则性初值的 Dunkl-Schrödinger 方程的正交 Strichartz 估计

Orthonormal Strichartz estimates for Dunkl-Schrödinger equation of initial data with Sobolev regularity

本文针对具有 Dunkl-Sobolev 正则性的初值，利用弱型正交估计、频率局部化与插值技术建立 Dunkl-Schrödinger 方程的正交 Strichartz 估计。