Problems in Euclidean harmonic analysis related to the geometry of curves and surfaces

与曲线和曲面几何相关的欧几里得调和分析问题

• 批准号: 230032422
• 负责人: Dr. Spyridon Dendrinos, since 1/2014
• 金额: --
• 依托单位: School of Mathematical Sciences
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 2016-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/230032422?language=en
• 关键词: Problems; Euclidean; harmonic; analysis; related

The purpose of this project is to investigate the properties of a number of operators that play a central role in harmonic analysis. These are the restriction of the Fourier transform, averaging operators, the restricted X-ray transform and maximal operators. What unifies the study of the above is the presence, in each case, of some underlying curve or surface whose geometric properties (e.g. the curvature) determine the mapping properties of the corresponding operators. In many cases it is also natural to aim for estimates that are uniform over large classes of the underlying varieties. These uniform estimates are optimal in a certain sense. In order to achieve our goals, we will need to further our understanding of issues such as the decay properties of the Fourier transform in relation to Newton polyhedral, geometric inequalities involving an important quantity called the affine arc length and their interplay with certain combinatorial techniques.

该项目的目的是调查许多在谐波分析中发挥核心作用的运营商的属性。这些是对傅立叶变换，平均操作员，受限制的X射线变换和最大运算符的限制。统一上述研究的是，在每种情况下，某些基础曲线或表面的存在（例如，曲率）决定了相应运算符的映射特性。在许多情况下，旨在实现在大类基础品种中均匀的估计值也很自然。这些统一的估计在某种意义上是最佳的。为了实现我们的目标，我们将需要进一步理解问题，例如与牛顿多面体，几何不平等相关的傅立叶变换的衰减特性，涉及一个重要数量，称为仿射弧长度及其与某些组合技术的相互作用。

项目成果

Uniform bounds for convolution and restricted X-ray transforms along degenerate curves

• DOI: 10.1016/j.jfa.2014.10.012
• 发表时间: 2015-02
• 期刊: arXiv: Classical Analysis and ODEs
• 作者: S. Dendrinos;Betsy Stovall