Factorization in Denjoy-Carleman classes associated to representations of $(\mathbb{R}^{d},+)$

函数的稀疏系统与拟解析类

Sparse systems of functions and quasi-analytic classes

本文刻画 Denjoy–Carleman 类的拟解析性，并依据连续统假设是否成立，完整判定各类是否承载稀疏函数系统，进而推广匿名预测器不存在性的结论。

Analogs of certain quasi-analiticity results on Riemannian symmetric spaces of noncompact type

An $L^2$ version of the celebrated Denjoy-Carleman theorem regarding quasi-analytic functions was proved by Chernoff \cite{CR} on $\mathbb R^d$ using iterates of the Laplacian. In $1934$ Ingham \cite{I} used the classical Denjoy-Carleman theorem to relate the decay of Fourier transform and quasi-analyticity of integrable functions on $\mathbb R$. In this paper we extend both these theorems to Riemannian symmetric spaces of noncompact type and show that the theorem of Ingham follows from that of Chernoff.

An exotic zoo of diffeomorphism groups on $\mathbb R^n$

Let $C^{[M]}$ be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight sequence $M=(M_k)$ is log-convex and has moderate growth. We prove that the groups ${\operatorname{Diff}}\mathcal{B}^{[M]}(\mathbb{R}^n)$, ${\operatorname{Diff}}W^{[M],p}(\mathbb{R}^n)$, ${\operatorname{Diff}}{\mathcal{S}}{}_{[L]}^{[M]}(\mathbb{R}^n)$, and ${\operatorname{Diff}}\mathcal{D}^{[M]}(\mathbb{R}^n)$ of $C^{[M]}$-diffeomorphisms on $\mathbb{R}^n$ which differ from the identity by a mapping in $\mathcal{B}^{[M]}$ (global Denjoy--Carleman), $W^{[M],p}$ (Sobolev-Denjoy-Carleman), ${\mathcal{S}…

Ultradifferentiable extension theorems: a survey

We survey ultradifferentiable extension theorems, i.e., quantitative versions of Whitney's classical extension theorem, with special emphasis on the existence of continuous linear extension operators. The focus is on Denjoy-Carleman classes for which we develop the theory from scratch and discuss important related concepts such as (non-)quasianalyticity. It allows us to give an efficient and, to a fair extent, elementary introduction to Braun-Meise-Taylor classes based on their representation as intersections and unions of Denjoy-Carleman classes.

On the projective description of spaces of ultradifferentiable functions of Roumieu type

We provide a projective description of the space $\mathcal{E}^{\{\mathfrak{M}\}}(Ω)$ of ultradifferentiable functions of Roumieu type, where $Ω$ is an arbitrary open set in $\mathbb{R}^d$ and $\mathfrak{M}$ is a weight matrix satisfying the analogue of Komatsu's condition $(M.2)'$. In particular, we obtain in a unified way projective descriptions of ultradifferentiable classes defined via a single weight sequence (Denjoy-Carleman approach) and via a weight function (Braun-Meise-Taylor approach) under considerably weaker assumptions than in earlier versions of these results.

光滑区域上层位势的Carleman因式分解

Carleman factorization of layer potentials on smooth domains

本文利用Carleman与Krein提示的因式分解结构，将光滑闭超曲面上Neumann-Poincaré算子的谱分析限定在$L^2$-空间框架中，并结合几何微局部分析技术，补充了光滑边界情形下层位势理论的最新进展。

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.

Roumieu 情形下 Borel 问题的最优解

On optimal solutions of the Borel problem in the Roumieu case

本文利用单参数权重序列族统一处理 Denjoy–Carleman 与 Braun–Meise–Taylor 超可微类，比较其 Borel 问题的最优解，并刻画 Roumieu 情形下的关系与等价条件。