Endpoint Behavior of Modulation Invariant Singular Integrals

调制不变奇异积分的端点行为

• 批准号: 1650810
• 负责人: Francesco DiPlinio
• 金额: $ 11.12万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-05-16 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1650810&HistoricalAwards=false
• 关键词: Endpoint; Behavior; Modulation; Invariant; Singular

Harmonic analysis studies how signals (functions) break up into a superposition of basic harmonics--signals with a well-specified duration, intensity and frequency--and how operations (filtering) applied to these components affect the reconstructed signal. Variants of this time-frequency decomposition process are performed in countless real-world applications, such as audio or image compression and filtering, image pattern recognition, data assimilation and denoising. One of the broad objectives of this project is the investigation of the theoretical feasibility threshold of the time-frequency techniques in terms of the relative size and smoothness of the input. An analogous procedure is adopted in tomographic imaging, where a solid body is reconstructed by means of sampling its density along penetrating waves, mathematically described as lines in three-dimensional space. This project will study mathematical toy models of sampling along lines or curves, whose theoretical understanding may play a significant role in the derivation of improved analytical image reconstruction methods. An integral component of the project is the training of graduate and undergraduate students within the active research group in harmonic analysis at Brown University, with the particular intent of attracting young and promising researchers to the field. The central objects of study of this project are modulation-invariant singular integrals and their behavior at or near the boundary of their known boundedness range. The model question, involving Carleson's maximal partial Fourier sum operator, is the characterization of the sharp integrability order sufficient for the almost-everywhere pointwise convergence of the Fourier series of a periodic function. The second, deeply related question concerns the extension of the Lacey-Thiele Holder-type estimates for the bilinear Hilbert transform to the boundary of the known range. Together with his collaborators, the principal investigator has recently obtained the current best results for both problems, relying in particular on a newly developed Calderon-Zygmund decomposition adapted to the modulation-invariant setting. It is expected that further developments of this technique will lead to additional improvements towards the solution of these two central questions, as well as of other significant open problems. A standout question is the extension of the known uniform estimates for the bilinear Hilbert transform to the full expected range of exponents, completing the original program of Calderon for the boundedness of the first commutator. Another central direction of the proposed investigation is the study of singular integral operators with rotational symmetries, a prime example of which is the Hilbert transform along a smooth vector field in the plane, by means of multiparameter time-frequency analysis techniques. Further improvements of the aforementioned techniques are also expected to impact on several questions concerning summability of multiple Fourier series.

谐波分析研究信号（函数）如何分解为基本谐波的叠加 - 信号 - 标志性持续时间，强度和频率以及应用于这些组件的操作（过滤）如何影响重建信号的操作（过滤）。这种时间频率分解过程的变体是在无数的现实世界应用中执行的，例如音频或图像压缩和滤波，图像模式识别，数据同化和脱氧。 该项目的广泛目标之一是研究时间频技术的理论可行性阈值，从输入的相对大小和平滑度来调查。在层析成像中采用了类似的过程，在层析成像成像中，通过对沿穿透波进行采样密度重建固体，数学描述为三维空间中的线。 该项目将研究沿线或曲线采样的数学玩具模型，其理论理解可能在改善分析图像重建方法的推导中起重要作用。 该项目不可或缺的组成部分是培训布朗大学主动研究小组中的研究生和本科生，特别是要吸引年轻和有前途的研究人员进入该领域。该项目的研究中心对象是调制不变的奇异积分及其在其已知界限范围的边界或附近的行为。涉及Carleson最大部分傅立叶总和运算符的模型问题是对周期性函数的傅立叶序列几乎均匀收敛的尖锐可集成性顺序的表征。第二个密切相关的问题涉及莱西 - 硫素持有人型的扩展，对双线性希尔伯特转换为已知范围的边界。与他的合作者一起，这位主要研究人员最近为这两个问题获得了当前的最佳结果，尤其依赖于新开发的Calderon-Zygmund分解，该分解适用于调制不变的设置。预计该技术的进一步发展将导致解决这两个核心问题以及其他重大开放问题的进一步改进。一个杰出的问题是，双线性希尔伯特转换为全预期的指数范围的已知统一估计值的扩展，完成了Calderon的原始程序，以实现第一个换向器的界限。拟议的研究的另一个核心方向是研究具有旋转对称性的奇异积分运算符，这是希尔伯特通过多吸光时频率分析技术沿平面矢量场的转换。预计上述技术的进一步改进也会影响有关多个傅立叶序列总结性的几个问题。