The image of the heat kernel transform on Riemannian symmetric spaces of the noncompact type

An uncertainty principle for spectral projections on rank one symmetric spaces of noncompact type

Let $G $ be a noncompact semisimple Lie group with finite centre. Let $X=G/K$ be the associated Riemannian symmetric space and assume that $X$ is of rank one. The spectral projections associated to the Laplace-Beltrami operator are given by $P_λf =f\ast Φ_λ$, where $Φ_λ$ are the elementary spherical functions on $X$. In this paper, we prove an Ingham type uncertainty principle for $P_λf$. Moreover, similar results are obtained in the case of spectral projections associated to Dunkl Laplacian.

A theorem of Chernoff on quasi-analytic functions for Riemannian symmetric spaces

An $L^2$ version of the classical Denjoy-Carleman theorem regarding quasi-analytic functions was proved by P. Chernoff on $\mathbb R^n$ using iterates of the Laplacian. We give a simple proof of this theorem which generalizes the result on $\mathbb R^n$ for any $p\in [1, 2]$. We then extend this result to Riemannian symmetric spaces of compact and noncompact type for $K$-biinvariant functions.

分数阶Sobolev映射到非紧覆盖空间的提升

Lifting of fractional Sobolev mappings to noncompact covering space

本文刻画了分数阶Sobolev映射到非紧覆盖空间的提升，并在$sp>1$及$sp\ge\dim\mathcal{M}$条件下获得最优非线性估计，同时给出相关函数空间和的非线性刻画。

On a theorem of Chernoff on rank one Riemannian symmetric spaces

In 1975, P.R. Chernoff used iterates of the Laplacian on $\mathbb{R}^n$ to prove an $L^2$ version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth function on $\mathbb{R}^n$ to be quasi-analytic. In this paper, we prove an exact analogue of Chernoff's theorem for all rank one Riemannian symmetric spaces (of noncompact and compact types) using iterates of the associated Laplace-Beltrami operators.